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Added new algorithm: cracking caesar cipher with the chi-squared test (TheAlgorithms#1950)
* added decrypt_caesar_with_chi_squared.py and ran all checks * Updated default parameters Removed mistake with mutable default arguments Co-authored-by: Christian Clauss <[email protected]> * Updated handling for optional arguments Co-authored-by: Christian Clauss <[email protected]> * Changed return statement to tuple Made function return a tuple instead of a list * Added more doctests * Fixed spelling mistakes * black . - reformatted decrypt_caesar_with_chi_squared.py * Updated if statements to fit the updated code * Minimized amount of lines in the code. Co-authored-by: Christian Clauss <[email protected]>
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‎ciphers/decrypt_caesar_with_chi_squared.py

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def decrypt_caesar_with_chi_squared(
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ciphertext: str,
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cipher_alphabet=None,
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frequencies_dict=None,
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case_sensetive: bool = False,
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) -> list:
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"""
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Basic Usage
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===========
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Arguments:
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* ciphertext (str): the text to decode (encoded with the caesar cipher)
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Optional Arguments:
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* cipher_alphabet (list): the alphabet used for the cipher (each letter is
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a string separated by commas)
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* frequencies_dict (dict): a dictionary of word frequencies where keys are
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the letters and values are a percentage representation of the frequency as
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a decimal/float
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* case_sensetive (bool): a boolean value: True if the case matters during
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decryption, False if it doesn't
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Returns:
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* A tuple in the form of:
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(
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most_likely_cipher,
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most_likely_cipher_chi_squared_value,
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decoded_most_likely_cipher
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)
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where...
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- most_likely_cipher is an integer representing the shift of the smallest
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chi-squared statistic (most likely key)
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- most_likely_cipher_chi_squared_value is a float representing the
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chi-squared statistic of the most likely shift
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- decoded_most_likely_cipher is a string with the decoded cipher
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(decoded by the most_likely_cipher key)
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The Chi-squared test
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====================
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The caesar cipher
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-----------------
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The caesar cipher is a very insecure encryption algorithm, however it has
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been used since Julius Caesar. The cipher is a simple substitution cipher
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where each character in the plain text is replaced by a character in the
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alphabet a certain number of characters after the original character. The
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number of characters away is called the shift or key. For example:
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Plain text: hello
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Key: 1
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Cipher text: ifmmp
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(each letter in hello has been shifted one to the right in the eng. alphabet)
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As you can imagine, this doesn't provide lots of security. In fact
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decrypting ciphertext by brute-force is extremely easy even by hand. However
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one way to do that is the chi-squared test.
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The chi-squared test
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-------------------
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Each letter in the english alphabet has a frequency, or the amount of times
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it shows up compared to other letters (usually expressed as a decimal
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representing the percentage likelihood). The most common letter in the
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english language is "e" with a frequency of 0.11162 or 11.162%. The test is
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completed in the following fashion.
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1. The ciphertext is decoded in a brute force way (every combination of the
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26 possible combinations)
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2. For every combination, for each letter in the combination, the average
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amount of times the letter should appear the message is calculated by
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multiplying the total number of characters by the frequency of the letter
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For example:
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In a message of 100 characters, e should appear around 11.162 times.
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3. Then, to calculate the margin of error (the amount of times the letter
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SHOULD appear with the amount of times the letter DOES appear), we use
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the chi-squared test. The following formula is used:
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Let:
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- n be the number of times the letter actually appears
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- p be the predicted value of the number of times the letter should
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appear (see #2)
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- let v be the chi-squared test result (referred to here as chi-squared
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value/statistic)
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(n - p)^2
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--------- = v
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p
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4. Each chi squared value for each letter is then added up to the total.
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The total is the chi-squared statistic for that encryption key.
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5. The encryption key with the lowest chi-squared value is the most likely
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to be the decoded answer.
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Further Reading
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================
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* http://practicalcryptography.com/cryptanalysis/text-characterisation/chi-squared-statistic/
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* https://en.wikipedia.org/wiki/Letter_frequency
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* https://en.wikipedia.org/wiki/Chi-squared_test
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* https://en.m.wikipedia.org/wiki/Caesar_cipher
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Doctests
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========
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>>> decrypt_caesar_with_chi_squared('dof pz aol jhlzhy jpwoly zv wvwbshy? pa pz avv lhzf av jyhjr!')
