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# Classical Cryptosystems

A convenient user interface to various classical ciphers. These include:

- affine cipher; see `AffineCryptosystem`
- Hill or matrix cipher; see `HillCryptosystem`
- shift cipher; see `ShiftCryptosystem`
- substitution cipher; see `SubstitutionCryptosystem`
- transposition cipher; see `TranspositionCryptosystem`
- Vigenere cipher; see `VigenereCryptosystem`

These classical cryptosystems support alphabets such as:

- the capital letters of the English alphabet; see `AlphabeticStrings()`
- the hexadecimal number system; see `HexadecimalStrings()`
- the binary number system; see `BinaryStrings()`
- the octal number system; see `OctalStrings()`
- the radix-64 number system; see `Radix64Strings()`

AUTHORS:

- David Kohel (2007): initial version with the Hill, substitution, transposition, and Vigenere cryptosystems.
- Minh Van Nguyen (2009-08): shift cipher, affine cipher

class `sage.crypto.classical.``AffineCryptosystem`(A)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create an affine cryptosystem.

  Let \(A = \{ a_0, a_1, a_2, \dots, a_{n-1} \}\) be a non-empty alphabet consisting of \(n\) unique elements. Define a mapping \(f : A \longrightarrow \ZZ / n\ZZ\) from the alphabet \(A\) to the set \(\ZZ / n\ZZ\) of integers modulo \(n\), given by \(f(a_i) = i\). Thus we can identify each element of the alphabet \(A\) with a unique integer \(0 \leq i < n\). A key of the affine cipher is an ordered pair of integers \((a, b) \in \ZZ / n\ZZ \times \ZZ / n\ZZ\) such that \(\gcd(a, n) = 1\). Therefore the key space is \(\ZZ / n\ZZ \times \ZZ / n\ZZ\). Since we assume that \(A\) does not have repeated elements, the mapping \(f : A \longrightarrow \ZZ/ n\ZZ\) is bijective. Encryption and decryption functions are both affine functions. Let \((a,b)\) be a secret key, i.e. an element of the key space, and let \(p\) be a plaintext character and consequently \(p \in \ZZ / n\ZZ\). Then the ciphertext character \(c\) corresponding to \(p\) is given by

  \[c \equiv ap + b \pmod{n}\]

  Similarly, given a ciphertext character \(c \in \ZZ / n\ZZ\) and a secret key \((a,b)\), we can recover the corresponding plaintext character as follows:

  \[p \equiv a^{-1} (c - b) \pmod{n}\]

  where \(a^{-1}\) is the inverse of \(a\) modulo \(n\). Use the bijection \(f : A \longrightarrow \ZZ / n\ZZ\) to convert \(c\) and \(p\) back to elements of the alphabet \(A\). Currently, only the following alphabet is supported for the affine cipher:

  - capital letters of the English alphabet as implemented in `AlphabeticStrings()`

  EXAMPLES:

  Encryption and decryption over the capital letters of the English alphabet:

      sage: A = AffineCryptosystem(AlphabeticStrings()); A
      Affine cryptosystem on Free alphabetic string monoid on A-Z
      sage: P = A.encoding("The affine cryptosystem generalizes the shift cipher.")
      sage: P
      THEAFFINECRYPTOSYSTEMGENERALIZESTHESHIFTCIPHER
      sage: a, b = (9, 13)
      sage: C = A.enciphering(a, b, P); C
      CYXNGGHAXFKVSCJTVTCXRPXAXKNIHEXTCYXTYHGCFHSYXK
      sage: A.deciphering(a, b, C)
      THEAFFINECRYPTOSYSTEMGENERALIZESTHESHIFTCIPHER
      sage: A.deciphering(a, b, C) == P
      True

  We can also use functional notation to work through the previous example:

      sage: A = AffineCryptosystem(AlphabeticStrings()); A
      Affine cryptosystem on Free alphabetic string monoid on A-Z
      sage: P = A.encoding("The affine cryptosystem generalizes the shift cipher.")
      sage: P
      THEAFFINECRYPTOSYSTEMGENERALIZESTHESHIFTCIPHER
      sage: a, b = (9, 13)
      sage: E = A(a, b); E
      Affine cipher on Free alphabetic string monoid on A-Z
      sage: C = E(P); C
      CYXNGGHAXFKVSCJTVTCXRPXAXKNIHEXTCYXTYHGCFHSYXK
      sage: aInv, bInv = A.inverse_key(a, b)
      sage: D = A(aInv, bInv); D
      Affine cipher on Free alphabetic string monoid on A-Z
      sage: D(C)
      THEAFFINECRYPTOSYSTEMGENERALIZESTHESHIFTCIPHER
      sage: D(C) == P
      True
      sage: D(C) == P == D(E(P))
      True

  Encrypting the ciphertext with the inverse key also produces the plaintext:

      sage: A = AffineCryptosystem(AlphabeticStrings())
      sage: P = A.encoding("Encrypt with inverse key.")
      sage: a, b = (11, 8)
      sage: C = A.enciphering(a, b, P)
      sage: P; C
      ENCRYPTWITHINVERSEKEY
      AVENMRJQSJHSVFANYAOAM
      sage: aInv, bInv = A.inverse_key(a, b)
      sage: A.enciphering(aInv, bInv, C)
      ENCRYPTWITHINVERSEKEY
      sage: A.enciphering(aInv, bInv, C) == P
      True

  For a secret key \((a,b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\), if \(a = 1\) then any affine cryptosystem with key \((1, b)\) for any \(b \in \ZZ/n\ZZ\) is a shift cryptosystem. Here is how we can create a Caesar cipher using an affine cipher:

      sage: caesar = AffineCryptosystem(AlphabeticStrings())
      sage: a, b = (1, 3)
      sage: P = caesar.encoding("abcdef"); P
      ABCDEF
      sage: C = caesar.enciphering(a, b, P); C
      DEFGHI
      sage: caesar.deciphering(a, b, C) == P
      True

  Any affine cipher with keys of the form \((a,0) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) is called a decimation cipher on the Roman alphabet, or decimation cipher for short:

      sage: A = AffineCryptosystem(AlphabeticStrings())
      sage: P = A.encoding("A decimation cipher is a specialized affine cipher.")
      sage: a, b = (17, 0)
      sage: C = A.enciphering(a, b, P)
      sage: P; C
      ADECIMATIONCIPHERISASPECIALIZEDAFFINECIPHER
      AZQIGWALGENIGVPQDGUAUVQIGAFGJQZAHHGNQIGVPQD
      sage: A.deciphering(a, b, C) == P
      True

  Generate a random key for encryption and decryption:

      sage: A = AffineCryptosystem(AlphabeticStrings())
      sage: P = A.encoding("An affine cipher with a random key.")
      sage: a, b = A.random_key()
      sage: C = A.enciphering(a, b, P)
      sage: A.deciphering(a, b, C) == P
      True

  TESTS:

  The binary number system is currently not a supported alphabet of this affine cryptosystem:

      sage: AffineCryptosystem(BinaryStrings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free binary string monoid) is not supported as a cipher domain of this affine cryptosystem.

  Nor are the octal, hexadecimal, and radix-64 number systems supported:

      sage: AffineCryptosystem(OctalStrings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free octal string monoid) is not supported as a cipher domain of this affine cryptosystem.
      sage: AffineCryptosystem(HexadecimalStrings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free hexadecimal string monoid) is not supported as a cipher domain of this affine cryptosystem.
      sage: AffineCryptosystem(Radix64Strings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free radix 64 string monoid) is not supported as a cipher domain of this affine cryptosystem.

  A secret key \((a,b)\) must be an element of \(\ZZ/n\ZZ \times \ZZ/n\ZZ\) with \(\gcd(a,n) = 1\). This rules out the case \(a = 0\) irrespective of the value of \(b\). For the upper-case letters of the English alphabet, where the alphabet size is \(n = 26\), \(a\) cannot take on any even value:

      sage: A = AffineCryptosystem(AlphabeticStrings())
      sage: A(0, 1)
      Traceback (most recent call last):
      ...
      ValueError: (a, b) = (0, 1) is outside the range of acceptable values for a key of this affine cryptosystem.
      sage: A(2, 1)
      Traceback (most recent call last):
      ...
      ValueError: (a, b) = (2, 1) is outside the range of acceptable values for a key of this affine cryptosystem.

  REFERENCES:

  [Sti06] | Douglas R. Stinson. Cryptography: Theory and Practice. 3rd edition, Chapman & Hall/CRC, 2006.

  `brute_force`(C, ranking='none')

    Attempt a brute force cryptanalysis of the ciphertext `C`.

    INPUT:

    - `C` – A ciphertext over one of the supported alphabets of this affine cryptosystem. See the class `AffineCryptosystem` for documentation on the supported alphabets.
    - `ranking` – (default `"none"`) the method to use for ranking all possible keys. If `ranking="none"`, then do not use any ranking function. The following ranking functions are supported:
      - `"chi_square"` – the chi-square ranking function as implemented in the method `rank_by_chi_square()`.
      - `"squared_differences"` – the squared differences ranking function as implemented in the method `rank_by_squared_differences()`.

    OUTPUT:

    - All the possible plaintext sequences corresponding to the ciphertext `C`. This method effectively uses all the possible keys in this affine cryptosystem to decrypt `C`. The method is also referred to as exhaustive key search. The output is a dictionary of key, candidate decipherment pairs.

    EXAMPLES:

    Cryptanalyze using all possible keys with the option `ranking="none"`:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 7)
        sage: P = A.encoding("Linear"); P
        LINEAR
        sage: C = A.enciphering(a, b, P)
        sage: L = A.brute_force(C)
        sage: sorted(L.items())[:26]  # display 26 candidate decipherments
        <BLANKLINE>
        [((1, 0), OFUTHG),
        ((1, 1), NETSGF),
        ((1, 2), MDSRFE),
        ((1, 3), LCRQED),
        ((1, 4), KBQPDC),
        ((1, 5), JAPOCB),
        ((1, 6), IZONBA),
        ((1, 7), HYNMAZ),
        ((1, 8), GXMLZY),
        ((1, 9), FWLKYX),
        ((1, 10), EVKJXW),
        ((1, 11), DUJIWV),
        ((1, 12), CTIHVU),
        ((1, 13), BSHGUT),
        ((1, 14), ARGFTS),
        ((1, 15), ZQFESR),
        ((1, 16), YPEDRQ),
        ((1, 17), XODCQP),
        ((1, 18), WNCBPO),
        ((1, 19), VMBAON),
        ((1, 20), ULAZNM),
        ((1, 21), TKZYML),
        ((1, 22), SJYXLK),
        ((1, 23), RIXWKJ),
        ((1, 24), QHWVJI),
        ((1, 25), PGVUIH)]

