Editing Hill cipher (section)

==Encryption==
Each letter is represented by a number [[modular arithmetic|modulo]] 26. Though this is not an essential feature of the cipher, this simple scheme is often used:
{| class="wikitable" style="text-align:center; font-size:90%;"
! Letter
|A||B||C||D||E||F||G||H||I||J||K||L||M||N||O||P||Q||R||S||T||U||V||W||X||Y||Z
|-
! Number
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19||20||21||22||23||24||25
|}
To encrypt a message, each block of ''n'' letters (considered as an ''n''-component [[vector space|vector]]) is multiplied by an [[Invertible matrix|invertible]] ''n'' &times; ''n'' [[matrix (mathematics)|matrix]], against [[modular arithmetic|modulus]] 26. To decrypt the message, each block is multiplied by the inverse of the matrix used for encryption.

The matrix used for encryption is the cipher [[key (cryptography)|key]], and it should be chosen randomly from the set of invertible ''n'' &times; ''n'' matrices ([[modular arithmetic|modulo]] 26). The cipher can, of course, be adapted to an alphabet with any number of letters; all arithmetic just needs to be done modulo the number of letters instead of modulo 26.

Consider the message 'ACT', and the key below (or GYB{{silver (color)|/}}NQK{{silver (color)|/}}URP in letters):
:<math>\begin{pmatrix} 6 & 24 & 1 \\ 13 & 16 & 10 \\ 20 & 17 & 15 \end{pmatrix}</math>
Since 'A' is 0, 'C' is 2 and 'T' is 19, the message is the vector:
:<math>\begin{pmatrix} 0 \\ 2 \\ 19 \end{pmatrix}</math>
Thus the enciphered vector is given by:
:<math>\begin{pmatrix} 6 & 24 & 1 \\ 13 & 16 & 10 \\ 20 & 17 & 15 \end{pmatrix} \begin{pmatrix} 0 \\ 2 \\ 19 \end{pmatrix} = \begin{pmatrix} 67 \\ 222 \\ 319 \end{pmatrix} \equiv \begin{pmatrix} 15 \\ 14 \\ 7 \end{pmatrix} \pmod{26}</math>
which corresponds to a [[ciphertext]] of 'POH'. Now, suppose that our message is instead 'CAT', or:
:<math>\begin{pmatrix} 2 \\ 0 \\ 19 \end{pmatrix}</math>
This time, the enciphered vector is given by:
:<math>\begin{pmatrix} 6 & 24 & 1 \\ 13 & 16 & 10 \\ 20 & 17 & 15 \end{pmatrix} \begin{pmatrix} 2 \\ 0 \\ 19 \end{pmatrix} = \begin{pmatrix} 31 \\ 216 \\ 325 \end{pmatrix} \equiv \begin{pmatrix} 5 \\ 8 \\ 13 \end{pmatrix} \pmod{26}</math>
which corresponds to a ciphertext of 'FIN'. Every letter has changed. The Hill cipher has achieved [[Claude Elwood Shannon|Shannon]]'s [[Confusion and diffusion|diffusion]], and an ''n''-dimensional Hill cipher can diffuse fully across ''n'' symbols at once.