cipherbreak.py

   1 """A set of functions to break the ciphers give in ciphers.py.
   2 """
   3
   4 import string
   5 import collections
   6 import norms
   7 import logging
   8 import random
   9 import math
  10 from itertools import starmap
  11 from segment import segment
  12 from multiprocessing import Pool
  13
  14 import matplotlib.pyplot as plt
  15
  16 # logging.basicConfig(filename="cipher.log", level=logging.INFO)
  17 # logger = logging.getLogger(__name__)
  18
  19 logger = logging.getLogger('cipherbreak')
  20 logger.setLevel(logging.WARNING)
  21 # logger.setLevel(logging.INFO)
  22 # logger.setLevel(logging.DEBUG)
  23
  24 # create the logging file handler
  25 fh = logging.FileHandler("cipher.log")
  26 formatter = logging.Formatter('%(asctime)s - %(name)s - %(levelname)s - %(message)s')
  27 fh.setFormatter(formatter)
  28
  29 # add handler to logger object
  30 logger.addHandler(fh)
  31
  32
  33 from cipher import *
  34 from language_models import *
  35
  36 # To time a run:
  37 #
  38 # import timeit
  39 # c5a = open('2012/5a.ciphertext', 'r').read()
  40 # timeit.timeit('keyword_break(c5a)', setup='gc.enable() ; from __main__ import c5a ; from cipher import keyword_break', number=1)
  41 # timeit.repeat('keyword_break_mp(c5a, chunksize=500)', setup='gc.enable() ; from __main__ import c5a ; from cipher import keyword_break_mp', repeat=5, number=1)
  42
  43
  44 def index_of_coincidence(text):
  45     stext = sanitise(text)
  46     counts = collections.Counter(stext)
  47     denom = len(stext) * (len(text) - 1) / 26
  48     return (
  49         sum(max(counts[l] * counts[l] - 1, 0) for l in string.ascii_lowercase)
  50         /
  51         denom
  52     )
  53
  54
  55 transpositions = collections.defaultdict(list)
  56 for word in keywords:
  57     transpositions[transpositions_of(word)] += [word]
  58
  59 def frequencies(text):
  60     """Count the number of occurrences of each character in text
  61
  62     >>> sorted(frequencies('abcdefabc').items())
  63     [('a', 2), ('b', 2), ('c', 2), ('d', 1), ('e', 1), ('f', 1)]
  64     >>> sorted(frequencies('the quick brown fox jumped over the lazy ' \
  65          'dog').items()) # doctest: +NORMALIZE_WHITESPACE
  66     [(' ', 8), ('a', 1), ('b', 1), ('c', 1), ('d', 2), ('e', 4), ('f', 1),
  67      ('g', 1), ('h', 2), ('i', 1), ('j', 1), ('k', 1), ('l', 1), ('m', 1),
  68      ('n', 1), ('o', 4), ('p', 1), ('q', 1), ('r', 2), ('t', 2), ('u', 2),
  69      ('v', 1), ('w', 1), ('x', 1), ('y', 1), ('z', 1)]
  70     >>> sorted(frequencies('The Quick BROWN fox jumped! over... the ' \
  71          '(9lazy) DOG').items()) # doctest: +NORMALIZE_WHITESPACE
  72     [(' ', 8), ('!', 1), ('(', 1), (')', 1), ('.', 3), ('9', 1), ('B', 1),
  73      ('D', 1), ('G', 1), ('N', 1), ('O', 2), ('Q', 1), ('R', 1), ('T', 1),
  74      ('W', 1), ('a', 1), ('c', 1), ('d', 1), ('e', 4), ('f', 1), ('h', 2),
  75      ('i', 1), ('j', 1), ('k', 1), ('l', 1), ('m', 1), ('o', 2), ('p', 1),
  76      ('r', 1), ('t', 1), ('u', 2), ('v', 1), ('x', 1), ('y', 1), ('z', 1)]
  77     >>> sorted(frequencies(sanitise('The Quick BROWN fox jumped! over... '\
  78          'the (9lazy) DOG')).items()) # doctest: +NORMALIZE_WHITESPACE
  79     [('a', 1), ('b', 1), ('c', 1), ('d', 2), ('e', 4), ('f', 1), ('g', 1),
  80      ('h', 2), ('i', 1), ('j', 1), ('k', 1), ('l', 1), ('m', 1), ('n', 1),
  81      ('o', 4), ('p', 1), ('q', 1), ('r', 2), ('t', 2), ('u', 2), ('v', 1),
  82      ('w', 1), ('x', 1), ('y', 1), ('z', 1)]
  83     >>> frequencies('abcdefabcdef')['x']
  84     0
  85     """
  86     return collections.Counter(c for c in text)
  87
  88
  89 def caesar_break(message, fitness=Pletters):
  90     """Breaks a Caesar cipher using frequency analysis
  91
  92     >>> caesar_break('ibxcsyorsaqcheyklxivoexlevmrimwxsfiqevvmihrsasrxliwyrh' \
  93           'ecjsppsamrkwleppfmergefifvmhixscsymjcsyqeoixlm') # doctest: +ELLIPSIS
  94     (4, -130.849989015...)
  95     >>> caesar_break('wxwmaxdgheetgwuxztgptedbgznitgwwhpguxyhkxbmhvvtlbhgtee' \
  96           'raxlmhiixweblmxgxwmhmaxybkbgztgwztsxwbgmxgmert') # doctest: +ELLIPSIS
  97     (19, -128.82410410...)
  98     >>> caesar_break('yltbbqnqnzvguvaxurorgenafsbezqvagbnornfgsbevpnaabjurer' \
  99           'svaquvzyvxrnznazlybequrvfohgriraabjtbaruraprur') # doctest: +ELLIPSIS
 100     (13, -126.25403935...)