    Use the chi-square ranking function, i.e. `ranking="chisquare"`:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 7)
        sage: P = A.encoding("Linear functions for encrypting and decrypting."); P
        LINEARFUNCTIONSFORENCRYPTINGANDDECRYPTING
        sage: C = A.enciphering(a, b, P)
        sage: Rank = A.brute_force(C, ranking="chisquare")
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((3, 7), LINEARFUNCTIONSFORENCRYPTINGANDDECRYPTING),
        ((23, 25), VYTCGPBMTENYSTOBSPCTEPIRNYTAGTDDCEPIRNYTA),
        ((1, 12), CTIHVUKDIBATLIXKLUHIBUPOATINVIEEHBUPOATIN),
        ((11, 15), HSRYELDAROVSWRQDWLYROLUBVSRIERTTYOLUBVSRI),
        ((25, 1), NWHIUVFMHOPWEHSFEVIHOVABPWHCUHLLIOVABPWHC),
        ((25, 7), TCNOABLSNUVCKNYLKBONUBGHVCNIANRROUBGHVCNI),
        ((15, 4), SHIBVOWZILEHDIJWDOBILOFYEHIRVIGGBLOFYEHIR),
        ((15, 23), PEFYSLTWFIBEAFGTALYFILCVBEFOSFDDYILCVBEFO),
        ((7, 10), IDUFHSYXUTEDNULYNSFUTSVGEDURHUMMFTSVGEDUR),
        ((19, 22), QVETRGABEFUVLENALGTEFGDSUVEHREMMTFGDSUVEH)]

    Use the squared differences ranking function, i.e. `ranking="squared_differences"`:

        sage: Rank = A.brute_force(C, ranking="squared_differences")
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((3, 7), LINEARFUNCTIONSFORENCRYPTINGANDDECRYPTING),
        ((23, 6), GJENRAMXEPYJDEZMDANEPATCYJELREOONPATCYJEL),
        ((23, 25), VYTCGPBMTENYSTOBSPCTEPIRNYTAGTDDCEPIRNYTA),
        ((19, 22), QVETRGABEFUVLENALGTEFGDSUVEHREMMTFGDSUVEH),
        ((19, 9), DIRGETNORSHIYRANYTGRSTQFHIRUERZZGSTQFHIRU),
        ((23, 18), KNIRVEQBITCNHIDQHERITEXGCNIPVISSRTEXGCNIP),
        ((17, 16), GHORBEIDOJMHFOVIFEROJETWMHOZBOAARJETWMHOZ),
        ((21, 14), AHEZRMOFEVQHTEBOTMZEVMNIQHEDREKKZVMNIQHED),
        ((1, 12), CTIHVUKDIBATLIXKLUHIBUPOATINVIEEHBUPOATIN),
        ((7, 18), SNEPRCIHEDONXEVIXCPEDCFQONEBREWWPDCFQONEB)]

    TESTS:

    Currently, the binary number system is not supported as an alphabet of this affine cryptosystem:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: BinStr = BinaryStrings()
        sage: C = BinStr.encoding("abc")
        sage: A.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: Ciphertext must be encoded using one of the supported cipher domains of this affine cryptosystem.

    Nor are the octal, hexadecimal, and radix-64 number systems supported:

        sage: OctStr = OctalStrings()
        sage: C = OctStr([1, 2, 3])
        sage: A.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: Ciphertext must be encoded using one of the supported cipher domains of this affine cryptosystem.
        sage: HexStr = HexadecimalStrings()
        sage: C = HexStr.encoding("abc")
        sage: A.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: Ciphertext must be encoded using one of the supported cipher domains of this affine cryptosystem.
        sage: RadStr = Radix64Strings()
        sage: C = RadStr([1, 2, 3])
        sage: A.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: Ciphertext must be encoded using one of the supported cipher domains of this affine cryptosystem.

    Only the chi-square and squared-differences ranking functions are currently supported. The keyword `ranking` must take on either of the values `"none"`, `"chisquare"` or `"squared_differences"`:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 7)
        sage: P = A.encoding("Linear")
        sage: C = A.enciphering(a, b, P)
        sage: A.brute_force(C, ranking="chi")
        Traceback (most recent call last):
        ...
        ValueError: Keyword 'ranking' must be either 'none', 'chisquare', or 'squared_differences'.
        sage: A.brute_force(C, ranking="")
        Traceback (most recent call last):
        ...
        ValueError: Keyword 'ranking' must be either 'none', 'chisquare', or 'squared_differences'.

  `deciphering`(a, b, C)

    Decrypt the ciphertext `C` with the key `(a, b)` using affine cipher decryption.

    INPUT:

    - `a, b` – a secret key belonging to the key space of this affine cipher. This key must be an element of \(\ZZ/n\ZZ \times \ZZ/n\ZZ\) such that \(\gcd(a,n) = 1\) with \(n\) being the size of the ciphertext and plaintext spaces.
    - `C` – a string of ciphertext; possibly an empty string. Characters in this string must be encoded using one of the supported alphabets. See the method `encoding()` for more information.

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

    Decryption over the capital letters of the English alphabet:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (5, 2)
        sage: P = A.encoding("Affine functions are linear functions.")
        sage: C = A.enciphering(a, b, P); C
        CBBQPWBYPMTQUPOCJWFQPWCJBYPMTQUPO
        sage: P == A.deciphering(a, b, C)
        True

    The previous example can also be worked through using functional notation:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (5, 2)
        sage: P = A.encoding("Affine functions are linear functions.")
        sage: E = A(a, b); E
        Affine cipher on Free alphabetic string monoid on A-Z
        sage: C = E(P); C
        CBBQPWBYPMTQUPOCJWFQPWCJBYPMTQUPO
        sage: aInv, bInv = A.inverse_key(a, b)
        sage: D = A(aInv, bInv); D
        Affine cipher on Free alphabetic string monoid on A-Z
        sage: D(C) == P
        True

    If the ciphertext is an empty string, then the plaintext is also an empty string regardless of the value of the secret key:

        sage: a, b = A.random_key()
        sage: A.deciphering(a, b, A.encoding(""))
        <BLANKLINE>
        sage: A.deciphering(a, b, A.encoding(" "))
        <BLANKLINE>

    TESTS:

    The key must be an ordered pair \((a,b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) with \(n\) being the size of the plaintext and ciphertext spaces. Furthermore, \(a\) must be relatively prime to \(n\), i.e. \(\gcd(a,n) = 1\):

        sage: A.deciphering(2, 6, P)
        Traceback (most recent call last):
        ...
        ValueError: (a, b) = (2, 6) is outside the range of acceptable values for a key of this affine cipher.

  `enciphering`(a, b, P)

    Encrypt the plaintext `P` with the key `(a, b)` using affine cipher encryption.

    INPUT:

    - `a, b` – a secret key belonging to the key space of this affine cipher. This key must be an element of \(\ZZ/n\ZZ \times \ZZ/n\ZZ\) such that \(\gcd(a,n) = 1\) with \(n\) being the size of the ciphertext and plaintext spaces.
    - `P` – a string of plaintext; possibly an empty string. Characters in this string must be encoded using one of the supported alphabets. See the method `encoding()` for more information.

    OUTPUT:

    - The ciphertext corresponding to the plaintext `P`.

    EXAMPLES:

    Encryption over the capital letters of the English alphabet:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 6)
        sage: P = A.encoding("Affine ciphers work with linear functions.")
        sage: A.enciphering(a, b, P)
        GVVETSMEZBSFIUWFKUELBNETSGFVOTMLEWTI

    Now work through the previous example using functional notation:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 6)
        sage: P = A.encoding("Affine ciphers work with linear functions.")
        sage: E = A(a, b); E
        Affine cipher on Free alphabetic string monoid on A-Z
        sage: E(P)
        GVVETSMEZBSFIUWFKUELBNETSGFVOTMLEWTI

    If the plaintext is an empty string, then the ciphertext is also an empty string regardless of the value of the secret key:

        sage: a, b = A.random_key()
        sage: A.enciphering(a, b, A.encoding(""))
        <BLANKLINE>
        sage: A.enciphering(a, b, A.encoding(" "))
        <BLANKLINE>

    TESTS:

    The key must be an ordered pair \((a,b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) with \(n\) being the size of the plaintext and ciphertext spaces. Furthermore, \(a\) must be relatively prime to \(n\), i.e. \(\gcd(a,n) = 1\):

        sage: A.enciphering(2, 6, P)
        Traceback (most recent call last):
        ...
        ValueError: (a, b) = (2, 6) is outside the range of acceptable values for a key of this affine cryptosystem.

  `encoding`(S)

    The encoding of the string `S` over the string monoid of this affine cipher. For example, if the string monoid of this cryptosystem is `AlphabeticStringMonoid`, then the encoding of `S` would be its upper-case equivalent stripped of all non-alphabetic characters. Only the following alphabet is supported for the affine cipher:

    - capital letters of the English alphabet as implemented in `AlphabeticStrings()`

    INPUT:

    - `S` – a string, possibly empty.

    OUTPUT:

    - The encoding of `S` over the string monoid of this cryptosystem. If `S` is an empty string, return an empty string.

    EXAMPLES:

    Encoding over the upper-case letters of the English alphabet:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: A.encoding("Affine cipher over capital letters of the English alphabet.")
        AFFINECIPHEROVERCAPITALLETTERSOFTHEENGLISHALPHABET

    The argument `S` can be an empty string, in which case an empty string is returned:

        sage: AffineCryptosystem(AlphabeticStrings()).encoding("")
        <BLANKLINE>

  `inverse_key`(a, b)

    The inverse key corresponding to the secret key \((a,b)\). If \(p\) is a plaintext character so that \(p \in \ZZ/n\ZZ\) and \(n\) is the alphabet size, then the ciphertext \(c\) corresponding to \(p\) is

    \[c \equiv ap + b \pmod{n}\]

    As \((a,b)\) is a key, then the multiplicative inverse \(a^{-1}\) exists and the original plaintext can be recovered as follows

    \[p \equiv a^{-1} (c - b) \pmod{n} \equiv a^{-1}c + a^{-1}(-b) \pmod{n}\]

    Therefore the ordered pair \((a^{-1}, -ba^{-1})\) is the inverse key corresponding to \((a,b)\).

    INPUT:

    - `a, b` – a secret key for this affine cipher. The ordered pair \((a,b)\) must be an element of \(\ZZ/n\ZZ \times \ZZ/n\ZZ\) such that \(\gcd(a,n) = 1\).

    OUTPUT:

    - The inverse key \((a^{-1}, -ba^{-1})\) corresponding to \((a,b)\).

    EXAMPLES:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (1, 2)
        sage: A.inverse_key(a, b)
        (1, 24)
        sage: A.inverse_key(3, 2)
        (9, 8)

    Suppose that the plaintext and ciphertext spaces are the capital letters of the English alphabet so that \(n = 26\). If \(\varphi(n)\) is the Euler phi function of \(n\), then there are \(\varphi(n)\) integers \(0 \leq a < n\) that are relatively prime to \(n\). For the capital letters of the English alphabet, there are 12 such integers relatively prime to \(n\):

        sage: euler_phi(A.alphabet_size())
        12

    And here is a list of those integers:

        sage: n = A.alphabet_size()
        sage: L = [i for i in xrange(n) if gcd(i, n) == 1]; L
        [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25]

    Then a secret key \((a,b)\) of this shift cryptosystem is such that \(a\) is an element of the list `L` in the last example. Any inverse key \((A, B)\) corresponding to \((a,b)\) is such that \(A\) is also in the list `L` above:

        sage: a, b = (3, 9)
        sage: a in L
        True
        sage: aInv, bInv = A.inverse_key(a, b)
        sage: aInv, bInv
        (9, 23)
        sage: aInv in L
        True

    TESTS:

    Any ordered pair of the form \((0, b)\) for any integer \(b\) cannot be a secret key of this affine cipher. Hence \((0, b)\) does not have a corresponding inverse key:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: A.inverse_key(0, 1)
        Traceback (most recent call last):
        ...
        ValueError: (a, b) = (0, 1) is outside the range of acceptable values for a key of this affine cipher.