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(7, 3129.228005747531, 'why is the caesar cipher so popular? it is too easy to crack!')
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>>> decrypt_caesar_with_chi_squared('crybd cdbsxq')
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(10, 233.35343938980898, 'short string')
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>>> decrypt_caesar_with_chi_squared(12)
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Traceback (most recent call last):
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AttributeError: 'int' object has no attribute 'lower'
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"""
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alphabet_letters = cipher_alphabet or [chr(i) for i in range(97, 123)]
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frequencies_dict = frequencies_dict or {}
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if frequencies_dict == {}:
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# Frequencies of letters in the english language (how much they show up)
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frequencies = {
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"a": 0.08497,
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"b": 0.01492,
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"c": 0.02202,
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"d": 0.04253,
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"e": 0.11162,
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"f": 0.02228,
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"g": 0.02015,
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"h": 0.06094,
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"i": 0.07546,
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"j": 0.00153,
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"k": 0.01292,
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"l": 0.04025,
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"m": 0.02406,
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"n": 0.06749,
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"o": 0.07507,
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"p": 0.01929,
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"q": 0.00095,
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"r": 0.07587,
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"s": 0.06327,
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"t": 0.09356,
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"u": 0.02758,
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"v": 0.00978,
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"w": 0.02560,
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"x": 0.00150,
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"y": 0.01994,
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"z": 0.00077,
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}
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else:
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# Custom frequencies dictionary
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frequencies = frequencies_dict
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if not case_sensetive:
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ciphertext = ciphertext.lower()
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# Chi squared statistic values
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chi_squared_statistic_values = {}
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# cycle through all of the shifts
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for shift in range(len(alphabet_letters)):
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decrypted_with_shift = ""
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# decrypt the message with the shift
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for letter in ciphertext:
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try:
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# Try to index the letter in the alphabet
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new_key = (alphabet_letters.index(letter) - shift) % len(
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alphabet_letters
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)
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decrypted_with_shift += alphabet_letters[new_key]
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except ValueError:
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# Append the character if it isn't in the alphabet
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decrypted_with_shift += letter
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chi_squared_statistic = 0
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# Loop through each letter in the decoded message with the shift
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for letter in decrypted_with_shift:
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if case_sensetive:
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if letter in frequencies:
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# Get the amount of times the letter occurs in the message
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occurrences = decrypted_with_shift.count(letter)
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# Get the excepcted amount of times the letter should appear based on letter frequencies
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expected = frequencies[letter] * occurrences
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# Complete the chi squared statistic formula
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chi_letter_value = ((occurrences - expected) ** 2) / expected
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# Add the margin of error to the total chi squared statistic
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chi_squared_statistic += chi_letter_value
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else:
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if letter.lower() in frequencies:
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# Get the amount of times the letter occurs in the message
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occurrences = decrypted_with_shift.count(letter)
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# Get the excepcted amount of times the letter should appear based on letter frequencies
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expected = frequencies[letter] * occurrences
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# Complete the chi squared statistic formula
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chi_letter_value = ((occurrences - expected) ** 2) / expected
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# Add the margin of error to the total chi squared statistic
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chi_squared_statistic += chi_letter_value
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# Add the data to the chi_squared_statistic_values dictionary
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chi_squared_statistic_values[shift] = [
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chi_squared_statistic,
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decrypted_with_shift,
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]
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# Get the most likely cipher by finding the cipher with the smallest chi squared statistic
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most_likely_cipher = min(
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chi_squared_statistic_values, key=chi_squared_statistic_values.get
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)
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# Get all the data from the most likely cipher (key, decoded message)
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most_likely_cipher_chi_squared_value = chi_squared_statistic_values[
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most_likely_cipher
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][0]
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decoded_most_likely_cipher = chi_squared_statistic_values[most_likely_cipher][1]
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# Return the data on the most likely shift
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return (
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most_likely_cipher,
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most_likely_cipher_chi_squared_value,
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decoded_most_likely_cipher,
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)

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