 101     """
 102     sanitised_message = sanitise(message)
 103     best_shift = 0
 104     best_fit = float('-inf')
 105     for shift in range(26):
 106         plaintext = caesar_decipher(sanitised_message, shift)
 107         fit = fitness(plaintext)
 108         logger.debug('Caesar break attempt using key {0} gives fit of {1} '
 109                      'and decrypt starting: {2}'.format(shift, fit,
 110                                                         plaintext[:50]))
 111         if fit > best_fit:
 112             best_fit = fit
 113             best_shift = shift
 114     logger.info('Caesar break best fit: key {0} gives fit of {1} and '
 115                 'decrypt starting: {2}'.format(best_shift, best_fit,
 116                     caesar_decipher(sanitised_message, best_shift)[:50]))
 117     return best_shift, best_fit
 118
 119 def affine_break(message, fitness=Pletters):
 120     """Breaks an affine cipher using frequency analysis
 121
 122     >>> affine_break('lmyfu bkuusd dyfaxw claol psfaom jfasd snsfg jfaoe ls ' \
 123           'omytd jlaxe mh jm bfmibj umis hfsul axubafkjamx. ls kffkxwsd jls ' \
 124           'ofgbjmwfkiu olfmxmtmwaokttg jlsx ls kffkxwsd jlsi zg tsxwjl. jlsx ' \
 125           'ls umfjsd jlsi zg hfsqysxog. ls dmmdtsd mx jls bats mh bkbsf. ls ' \
 126           'bfmctsd kfmyxd jls lyj, mztanamyu xmc jm clm cku tmmeaxw kj lai ' \
 127           'kxd clm ckuxj.') # doctest: +ELLIPSIS
 128     ((15, 22, True), -340.601181913...)
 129     """
 130     sanitised_message = sanitise(message)
 131     best_multiplier = 0
 132     best_adder = 0
 133     best_one_based = True
 134     best_fit = float("-inf")
 135     for one_based in [True, False]:
 136         for multiplier in [x for x in range(1, 26, 2) if x != 13]:
 137             for adder in range(26):
 138                 plaintext = affine_decipher(sanitised_message,
 139                                             multiplier, adder, one_based)
 140                 fit = fitness(plaintext)
 141                 logger.debug('Affine break attempt using key {0}x+{1} ({2}) '
 142                              'gives fit of {3} and decrypt starting: {4}'.
 143                              format(multiplier, adder, one_based, fit,
 144                                     plaintext[:50]))
 145                 if fit > best_fit:
 146                     best_fit = fit
 147                     best_multiplier = multiplier
 148                     best_adder = adder
 149                     best_one_based = one_based
 150     logger.info('Affine break best fit with key {0}x+{1} ({2}) gives fit of '
 151                 '{3} and decrypt starting: {4}'.format(
 152                     best_multiplier, best_adder, best_one_based, best_fit,
 153                     affine_decipher(sanitised_message, best_multiplier,
 154                                     best_adder, best_one_based)[:50]))
 155     return (best_multiplier, best_adder, best_one_based), best_fit
 156
 157 def keyword_break(message, wordlist=keywords, fitness=Pletters):
 158     """Breaks a keyword substitution cipher using a dictionary and
 159     frequency analysis.
 160
 161     >>> keyword_break(keyword_encipher('this is a test message for the ' \
 162           'keyword decipherment', 'elephant', KeywordWrapAlphabet.from_last), \
 163           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS
 164     (('elephant', <KeywordWrapAlphabet.from_last: 2>), -52.834575011...)
 165     """
 166     best_keyword = ''
 167     best_wrap_alphabet = True
 168     best_fit = float("-inf")
 169     for wrap_alphabet in KeywordWrapAlphabet:
 170         for keyword in wordlist:
 171             plaintext = keyword_decipher(message, keyword, wrap_alphabet)
 172             fit = fitness(plaintext)
 173             logger.debug('Keyword break attempt using key {0} (wrap={1}) '
 174                          'gives fit of {2} and decrypt starting: {3}'.format(
 175                              keyword, wrap_alphabet, fit,
 176                              sanitise(plaintext)[:50]))
 177             if fit > best_fit:
 178                 best_fit = fit
 179                 best_keyword = keyword
 180                 best_wrap_alphabet = wrap_alphabet
 181     logger.info('Keyword break best fit with key {0} (wrap={1}) gives fit of '
 182                 '{2} and decrypt starting: {3}'.format(best_keyword,
 183                     best_wrap_alphabet, best_fit, sanitise(
 184                         keyword_decipher(message, best_keyword,
 185                                          best_wrap_alphabet))[:50]))
 186     return (best_keyword, best_wrap_alphabet), best_fit
 187
 188 def keyword_break_mp(message, wordlist=keywords, fitness=Pletters,
 189                      number_of_solutions=1, chunksize=500):
 190     """Breaks a keyword substitution cipher using a dictionary and
 191     frequency analysis
 192
 193     >>> keyword_break_mp(keyword_encipher('this is a test message for the ' \
 194           'keyword decipherment', 'elephant', KeywordWrapAlphabet.from_last), \
 195           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS
 196     (('elephant', <KeywordWrapAlphabet.from_last: 2>), -52.834575011...)
 197     >>> keyword_break_mp(keyword_encipher('this is a test message for the ' \
 198           'keyword decipherment', 'elephant', KeywordWrapAlphabet.from_last), \
 199           wordlist=['cat', 'elephant', 'kangaroo'], \
 200           number_of_solutions=2) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 201     [(('elephant', <KeywordWrapAlphabet.from_last: 2>), -52.834575011...),
 202     (('elephant', <KeywordWrapAlphabet.from_largest: 3>), -52.834575011...)]