  `random_key`()

    Generate a random key within the key space of this affine cipher. The generated secret key is an ordered pair \((a, b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) with \(n\) being the size of the cipher domain and \(\gcd(a, n) = 1\). Let \(\varphi(n)\) denote the Euler phi function of \(n\). Then the affine cipher has \(n \cdot \varphi(n)\) possible keys (see page 10 of [Sti06]).

    OUTPUT:

    - A random key within the key space of this affine cryptosystem. The output key is an ordered pair \((a,b)\).

    EXAMPLES:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: A.random_key()  # random
        (17, 25)

    If \((a,b)\) is a secret key and \(n\) is the size of the plaintext and ciphertext alphabets, then \(\gcd(a, n) = 1\):

        sage: a, b = A.random_key()
        sage: n = A.alphabet_size()
        sage: gcd(a, n)
        1

  `rank_by_chi_square`(C, pdict)

    Use the chi-square statistic to rank all possible keys. Currently, this method only applies to the capital letters of the English alphabet.

    ALGORITHM:

    Consider a non-empty alphabet \(A\) consisting of \(n\) elements, and let \(C\) be a ciphertext encoded using elements of \(A\). The plaintext \(P\) corresponding to \(C\) is also encoded using elements of \(A\). Let \(M\) be a candidate decipherment of \(C\), i.e. \(M\) is the result of attempting to decrypt \(C\) using a key \((a,b)\) which is not necessarily the same key used to encrypt \(P\). Suppose \(F_A(e)\) is the characteristic frequency probability of \(e \in A\) and let \(F_M(e)\) be the message frequency probability with respect to \(M\). The characteristic frequency probability distribution of an alphabet is the expected frequency probability distribution for that alphabet. The message frequency probability distribution of \(M\) provides a distribution of the ratio of character occurrences over message length. One can interpret the characteristic frequency probability \(F_A(e)\) as the expected probability, while the message frequency probability \(F_M(e)\) is the observed probability. If \(M\) is of length \(L\), then the observed frequency of \(e \in A\) is

    \[O_M(e) = F_M(e) \cdot L\]

    and the expected frequency of \(e \in A\) is

    \[E_A(e) = F_A(e) \cdot L\]

    The chi-square rank \(R_{\chi^2}(M)\) of \(M\) corresponding to a key \((a,b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) is given by

    \[R_{\chi^2}(M) = \sum_{e \in A} \frac {\big( O_M(e) - E_A(e) \big)^2} {E_A(e)}\]

    Cryptanalysis by exhaustive key search produces a candidate decipherment \(M_{a,b}\) for each possible key \((a,b)\). For a set \(D = \big\{M_{a_1,b_1}, M_{a_2,b_2}, \dots, M_{a_k,b_k} \big\}\) of all candidate decipherments corresponding to a ciphertext \(C\), the smaller is the rank \(R_{\chi^2}(M_{a_i,b_i})\) the more likely that \((a_i,b_i)\) is the secret key. This key ranking method is based on the Pearson chi-square test [PearsonTest09].

    INPUT:

    - `C` – The ciphertext, a non-empty string. The ciphertext must be encoded using the upper-case letters of the English alphabet.
    - `pdict` – A dictionary of key, possible plaintext pairs. This should be the output of `brute_force()` with `ranking="none"`.

    OUTPUT:

    - A list ranking the most likely keys first. Each element of the list is a tuple of key, possible plaintext pairs.

    EXAMPLES:

    Use the chi-square statistic to rank all possible keys and their corresponding decipherment:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 7)
        sage: P = A.encoding("Line.")
        sage: C = A.enciphering(a, b, P)
        sage: Plist = A.brute_force(C)
        sage: Rank = A.rank_by_chi_square(C, Plist)
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((1, 1), NETS),
        ((3, 7), LINE),
        ((17, 20), STAD),
        ((5, 2), SLOT),
        ((5, 5), HADI),
        ((9, 25), TSLI),
        ((17, 15), DELO),
        ((15, 6), ETUN),
        ((21, 8), ELID),
        ((7, 17), HCTE)]

    As more ciphertext is available, the reliability of the chi-square ranking function increases:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (11, 24)
        sage: P = A.encoding("Longer message is more information for cryptanalysis.")
        sage: C = A.enciphering(a, b, P)
        sage: Plist = A.brute_force(C)
        sage: Rank = A.rank_by_chi_square(C, Plist)
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((11, 24), LONGERMESSAGEISMOREINFORMATIONFORCRYPTANALYSIS),
        ((17, 9), INURFSBFLLHRFDLBNSFDUYNSBHEDNUYNSTSVGEHUHIVLDL),
        ((9, 18), RMFIUHYUOOSIUWOYMHUWFBMHYSVWMFBMHGHETVSFSREOWO),
        ((15, 12), VSTACPUCOOGACYOUSPCYTBSPUGNYSTBSPEPIRNGTGVIOYO),
        ((3, 22), PAFOYLKYGGSOYEGKALYEFTALKSBEAFTALILCVBSFSPCGEG),
        ((25, 3), OHSRNADNPPFRNVPDHANVSCHADFEVHSCHAJABWEFSFOBPVP),
        ((7, 25), GHYNVIPVRRLNVFRPHIVFYEHIPLAFHYEHIDITQALYLGTRFR),
        ((5, 2), NEHCIVKISSUCIWSKEVIWHFEVKUPWEHFEVOVABPUHUNASWS),
        ((15, 25), IFGNPCHPBBTNPLBHFCPLGOFCHTALFGOFCRCVEATGTIVBLB),
        ((9, 6), BWPSERIEYYCSEGYIWREGPLWRICFGWPLWRQRODFCPCBOYGY)]

    TESTS:

    The ciphertext cannot be an empty string:

        sage: A.rank_by_chi_square("", Plist)
        Traceback (most recent call last):
        ...
        AttributeError: 'str' object has no attribute 'parent'
        sage: A.rank_by_chi_square(A.encoding(""), Plist)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.
        sage: A.rank_by_chi_square(A.encoding(" "), Plist)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.

    The ciphertext must be encoded using the capital letters of the English alphabet as implemented in `AlphabeticStrings()`:

        sage: H = HexadecimalStrings()
        sage: A.rank_by_chi_square(H.encoding("shift"), Plist)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.
        sage: B = BinaryStrings()
        sage: A.rank_by_chi_square(B.encoding("shift"), Plist)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.

    The dictionary `pdict` cannot be empty:

        sage: A.rank_by_chi_square(C, {})
        Traceback (most recent call last):
        ...
        KeyError: (1, 0)

  `rank_by_squared_differences`(C, pdict)

    Use the squared-differences measure to rank all possible keys. Currently, this method only applies to the capital letters of the English alphabet.

    ALGORITHM:

    Consider a non-empty alphabet \(A\) consisting of \(n\) elements, and let \(C\) be a ciphertext encoded using elements of \(A\). The plaintext \(P\) corresponding to \(C\) is also encoded using elements of \(A\). Let \(M\) be a candidate decipherment of \(C\), i.e. \(M\) is the result of attempting to decrypt \(C\) using a key \((a,b)\) which is not necessarily the same key used to encrypt \(P\). Suppose \(F_A(e)\) is the characteristic frequency probability of \(e \in A\) and let \(F_M(e)\) be the message frequency probability with respect to \(M\). The characteristic frequency probability distribution of an alphabet is the expected frequency probability distribution for that alphabet. The message frequency probability distribution of \(M\) provides a distribution of the ratio of character occurrences over message length. One can interpret the characteristic frequency probability \(F_A(e)\) as the expected probability, while the message frequency probability \(F_M(e)\) is the observed probability. If \(M\) is of length \(L\), then the observed frequency of \(e \in A\) is

    \[O_M(e) = F_M(e) \cdot L\]

    and the expected frequency of \(e \in A\) is

    \[E_A(e) = F_A(e) \cdot L\]

    The squared-differences, or residual sum of squares, rank \(R_{RSS}(M)\) of \(M\) corresponding to a key \((a,b) \in \ZZ/n\ZZ \times \ZZ/n\ZZ\) is given by

    \[R_{RSS}(M) = \sum_{e \in A} \big( O_M(e) - E_A(e) \big)^2\]

    Cryptanalysis by exhaustive key search produces a candidate decipherment \(M_{a,b}\) for each possible key \((a,b)\). For a set \(D = \big\{M_{a_1,b_1}, M_{a_2,b_2}, \dots, M_{a_k,b_k} \big\}\) of all candidate decipherments corresponding to a ciphertext \(C\), the smaller is the rank \(R_{RSS}(M_{a_i,b_i})\) the more likely that \((a_i,b_i)\) is the secret key. This key ranking method is based on the residual sum of squares measure [RSS09].

    INPUT:

    - `C` – The ciphertext, a non-empty string. The ciphertext must be encoded using the upper-case letters of the English alphabet.
    - `pdict` – A dictionary of key, possible plaintext pairs. This should be the output of `brute_force()` with `ranking="none"`.

    OUTPUT:

    - A list ranking the most likely keys first. Each element of the list is a tuple of key, possible plaintext pairs.

    EXAMPLES:

    Use the method of squared differences to rank all possible keys and their corresponding decipherment:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (3, 7)
        sage: P = A.encoding("Line.")
        sage: C = A.enciphering(a, b, P)
        sage: Plist = A.brute_force(C)
        sage: Rank = A.rank_by_squared_differences(C, Plist)
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((1, 1), NETS),
        ((15, 6), ETUN),
        ((7, 17), HCTE),
        ((3, 7), LINE),
        ((17, 15), DELO),
        ((9, 4), EDWT),
        ((9, 9), POHE),
        ((21, 8), ELID),
        ((17, 20), STAD),
        ((7, 18), SNEP)]

    As more ciphertext is available, the reliability of the squared-differences ranking function increases:

        sage: A = AffineCryptosystem(AlphabeticStrings())
        sage: a, b = (11, 24)
        sage: P = A.encoding("Longer message is more information for cryptanalysis.")
        sage: C = A.enciphering(a, b, P)
        sage: Plist = A.brute_force(C)
        sage: Rank = A.rank_by_squared_differences(C, Plist)
        sage: Rank[:10]  # display only the top 10 candidate keys
        <BLANKLINE>
        [((11, 24), LONGERMESSAGEISMOREINFORMATIONFORCRYPTANALYSIS),
        ((9, 14), DYRUGTKGAAEUGIAKYTGIRNYTKEHIYRNYTSTQFHEREDQAIA),
        ((23, 24), DSNEUHIUMMAEUOMISHUONZSHIAROSNZSHKHQXRANADQMOM),
        ((23, 1), ETOFVIJVNNBFVPNJTIVPOATIJBSPTOATILIRYSBOBERNPN),
        ((21, 16), VEBGANYAQQOGAMQYENAMBDENYOTMEBDENUNIHTOBOVIQMQ),
        ((7, 12), TULAIVCIEEYAISECUVISLRUVCYNSULRUVQVGDNYLYTGESE),
        ((5, 20), ZQTOUHWUEEGOUIEWQHUITRQHWGBIQTRQHAHMNBGTGZMEIE),
        ((21, 8), JSPUOBMOEECUOAEMSBOAPRSBMCHASPRSBIBWVHCPCJWEAE),
        ((25, 7), SLWVREHRTTJVRZTHLERZWGLEHJIZLWGLENEFAIJWJSFTZT),
        ((25, 15), ATEDZMPZBBRDZHBPTMZHEOTMPRQHTEOTMVMNIQRERANBHB)]

    TESTS:

    The ciphertext cannot be an empty string:

        sage: A.rank_by_squared_differences("", Plist)
        Traceback (most recent call last):
        ...
        AttributeError: 'str' object has no attribute 'parent'
        sage: A.rank_by_squared_differences(A.encoding(""), Plist)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.
        sage: A.rank_by_squared_differences(A.encoding(" "), Plist)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.

