slides/caesar-break.html

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  52
  53 # Breaking caesar ciphers
  54
  55 ![center-aligned Caesar wheel](caesarwheel1.gif)
  56
  57 ---
  58
  59 layout: true
  60
  61 .indexlink[[Index](index.html)]
  62
  63 ---
  64
  65 # Human vs Machine
  66
  67 Slow but clever vs Dumb but fast
  68
  69 ## Human approach
  70
  71 Ciphertext | Plaintext
  72 ---|---
  73 ![left-aligned Ciphertext frequencies](c1a_frequency_histogram.png) | ![left-aligned English frequencies](english_frequency_histogram.png)
  74
  75 ---
  76
  77 # Human vs machine
  78
  79 ## Machine approach
  80
  81 Brute force.
  82
  83 Try all keys.
  84
  85 * How many keys to try?
  86
  87 ## Basic idea
  88
  89 ```
  90 for each key:
  91     decipher with this key
  92     how close is it to English?
  93     remember the best key
  94 ```
  95
  96 What steps do we know how to do?
  97
  98 ---
  99 # How close is it to English?
 100
 101 What does English look like?
 102
 103 * We need a model of English.
 104
 105 How do we define "closeness"?
 106
 107 ## Here begineth the yak shaving
 108
 109 ---
 110
 111 # What does English look like?
 112
 113 ## Abstraction: frequency of letter counts
 114
 115 Letter | Count
 116 -------|------
 117 a | 489107
 118 b | 92647
 119 c | 140497
 120 d | 267381
 121 e | 756288
 122 . | .
 123 . | .
 124 . | .
 125 z | 3575
 126
 127 Use this to predict the probability of each letter, and hence the probability of a sequence of letters.
 128
 129 ---
 130
 131 # An infinite number of monkeys
 132
 133 What is the probability that this string of letters is a sample of English?
 134
 135 ## Naive Bayes, or the bag of letters
 136
 137 Ignore letter order, just treat each letter individually.
 138
 139 Probability of a text is `\( \prod_i p_i \)`
 140
 141 Letter      | h       | e       | l       | l       | o       | hello
 142 ------------|---------|---------|---------|---------|---------|-------
 143 Probability | 0.06645 | 0.12099 | 0.04134 | 0.04134 | 0.08052 | 1.10648239 × 10<sup>-6</sup>
 144
 145 Letter      | i       | f       | m       | m       | p       | ifmmp
 146 ------------|---------|---------|---------|---------|---------|-------
 147 Probability | 0.06723 | 0.02159 | 0.02748 | 0.02748 | 0.01607 | 1.76244520 × 10<sup>-8</sup>
 148
 149 (Implmentation issue: this can often underflow, so get in the habit of rephrasing it as `\( \sum_i \log p_i \)`)
 150
 151 Letter      | h       | e       | l       | l       | o       | hello
 152 ------------|---------|---------|---------|---------|---------|-------
 153 Probability | -1.1774 | -0.9172 | -1.3836 | -1.3836 | -1.0940 | -5.956055
 154
 155
 156 ---
 157
 158 # Frequencies of English
 159
 160 But before then how do we count the letters?
 161
 162 * Read a file into a string
 163 ```python
 164 open()
 165 .read()
 166 ```
 167 * Count them
 168 ```python
 169 import collections
 170 collections.Counter()
 171 ```
 172
 173 Create the `language_models.py` file for this.
 174
 175 ---
 176
 177 # Canonical forms
 178
 179 Counting letters in _War and Peace_ gives all manner of junk.
 180
 181 * Convert the text in canonical form (lower case, accents removed, non-letters stripped) before counting
 182
 183 ```python
 184 [l.lower() for l in text if ...]
 185 ```
 186 ---
 187
 188 # Accents
 189
 190 ```python
 191 >>> 'é' in string.ascii_letters
 192 >>> 'e' in string.ascii_letters
 193 ```
 194
 195 ## Unicode, combining codepoints, and normal forms
 196
 197 Text encodings will bite you when you least expect it.
 198
 199 - **é** : LATIN SMALL LETTER E WITH ACUTE (U+00E9)
 200
 201 - **e** + **&nbsp;&#x301;** : LATIN SMALL LETTER E (U+0065) + COMBINING ACUTE ACCENT (U+0301)
 202
 203 * urlencoding is the other pain point.