 203     """
 204     with Pool() as pool:
 205         helper_args = [(message, word, wrap, fitness)
 206                        for word in wordlist
 207                        for wrap in KeywordWrapAlphabet]
 208         # Gotcha: the helper function here needs to be defined at the top level
 209         #   (limitation of Pool.starmap)
 210         breaks = pool.starmap(keyword_break_worker, helper_args, chunksize)
 211         if number_of_solutions == 1:
 212             return max(breaks, key=lambda k: k[1])
 213         else:
 214             return sorted(breaks, key=lambda k: k[1], reverse=True)[:number_of_solutions]
 215
 216 def keyword_break_worker(message, keyword, wrap_alphabet, fitness):
 217     plaintext = keyword_decipher(message, keyword, wrap_alphabet)
 218     fit = fitness(plaintext)
 219     logger.debug('Keyword break attempt using key {0} (wrap={1}) gives fit of '
 220                  '{2} and decrypt starting: {3}'.format(keyword,
 221                      wrap_alphabet, fit, sanitise(plaintext)[:50]))
 222     return (keyword, wrap_alphabet), fit
 223
 224 # def monoalphabetic_break_hillclimbing(message, max_iterations=10000000,
 225 #         alphabet=None, fitness=Pletters):
 226 #     ciphertext = unaccent(message).lower()
 227 #     if not alphabet:
 228 #         alphabet = list(string.ascii_lowercase)
 229 #         random.shuffle(alphabet)
 230 #         alphabet = cat(alphabet)
 231 #     return monoalphabetic_break_hillclimbing_worker(ciphertext, alphabet,
 232 #                                                     max_iterations, fitness)
 233
 234 # def monoalphabetic_break_hillclimbing_mp(message, workers=10,
 235 #         max_iterations = 10000000, alphabet=None, fitness=Pletters, chunksize=1):
 236 #     worker_args = []
 237 #     ciphertext = unaccent(message).lower()
 238 #     for i in range(workers):
 239 #         if alphabet:
 240 #             this_alphabet = alphabet
 241 #         else:
 242 #             this_alphabet = list(string.ascii_lowercase)
 243 #             random.shuffle(this_alphabet)
 244 #             this_alphabet = cat(this_alphabet)
 245 #         worker_args.append((ciphertext, this_alphabet, max_iterations, fitness))
 246 #     with Pool() as pool:
 247 #         breaks = pool.starmap(monoalphabetic_break_hillclimbing_worker,
 248 #                               worker_args, chunksize)
 249 #     return max(breaks, key=lambda k: k[1])
 250
 251 # def monoalphabetic_break_hillclimbing_worker(message, alphabet,
 252 #         max_iterations, fitness):
 253 #     def swap(letters, i, j):
 254 #         if i > j:
 255 #             i, j = j, i
 256 #         if i == j:
 257 #             return letters
 258 #         else:
 259 #             return (letters[:i] + letters[j] + letters[i+1:j] + letters[i] +
 260 #                     letters[j+1:])
 261 #     best_alphabet = alphabet
 262 #     best_fitness = float('-inf')
 263 #     for i in range(max_iterations):
 264 #         alphabet = swap(best_alphabet, random.randrange(26), random.randrange(26))
 265 #         cipher_translation = ''.maketrans(string.ascii_lowercase, alphabet)
 266 #         plaintext = message.translate(cipher_translation)
 267 #         if fitness(plaintext) > best_fitness:
 268 #             best_fitness = fitness(plaintext)
 269 #             best_alphabet = alphabet
 270 #             print(i, best_alphabet, best_fitness, plaintext[:50])
 271 #     return best_alphabet, best_fitness
 272
 273
 274 def monoalphabetic_break_hillclimbing(message,
 275                               max_iterations=20000,
 276                               plain_alphabet=None,
 277                               cipher_alphabet=None,
 278                               fitness=Pletters, chunksize=1):
 279     return simulated_annealing_break(message,
 280                               workers=1,
 281                               initial_temperature=0,
 282                               max_iterations=max_iterations,
 283                               plain_alphabet=plain_alphabet,
 284                               cipher_alphabet=cipher_alphabet,
 285                               fitness=fitness, chunksize=chunksize)
 286
 287
 288 def monoalphabetic_break_hillclimbing_mp(message,
 289                               workers=10,
 290                               max_iterations=20000,
 291                               plain_alphabet=None,
 292                               cipher_alphabet=None,
 293                               fitness=Pletters, chunksize=1):
 294     return simulated_annealing_break(message,
 295                               workers=workers,
 296                               initial_temperature=0,
 297                               max_iterations=max_iterations,
 298                               plain_alphabet=plain_alphabet,
 299                               cipher_alphabet=cipher_alphabet,
 300                               fitness=fitness, chunksize=chunksize)
 301
 302
 303 def simulated_annealing_break(message, workers=10,
 304                               initial_temperature=200,
 305                               max_iterations=20000,
 306                               plain_alphabet=None,
 307                               cipher_alphabet=None,
 308                               fitness=Pletters, chunksize=1):
 309     worker_args = []
 310     ciphertext = sanitise(message)
 311     for i in range(workers):
 312         if not plain_alphabet:
 313             plain_alphabet = string.ascii_lowercase
 314         if not cipher_alphabet:
 315             cipher_alphabet = list(string.ascii_lowercase)
 316             random.shuffle(cipher_alphabet)
 317             cipher_alphabet = cat(cipher_alphabet)
 318         worker_args.append((ciphertext, plain_alphabet, cipher_alphabet,
 319                             initial_temperature, max_iterations, fitness))
 320     with Pool() as pool:
 321         breaks = pool.starmap(simulated_annealing_break_worker,
 322                               worker_args, chunksize)
 323     return max(breaks, key=lambda k: k[1])
 324
 325
 326 def simulated_annealing_break_worker(message, plain_alphabet, cipher_alphabet,
 327                                      t0, max_iterations, fitness):
 328     def swap(letters, i, j):
 329         if i > j:
 330             i, j = j, i
 331         if i == j:
 332             return letters
 333         else:
 334             return (letters[:i] + letters[j] + letters[i+1:j] + letters[i] +
 335                     letters[j+1:])
 336
 337     temperature = t0
 338
 339     dt = t0 / (0.9 * max_iterations)
 340
 341     current_alphabet = cipher_alphabet
 342     alphabet = current_alphabet
 343     cipher_translation = ''.maketrans(current_alphabet, plain_alphabet)
 344     plaintext = message.translate(cipher_translation)
 345     current_fitness = fitness(plaintext)
 346
 347     best_alphabet = current_alphabet
 348     best_fitness = current_fitness
 349     best_plaintext = plaintext
 350
 351     # print('starting for', max_iterations)
 352     for i in range(max_iterations):
 353         swap_a = random.randrange(26)
 354         swap_b = (swap_a + int(random.gauss(0, 4))) % 26
 355         alphabet = swap(current_alphabet, swap_a, swap_b)
 356         cipher_translation = ''.maketrans(alphabet, plain_alphabet)
 357         plaintext = message.translate(cipher_translation)
 358         new_fitness = fitness(plaintext)
 359         try:
 360             sa_chance = math.exp((new_fitness - current_fitness) / temperature)
 361         except (OverflowError, ZeroDivisionError):
 362             # print('exception triggered: new_fit {}, current_fit {}, temp {}'.format(new_fitness, current_fitness, temperature))
 363             sa_chance = 0
 364         if (new_fitness > current_fitness or random.random() < sa_chance):
 365             # logger.debug('Simulated annealing: iteration {}, temperature {}, '
 366             #     'current alphabet {}, current_fitness {}, '
 367             #     'best_plaintext {}'.format(i, temperature, current_alphabet,
 368             #     current_fitness, best_plaintext[:50]))
 369
 370             # logger.debug('new_fit {}, current_fit {}, temp {}, sa_chance {}'.format(new_fitness, current_fitness, temperature, sa_chance))
 371             current_fitness = new_fitness
 372             current_alphabet = alphabet
 373
 374         if current_fitness > best_fitness:
 375             best_alphabet = current_alphabet
 376             best_fitness = current_fitness
 377             best_plaintext = plaintext
 378         if i % 500 == 0:
 379             logger.debug('Simulated annealing: iteration {}, temperature {}, '
 380                 'current alphabet {}, current_fitness {}, '
 381                 'best_plaintext {}'.format(i, temperature, current_alphabet,
 382                 current_fitness, plaintext[:50]))
 383         temperature = max(temperature - dt, 0.001)
 384
 385     return best_alphabet, best_fitness # current_alphabet, current_fitness
 386
 387
 388 def vigenere_keyword_break_mp(message, wordlist=keywords, fitness=Pletters,
 389                               chunksize=500):
 390     """Breaks a vigenere cipher using a dictionary and frequency analysis.
