Editing Vigenère cipher (section)

== Algebraic description ==
Vigenère can also be described algebraically. If the letters <code>A</code>–<code>Z</code> are taken to be the numbers 0–25 (<math>A \,\widehat{=}\, 0</math>, <math>B \,\widehat{=}\, 1</math>, etc.), and addition is performed [[modular arithmetic|modulo]] 26, Vigenère encryption <math>E</math> using the key <math>K</math> can be written as

:<math>C_i = E_K(M_i) = (M_i+K_i) \bmod 26</math>

and decryption <math>D</math> using the key <math>K</math> as

:<math>M_i = D_K(C_i) = (C_i-K_i) \bmod 26,</math>

in which <math>M = M_1 \dots M_n</math> is the message, <math>C = C_1 \dots C_n</math> is the ciphertext and <math>K = K_1 \dots K_n</math> is the key obtained by repeating the keyword <math>\lceil n / m \rceil</math> times in which <math>m</math> is the keyword length.

Thus, by using the previous example, to encrypt <math>A \,\widehat{=}\, 0</math> with key letter <math>L \,\widehat{=}\, 11</math> the calculation would result in <math>11 \,\widehat{=}\, L</math>.
:<math>11 = (0+11) \bmod 26</math>

Therefore, to decrypt <math>R \,\widehat{=}\, 17</math> with key letter <math>E \,\widehat{=}\, 4</math>, the calculation would result in <math>13 \,\widehat{=}\, N</math>.
:<math>13 = (17-4) \bmod 26</math>

In general, if <math>\Sigma</math> is the alphabet of length <math>\ell</math>, and <math>m</math> is the length of key, Vigenère encryption and decryption can be written:

:<math>C_i = E_K(M_i) = (M_i+K_{(i \bmod m)}) \bmod \ell,</math>

:<math>M_i = D_K(C_i) = (C_i-K_{(i \bmod m)}) \bmod \ell.</math>

<math>M_i</math> denotes the offset of the ''i''-th character of the plaintext <math>M</math> in the alphabet <math>\Sigma</math>. For example, by taking the 26 English characters as the alphabet <math>\Sigma = (A,B,C,\ldots,X,Y,Z)</math>, the offset of A is 0, the offset of B is 1 etc. <math>C_i</math> and <math>K_i</math> are similar.