    The ciphertext must be encoded using the capital letters of the English alphabet as implemented in `AlphabeticStrings()`:

        sage: H = HexadecimalStrings()
        sage: A.rank_by_squared_differences(H.encoding("line"), Plist)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.
        sage: B = BinaryStrings()
        sage: A.rank_by_squared_differences(B.encoding("line"), Plist)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.

    The dictionary `pdict` cannot be empty:

        sage: A.rank_by_squared_differences(C, {})
        Traceback (most recent call last):
        ...
        KeyError: (1, 0)

class `sage.crypto.classical.``HillCryptosystem`(S, m)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create a Hill cryptosystem defined by the \(m\) x \(m\) matrix space over \(\ZZ / N \ZZ\), where \(N\) is the alphabet size of the string monoid `S`.

  INPUT:

  - `S` - a string monoid over some alphabet
  - `m` - integer \(> 0\); the block length of matrices that specify block permutations

  OUTPUT:

  - A Hill cryptosystem of block length `m` over the alphabet `S`.

  EXAMPLES:

      sage: S = AlphabeticStrings()
      sage: E = HillCryptosystem(S,3)
      sage: E
      Hill cryptosystem on Free alphabetic string monoid on A-Z of block length 3
      sage: R = IntegerModRing(26)
      sage: M = MatrixSpace(R,3,3)
      sage: A = M([[1,0,1],[0,1,1],[2,2,3]])
      sage: A
      [1 0 1]
      [0 1 1]
      [2 2 3]
      sage: e = E(A)
      sage: e
      Hill cipher on Free alphabetic string monoid on A-Z of block length 3
      sage: e(S("LAMAISONBLANCHE"))
      JYVKSKQPELAYKPV

  TESTS:

      sage: S = AlphabeticStrings()
      sage: E = HillCryptosystem(S,3)
      sage: E == loads(dumps(E))
      True

  `block_length`()

    The row or column dimension of a matrix specifying a block permutation. Encryption and decryption keys of a Hill cipher are square matrices, i.e. the row and column dimensions of an encryption or decryption key are the same. This row/column dimension is referred to as the block length.

    OUTPUT:

    - The block length of an encryption/decryption key.

    EXAMPLES:

        sage: A = AlphabeticStrings()
        sage: n = randint(1, A.ngens() - 1)
        sage: H = HillCryptosystem(A, n)
        sage: H.block_length() == n
        True

  `deciphering`(A, C)

    Decrypt the ciphertext `C` using the key `A`.

    INPUT:

    - `A` - a key within the key space of this Hill cipher
    - `C` - a string (possibly empty) over the string monoid of this Hill cipher

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

        sage: H = HillCryptosystem(AlphabeticStrings(), 3)
        sage: K = H.random_key()
        sage: M = H.encoding("Good day, mate! How ya going?")
        sage: H.deciphering(K, H.enciphering(K, M)) == M
        True

  `enciphering`(A, M)

    Encrypt the plaintext `M` using the key `A`.

    INPUT:

    - `A` - a key within the key space of this Hill cipher
    - `M` - a string (possibly empty) over the string monoid of this Hill cipher.

    OUTPUT:

    - The ciphertext corresponding to the plaintext `M`.

    EXAMPLES:

        sage: H = HillCryptosystem(AlphabeticStrings(), 3)
        sage: K = H.random_key()
        sage: M = H.encoding("Good day, mate! How ya going?")
        sage: H.deciphering(K, H.enciphering(K, M)) == M
        True

  `encoding`(M)

    The encoding of the string `M` over the string monoid of this Hill cipher. For example, if the string monoid of this Hill cipher is `AlphabeticStringMonoid`, then the encoding of `M` would be its upper-case equivalent stripped of all non-alphabetic characters.

    INPUT:

    - `M` - a string, possibly empty

    OUTPUT:

    - The encoding of `M` over the string monoid of this Hill cipher.

    EXAMPLES:

        sage: M = "The matrix cipher by Lester S. Hill."
        sage: A = AlphabeticStrings()
        sage: H = HillCryptosystem(A, 7)
        sage: H.encoding(M) == A.encoding(M)
        True

  `inverse_key`(A)

    The inverse key corresponding to the key `A`.

    INPUT:

    - `A` - an invertible matrix of the key space of this Hill cipher

    OUTPUT:

    - The inverse matrix of `A`.

    EXAMPLES:

        sage: S = AlphabeticStrings()
        sage: E = HillCryptosystem(S,3)
        sage: A = E.random_key()
        sage: B = E.inverse_key(A)
        sage: M = S("LAMAISONBLANCHE")
        sage: e = E(A)
        sage: c = E(B)
        sage: c(e(M))
        LAMAISONBLANCHE

  `random_key`()

    A random key within the key space of this Hill cipher. That is, generate a random \(m\) x \(m\) matrix to be used as a block permutation, where \(m\) is the block length of this Hill cipher. If \(n\) is the size of the cryptosystem alphabet, then there are \(n^{m^2}\) possible keys. However the number of valid keys, i.e. invertible \(m\) x \(m\) square matrices, is smaller than \(n^{m^2}\).

    OUTPUT:

    - A random key within the key space of this Hill cipher.

    EXAMPLES:

        sage: A = AlphabeticStrings()
        sage: n = 3
        sage: H = HillCryptosystem(A, n)
        sage: K = H.random_key()
        sage: Ki = H.inverse_key(K)
        sage: M = "LAMAISONBLANCHE"
        sage: e = H(K)
        sage: d = H(Ki)
        sage: d(e(A(M))) == A(M)
        True

class `sage.crypto.classical.``ShiftCryptosystem`(A)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create a shift cryptosystem.

  Let \(A = \{ a_0, a_1, a_2, \dots, a_{n-1} \}\) be a non-empty alphabet consisting of \(n\) unique elements. Define a mapping \(f : A \longrightarrow \ZZ/ n\ZZ\) from the alphabet \(A\) to the set \(\ZZ / n\ZZ\) of integers modulo \(n\), given by \(f(a_i) = i\). Thus we can identify each element of the alphabet \(A\) with a unique integer \(0 \leq i < n\). A key of the shift cipher is an integer \(0 \leq k < n\). Therefore the key space is \(\ZZ / n\ZZ\). Since we assume that \(A\) does not have repeated elements, the mapping \(f : A \longrightarrow \ZZ/ n\ZZ\) is bijective. Encryption works by moving along the alphabet by \(k\) positions, with wrap around. Decryption reverses the process by moving backwards by \(k\) positions, with wrap around. More generally, let \(k\) be a secret key, i.e. an element of the key space, and let \(p\) be a plaintext character and consequently \(p \in \ZZ / n\ZZ\). Then the ciphertext character \(c\) corresponding to \(p\) is given by

  \[c \equiv p + k \pmod{n}\]

  Similarly, given a ciphertext character \(c \in \ZZ / n\ZZ\) and a secret key \(k\), we can recover the corresponding plaintext character as follows:

  \[p \equiv c - k \pmod{n}\]

  Use the bijection \(f : A \longrightarrow \ZZ/ n\ZZ\) to convert \(c\) and \(p\) back to elements of the alphabet \(A\). Currently, the following alphabets are supported for the shift cipher:

  - capital letters of the English alphabet as implemented in `AlphabeticStrings()`
  - the alphabet consisting of the hexadecimal number system as implemented in `HexadecimalStrings()`
  - the alphabet consisting of the binary number system as implemented in `BinaryStrings()`

  EXAMPLES:

  Some examples illustrating encryption and decryption over various alphabets. Here is an example over the upper-case letters of the English alphabet:

      sage: S = ShiftCryptosystem(AlphabeticStrings()); S
      Shift cryptosystem on Free alphabetic string monoid on A-Z
      sage: P = S.encoding("The shift cryptosystem generalizes the Caesar cipher.")
      sage: P
      THESHIFTCRYPTOSYSTEMGENERALIZESTHECAESARCIPHER
      sage: K = 7
      sage: C = S.enciphering(K, P); C
      AOLZOPMAJYFWAVZFZALTNLULYHSPGLZAOLJHLZHYJPWOLY
      sage: S.deciphering(K, C)
      THESHIFTCRYPTOSYSTEMGENERALIZESTHECAESARCIPHER
      sage: S.deciphering(K, C) == P
      True

  The previous example can also be done as follows:

      sage: S = ShiftCryptosystem(AlphabeticStrings())
      sage: P = S.encoding("The shift cryptosystem generalizes the Caesar cipher.")
      sage: K = 7
      sage: E = S(K); E
      Shift cipher on Free alphabetic string monoid on A-Z
      sage: C = E(P); C
      AOLZOPMAJYFWAVZFZALTNLULYHSPGLZAOLJHLZHYJPWOLY
      sage: D = S(S.inverse_key(K)); D
      Shift cipher on Free alphabetic string monoid on A-Z
      sage: D(C) == P
      True
      sage: D(C) == P == D(E(P))
      True

  Over the hexadecimal number system:

      sage: S = ShiftCryptosystem(HexadecimalStrings()); S
      Shift cryptosystem on Free hexadecimal string monoid
      sage: P = S.encoding("Encryption & decryption shifts along the alphabet."); P
      456e6372797074696f6e20262064656372797074696f6e2073686966747320616c6f6e672074686520616c7068616265742e
      sage: K = 5
      sage: C = S.enciphering(K, P); C
      9ab3b8c7cec5c9beb4b3757b75b9bab8c7cec5c9beb4b375c8bdbebbc9c875b6b1b4b3bc75c9bdba75b6b1c5bdb6b7bac973
      sage: S.deciphering(K, C)
      456e6372797074696f6e20262064656372797074696f6e2073686966747320616c6f6e672074686520616c7068616265742e
      sage: S.deciphering(K, C) == P
      True

  And over the binary number system:

      sage: S = ShiftCryptosystem(BinaryStrings()); S
      Shift cryptosystem on Free binary string monoid
      sage: P = S.encoding("The binary alphabet is very insecure."); P
      01010100011010000110010100100000011000100110100101101110011000010111001001111001001000000110000101101100011100000110100001100001011000100110010101110100001000000110100101110011001000000111011001100101011100100111100100100000011010010110111001110011011001010110001101110101011100100110010100101110
      sage: K = 1
      sage: C = S.enciphering(K, P); C
      10101011100101111001101011011111100111011001011010010001100111101000110110000110110111111001111010010011100011111001011110011110100111011001101010001011110111111001011010001100110111111000100110011010100011011000011011011111100101101001000110001100100110101001110010001010100011011001101011010001
      sage: S.deciphering(K, C)
      01010100011010000110010100100000011000100110100101101110011000010111001001111001001000000110000101101100011100000110100001100001011000100110010101110100001000000110100101110011001000000111011001100101011100100111100100100000011010010110111001110011011001010110001101110101011100100110010100101110
      sage: S.deciphering(K, C) == P
      True

  A shift cryptosystem with key \(k = 3\) is commonly referred to as the Caesar cipher. Create a Caesar cipher over the upper-case letters of the English alphabet:

      sage: caesar = ShiftCryptosystem(AlphabeticStrings())
      sage: K = 3
      sage: P = caesar.encoding("abcdef"); P
      ABCDEF
      sage: C = caesar.enciphering(K, P); C
      DEFGHI
      sage: caesar.deciphering(K, C) == P
      True

  Generate a random key for encryption and decryption:

      sage: S = ShiftCryptosystem(AlphabeticStrings())
      sage: P = S.encoding("Shift cipher with a random key.")
      sage: K = S.random_key()
      sage: C = S.enciphering(K, P)
      sage: S.deciphering(K, C) == P
      True

  Decrypting with the key `K` is equivalent to encrypting with its corresponding inverse key:

      sage: S.enciphering(S.inverse_key(K), C) == P
      True

  TESTS:

  Currently, the octal number system is not supported as an alphabet for this shift cryptosystem:

      sage: ShiftCryptosystem(OctalStrings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free octal string monoid) is not supported as a cipher domain of this shift cryptosystem.