 391
 392     >>> vigenere_keyword_break_mp(vigenere_encipher(sanitise('this is a test ' \
 393              'message for the vigenere decipherment'), 'cat'), \
 394              wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS
 395     ('cat', -52.9472712...)
 396     """
 397     with Pool() as pool:
 398         helper_args = [(message, word, fitness)
 399                        for word in wordlist]
 400         # Gotcha: the helper function here needs to be defined at the top level
 401         #   (limitation of Pool.starmap)
 402         breaks = pool.starmap(vigenere_keyword_break_worker, helper_args,
 403                               chunksize)
 404         return max(breaks, key=lambda k: k[1])
 405 vigenere_keyword_break = vigenere_keyword_break_mp
 406
 407 def vigenere_keyword_break_worker(message, keyword, fitness):
 408     plaintext = vigenere_decipher(message, keyword)
 409     fit = fitness(plaintext)
 410     logger.debug('Vigenere keyword break attempt using key {0} gives fit of '
 411                  '{1} and decrypt starting: {2}'.format(keyword,
 412                      fit, sanitise(plaintext)[:50]))
 413     return keyword, fit
 414
 415
 416 def vigenere_frequency_break(message, max_key_length=20, fitness=Pletters):
 417     """Breaks a Vigenere cipher with frequency analysis
 418
 419     >>> vigenere_frequency_break(vigenere_encipher(sanitise("It is time to " \
 420             "run. She is ready and so am I. I stole Daniel's pocketbook this " \
 421             "afternoon when he left his jacket hanging on the easel in the " \
 422             "attic. I jump every time I hear a footstep on the stairs, " \
 423             "certain that the theft has been discovered and that I will " \
 424             "be caught. The SS officer visits less often now that he is " \
 425             "sure"), 'florence')) # doctest: +ELLIPSIS
 426     ('florence', -307.5473096...)
 427     """
 428     def worker(message, key_length, fitness):
 429         splits = every_nth(sanitised_message, key_length)
 430         key = cat([unpos(caesar_break(s)[0]) for s in splits])
 431         plaintext = vigenere_decipher(message, key)
 432         fit = fitness(plaintext)
 433         return key, fit
 434     sanitised_message = sanitise(message)
 435     results = starmap(worker, [(sanitised_message, i, fitness)
 436                                for i in range(1, max_key_length+1)])
 437     return max(results, key=lambda k: k[1])
 438
 439
 440 def beaufort_sub_break(message, fitness=Pletters):
 441     """Breaks one chunk of a Beaufort cipher with frequency analysis
 442
 443     >>> beaufort_sub_break('samwpplggnnmmyaazgympjapopnwiywwomwspgpjmefwmawx' \
 444       'jafjhxwwwdigxshnlywiamhyshtasxptwueahhytjwsn') # doctest: +ELLIPSIS
 445     (0, -117.4492...)
 446     >>> beaufort_sub_break('eyprzjjzznxymrygryjqmqhznjrjjapenejznawngnnezgza' \
 447       'dgndknaogpdjneadadazlhkhxkryevrronrmdjnndjlo') # doctest: +ELLIPSIS
 448     (17, -114.9598...)
 449     """
 450     best_shift = 0
 451     best_fit = float('-inf')
 452     for key in range(26):
 453         plaintext = [unpos(key - pos(l)) for l in message]
 454         fit = fitness(plaintext)
 455         logger.debug('Beaufort sub break attempt using key {0} gives fit of {1} '
 456                      'and decrypt starting: {2}'.format(key, fit,
 457                                                         plaintext[:50]))
 458         if fit > best_fit:
 459             best_fit = fit
 460             best_key = key
 461     logger.info('Beaufort sub break best fit: key {0} gives fit of {1} and '
 462                 'decrypt starting: {2}'.format(best_key, best_fit,
 463                     cat([unpos(best_key - pos(l)) for l in message[:50]])))
 464     return best_key, best_fit
 465
 466
 467 def beaufort_frequency_break(message, max_key_length=20, fitness=Pletters):
 468     """Breaks a Beaufort cipher with frequency analysis
 469
 470     >>> beaufort_frequency_break(beaufort_encipher(sanitise("It is time to " \
 471             "run. She is ready and so am I. I stole Daniel's pocketbook this " \
 472             "afternoon when he left his jacket hanging on the easel in the " \
 473             "attic. I jump every time I hear a footstep on the stairs, " \
 474             "certain that the theft has been discovered and that I will " \
 475             "be caught. The SS officer visits less often now " \
 476             "that he is sure"), 'florence')) # doctest: +ELLIPSIS
 477     ('florence', -307.5473096791...)
