# Affine ciphers
+a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z
+--|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
+b | e | h | k | n | q | t | w | z | c | f | i | l | o | r | u | x | a | d | g | j | m | p | s | v | y
+
+An extension of Caesar ciphers
+
+* Count the gaps in the letters.
+
+---
+# How affine ciphers work
+
+_ciphertext_letter_ =_plaintext_letter_ × a + b
+
+* Convert letters to numbers
+* Take the total modulus 26
+
+# Enciphering is easy
+
+* Build the `affine_encipher()` function
+
+---
+
+# Deciphering affine ciphers is harder
+
+`$$p = \frac{c - b}{a}$$`
+
+But modular division is hard!
+
+
+---
+
## Explanation of extended Euclid's algorithm from [Programming with finite fields](http://jeremykun.com/2014/03/13/programming-with-finite-fields/)
**Definition:** An element _d_ is called a greatest common divisor (gcd) of _a, b_ if it divides both _a_ and _b_, and for every other _z_ dividing both _a_ and _b_, _z_ divides _d_.
--- /dev/null
+<!DOCTYPE html>
+<html>
+ <head>
+ <title>Alternative plaintext scoring</title>
+ <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
+ <style type="text/css">
+ /* Slideshow styles */
+ body {
+ font-size: 20px;
+ }
+ h1, h2, h3 {
+ font-weight: 400;
+ margin-bottom: 0;
+ }
+ h1 { font-size: 3em; }
+ h2 { font-size: 2em; }
+ h3 { font-size: 1.6em; }
+ a, a > code {
+ text-decoration: none;
+ }
+ code {
+ -moz-border-radius: 5px;
+ -web-border-radius: 5px;
+ background: #e7e8e2;
+ border-radius: 5px;
+ font-size: 16px;
+ }
+ .plaintext {
+ background: #272822;
+ color: #80ff80;
+ text-shadow: 0 0 20px #333;
+ padding: 2px 5px;
+ }
+ .ciphertext {
+ background: #272822;
+ color: #ff6666;
+ text-shadow: 0 0 20px #333;
+ padding: 2px 5px;
+ }
+ .float-right {
+ float: right;
+ }
+ </style>
+ </head>
+ <body>
+ <textarea id="source">
+
+# Alternative plaintext scoring methods
+
+---
+
+# Back to frequency of letter counts
+
+Letter | Count
+-------|------
+a | 489107
+b | 92647
+c | 140497
+d | 267381
+e | 756288
+. | .
+. | .
+. | .
+z | 3575
+
+Another way of thinking about this is a 26-dimensional vector.
+
+Create a vector of our text, and one of idealised English.
+
+The distance between the vectors is how far from English the text is.
+
+---
+
+# Vector distances
+
+.float-right[]
+
+Several different distance measures (__metrics__, also called __norms__):
+
+* L<sub>2</sub> norm (Euclidean distance):
+`\(\|\mathbf{a} - \mathbf{b}\| = \sqrt{\sum_i (\mathbf{a}_i - \mathbf{b}_i)^2} \)`
+
+* L<sub>1</sub> norm (Manhattan distance, taxicab distance):
+`\(\|\mathbf{a} - \mathbf{b}\| = \sum_i |\mathbf{a}_i - \mathbf{b}_i| \)`
+
+* L<sub>3</sub> norm:
+`\(\|\mathbf{a} - \mathbf{b}\| = \sqrt[3]{\sum_i |\mathbf{a}_i - \mathbf{b}_i|^3} \)`
+
+The higher the power used, the more weight is given to the largest differences in components.
+
+(Extends out to:
+
+* L<sub>0</sub> norm (Hamming distance):
+`$$\|\mathbf{a} - \mathbf{b}\| = \sum_i \left\{
+\begin{matrix} 1 &\mbox{if}\ \mathbf{a}_i \neq \mathbf{b}_i , \\
+ 0 &\mbox{if}\ \mathbf{a}_i = \mathbf{b}_i \end{matrix} \right. $$`
+
+* L<sub>∞</sub> norm:
+`\(\|\mathbf{a} - \mathbf{b}\| = \max_i{(\mathbf{a}_i - \mathbf{b}_i)} \)`
+
+neither of which will be that useful here, but they keep cropping up.)
+---
+
+# Normalisation of vectors
+
+Frequency distributions drawn from different sources will have different lengths. For a fair comparison we need to scale them.
+
+* Eucliean scaling (vector with unit length): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \sqrt{\mathbf{x}_1^2 + \mathbf{x}_2^2 + \mathbf{x}_3^2 + \dots } }$$`
+
+* Normalisation (components of vector sum to 1): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }$$`
+
+---
+
+# Angle, not distance
+
+Rather than looking at the distance between the vectors, look at the angle between them.