  Nor is the radix-64 number system supported:

      sage: ShiftCryptosystem(Radix64Strings())
      Traceback (most recent call last):
      ...
      TypeError: A (= Free radix 64 string monoid) is not supported as a cipher domain of this shift cryptosystem.

  Testing of dumping and loading objects:

      sage: SA = ShiftCryptosystem(AlphabeticStrings())
      sage: SA == loads(dumps(SA))
      True
      sage: SH = ShiftCryptosystem(HexadecimalStrings())
      sage: SH == loads(dumps(SH))
      True
      sage: SB = ShiftCryptosystem(BinaryStrings())
      sage: SB == loads(dumps(SB))
      True

  The key `K` must satisfy the inequality \(0 \leq K < n\) with \(n\) being the size of the plaintext, ciphertext, and key spaces. For the shift cryptosystem, all these spaces are the same alphabet. This inequality must be satisfied for each of the supported alphabets. The capital letters of the English alphabet:

      sage: S = ShiftCryptosystem(AlphabeticStrings())
      sage: S(2 + S.alphabet_size())
      Traceback (most recent call last):
      ...
      ValueError: K (=28) is outside the range of acceptable values for a key of this shift cryptosystem.
      sage: S(-2)
      Traceback (most recent call last):
      ...
      ValueError: K (=-2) is outside the range of acceptable values for a key of this shift cryptosystem.

  The hexadecimal number system:

      sage: S = ShiftCryptosystem(HexadecimalStrings())
      sage: S(1 + S.alphabet_size())
      Traceback (most recent call last):
      ...
      ValueError: K (=17) is outside the range of acceptable values for a key of this shift cryptosystem.
      sage: S(-1)
      Traceback (most recent call last):
      ...
      ValueError: K (=-1) is outside the range of acceptable values for a key of this shift cryptosystem.

  The binary number system:

      sage: S = ShiftCryptosystem(BinaryStrings())
      sage: S(1 + S.alphabet_size())
      Traceback (most recent call last):
      ...
      ValueError: K (=3) is outside the range of acceptable values for a key of this shift cryptosystem.
      sage: S(-2)
      Traceback (most recent call last):
      ...
      ValueError: K (=-2) is outside the range of acceptable values for a key of this shift cryptosystem.

  `brute_force`(C, ranking='none')

    Attempt a brute force cryptanalysis of the ciphertext `C`.

    INPUT:

    - `C` – A ciphertext over one of the supported alphabets of this shift cryptosystem. See the class `ShiftCryptosystem` for documentation on the supported alphabets.
    - `ranking` – (default `"none"`) the method to use for ranking all possible keys. If `ranking="none"`, then do not use any ranking function. The following ranking functions are supported:
      - `"chisquare"` – the chi-square ranking function as implemented in the method `rank_by_chi_square()`.
      - `"squared_differences"` – the squared differences ranking function as implemented in the method `rank_by_squared_differences()`.

    OUTPUT:

    - All the possible plaintext sequences corresponding to the ciphertext `C`. This method effectively uses all the possible keys in this shift cryptosystem to decrypt `C`. The method is also referred to as exhaustive key search. The output is a dictionary of key, plaintext pairs.

    EXAMPLES:

    Cryptanalyze using all possible keys for various alphabets. Over the upper-case letters of the English alphabet:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("The shift cryptosystem generalizes the Caesar cipher.")
        sage: K = 7
        sage: C = S.enciphering(K, P)
        sage: Dict = S.brute_force(C)
        sage: for k in xrange(len(Dict)):
        ...       if Dict[k] == P:
        ...           print "key =", k
        ...
        key = 7

    Over the hexadecimal number system:

        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: P = S.encoding("Encryption & decryption shifts along the alphabet.")
        sage: K = 5
        sage: C = S.enciphering(K, P)
        sage: Dict = S.brute_force(C)
        sage: for k in xrange(len(Dict)):
        ...       if Dict[k] == P:
        ...           print "key =", k
        ...
        key = 5

    And over the binary number system:

        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: P = S.encoding("The binary alphabet is very insecure.")
        sage: K = 1
        sage: C = S.enciphering(K, P)
        sage: Dict = S.brute_force(C)
        sage: for k in xrange(len(Dict)):
        ...       if Dict[k] == P:
        ...           print "key =", k
        ...
        key = 1

    Don’t use any ranking functions, i.e. `ranking="none"`:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("Shifting using modular arithmetic.")
        sage: K = 8
        sage: C = S.enciphering(K, P)
        sage: pdict = S.brute_force(C)
        sage: sorted(pdict.items())
        <BLANKLINE>
        [(0, APQNBQVOCAQVOUWLCTIZIZQBPUMBQK),
        (1, ZOPMAPUNBZPUNTVKBSHYHYPAOTLAPJ),
        (2, YNOLZOTMAYOTMSUJARGXGXOZNSKZOI),
        (3, XMNKYNSLZXNSLRTIZQFWFWNYMRJYNH),
        (4, WLMJXMRKYWMRKQSHYPEVEVMXLQIXMG),
        (5, VKLIWLQJXVLQJPRGXODUDULWKPHWLF),
        (6, UJKHVKPIWUKPIOQFWNCTCTKVJOGVKE),
        (7, TIJGUJOHVTJOHNPEVMBSBSJUINFUJD),
        (8, SHIFTINGUSINGMODULARARITHMETIC),
        (9, RGHESHMFTRHMFLNCTKZQZQHSGLDSHB),
        (10, QFGDRGLESQGLEKMBSJYPYPGRFKCRGA),
        (11, PEFCQFKDRPFKDJLARIXOXOFQEJBQFZ),
        (12, ODEBPEJCQOEJCIKZQHWNWNEPDIAPEY),
        (13, NCDAODIBPNDIBHJYPGVMVMDOCHZODX),
        (14, MBCZNCHAOMCHAGIXOFULULCNBGYNCW),
        (15, LABYMBGZNLBGZFHWNETKTKBMAFXMBV),
        (16, KZAXLAFYMKAFYEGVMDSJSJALZEWLAU),
        (17, JYZWKZEXLJZEXDFULCRIRIZKYDVKZT),
        (18, IXYVJYDWKIYDWCETKBQHQHYJXCUJYS),
        (19, HWXUIXCVJHXCVBDSJAPGPGXIWBTIXR),
        (20, GVWTHWBUIGWBUACRIZOFOFWHVASHWQ),
        (21, FUVSGVATHFVATZBQHYNENEVGUZRGVP),
        (22, ETURFUZSGEUZSYAPGXMDMDUFTYQFUO),
        (23, DSTQETYRFDTYRXZOFWLCLCTESXPETN),
        (24, CRSPDSXQECSXQWYNEVKBKBSDRWODSM),
        (25, BQROCRWPDBRWPVXMDUJAJARCQVNCRL)]

    Use the chi-square ranking function, i.e. `ranking="chisquare"`:

        sage: S.brute_force(C, ranking="chisquare")
        <BLANKLINE>
        [(8, SHIFTINGUSINGMODULARARITHMETIC),
        (14, MBCZNCHAOMCHAGIXOFULULCNBGYNCW),
        (20, GVWTHWBUIGWBUACRIZOFOFWHVASHWQ),
        (13, NCDAODIBPNDIBHJYPGVMVMDOCHZODX),
        (1, ZOPMAPUNBZPUNTVKBSHYHYPAOTLAPJ),
        (23, DSTQETYRFDTYRXZOFWLCLCTESXPETN),
        (10, QFGDRGLESQGLEKMBSJYPYPGRFKCRGA),
        (6, UJKHVKPIWUKPIOQFWNCTCTKVJOGVKE),
        (22, ETURFUZSGEUZSYAPGXMDMDUFTYQFUO),
        (15, LABYMBGZNLBGZFHWNETKTKBMAFXMBV),
        (12, ODEBPEJCQOEJCIKZQHWNWNEPDIAPEY),
        (21, FUVSGVATHFVATZBQHYNENEVGUZRGVP),
        (16, KZAXLAFYMKAFYEGVMDSJSJALZEWLAU),
        (25, BQROCRWPDBRWPVXMDUJAJARCQVNCRL),
        (9, RGHESHMFTRHMFLNCTKZQZQHSGLDSHB),
        (24, CRSPDSXQECSXQWYNEVKBKBSDRWODSM),
        (3, XMNKYNSLZXNSLRTIZQFWFWNYMRJYNH),
        (5, VKLIWLQJXVLQJPRGXODUDULWKPHWLF),
        (7, TIJGUJOHVTJOHNPEVMBSBSJUINFUJD),
        (2, YNOLZOTMAYOTMSUJARGXGXOZNSKZOI),
        (18, IXYVJYDWKIYDWCETKBQHQHYJXCUJYS),
        (4, WLMJXMRKYWMRKQSHYPEVEVMXLQIXMG),
        (11, PEFCQFKDRPFKDJLARIXOXOFQEJBQFZ),
        (19, HWXUIXCVJHXCVBDSJAPGPGXIWBTIXR),
        (0, APQNBQVOCAQVOUWLCTIZIZQBPUMBQK),
        (17, JYZWKZEXLJZEXDFULCRIRIZKYDVKZT)]

    Use the squared differences ranking function, i.e. `ranking="squared_differences"`:

        sage: S.brute_force(C, ranking="squared_differences")
        <BLANKLINE>
        [(8, SHIFTINGUSINGMODULARARITHMETIC),
        (23, DSTQETYRFDTYRXZOFWLCLCTESXPETN),
        (12, ODEBPEJCQOEJCIKZQHWNWNEPDIAPEY),
        (2, YNOLZOTMAYOTMSUJARGXGXOZNSKZOI),
        (9, RGHESHMFTRHMFLNCTKZQZQHSGLDSHB),
        (7, TIJGUJOHVTJOHNPEVMBSBSJUINFUJD),
        (21, FUVSGVATHFVATZBQHYNENEVGUZRGVP),
        (22, ETURFUZSGEUZSYAPGXMDMDUFTYQFUO),
        (1, ZOPMAPUNBZPUNTVKBSHYHYPAOTLAPJ),
        (16, KZAXLAFYMKAFYEGVMDSJSJALZEWLAU),
        (20, GVWTHWBUIGWBUACRIZOFOFWHVASHWQ),
        (24, CRSPDSXQECSXQWYNEVKBKBSDRWODSM),
        (14, MBCZNCHAOMCHAGIXOFULULCNBGYNCW),
        (13, NCDAODIBPNDIBHJYPGVMVMDOCHZODX),
        (3, XMNKYNSLZXNSLRTIZQFWFWNYMRJYNH),
        (10, QFGDRGLESQGLEKMBSJYPYPGRFKCRGA),
        (15, LABYMBGZNLBGZFHWNETKTKBMAFXMBV),
        (6, UJKHVKPIWUKPIOQFWNCTCTKVJOGVKE),
        (11, PEFCQFKDRPFKDJLARIXOXOFQEJBQFZ),
        (25, BQROCRWPDBRWPVXMDUJAJARCQVNCRL),
        (17, JYZWKZEXLJZEXDFULCRIRIZKYDVKZT),
        (19, HWXUIXCVJHXCVBDSJAPGPGXIWBTIXR),
        (4, WLMJXMRKYWMRKQSHYPEVEVMXLQIXMG),
        (0, APQNBQVOCAQVOUWLCTIZIZQBPUMBQK),
        (18, IXYVJYDWKIYDWCETKBQHQHYJXCUJYS),
        (5, VKLIWLQJXVLQJPRGXODUDULWKPHWLF)]

    TESTS:

    Currently, the octal number system is not supported as an alphabet for this shift cryptosystem:

        sage: SA = ShiftCryptosystem(AlphabeticStrings())
        sage: OctStr = OctalStrings()
        sage: C = OctStr([1, 2, 3])
        sage: SA.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: ciphertext must be encoded using one of the supported cipher domains of this shift cryptosystem.

