Affine cipher

The affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. The formula used means that each letter encrypts to one other letter, and back again, meaning the cipher is essentially a standard substitution cipher with a rule governing which letter goes to which. As such, it has the weaknesses of all substitution ciphers. Each letter is enciphered with the function $(ax + b) mod 26$ , where $b$ is the magnitude of the shift.

In the affine cipher the letters of an alphabet of size $m$ are first mapped to the integers in the range $0 \dots m - 1$ . It then uses modular arithmetic to transform the integer that each plaintext letter corresponds to into another integer that correspond to a ciphertext letter. The encryption function for a single letter is

where modulus $m$ is the size of the alphabet and $a$ and $b$ are the key of the cipher. The value $a$ must be chosen such that $a$ and $m$ are coprime. The decryption function is

where $a -1$ is the modular multiplicative inverse of $a$ modulo $m$ . I.e., it satisfies the equation

The multiplicative inverse of $a$ only exists if $a$ and $m$ are coprime. Hence without the restriction on $a$ decryption might not be possible. It can be shown as follows that decryption function is the inverse of the encryption function,

Since the affine cipher is still a monoalphabetic substitution cipher, it inherits the weaknesses of that class of ciphers. The Affine cipher is a Caesar cipher when $a = 1$ since the encrypting function simply reduces to a linear shift.

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