Affine Cipher - Description

Description

In the affine cipher the letters of an alphabet of size are first mapped to the integers in the range . 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 is the size of the alphabet and and are the key of the cipher. The value must be chosen such that and are coprime. The decryption function is

where is the modular multiplicative inverse of modulo . I.e., it satisfies the equation

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

\begin{align} \mbox{D}(\mbox{E}(x)) &= a^{-1}(\mbox{E}(x)-b)\mod{m}\\ &= a^{-1}(((ax+b)\mod{m})-b)\mod{m} \\ &= a^{-1}(ax+b-b)\mod{m} \\ &= a^{-1}ax \mod{m}\\ &= x\mod{m}. \end{align}

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