    Nor is the radix-64 alphabet supported:

        sage: Rad64 = Radix64Strings()
        sage: C = Rad64([1, 2, 3])
        sage: SA.brute_force(C)
        Traceback (most recent call last):
        ...
        TypeError: ciphertext must be encoded using one of the supported cipher domains of this shift cryptosystem.

  `deciphering`(K, C)

    Decrypt the ciphertext `C` with the key `K` using shift cipher decryption.

    INPUT:

    - `K` – a secret key; a key belonging to the key space of this shift cipher. This key is an integer \(k\) satisfying the inequality \(0 \leq k < n\), where \(n\) is the size of the cipher domain.
    - `C` – a string of ciphertext; possibly an empty string. Characters in this string must be encoded using one of the supported alphabets. See the method `encoding()` for more information.

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

    Let’s perform decryption over the supported alphabets. Here is decryption over the capital letters of the English alphabet:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("Stop shifting me."); P
        STOPSHIFTINGME
        sage: K = 13
        sage: C = S.enciphering(K, P); C
        FGBCFUVSGVATZR
        sage: S.deciphering(K, C) == P
        True

    Decryption over the hexadecimal number system:

        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: P = S.encoding("Shift me now."); P
        5368696674206d65206e6f772e
        sage: K = 7
        sage: C = S.enciphering(K, P); C
        cadfd0ddeb97d4dc97d5d6ee95
        sage: S.deciphering(K, C) == P
        True

    Decryption over the binary number system:

        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: P = S.encoding("OK, enough shifting."); P
        0100111101001011001011000010000001100101011011100110111101110101011001110110100000100000011100110110100001101001011001100111010001101001011011100110011100101110
        sage: K = 1
        sage: C = S.enciphering(K, P); C
        1011000010110100110100111101111110011010100100011001000010001010100110001001011111011111100011001001011110010110100110011000101110010110100100011001100011010001
        sage: S.deciphering(K, C) == P
        True

  `enciphering`(K, P)

    Encrypt the plaintext `P` with the key `K` using shift cipher encryption.

    INPUT:

    - `K` – a key belonging to the key space of this shift cipher. This key is an integer \(k\) satisfying the inequality \(0 \leq k < n\), where \(n\) is the size of the cipher domain.
    - `P` – a string of plaintext; possibly an empty string. Characters in this string must be encoded using one of the supported alphabets. See the method `encoding()` for more information.

    OUTPUT:

    - The ciphertext corresponding to the plaintext `P`.

    EXAMPLES:

    Let’s perform encryption over the supported alphabets. Here is encryption over the capital letters of the English alphabet:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("Shift your gear."); P
        SHIFTYOURGEAR
        sage: K = 3
        sage: S.enciphering(K, P)
        VKLIWBRXUJHDU

    Encryption over the hexadecimal number system:

        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: P = S.encoding("Capitalize with the shift key."); P
        4361706974616c697a65207769746820746865207368696674206b65792e
        sage: K = 5
        sage: S.enciphering(K, P)
        98b6c5bec9b6b1becfba75ccbec9bd75c9bdba75c8bdbebbc975b0bace73

    Encryption over the binary number system:

        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: P = S.encoding("Don't shift."); P
        010001000110111101101110001001110111010000100000011100110110100001101001011001100111010000101110
        sage: K = 1
        sage: S.enciphering(K, P)
        101110111001000010010001110110001000101111011111100011001001011110010110100110011000101111010001

  `encoding`(S)

    The encoding of the string `S` over the string monoid of this shift cipher. For example, if the string monoid of this cryptosystem is `AlphabeticStringMonoid`, then the encoding of `S` would be its upper-case equivalent stripped of all non-alphabetic characters. The following alphabets are supported for the shift cipher:

    - capital letters of the English alphabet as implemented in `AlphabeticStrings()`
    - the alphabet consisting of the hexadecimal number system as implemented in `HexadecimalStrings()`
    - the alphabet consisting of the binary number system as implemented in `BinaryStrings()`

    INPUT:

    - `S` – a string, possibly empty.

    OUTPUT:

    - The encoding of `S` over the string monoid of this cryptosystem. If `S` is an empty string, return an empty string.

    EXAMPLES:

    Encoding over the upper-case letters of the English alphabet:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: S.encoding("Shift cipher on capital letters of the English alphabet.")
        SHIFTCIPHERONCAPITALLETTERSOFTHEENGLISHALPHABET

    Encoding over the binary system:

        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: S.encoding("Binary")
        010000100110100101101110011000010111001001111001

    Encoding over the hexadecimal system:

        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: S.encoding("Over hexadecimal system.")
        4f7665722068657861646563696d616c2073797374656d2e

    The argument `S` can be an empty string, in which case an empty string is returned:

        sage: ShiftCryptosystem(AlphabeticStrings()).encoding("")
        <BLANKLINE>
        sage: ShiftCryptosystem(HexadecimalStrings()).encoding("")
        <BLANKLINE>
        sage: ShiftCryptosystem(BinaryStrings()).encoding("")
        <BLANKLINE>

  `inverse_key`(K)

    The inverse key corresponding to the key `K`. For the shift cipher, the inverse key corresponding to `K` is \(-K \bmod n\), where \(n > 0\) is the size of the cipher domain, i.e. the plaintext/ciphertext space. A key \(k\) of the shift cipher is an integer \(0 \leq k < n\). The key \(k = 0\) has no effect on either the plaintext or the ciphertext.

    INPUT:

    - `K` – a key for this shift cipher. This must be an integer \(k\) such that \(0 \leq k < n\), where \(n\) is the size of the cipher domain.

    OUTPUT:

    - The inverse key corresponding to `K`.

    EXAMPLES:

    Some random keys and their respective inverse keys:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: key = S.random_key(); key  # random
        2
        sage: S.inverse_key(key)         # random
        24
        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: key = S.random_key(); key  # random
        12
        sage: S.inverse_key(key)         # random
        4
        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: key = S.random_key(); key  # random
        1
        sage: S.inverse_key(key)         # random
        1
        sage: key = S.random_key(); key  # random
        0
        sage: S.inverse_key(key)         # random
        0

    Regardless of the value of a key, the addition of the key and its inverse must be equal to the alphabet size. This relationship holds exactly when the value of the key is non-zero:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: K = S.random_key()
        sage: while K == 0:
        ...       K = S.random_key()
        ...
        sage: invK = S.inverse_key(K)
        sage: K + invK == S.alphabet_size()
        True
        sage: invK + K == S.alphabet_size()
        True
        sage: K = S.random_key()
        sage: while K != 0:
        ...       K = S.random_key()
        ...
        sage: invK = S.inverse_key(K)
        sage: K + invK != S.alphabet_size()
        True
        sage: K; invK
        0
        0

    TESTS:

    The key `K` must satisfy the inequality \(0 \leq K < n\) with \(n\) being the size of the plaintext, ciphertext, and key spaces. For the shift cryptosystem, all these spaces are the same alphabet. This inequality must be satisfied for each of the supported alphabets. The capital letters of the English alphabet:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: S.inverse_key(S.alphabet_size())
        Traceback (most recent call last):
        ...
        ValueError: K (=26) is outside the range of acceptable values for a key of this shift cryptosystem.
        sage: S.inverse_key(-1)
        Traceback (most recent call last):
        ...
        ValueError: K (=-1) is outside the range of acceptable values for a key of this shift cryptosystem.

    The hexadecimal number system:

        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: S.inverse_key(S.alphabet_size())
        Traceback (most recent call last):
        ...
        ValueError: K (=16) is outside the range of acceptable values for a key of this shift cryptosystem.
        sage: S.inverse_key(-1)
        Traceback (most recent call last):
        ...
        ValueError: K (=-1) is outside the range of acceptable values for a key of this shift cryptosystem.

    The binary number system:

        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: S.inverse_key(S.alphabet_size())
        Traceback (most recent call last):
        ...
        ValueError: K (=2) is outside the range of acceptable values for a key of this shift cryptosystem.
        sage: S.inverse_key(-1)
        Traceback (most recent call last):
        ...
        ValueError: K (=-1) is outside the range of acceptable values for a key of this shift cryptosystem.

  `random_key`()

    Generate a random key within the key space of this shift cipher. The generated key is an integer \(0 \leq k < n\) with \(n\) being the size of the cipher domain. Thus there are \(n\) possible keys in the key space, which is the set \(\ZZ / n\ZZ\). The key \(k = 0\) has no effect on either the plaintext or the ciphertext.

    OUTPUT:

    - A random key within the key space of this shift cryptosystem.

    EXAMPLES:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: S.random_key()  # random
        18
        sage: S = ShiftCryptosystem(BinaryStrings())
        sage: S.random_key()  # random
        0
        sage: S = ShiftCryptosystem(HexadecimalStrings())
        sage: S.random_key()  # random
        5

    Regardless of the value of a key, the addition of the key and its inverse must be equal to the alphabet size. This relationship holds exactly when the value of the key is non-zero:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: K = S.random_key()
        sage: while K == 0:
        ...       K = S.random_key()
        ...
        sage: invK = S.inverse_key(K)
        sage: K + invK == S.alphabet_size()
        True
        sage: invK + K == S.alphabet_size()
        True
        sage: K = S.random_key()
        sage: while K != 0:
        ...       K = S.random_key()
        ...
        sage: invK = S.inverse_key(K)
        sage: K + invK != S.alphabet_size()
        True
        sage: K; invK
        0
        0

  `rank_by_chi_square`(C, pdict)

    Use the chi-square statistic to rank all possible keys. Currently, this method only applies to the capital letters of the English alphabet.