 478     """
 479     def worker(message, key_length, fitness):
 480         splits = every_nth(message, key_length)
 481         key = cat([unpos(beaufort_sub_break(s)[0]) for s in splits])
 482         plaintext = beaufort_decipher(message, key)
 483         fit = fitness(plaintext)
 484         return key, fit
 485     sanitised_message = sanitise(message)
 486     results = starmap(worker, [(sanitised_message, i, fitness)
 487                                for i in range(1, max_key_length+1)])
 488     return max(results, key=lambda k: k[1])
 489
 490
 491 def beaufort_variant_frequency_break(message, max_key_length=20, fitness=Pletters):
 492     """Breaks a Beaufort cipher with frequency analysis
 493
 494     >>> beaufort_variant_frequency_break(beaufort_variant_encipher(sanitise("It is time to " \
 495             "run. She is ready and so am I. I stole Daniel's pocketbook this " \
 496             "afternoon when he left his jacket hanging on the easel in the " \
 497             "attic. I jump every time I hear a footstep on the stairs, " \
 498             "certain that the theft has been discovered and that I will " \
 499             "be caught. The SS officer visits less often now " \
 500             "that he is sure"), 'florence')) # doctest: +ELLIPSIS
 501     ('florence', -307.5473096791...)
 502     """
 503     def worker(message, key_length, fitness):
 504         splits = every_nth(sanitised_message, key_length)
 505         key = cat([unpos(-caesar_break(s)[0]) for s in splits])
 506         plaintext = beaufort_variant_decipher(message, key)
 507         fit = fitness(plaintext)
 508         return key, fit
 509     sanitised_message = sanitise(message)
 510     results = starmap(worker, [(sanitised_message, i, fitness)
 511                                for i in range(1, max_key_length+1)])
 512     return max(results, key=lambda k: k[1])
 513
 514 def polybius_break_mp(message, column_labels, row_labels,
 515                       letters_to_merge=None,
 516                       wordlist=keywords, fitness=Pletters,
 517                       number_of_solutions=1, chunksize=500):
 518     """Breaks a Polybius substitution cipher using a dictionary and
 519     frequency analysis
 520
 521     >>> polybius_break_mp(polybius_encipher('this is a test message for the ' \
 522           'polybius decipherment', 'elephant', 'abcde', 'abcde'), \
 523           'abcde', 'abcde', \
 524           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 525     (('elephant', <KeywordWrapAlphabet.from_a: 1>, 'abcde', 'abcde', False), \
 526     -54.53880...)
 527     >>> polybius_break_mp(polybius_encipher('this is a test message for the ' \
 528           'polybius decipherment', 'elephant', 'abcde', 'abcde', column_first=True), \
 529           'abcde', 'abcde', \
 530           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 531     (('elephant', <KeywordWrapAlphabet.from_a: 1>, 'abcde', 'abcde', True), \
 532     -54.53880...)
 533     >>> polybius_break_mp(polybius_encipher('this is a test message for the ' \
 534           'polybius decipherment', 'elephant', 'abcde', 'abcde', column_first=False), \
 535           'abcde', 'abcde', \
 536           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 537     (('elephant', <KeywordWrapAlphabet.from_a: 1>, 'abcde', 'abcde', False), \
 538     -54.53880...)
 539     >>> polybius_break_mp(polybius_encipher('this is a test message for the ' \
 540           'polybius decipherment', 'elephant', 'abcde', 'pqrst', column_first=True), \
 541           'abcde', 'pqrst', \
 542           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 543     (('elephant', <KeywordWrapAlphabet.from_a: 1>, 'abcde', 'pqrst', True), \
 544     -54.53880...)
 545     """
 546     if letters_to_merge is None:
 547         letters_to_merge = {'j': 'i'}
 548     with Pool() as pool:
 549         helper_args = [(message, word, wrap,
 550                         column_labels, row_labels, column_first,
 551                         letters_to_merge,
 552                         fitness)
 553                        for word in wordlist
 554                        for wrap in KeywordWrapAlphabet
 555                        for column_first in [False, True]]
 556         # Gotcha: the helper function here needs to be defined at the top level
 557         #   (limitation of Pool.starmap)
 558         breaks = pool.starmap(polybius_break_worker, helper_args, chunksize)
 559         if number_of_solutions == 1:
 560             return max(breaks, key=lambda k: k[1])
 561         else:
 562             return sorted(breaks, key=lambda k: k[1], reverse=True)[:number_of_solutions]
 563
 564 def polybius_break_worker(message, keyword, wrap_alphabet,
 565                           column_order, row_order, column_first,
 566                           letters_to_merge,
 567                           fitness):
 568     plaintext = polybius_decipher(message, keyword,
 569                                   column_order, row_order,
 570                                   column_first=column_first,
 571                                   letters_to_merge=letters_to_merge,
 572                                   wrap_alphabet=wrap_alphabet)
 573     if plaintext:
 574         fit = fitness(plaintext)
 575     else:
 576         fit = float('-inf')
 577     logger.debug('Polybius break attempt using key {0} (wrap={1}, merging {2}), '
 578                  'columns as {3}, rows as {4} (column_first={5}) '
 579                  'gives fit of {6} and decrypt starting: '
 580                  '{7}'.format(keyword, wrap_alphabet, letters_to_merge,
 581                               column_order, row_order, column_first,
 582                               fit, sanitise(plaintext)[:50]))
 583     return (keyword, wrap_alphabet, column_order, row_order, column_first), fit
 584
 585
 586 def column_transposition_break_mp(message, translist=transpositions,
 587                                   fitness=Pbigrams, chunksize=500):
 588     """Breaks a column transposition cipher using a dictionary and
 589     n-gram frequency analysis
 590
 591     >>> column_transposition_break_mp(column_transposition_encipher(sanitise( \
 592             "It is a truth universally acknowledged, that a single man in \
 593              possession of a good fortune, must be in want of a wife. However \
 594              little known the feelings or views of such a man may be on his \
 595              first entering a neighbourhood, this truth is so well fixed in \
 596              the minds of the surrounding families, that he is considered the \
 597              rightful property of some one or other of their daughters."), \
 598         'encipher'), \
 599         translist={(2, 0, 5, 3, 1, 4, 6): ['encipher'], \
 600                    (5, 0, 6, 1, 3, 4, 2): ['fourteen'], \
 601                    (6, 1, 0, 4, 5, 3, 2): ['keyword']}) # doctest: +ELLIPSIS
 602     (((2, 0, 5, 3, 1, 4, 6), False, False), -709.4646722...)
