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# Cryptosystems

This module contains base classes for various cryptosystems, including symmetric key and public-key cryptosystems. The classes defined in this module should not be called directly. It is the responsibility of child classes to implement specific cryptosystems. Take for example the Hill or matrix cryptosystem as implemented in `HillCryptosystem`. It is a symmetric key cipher so `HillCryptosystem` is a child class of `SymmetricKeyCryptosystem`, which in turn is a child class of `Cryptosystem`. The following diagram shows the inheritance relationship of particular cryptosystems:

    Cryptosystem
    + SymmetricKeyCryptosystem
    | + HillCryptosystem
    | + LFSRCryptosystem
    | + ShiftCryptosystem
    | + ShrinkingGeneratorCryptosystem
    | + SubstitutionCryptosystem
    | + TranspositionCryptosystem
    | + VigenereCryptosystem
    + PublicKeyCryptosystem

class `sage.crypto.cryptosystem.``Cryptosystem`(plaintext_space, ciphertext_space, key_space, block_length=1, period=None)

  Bases: `sage.structure.parent_old.Parent`, `sage.structure.parent.Set_generic`

  A base cryptosystem class. This is meant to be extended by other specialized child classes that implement specific cryptosystems. A cryptosystem is a pair of maps

  \[E : {\mathcal K} \rightarrow {\rm Hom}({\mathcal M},{\mathcal C})\]
  \[D : {\mathcal K} \rightarrow {\rm Hom}({\mathcal C},{\mathcal M})\]

  where \({\mathcal K}\) is the key space, \({\mathcal M}\) is the plaintext or message space, and \({\mathcal C}\) is the ciphertext space. In many instances \({\mathcal M} = {\mathcal C}\) and the images will lie in \({\rm Aut}({\mathcal M})\). An element of the image of \(E\) is called a cipher.

  We may assume that \(E\) and \(D\) are injective, hence identify a key \(K\) in \({\mathcal K}\) with its image \(E_K := E(K)\) in \(\mathrm{Hom}({\mathcal M},{\mathcal C})\).

  The cryptosystem has the property that for every encryption key \(K_1\) there is a decryption key \(K_2\) such that \(D_{K_2} \circ E_{K_1}\). A cryptosystem with the property that \(K := K_2 = K_1\), is called a symmetric cryptosystem. Otherwise, if the key \(K_2 \ne K_1\), nor is \(K_2\) easily derived from \(K_1\), we call the cryptosystem asymmetric or public key. In that case, \(K_1\) is called the public key and \(K_2\) is called the private key.

  INPUT:

  - `plaintext_space` – the plaintext alphabet.
  - `ciphertext_space` – the ciphertext alphabet.
  - `key_space` – the key alphabet.
  - `block_length` – (default: 1) the block length.
  - `period` – (default: `None`) the period.

  EXAMPLES:

  Various classical cryptosystems:

      sage: ShiftCryptosystem(AlphabeticStrings())
      Shift cryptosystem on Free alphabetic string monoid on A-Z
      sage: SubstitutionCryptosystem(HexadecimalStrings())
      Substitution cryptosystem on Free hexadecimal string monoid
      sage: HillCryptosystem(BinaryStrings(), 3)
      Hill cryptosystem on Free binary string monoid of block length 3
      sage: TranspositionCryptosystem(OctalStrings(), 5)
      Transposition cryptosystem on Free octal string monoid of block length 5
      sage: VigenereCryptosystem(Radix64Strings(), 7)
      Vigenere cryptosystem on Free radix 64 string monoid of period 7

  `block_length`()

    Return the block length of this cryptosystem. For some cryptosystems this is not relevant, in which case the block length defaults to 1.

    EXAMPLES:

    The block lengths of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).block_length()
        1
        sage: SubstitutionCryptosystem(HexadecimalStrings()).block_length()
        1
        sage: HillCryptosystem(BinaryStrings(), 3).block_length()
        3
        sage: TranspositionCryptosystem(OctalStrings(), 5).block_length()
        5
        sage: VigenereCryptosystem(Radix64Strings(), 7).block_length()
        1

  `cipher_codomain`()

    Return the alphabet used by this cryptosystem for encoding ciphertexts. This is the same as the ciphertext space.