    ALGORITHM:

    Consider a non-empty alphabet \(A\) consisting of \(n\) elements, and let \(C\) be a ciphertext encoded using elements of \(A\). The plaintext \(P\) corresponding to \(C\) is also encoded using elements of \(A\). Let \(M\) be a candidate decipherment of \(C\), i.e. \(M\) is the result of attempting to decrypt \(C\) using a key \(k \in \ZZ/n\ZZ\) which is not necessarily the same key used to encrypt \(P\). Suppose \(F_A(e)\) is the characteristic frequency probability of \(e \in A\) and let \(F_M(e)\) be the message frequency probability with respect to \(M\). The characteristic frequency probability distribution of an alphabet is the expected frequency probability distribution for that alphabet. The message frequency probability distribution of \(M\) provides a distribution of the ratio of character occurrences over message length. One can interpret the characteristic frequency probability \(F_A(e)\) as the expected probability, while the message frequency probability \(F_M(e)\) is the observed probability. If \(M\) is of length \(L\), then the observed frequency of \(e \in A\) is

    \[O_M(e) = F_M(e) \cdot L\]

    and the expected frequency of \(e \in A\) is

    \[E_A(e) = F_A(e) \cdot L\]

    The chi-square rank \(R_{\chi^2}(M)\) of \(M\) corresponding to a key \(k \in \ZZ/n\ZZ\) is given by

    \[R_{\chi^2}(M) = \sum_{e \in A} \frac {\big( O_M(e) - E_A(e) \big)^2} {E_A(e)}\]

    Cryptanalysis by exhaustive key search produces a candidate decipherment \(M_{k}\) for each possible key \(k \in \ZZ/n\ZZ\). For a set \(D = \big\{M_{k_1}, M_{k_2}, \dots, M_{k_r} \big\}\) of all candidate decipherments corresponding to a ciphertext \(C\), the smaller is the rank \(R_{\chi^2}(M_{k_i})\) the more likely that \(k_i\) is the secret key. This key ranking method is based on the Pearson chi-square test [PearsonTest09].

    INPUT:

    - `C` – The ciphertext, a non-empty string. The ciphertext must be encoded using the upper-case letters of the English alphabet.
    - `pdict` – A dictionary of key, possible plaintext pairs. This should be the output of `brute_force()` with `ranking="none"`.

    OUTPUT:

    - A list ranking the most likely keys first. Each element of the list is a tuple of key, possible plaintext pairs.

    EXAMPLES:

    Use the chi-square statistic to rank all possible keys and their corresponding decipherment:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("Shi."); P
        SHI
        sage: K = 5
        sage: C = S.enciphering(K, P)
        sage: Pdict = S.brute_force(C)
        sage: S.rank_by_chi_square(C, Pdict)
        <BLANKLINE>
        [(9, ODE),
        (5, SHI),
        (20, DST),
        (19, ETU),
        (21, CRS),
        (10, NCD),
        (25, YNO),
        (6, RGH),
        (12, LAB),
        (8, PEF),
        (1, WLM),
        (11, MBC),
        (18, FUV),
        (17, GVW),
        (2, VKL),
        (4, TIJ),
        (3, UJK),
        (0, XMN),
        (16, HWX),
        (15, IXY),
        (23, APQ),
        (24, ZOP),
        (22, BQR),
        (7, QFG),
        (13, KZA),
        (14, JYZ)]

    As more ciphertext is available, the reliability of the chi-square ranking function increases:

        sage: P = S.encoding("Shift cipher."); P
        SHIFTCIPHER
        sage: C = S.enciphering(K, P)
        sage: Pdict = S.brute_force(C)
        sage: S.rank_by_chi_square(C, Pdict)
        <BLANKLINE>
        [(5, SHIFTCIPHER),
        (9, ODEBPYELDAN),
        (18, FUVSGPVCURE),
        (2, VKLIWFLSKHU),
        (20, DSTQENTASPC),
        (19, ETURFOUBTQD),
        (21, CRSPDMSZROB),
        (6, RGHESBHOGDQ),
        (7, QFGDRAGNFCP),
        (12, LABYMVBIAXK),
        (17, GVWTHQWDVSF),
        (24, ZOPMAJPWOLY),
        (1, WLMJXGMTLIV),
        (0, XMNKYHNUMJW),
        (11, MBCZNWCJBYL),
        (8, PEFCQZFMEBO),
        (25, YNOLZIOVNKX),
        (10, NCDAOXDKCZM),
        (3, UJKHVEKRJGT),
        (4, TIJGUDJQIFS),
        (22, BQROCLRYQNA),
        (16, HWXUIRXEWTG),
        (15, IXYVJSYFXUH),
        (14, JYZWKTZGYVI),
        (13, KZAXLUAHZWJ),
        (23, APQNBKQXPMZ)]

    TESTS:

    The ciphertext cannot be an empty string:

        sage: S.rank_by_chi_square("", Pdict)
        Traceback (most recent call last):
        ...
        AttributeError: 'str' object has no attribute 'parent'
        sage: S.rank_by_chi_square(S.encoding(""), Pdict)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.
        sage: S.rank_by_chi_square(S.encoding(" "), Pdict)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.

    The ciphertext must be encoded using the capital letters of the English alphabet as implemented in `AlphabeticStrings()`:

        sage: H = HexadecimalStrings()
        sage: S.rank_by_chi_square(H.encoding("shift"), Pdict)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.
        sage: B = BinaryStrings()
        sage: S.rank_by_chi_square(B.encoding("shift"), Pdict)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.

    The dictionary `pdict` cannot be empty:

        sage: S.rank_by_chi_square(C, {})
        Traceback (most recent call last):
        ...
        KeyError: 0

    REFERENCES:

    [PearsonTest09] | (1, 2) Pearson chi-square test. Wikipedia, accessed 13th October 2009.

  `rank_by_squared_differences`(C, pdict)

    Use the squared-differences measure to rank all possible keys. Currently, this method only applies to the capital letters of the English alphabet.

    ALGORITHM:

    Consider a non-empty alphabet \(A\) consisting of \(n\) elements, and let \(C\) be a ciphertext encoded using elements of \(A\). The plaintext \(P\) corresponding to \(C\) is also encoded using elements of \(A\). Let \(M\) be a candidate decipherment of \(C\), i.e. \(M\) is the result of attempting to decrypt \(C\) using a key \(k \in \ZZ/n\ZZ\) which is not necessarily the same key used to encrypt \(P\). Suppose \(F_A(e)\) is the characteristic frequency probability of \(e \in A\) and let \(F_M(e)\) be the message frequency probability with respect to \(M\). The characteristic frequency probability distribution of an alphabet is the expected frequency probability distribution for that alphabet. The message frequency probability distribution of \(M\) provides a distribution of the ratio of character occurrences over message length. One can interpret the characteristic frequency probability \(F_A(e)\) as the expected probability, while the message frequency probability \(F_M(e)\) is the observed probability. If \(M\) is of length \(L\), then the observed frequency of \(e \in A\) is

    \[O_M(e) = F_M(e) \cdot L\]

    and the expected frequency of \(e \in A\) is

    \[E_A(e) = F_A(e) \cdot L\]

    The squared-differences, or residual sum of squares, rank \(R_{RSS}(M)\) of \(M\) corresponding to a key \(k \in \ZZ/n\ZZ\) is given by

    \[R_{RSS}(M) = \sum_{e \in A} \big( O_M(e) - E_A(e) \big)^2\]

    Cryptanalysis by exhaustive key search produces a candidate decipherment \(M_{k}\) for each possible key \(k \in \ZZ/n\ZZ\). For a set \(D = \big\{M_{k_1}, M_{k_2}, \dots, M_{k_r} \big\}\) of all candidate decipherments corresponding to a ciphertext \(C\), the smaller is the rank \(R_{RSS}(M_{k_i})\) the more likely that \(k_i\) is the secret key. This key ranking method is based on the residual sum of squares measure [RSS09].

    INPUT:

    - `C` – The ciphertext, a non-empty string. The ciphertext must be encoded using the upper-case letters of the English alphabet.
    - `pdict` – A dictionary of key, possible plaintext pairs. This should be the output of `brute_force()` with `ranking="none"`.

    OUTPUT:

    - A list ranking the most likely keys first. Each element of the list is a tuple of key, possible plaintext pairs.

    EXAMPLES:

    Use the method of squared differences to rank all possible keys and their corresponding decipherment:

        sage: S = ShiftCryptosystem(AlphabeticStrings())
        sage: P = S.encoding("Shi."); P
        SHI
        sage: K = 5
        sage: C = S.enciphering(K, P)
        sage: Pdict = S.brute_force(C)
        sage: S.rank_by_squared_differences(C, Pdict)
        <BLANKLINE>
        [(19, ETU),
        (9, ODE),
        (20, DST),
        (5, SHI),
        (8, PEF),
        (4, TIJ),
        (25, YNO),
        (21, CRS),
        (6, RGH),
        (10, NCD),
        (12, LAB),
        (23, APQ),
        (24, ZOP),
        (0, XMN),
        (13, KZA),
        (15, IXY),
        (1, WLM),
        (16, HWX),
        (22, BQR),
        (11, MBC),
        (18, FUV),
        (2, VKL),
        (17, GVW),
        (7, QFG),
        (3, UJK),
        (14, JYZ)]

    As more ciphertext is available, the reliability of the squared differences ranking function increases:

        sage: P = S.encoding("Shift cipher."); P
        SHIFTCIPHER
        sage: C = S.enciphering(K, P)
        sage: Pdict = S.brute_force(C)
        sage: S.rank_by_squared_differences(C, Pdict)
        <BLANKLINE>
        [(20, DSTQENTASPC),
        (5, SHIFTCIPHER),
        (9, ODEBPYELDAN),
        (19, ETURFOUBTQD),
        (6, RGHESBHOGDQ),
        (16, HWXUIRXEWTG),
        (8, PEFCQZFMEBO),
        (21, CRSPDMSZROB),
        (22, BQROCLRYQNA),
        (25, YNOLZIOVNKX),
        (3, UJKHVEKRJGT),
        (18, FUVSGPVCURE),
        (4, TIJGUDJQIFS),
        (10, NCDAOXDKCZM),
        (7, QFGDRAGNFCP),
        (24, ZOPMAJPWOLY),
        (2, VKLIWFLSKHU),
        (12, LABYMVBIAXK),
        (17, GVWTHQWDVSF),
        (1, WLMJXGMTLIV),
        (13, KZAXLUAHZWJ),
        (0, XMNKYHNUMJW),
        (15, IXYVJSYFXUH),
        (14, JYZWKTZGYVI),
        (11, MBCZNWCJBYL),
        (23, APQNBKQXPMZ)]

    TESTS:

    The ciphertext cannot be an empty string:

        sage: S.rank_by_squared_differences("", Pdict)
        Traceback (most recent call last):
        ...
        AttributeError: 'str' object has no attribute 'parent'
        sage: S.rank_by_squared_differences(S.encoding(""), Pdict)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.
        sage: S.rank_by_squared_differences(S.encoding(" "), Pdict)
        Traceback (most recent call last):
        ...
        ValueError: The ciphertext must be a non-empty string.

    The ciphertext must be encoded using the capital letters of the English alphabet as implemented in `AlphabeticStrings()`:

        sage: H = HexadecimalStrings()
        sage: S.rank_by_squared_differences(H.encoding("shift"), Pdict)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.
        sage: B = BinaryStrings()
        sage: S.rank_by_squared_differences(B.encoding("shift"), Pdict)
        Traceback (most recent call last):
        ...
        TypeError: The ciphertext must be capital letters of the English alphabet.

    The dictionary `pdict` cannot be empty:

        sage: S.rank_by_squared_differences(C, {})
        Traceback (most recent call last):
        ...
        KeyError: 0

    REFERENCES:

    [RSS09] | (1, 2) Residual sum of squares. Wikipedia, accessed 13th October 2009.

class `sage.crypto.classical.``SubstitutionCryptosystem`(S)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create a substitution cryptosystem.