+
+.float-right[]
+
+Vector dot product shows how much of one vector lies in the direction of another:
+`\( \mathbf{A} \bullet \mathbf{B} =
+\| \mathbf{A} \| \cdot \| \mathbf{B} \| \cos{\theta} \)`
+
+But,
+`\( \mathbf{A} \bullet \mathbf{B} = \sum_i \mathbf{A}_i \cdot \mathbf{B}_i \)`
+and `\( \| \mathbf{A} \| = \sum_i \mathbf{A}_i^2 \)`
+
+A bit of rearranging give the cosine simiarity:
+`$$ \cos{\theta} = \frac{ \mathbf{A} \bullet \mathbf{B} }{ \| \mathbf{A} \| \cdot \| \mathbf{B} \| } =
+\frac{\sum_i \mathbf{A}_i \cdot \mathbf{B}_i}{\sum_i \mathbf{A}_i^2 \times \sum_i \mathbf{B}_i^2} $$`
+
+This is independent of vector lengths!
+
+Cosine similarity is 1 if in parallel, 0 if perpendicular, -1 if antiparallel.
+
+---
+
+# Which is best?
+
+ | Euclidean | Normalised
+---|-----------|------------
+L1 | x | x
+L2 | x | x
+L3 | x | x
+Cosine | x | x
+
+And the probability measure!
+
+* Nine different ways of measuring fitness.
+
+## Computing is an empircal science
+
+Let's do some experiments to find the best solution!
+
+---
+
+# Experimental harness
+
+## Step 1: build some other scoring functions
+
+We need a way of passing the different functions to the keyfinding function.
+
+## Step 2: find the best scoring function
+
+Try them all on random ciphertexts, see which one works best.
+
+---
+
+# Functions are values!
+
+```python
+>>> Pletters
+<function Pletters at 0x7f60e6d9c4d0>
+```
+
+```python
+def caesar_break(message, fitness=Pletters):
+ """Breaks a Caesar cipher using frequency analysis
+...
+ for shift in range(26):
+ plaintext = caesar_decipher(message, shift)
+ fit = fitness(plaintext)
+```
+
+---
+
+# Changing the comparison function
+
+* Must be a function that takes a text and returns a score
+ * Better fit must give higher score, opposite of the vector distance norms
+
+```python
+def make_frequency_compare_function(target_frequency, frequency_scaling, metric, invert):
+ def frequency_compare(text):
+ ...
+ return score
+ return frequency_compare
+```
+
+---
+
+# Data-driven processing
+
+```python
+metrics = [{'func': norms.l1, 'invert': True, 'name': 'l1'},
+ {'func': norms.l2, 'invert': True, 'name': 'l2'},
+ {'func': norms.l3, 'invert': True, 'name': 'l3'},
+ {'func': norms.cosine_similarity, 'invert': False, 'name': 'cosine_similarity'}]
+scalings = [{'corpus_frequency': normalised_english_counts,
+ 'scaling': norms.normalise,
+ 'name': 'normalised'},
+ {'corpus_frequency': euclidean_scaled_english_counts,
+ 'scaling': norms.euclidean_scale,
+ 'name': 'euclidean_scaled'}]
+```
+
+Use this to make all nine scoring functions.
+
+
+ </textarea>
+ <script src="http://gnab.github.io/remark/downloads/remark-0.6.0.min.js" type="text/javascript">
+ </script>
+
+ <script type="text/javascript"
+ src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML&delayStartupUntil=configured"></script>
+
+ <script type="text/javascript">
+ var slideshow = remark.create({ ratio: "16:9" });
+
+ // Setup MathJax
+ MathJax.Hub.Config({
+ tex2jax: {
+ skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
+ }
+ });
+ MathJax.Hub.Queue(function() {
+ $(MathJax.Hub.getAllJax()).map(function(index, elem) {
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+ }).parent().addClass('has-jax');
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+</html>
'and decrypt starting: {2}'.format(shift, fit, plaintext[:50]))
```
- * Yes, it's ugly.
+* Yes, it's ugly.
- Use `logger.setLevel()` to change the level: CRITICAL, ERROR, WARNING, INFO, DEBUG
+Use `logger.setLevel()` to change the level: CRITICAL, ERROR, WARNING, INFO, DEBUG
----
-
-# Back to frequency of letter counts
-
-Letter | Count
--------|------
-a | 489107
-b | 92647
-c | 140497
-d | 267381
-e | 756288
-. | .
-. | .
-. | .
-z | 3575
-
-Another way of thinking about this is a 26-dimensional vector.
-
-Create a vector of our text, and one of idealised English.
-
-The distance between the vectors is how far from English the text is.
-
----
-
-# Vector distances
-
-.float-right[]
-
-Several different distance measures (__metrics__, also called __norms__):
-
-* L<sub>2</sub> norm (Euclidean distance):
-`\(\|\mathbf{a} - \mathbf{b}\| = \sqrt{\sum_i (\mathbf{a}_i - \mathbf{b}_i)^2} \)`
-
-* L<sub>1</sub> norm (Manhattan distance, taxicab distance):
-`\(\|\mathbf{a} - \mathbf{b}\| = \sum_i |\mathbf{a}_i - \mathbf{b}_i| \)`
-
-* L<sub>3</sub> norm:
-`\(\|\mathbf{a} - \mathbf{b}\| = \sqrt[3]{\sum_i |\mathbf{a}_i - \mathbf{b}_i|^3} \)`
-
-The higher the power used, the more weight is given to the largest differences in components.