 603     >>> column_transposition_break_mp(column_transposition_encipher(sanitise( \
 604             "It is a truth universally acknowledged, that a single man in \
 605              possession of a good fortune, must be in want of a wife. However \
 606              little known the feelings or views of such a man may be on his \
 607              first entering a neighbourhood, this truth is so well fixed in \
 608              the minds of the surrounding families, that he is considered the \
 609              rightful property of some one or other of their daughters."), \
 610         'encipher'), \
 611         translist={(2, 0, 5, 3, 1, 4, 6): ['encipher'], \
 612                    (5, 0, 6, 1, 3, 4, 2): ['fourteen'], \
 613                    (6, 1, 0, 4, 5, 3, 2): ['keyword']}, \
 614         fitness=Ptrigrams) # doctest: +ELLIPSIS
 615     (((2, 0, 5, 3, 1, 4, 6), False, False), -997.0129085...)
 616     """
 617     with Pool() as pool:
 618         helper_args = [(message, trans, fillcolumnwise, emptycolumnwise,
 619                         fitness)
 620                        for trans in translist
 621                        for fillcolumnwise in [True, False]
 622                        for emptycolumnwise in [True, False]]
 623         # Gotcha: the helper function here needs to be defined at the top level
 624         #   (limitation of Pool.starmap)
 625         breaks = pool.starmap(column_transposition_break_worker,
 626                               helper_args, chunksize)
 627         return max(breaks, key=lambda k: k[1])
 628 column_transposition_break = column_transposition_break_mp
 629
 630 def column_transposition_break_worker(message, transposition,
 631         fillcolumnwise, emptycolumnwise, fitness):
 632     plaintext = column_transposition_decipher(message, transposition,
 633         fillcolumnwise=fillcolumnwise, emptycolumnwise=emptycolumnwise)
 634     fit = fitness(sanitise(plaintext))
 635     logger.debug('Column transposition break attempt using key {0} '
 636                          'gives fit of {1} and decrypt starting: {2}'.format(
 637                              transposition, fit,
 638                              sanitise(plaintext)[:50]))
 639     return (transposition, fillcolumnwise, emptycolumnwise), fit
 640
 641
 642 def scytale_break_mp(message, max_key_length=20,
 643                      fitness=Pbigrams, chunksize=500):
 644     """Breaks a scytale cipher using a range of lengths and
 645     n-gram frequency analysis
 646
 647     >>> scytale_break_mp(scytale_encipher(sanitise( \
 648             "It is a truth universally acknowledged, that a single man in \
 649              possession of a good fortune, must be in want of a wife. However \
 650              little known the feelings or views of such a man may be on his \
 651              first entering a neighbourhood, this truth is so well fixed in \
 652              the minds of the surrounding families, that he is considered the \
 653              rightful property of some one or other of their daughters."), \
 654         5)) # doctest: +ELLIPSIS
 655     (5, -709.4646722...)
 656     >>> scytale_break_mp(scytale_encipher(sanitise( \
 657             "It is a truth universally acknowledged, that a single man in \
 658              possession of a good fortune, must be in want of a wife. However \
 659              little known the feelings or views of such a man may be on his \
 660              first entering a neighbourhood, this truth is so well fixed in \
 661              the minds of the surrounding families, that he is considered the \
 662              rightful property of some one or other of their daughters."), \
 663         5), \
 664         fitness=Ptrigrams) # doctest: +ELLIPSIS
 665     (5, -997.0129085...)
 666     """
 667     with Pool() as pool:
 668         helper_args = [(message, trans, False, True, fitness)
 669             for trans in
 670                 [[col for col in range(math.ceil(len(message)/rows))]
 671                     for rows in range(1,max_key_length+1)]]
 672         # Gotcha: the helper function here needs to be defined at the top level
 673         #   (limitation of Pool.starmap)
 674         breaks = pool.starmap(column_transposition_break_worker,
 675                               helper_args, chunksize)
 676         best = max(breaks, key=lambda k: k[1])
 677         return math.trunc(len(message) / len(best[0][0])), best[1]
 678 scytale_break = scytale_break_mp
 679
 680
 681 def railfence_break(message, max_key_length=20,
 682                      fitness=Pletters, chunksize=500):
 683     """Breaks a railfence cipher using a matrix of given rank and letter frequencies
 684
 685
 686     """
 687
 688     sanitised_message = sanitise(message)
 689     results = starmap(worker, [(sanitised_message, i, fitness)
 690                                for i in range(2, max_key_length+1)])
 691     return max(results, key=lambda k: k[1])
 692
 693
 694 def railfence_break(message, max_key_length=20,
 695                      fitness=Pbigrams, chunksize=500):
 696     """Breaks a railfence cipher using a range of lengths and
 697     n-gram frequency analysis
 698
 699     >>> railfence_break(railfence_encipher(sanitise( \
 700             "It is a truth universally acknowledged, that a single man in \
 701              possession of a good fortune, must be in want of a wife. However \
 702              little known the feelings or views of such a man may be on his \
 703              first entering a neighbourhood, this truth is so well fixed in \
 704              the minds of the surrounding families, that he is considered the \
 705              rightful property of some one or other of their daughters."), \
 706         7)) # doctest: +ELLIPSIS
 707     (7, -709.46467226...)
 708     >>> railfence_break(railfence_encipher(sanitise( \
 709             "It is a truth universally acknowledged, that a single man in \
 710              possession of a good fortune, must be in want of a wife. However \
 711              little known the feelings or views of such a man may be on his \
 712              first entering a neighbourhood, this truth is so well fixed in \
 713              the minds of the surrounding families, that he is considered the \
 714              rightful property of some one or other of their daughters."), \
 715         7), \
 716         fitness=Ptrigrams) # doctest: +ELLIPSIS
 717     (7, -997.0129085...)