  INPUT:

  - `S` - a string monoid over some alphabet

  OUTPUT:

  - A substitution cryptosystem over the alphabet `S`.

  EXAMPLES:

      sage: M = AlphabeticStrings()
      sage: E = SubstitutionCryptosystem(M)
      sage: E
      Substitution cryptosystem on Free alphabetic string monoid on A-Z
      sage: K = M([ 25-i for i in range(26) ])
      sage: K
      ZYXWVUTSRQPONMLKJIHGFEDCBA
      sage: e = E(K)
      sage: m = M("THECATINTHEHAT")
      sage: e(m)
      GSVXZGRMGSVSZG

  TESTS:

      sage: M = AlphabeticStrings()
      sage: E = SubstitutionCryptosystem(M)
      sage: E == loads(dumps(E))
      True

  `deciphering`(K, C)

    Decrypt the ciphertext `C` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this substitution cipher
    - `C` - a string (possibly empty) over the string monoid of this cryptosystem.

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

        sage: S = SubstitutionCryptosystem(AlphabeticStrings())
        sage: K = S.random_key()
        sage: M = S.encoding("Don't substitute me!")
        sage: S.deciphering(K, S.enciphering(K, M)) == M
        True

  `enciphering`(K, M)

    Encrypt the plaintext `M` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this substitution cipher
    - `M` - a string (possibly empty) over the string monoid of this cryptosystem.

    OUTPUT:

    - The ciphertext corresponding to the plaintext `M`.

    EXAMPLES:

        sage: S = SubstitutionCryptosystem(AlphabeticStrings())
        sage: K = S.random_key()
        sage: M = S.encoding("Don't substitute me.")
        sage: S.deciphering(K, S.enciphering(K, M)) == M
        True

  `encoding`(M)

    The encoding of the string `M` over the string monoid of this substitution cipher. For example, if the string monoid of this cryptosystem is `AlphabeticStringMonoid`, then the encoding of `M` would be its upper-case equivalent stripped of all non-alphabetic characters.

    INPUT:

    - `M` - a string, possibly empty

    OUTPUT:

    - The encoding of `M` over the string monoid of this cryptosystem.

    EXAMPLES:

        sage: M = "Peter Pan(ning) for gold."
        sage: A = AlphabeticStrings()
        sage: S = SubstitutionCryptosystem(A)
        sage: S.encoding(M) == A.encoding(M)
        True

  `inverse_key`(K)

    The inverse key corresponding to the key `K`. The specified key is a permutation of the cryptosystem alphabet.

    INPUT:

    - `K` - a key belonging to the key space of this cryptosystem

    OUTPUT:

    - The inverse key of `K`.

    EXAMPLES:

        sage: S = AlphabeticStrings()
        sage: E = SubstitutionCryptosystem(S)
        sage: K = E.random_key()
        sage: L = E.inverse_key(K)
        sage: M = S("THECATINTHEHAT")
        sage: e = E(K)
        sage: c = E(L)
        sage: c(e(M))
        THECATINTHEHAT

  `random_key`()

    Generate a random key within the key space of this substitution cipher. The generated key is a permutation of the cryptosystem alphabet. Let \(n\) be the length of the alphabet. Then there are \(n!\) possible keys in the key space.

    OUTPUT:

    - A random key within the key space of this cryptosystem.

    EXAMPLES:

        sage: A = AlphabeticStrings()
        sage: S = SubstitutionCryptosystem(A)
        sage: K = S.random_key()
        sage: Ki = S.inverse_key(K)
        sage: M = "THECATINTHEHAT"
        sage: e = S(K)
        sage: d = S(Ki)
        sage: d(e(A(M))) == A(M)
        True

class `sage.crypto.classical.``TranspositionCryptosystem`(S, n)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create a transposition cryptosystem of block length `n`.

  INPUT:

  - `S` - a string monoid over some alphabet
  - `n` - integer \(> 0\); a block length of a block permutation

  OUTPUT:

  - A transposition cryptosystem of block length `n` over the alphabet `S`.

  EXAMPLES:

      sage: S = AlphabeticStrings()
      sage: E = TranspositionCryptosystem(S,14)
      sage: E
      Transposition cryptosystem on Free alphabetic string monoid on A-Z of block length 14
      sage: K = [ 14-i for i in range(14) ]
      sage: K
      [14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
      sage: e = E(K)
      sage: e(S("THECATINTHEHAT"))
      TAHEHTNITACEHT

  TESTS:

      sage: S = AlphabeticStrings()
      sage: E = TranspositionCryptosystem(S,14)
      sage: E == loads(dumps(E))
      True

  `deciphering`(K, C)

    Decrypt the ciphertext `C` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this transposition cipher
    - `C` - a string (possibly empty) over the string monoid of this cryptosystem.

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

        sage: T = TranspositionCryptosystem(AlphabeticStrings(), 14)
        sage: K = T.random_key()
        sage: M = T.encoding("The cat in the hat.")
        sage: T.deciphering(K, T.enciphering(K, M)) == M
        True

  `enciphering`(K, M)

    Encrypt the plaintext `M` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this transposition cipher
    - `M` - a string (possibly empty) over the string monoid of this cryptosystem

    OUTPUT:

    - The ciphertext corresponding to the plaintext `M`.

    EXAMPLES:

        sage: T = TranspositionCryptosystem(AlphabeticStrings(), 14)
        sage: K = T.random_key()
        sage: M = T.encoding("The cat in the hat.")
        sage: T.deciphering(K, T.enciphering(K, M)) == M
        True

  `encoding`(M)

    The encoding of the string `M` over the string monoid of this transposition cipher. For example, if the string monoid of this cryptosystem is `AlphabeticStringMonoid`, then the encoding of `M` would be its upper-case equivalent stripped of all non-alphabetic characters.

    INPUT:

    - `M` - a string, possibly empty

    OUTPUT:

    - The encoding of `M` over the string monoid of this cryptosystem.

    EXAMPLES:

        sage: M = "Transposition cipher is not about matrix transpose."
        sage: A = AlphabeticStrings()
        sage: T = TranspositionCryptosystem(A, 11)
        sage: T.encoding(M) == A.encoding(M)
        True

  `inverse_key`(K, check=True)

    The inverse key corresponding to the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this transposition cipher
    - `check` - bool (default: `True`); check that `K` belongs to the key space of this cryptosystem.

    OUTPUT:

    - The inverse key corresponding to `K`.

    EXAMPLES:

        sage: S = AlphabeticStrings()
        sage: E = TranspositionCryptosystem(S, 14)
        sage: K = E.random_key()
        sage: Ki = E.inverse_key(K)
        sage: e = E(K)
        sage: d = E(Ki)
        sage: M = "THECATINTHEHAT"
        sage: C = e(S(M))
        sage: d(S(C)) == S(M)
        True

  `random_key`()

    Generate a random key within the key space of this transposition cryptosystem. Let \(n > 0\) be the block length of this cryptosystem. Then there are \(n!\) possible keys.

    OUTPUT:

    - A random key within the key space of this cryptosystem.

    EXAMPLES:

        sage: S = AlphabeticStrings()
        sage: E = TranspositionCryptosystem(S, 14)
        sage: K = E.random_key()
        sage: Ki = E.inverse_key(K)
        sage: e = E(K)
        sage: d = E(Ki)
        sage: M = "THECATINTHEHAT"
        sage: C = e(S(M))
        sage: d(S(C)) == S(M)
        True

class `sage.crypto.classical.``VigenereCryptosystem`(S, n)

  Bases: `sage.crypto.cryptosystem.SymmetricKeyCryptosystem`

  Create a Vigenere cryptosystem of block length `n`.

  INPUT:

  - `S`– a string monoid over some alphabet
  - `n` - integer \(> 0\); block length of an encryption/decryption key

  OUTPUT:

  - A Vigenere cryptosystem of block length `n` over the alphabet `S`.

  EXAMPLES:

      sage: S = AlphabeticStrings()
      sage: E = VigenereCryptosystem(S,14)
      sage: E
      Vigenere cryptosystem on Free alphabetic string monoid on A-Z of period 14
      sage: K = S('ABCDEFGHIJKLMN')
      sage: K
      ABCDEFGHIJKLMN
      sage: e = E(K)
      sage: e
      Cipher on Free alphabetic string monoid on A-Z
      sage: e(S("THECATINTHEHAT"))
      TIGFEYOUBQOSMG

  TESTS:

      sage: S = AlphabeticStrings()
      sage: E = VigenereCryptosystem(S,14)
      sage: E == loads(dumps(E))
      True

  `deciphering`(K, C)

    Decrypt the ciphertext `C` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this Vigenere cipher
    - `C` - a string (possibly empty) over the string monoid of this cryptosystem

    OUTPUT:

    - The plaintext corresponding to the ciphertext `C`.

    EXAMPLES:

        sage: V = VigenereCryptosystem(AlphabeticStrings(), 24)
        sage: K = V.random_key()
        sage: M = V.encoding("Jack and Jill went up the hill.")
        sage: V.deciphering(K, V.enciphering(K, M)) == M
        True

  `enciphering`(K, M)

    Encrypt the plaintext `M` using the key `K`.

    INPUT:

    - `K` - a key belonging to the key space of this Vigenere cipher
    - `M` - a string (possibly empty) over the string monoid of this cryptosystem

    OUTPUT:

    - The ciphertext corresponding to the plaintext `M`.

    EXAMPLES:

        sage: V = VigenereCryptosystem(AlphabeticStrings(), 24)
        sage: K = V.random_key()
        sage: M = V.encoding("Jack and Jill went up the hill.")
        sage: V.deciphering(K, V.enciphering(K, M)) == M
        True

  `encoding`(M)

    The encoding of the string `M` over the string monoid of this Vigenere cipher. For example, if the string monoid of this cryptosystem is `AlphabeticStringMonoid`, then the encoding of `M` would be its upper-case equivalent stripped of all non-alphabetic characters.

    INPUT:

    - `M` - a string, possibly empty

    OUTPUT:

    - The encoding of `M` over the string monoid of this cryptosystem.

    EXAMPLES:

        sage: A = AlphabeticStrings()
        sage: V = VigenereCryptosystem(A, 24)
        sage: M = "Jack and Jill went up the hill."
        sage: V.encoding(M) == A.encoding(M)
        True

  `inverse_key`(K)

    The inverse key corresponding to the key `K`.

    INPUT:

    - `K` - a key within the key space of this Vigenere cryptosystem

    OUTPUT:

    - The inverse key corresponding to `K`.

    EXAMPLES:

        sage: S = AlphabeticStrings()
        sage: E = VigenereCryptosystem(S,14)
        sage: K = E.random_key()
        sage: L = E.inverse_key(K)
        sage: M = S("THECATINTHEHAT")
        sage: e = E(K)
        sage: c = E(L)
        sage: c(e(M))
        THECATINTHEHAT

  `random_key`()

    Generate a random key within the key space of this Vigenere cryptosystem. Let \(n > 0\) be the length of the cryptosystem alphabet and let \(m > 0\) be the block length of this cryptosystem. Then there are \(n^m\) possible keys.

    OUTPUT:

    - A random key within the key space of this cryptosystem.

    EXAMPLES:

        sage: A = AlphabeticStrings()
        sage: V = VigenereCryptosystem(A, 14)
        sage: M = "THECATINTHEHAT"
        sage: K = V.random_key()
        sage: Ki = V.inverse_key(K)
        sage: e = V(K)
        sage: d = V(Ki)
        sage: d(e(A(M))) == A(M)
        True

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