 204
 205 ---
 206
 207 # Five minutes on StackOverflow later...
 208
 209 ```python
 210 def unaccent(text):
 211     """Remove all accents from letters.
 212     It does this by converting the unicode string to decomposed compatibility
 213     form, dropping all the combining accents, then re-encoding the bytes.
 214
 215     >>> unaccent('hello')
 216     'hello'
 217     >>> unaccent('HELLO')
 218     'HELLO'
 219     >>> unaccent('héllo')
 220     'hello'
 221     >>> unaccent('héllö')
 222     'hello'
 223     >>> unaccent('HÉLLÖ')
 224     'HELLO'
 225     """
 226     return unicodedata.normalize('NFKD', text).\
 227         encode('ascii', 'ignore').\
 228         decode('utf-8')
 229 ```
 230
 231 ---
 232
 233 # Find the frequencies of letters in English
 234
 235 1. Read from `shakespeare.txt`, `sherlock-holmes.txt`, and `war-and-peace.txt`.
 236 2. Find the frequencies (`.update()`)
 237 3. Sort by count
 238 4. Write counts to `count_1l.txt` (`'text{}\n'.format()`)
 239
 240 ---
 241
 242 # Reading letter probabilities
 243
 244 1. Load the file `count_1l.txt` into a dict, with letters as keys.
 245
 246 2. Normalise the counts (components of vector sum to 1): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }$$`
 247     * Return a new dict
 248     * Remember the doctest!
 249
 250 3. Create a dict `Pl` that gives the log probability of a letter
 251
 252 4. Create a function `Pletters` that gives the probability of an iterable of letters
 253     * What preconditions should this function have?
 254     * Remember the doctest!
 255
 256 ---
 257
 258 # Breaking caesar ciphers
 259
 260 ## Remember the basic idea
 261
 262 ```
 263 for each key:
 264     decipher with this key
 265     how close is it to English?
 266     remember the best key
 267 ```
 268
 269 Try it on the text in `2013/1a.ciphertext`. Does it work?
 270
 271 ---
 272
 273 # Aside: Logging
 274
 275 Better than scattering `print()`statements through your code
 276
 277 ```python
 278 import logging
 279
 280 logger = logging.getLogger(__name__)
 281 logger.addHandler(logging.FileHandler('cipher.log'))
 282 logger.setLevel(logging.WARNING)
 283
 284         logger.debug('Caesar break attempt using key {0} gives fit of {1} '
 285                       'and decrypt starting: {2}'.format(shift, fit, plaintext[:50]))
 286
 287 ```
 288 * Yes, it's ugly.
 289
 290 Use `logger.setLevel()` to change the level: CRITICAL, ERROR, WARNING, INFO, DEBUG
 291
 292 Use `logger.debug()`, `logger.info()`, etc. to log a message.
 293
 294 ---
 295
 296 # Homework: how much ciphertext do we need?
 297
 298 ## Let's do an experiment to find out
 299
 300 1. Load the whole corpus into a string (sanitised)
 301 2. Select a random chunk of plaintext and a random key
 302 3. Encipher the text
 303 4. Score 1 point if `caesar_cipher_break()` recovers the correct key
 304 5. Repeat many times and with many plaintext lengths
 305
 306 ```python
 307 import csv
 308
 309 def show_results():
 310     with open('caesar_break_parameter_trials.csv', 'w') as f:
 311         writer = csv.DictWriter(f, ['name'] + message_lengths,
 312             quoting=csv.QUOTE_NONNUMERIC)
 313         writer.writeheader()
 314         for scoring in sorted(scores.keys()):
 315             scores[scoring]['name'] = scoring
 316             writer.writerow(scores[scoring])
 317 ```
 318
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