 718     """
 719     def worker(message, height, fitness):
 720         plaintext = railfence_decipher(message, height)
 721         fit = fitness(plaintext)
 722         return height, fit
 723
 724     sanitised_message = sanitise(message)
 725     results = starmap(worker, [(sanitised_message, i, fitness)
 726                                for i in range(2, max_key_length+1)])
 727     return max(results, key=lambda k: k[1])
 728
 729 def amsco_break(message, translist=transpositions, patterns = [(1, 2), (2, 1)],
 730                                   fillstyles = [AmscoFillStyle.continuous,
 731                                                 AmscoFillStyle.same_each_row,
 732                                                 AmscoFillStyle.reverse_each_row],
 733                                   fitness=Pbigrams,
 734                                   chunksize=500):
 735     """Breaks an AMSCO transposition cipher using a dictionary and
 736     n-gram frequency analysis
 737
 738     >>> amsco_break(amsco_transposition_encipher(sanitise( \
 739             "It is a truth universally acknowledged, that a single man in \
 740              possession of a good fortune, must be in want of a wife. However \
 741              little known the feelings or views of such a man may be on his \
 742              first entering a neighbourhood, this truth is so well fixed in \
 743              the minds of the surrounding families, that he is considered the \
 744              rightful property of some one or other of their daughters."), \
 745         'encipher'), \
 746         translist={(2, 0, 5, 3, 1, 4, 6): ['encipher'], \
 747                    (5, 0, 6, 1, 3, 4, 2): ['fourteen'], \
 748                    (6, 1, 0, 4, 5, 3, 2): ['keyword']}, \
 749         patterns=[(1, 2)]) # doctest: +ELLIPSIS
 750     (((2, 0, 5, 3, 1, 4, 6), (1, 2), <AmscoFillStyle.continuous: 1>), -709.4646722...)
 751     >>> amsco_break(amsco_transposition_encipher(sanitise( \
 752             "It is a truth universally acknowledged, that a single man in \
 753              possession of a good fortune, must be in want of a wife. However \
 754              little known the feelings or views of such a man may be on his \
 755              first entering a neighbourhood, this truth is so well fixed in \
 756              the minds of the surrounding families, that he is considered the \
 757              rightful property of some one or other of their daughters."), \
 758         'encipher', fillpattern=(2, 1)), \
 759         translist={(2, 0, 5, 3, 1, 4, 6): ['encipher'], \
 760                    (5, 0, 6, 1, 3, 4, 2): ['fourteen'], \
 761                    (6, 1, 0, 4, 5, 3, 2): ['keyword']}, \
 762         patterns=[(1, 2), (2, 1)], fitness=Ptrigrams) # doctest: +ELLIPSIS
 763     (((2, 0, 5, 3, 1, 4, 6), (2, 1), <AmscoFillStyle.continuous: 1>), -997.0129085...)
 764     """
 765     with Pool() as pool:
 766         helper_args = [(message, trans, pattern, fillstyle, fitness)
 767                        for trans in translist
 768                        for pattern in patterns
 769                        for fillstyle in fillstyles]
 770         # Gotcha: the helper function here needs to be defined at the top level
 771         #   (limitation of Pool.starmap)
 772         breaks = pool.starmap(amsco_break_worker, helper_args, chunksize)
 773         return max(breaks, key=lambda k: k[1])
 774
 775 def amsco_break_worker(message, transposition,
 776         pattern, fillstyle, fitness):
 777     plaintext = amsco_transposition_decipher(message, transposition,
 778         fillpattern=pattern, fillstyle=fillstyle)
 779     fit = fitness(sanitise(plaintext))
 780     logger.debug('AMSCO transposition break attempt using key {0} and pattern'
 781                          '{1} ({2}) gives fit of {3} and decrypt starting: '
 782                          '{4}'.format(
 783                              transposition, pattern, fillstyle, fit,
 784                              sanitise(plaintext)[:50]))
 785     return (transposition, pattern, fillstyle), fit
 786
 787
 788 def hill_break(message, matrix_size=2, fitness=Pletters,
 789     number_of_solutions=1, chunksize=500):
 790
 791     all_matrices = [np.matrix(list(m))
 792         for m in itertools.product([list(r)
 793             for r in itertools.product(range(26), repeat=matrix_size)],
 794         repeat=matrix_size)]
 795     valid_matrices = [m for m, d in
 796         zip(all_matrices, (int(round(linalg.det(m))) for m in all_matrices))
 797                   if d != 0
 798                   if d % 2 != 0
 799                   if d % 13 != 0 ]
 800     with Pool() as pool:
 801         helper_args = [(message, matrix, fitness)
 802                        for matrix in valid_matrices]
 803         # Gotcha: the helper function here needs to be defined at the top level
 804         #   (limitation of Pool.starmap)
 805         breaks = pool.starmap(hill_break_worker, helper_args, chunksize)
 806         if number_of_solutions == 1:
 807             return max(breaks, key=lambda k: k[1])
 808         else:
 809             return sorted(breaks, key=lambda k: k[1], reverse=True)[:number_of_solutions]
 810
 811 def hill_break_worker(message, matrix, fitness):
 812     plaintext = hill_decipher(matrix, message)
 813     fit = fitness(plaintext)
 814     logger.debug('Hill cipher break attempt using key {0} gives fit of '
 815                  '{1} and decrypt starting: {2}'.format(matrix,
 816                      fit, sanitise(plaintext)[:50]))
 817     return matrix, fit
 818
 819 def bifid_break_mp(message, wordlist=keywords, fitness=Pletters, max_period=10,
 820                      number_of_solutions=1, chunksize=500):
 821     """Breaks a keyword substitution cipher using a dictionary and
 822     frequency analysis
 823
 824     >>> bifid_break_mp(bifid_encipher('this is a test message for the ' \
 825           'keyword decipherment', 'elephant', wrap_alphabet=KeywordWrapAlphabet.from_last), \
 826           wordlist=['cat', 'elephant', 'kangaroo']) # doctest: +ELLIPSIS
 827     (('elephant', <KeywordWrapAlphabet.from_last: 2>, 0), -52.834575011...)
 828     >>> bifid_break_mp(bifid_encipher('this is a test message for the ' \
 829           'keyword decipherment', 'elephant', wrap_alphabet=KeywordWrapAlphabet.from_last), \
 830           wordlist=['cat', 'elephant', 'kangaroo'], \
 831           number_of_solutions=2) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
 832     [(('elephant', <KeywordWrapAlphabet.from_last: 2>, 0), -52.834575011...),
 833     (('elephant', <KeywordWrapAlphabet.from_largest: 3>, 0), -52.834575011...)]