    EXAMPLES:

    The cipher codomains, or ciphertext spaces, of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).cipher_codomain()
        Free alphabetic string monoid on A-Z
        sage: SubstitutionCryptosystem(HexadecimalStrings()).cipher_codomain()
        Free hexadecimal string monoid
        sage: HillCryptosystem(BinaryStrings(), 3).cipher_codomain()
        Free binary string monoid
        sage: TranspositionCryptosystem(OctalStrings(), 5).cipher_codomain()
        Free octal string monoid
        sage: VigenereCryptosystem(Radix64Strings(), 7).cipher_codomain()
        Free radix 64 string monoid

  `cipher_domain`()

    Return the alphabet used by this cryptosystem for encoding plaintexts. This is the same as the plaintext space.

    EXAMPLES:

    The cipher domains, or plaintext spaces, of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).cipher_domain()
        Free alphabetic string monoid on A-Z
        sage: SubstitutionCryptosystem(HexadecimalStrings()).cipher_domain()
        Free hexadecimal string monoid
        sage: HillCryptosystem(BinaryStrings(), 3).cipher_domain()
        Free binary string monoid
        sage: TranspositionCryptosystem(OctalStrings(), 5).cipher_domain()
        Free octal string monoid
        sage: VigenereCryptosystem(Radix64Strings(), 7).cipher_domain()
        Free radix 64 string monoid

  `ciphertext_space`()

    Return the ciphertext alphabet of this cryptosystem.

    EXAMPLES:

    The ciphertext spaces of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).ciphertext_space()
        Free alphabetic string monoid on A-Z
        sage: SubstitutionCryptosystem(HexadecimalStrings()).ciphertext_space()
        Free hexadecimal string monoid
        sage: HillCryptosystem(BinaryStrings(), 3).ciphertext_space()
        Free binary string monoid
        sage: TranspositionCryptosystem(OctalStrings(), 5).ciphertext_space()
        Free octal string monoid
        sage: VigenereCryptosystem(Radix64Strings(), 7).ciphertext_space()
        Free radix 64 string monoid

  `key_space`()

    Return the alphabet used by this cryptosystem for encoding keys.

    EXAMPLES:

    The key spaces of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).key_space()
        Ring of integers modulo 26
        sage: SubstitutionCryptosystem(HexadecimalStrings()).key_space()
        Free hexadecimal string monoid
        sage: HillCryptosystem(BinaryStrings(), 3).key_space()
        Full MatrixSpace of 3 by 3 dense matrices over Ring of integers modulo 2
        sage: TranspositionCryptosystem(OctalStrings(), 5).key_space()
        Symmetric group of order 5! as a permutation group
        sage: VigenereCryptosystem(Radix64Strings(), 7).key_space()
        Free radix 64 string monoid

  `period`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `plaintext_space`()

    Return the plaintext alphabet of this cryptosystem.

    EXAMPLES:

    The plaintext spaces of various classical cryptosystems:

        sage: ShiftCryptosystem(AlphabeticStrings()).plaintext_space()
        Free alphabetic string monoid on A-Z
        sage: SubstitutionCryptosystem(HexadecimalStrings()).plaintext_space()
        Free hexadecimal string monoid
        sage: HillCryptosystem(BinaryStrings(), 3).plaintext_space()
        Free binary string monoid
        sage: TranspositionCryptosystem(OctalStrings(), 5).plaintext_space()
        Free octal string monoid
        sage: VigenereCryptosystem(Radix64Strings(), 7).plaintext_space()
        Free radix 64 string monoid

class `sage.crypto.cryptosystem.``PublicKeyCryptosystem`(plaintext_space, ciphertext_space, key_space, block_length=1, period=None)

  Bases: `sage.crypto.cryptosystem.Cryptosystem`

  The base class for asymmetric or public-key cryptosystems.

class `sage.crypto.cryptosystem.``SymmetricKeyCryptosystem`(plaintext_space, ciphertext_space, key_space, block_length=1, period=None)

  Bases: `sage.crypto.cryptosystem.Cryptosystem`

  The base class for symmetric key, or secret key, cryptosystems.

  `alphabet_size`()

    Return the number of elements in the alphabet of this cryptosystem. This only applies to any cryptosystem whose plaintext and ciphertext spaces are the same alphabet.

    EXAMPLES:

        sage: ShiftCryptosystem(AlphabeticStrings()).alphabet_size()
        26
        sage: ShiftCryptosystem(BinaryStrings()).alphabet_size()
        2
        sage: ShiftCryptosystem(HexadecimalStrings()).alphabet_size()
        16
        sage: SubstitutionCryptosystem(OctalStrings()).alphabet_size()
        8
        sage: SubstitutionCryptosystem(Radix64Strings()).alphabet_size()
        64

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