-
-(Extends out to:
-
-* L<sub>0</sub> norm (Hamming distance):
-`$$\|\mathbf{a} - \mathbf{b}\| = \sum_i \left\{
-\begin{matrix} 1 &\mbox{if}\ \mathbf{a}_i \neq \mathbf{b}_i , \\
- 0 &\mbox{if}\ \mathbf{a}_i = \mathbf{b}_i \end{matrix} \right. $$`
-
-* L<sub>∞</sub> norm:
-`\(\|\mathbf{a} - \mathbf{b}\| = \max_i{(\mathbf{a}_i - \mathbf{b}_i)} \)`
-
-neither of which will be that useful here, but they keep cropping up.)
----
-
-# Normalisation of vectors
-
-Frequency distributions drawn from different sources will have different lengths. For a fair comparison we need to scale them.
-
-* Eucliean scaling (vector with unit length): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \sqrt{\mathbf{x}_1^2 + \mathbf{x}_2^2 + \mathbf{x}_3^2 + \dots } }$$`
-
-* Normalisation (components of vector sum to 1): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }$$`
-
----
-
-# Angle, not distance
-
-Rather than looking at the distance between the vectors, look at the angle between them.
-
-.float-right[]
-
-Vector dot product shows how much of one vector lies in the direction of another:
-`\( \mathbf{A} \bullet \mathbf{B} =
-\| \mathbf{A} \| \cdot \| \mathbf{B} \| \cos{\theta} \)`
-
-But,
-`\( \mathbf{A} \bullet \mathbf{B} = \sum_i \mathbf{A}_i \cdot \mathbf{B}_i \)`
-and `\( \| \mathbf{A} \| = \sum_i \mathbf{A}_i^2 \)`
-
-A bit of rearranging give the cosine simiarity:
-`$$ \cos{\theta} = \frac{ \mathbf{A} \bullet \mathbf{B} }{ \| \mathbf{A} \| \cdot \| \mathbf{B} \| } =
-\frac{\sum_i \mathbf{A}_i \cdot \mathbf{B}_i}{\sum_i \mathbf{A}_i^2 \times \sum_i \mathbf{B}_i^2} $$`
-
-This is independent of vector lengths!
-
-Cosine similarity is 1 if in parallel, 0 if perpendicular, -1 if antiparallel.
-
----
-
-# Which is best?
-
- | Euclidean | Normalised
----|-----------|------------
-L1 | x | x
-L2 | x | x
-L3 | x | x
-Cosine | x | x
-
-And the probability measure!
-
-* Nine different ways of measuring fitness.
-
-## Computing is an empircal science
-
-Let's do some experiments to find the best solution!
+Use `logger.debug()`, `logger.info()`, etc. to log a message.
---
-# Experimental harness
-
-## Step 1: build some other scoring functions
-
-We need a way of passing the different functions to the keyfinding function.
-
-## Step 2: find the best scoring function
-
-Try them all on random ciphertexts, see which one works best.
-
----
+# How much ciphertext do we need?
-# Functions are values!
-
-```python
->>> Pletters
-<function Pletters at 0x7f60e6d9c4d0>
-```
-
-```python
-def caesar_break(message, fitness=Pletters):
- """Breaks a Caesar cipher using frequency analysis
-...
- for shift in range(26):
- plaintext = caesar_decipher(message, shift)
- fit = fitness(plaintext)
-```
-
----
-
-# Changing the comparison function
-
-* Must be a function that takes a text and returns a score
- * Better fit must give higher score, opposite of the vector distance norms
-
-```python
-def make_frequency_compare_function(target_frequency, frequency_scaling, metric, invert):
- def frequency_compare(text):
- ...
- return score
- return frequency_compare
-```
-
----
-
-# Data-driven processing
-
-```python
-metrics = [{'func': norms.l1, 'invert': True, 'name': 'l1'},
- {'func': norms.l2, 'invert': True, 'name': 'l2'},
- {'func': norms.l3, 'invert': True, 'name': 'l3'},
- {'func': norms.cosine_similarity, 'invert': False, 'name': 'cosine_similarity'}]
-scalings = [{'corpus_frequency': normalised_english_counts,
- 'scaling': norms.normalise,
- 'name': 'normalised'},
- {'corpus_frequency': euclidean_scaled_english_counts,
- 'scaling': norms.euclidean_scale,
- 'name': 'euclidean_scaled'}]
-```
+## Let's do an experiment to find out
-Use this to make all nine scoring functions.
+1. Load the whole corpus into a string (sanitised)
+2. Select a random chunk of plaintext and a random key
+3. Encipher the text
+4. Score 1 point if `caesar_cipher_break()` recovers the correct key
+5. Repeat many times and with many plaintext lengths
</textarea>
projects / cipher-training.git / commitdiff