 834     """
 835     with Pool() as pool:
 836         helper_args = [(message, word, wrap, period, fitness)
 837                        for word in wordlist
 838                        for wrap in KeywordWrapAlphabet
 839                        for period in range(max_period+1)]
 840         # Gotcha: the helper function here needs to be defined at the top level
 841         #   (limitation of Pool.starmap)
 842         breaks = pool.starmap(bifid_break_worker, helper_args, chunksize)
 843         if number_of_solutions == 1:
 844             return max(breaks, key=lambda k: k[1])
 845         else:
 846             return sorted(breaks, key=lambda k: k[1], reverse=True)[:number_of_solutions]
 847
 848 def bifid_break_worker(message, keyword, wrap_alphabet, period, fitness):
 849     plaintext = bifid_decipher(message, keyword, wrap_alphabet, period=period)
 850     fit = fitness(plaintext)
 851     logger.debug('Keyword break attempt using key {0} (wrap={1}) gives fit of '
 852                  '{2} and decrypt starting: {3}'.format(keyword,
 853                      wrap_alphabet, fit, sanitise(plaintext)[:50]))
 854     return (keyword, wrap_alphabet, period), fit
 855
 856
 857 def autokey_sa_break( message
 858                     , min_keylength=2
 859                     , max_keylength=20
 860                     , workers=10
 861                     , initial_temperature=200
 862                     , max_iterations=20000
 863                     , fitness=Pletters
 864                     , chunksize=1
 865                     , result_count=1
 866                     ):
 867     """Break an autokey cipher by simulated annealing
 868     """
 869     worker_args = []
 870     ciphertext = sanitise(message)
 871     for keylength in range(min_keylength, max_keylength+1):
 872         for i in range(workers):
 873             key = cat(random.choice(string.ascii_lowercase) for _ in range(keylength))
 874             worker_args.append((ciphertext, key,
 875                             initial_temperature, max_iterations, fitness))
 876
 877     with Pool() as pool:
 878         breaks = pool.starmap(autokey_sa_break_worker,
 879                               worker_args, chunksize)
 880     if result_count <= 1:
 881         return max(breaks, key=lambda k: k[1])
 882     else:
 883         return sorted(set(breaks), key=lambda k: k[1], reverse=True)[:result_count]
 884
 885
 886 def autokey_sa_break_worker(message, key,
 887                                      t0, max_iterations, fitness):
 888
 889     temperature = t0
 890
 891     dt = t0 / (0.9 * max_iterations)
 892
 893     plaintext = autokey_decipher(message, key)
 894     current_fitness = fitness(plaintext)
 895     current_key = key
 896
 897     best_key = current_key
 898     best_fitness = current_fitness
 899     best_plaintext = plaintext
 900
 901     # print('starting for', max_iterations)
 902     for i in range(max_iterations):
 903         swap_pos = random.randrange(len(current_key))
 904         swap_char = random.choice(string.ascii_lowercase)
 905
 906         new_key = current_key[:swap_pos] + swap_char + current_key[swap_pos+1:]
 907
 908         plaintext = autokey_decipher(message, new_key)
 909         new_fitness = fitness(plaintext)
 910         try:
 911             sa_chance = math.exp((new_fitness - current_fitness) / temperature)
 912         except (OverflowError, ZeroDivisionError):
 913             # print('exception triggered: new_fit {}, current_fit {}, temp {}'.format(new_fitness, current_fitness, temperature))
 914             sa_chance = 0
 915         if (new_fitness > current_fitness or random.random() < sa_chance):
 916             # logger.debug('Simulated annealing: iteration {}, temperature {}, '
 917             #     'current alphabet {}, current_fitness {}, '
 918             #     'best_plaintext {}'.format(i, temperature, current_alphabet,
 919             #     current_fitness, best_plaintext[:50]))
 920
 921             # logger.debug('new_fit {}, current_fit {}, temp {}, sa_chance {}'.format(new_fitness, current_fitness, temperature, sa_chance))
 922 #             print(new_fitness, new_key, plaintext[:100])
 923             current_fitness = new_fitness
 924             current_key = new_key
 925
 926         if current_fitness > best_fitness:
 927             best_key = current_key
 928             best_fitness = current_fitness
 929             best_plaintext = plaintext
 930         if i % 500 == 0:
 931             logger.debug('Simulated annealing: iteration {}, temperature {}, '
 932                 'current key {}, current_fitness {}, '
 933                 'best_plaintext {}'.format(i, temperature, current_key,
 934                 current_fitness, plaintext[:50]))
 935         temperature = max(temperature - dt, 0.001)
 936
 937 #     print(best_key, best_fitness, best_plaintext[:70])
 938     return best_key, best_fitness # current_alphabet, current_fitness
 939
 940
 941 def pocket_enigma_break_by_crib(message, wheel_spec, crib, crib_position):
 942     """Break a pocket enigma using a crib (some plaintext that's expected to
 943     be in a certain position). Returns a list of possible starting wheel
 944     positions that could produce the crib.
 945
 946     >>> pocket_enigma_break_by_crib('kzpjlzmoga', 1, 'h', 0)
 947     ['a', 'f', 'q']
 948     >>> pocket_enigma_break_by_crib('kzpjlzmoga', 1, 'he', 0)
 949     ['a']
 950     >>> pocket_enigma_break_by_crib('kzpjlzmoga', 1, 'll', 2)
 951     ['a']
 952     >>> pocket_enigma_break_by_crib('kzpjlzmoga', 1, 'l', 2)
 953     ['a']
 954     >>> pocket_enigma_break_by_crib('kzpjlzmoga', 1, 'l', 3)
 955     ['a', 'j', 'n']
 956     >>> pocket_enigma_break_by_crib('aaaaa', 1, 'l', 3)
 957     []
 958     """
 959     pe = PocketEnigma(wheel=wheel_spec)
 960     possible_positions = []
 961     for p in string.ascii_lowercase:
 962         pe.set_position(p)
 963         plaintext = pe.decipher(message)
 964         if plaintext[crib_position:crib_position+len(crib)] == crib:
 965             possible_positions += [p]
 966     return possible_positions
 967
 968
 969 def plot_frequency_histogram(freqs, sort_key=None):
 970     x = range(len(freqs))
 971     y = [freqs[l] for l in sorted(freqs, key=sort_key)]
 972     f = plt.figure()
 973     ax = f.add_axes([0.1, 0.1, 0.9, 0.9])
 974     ax.bar(x, y, align='center')
 975     ax.set_xticks(x)
 976     ax.set_xticklabels(sorted(freqs, key=sort_key))
 977     f.show()
 978
 979
 980 if __name__ == "__main__":
 981     import doctest
 982     doctest.testmod()