Affine Cipher

Affine cipher is the name given to a substitution cipher whose key consists of 2 coefficients A and B constituting the parameters of a mathematical linear function $ f=Ax+B $ (called affine).

Encryption uses a classic alphabet, and two integers, called coefficients or keys A and B, these are the parameters of the affine function Ax+B (which is a straight line/linear equation).

For each letter of the alphabet is associated to the value of its position in the alphabet (starting at 0).

For each letter of value $ x $ of the plain text, is associated a value $ y $, result of the affine function $ y = A \times x + B \mod 26 $ (with $ 26 $ the alphabet size). Each computed value $ y $ corresponds to a letter with the same position in the alphabet, it is the ciphered letter. The Affine ciphertext is the replacement of all the letters by the new ones.

Affine decryption requires to know the two keys A and B (the one from encryption) and the used alphabet.

For each letter of the alphabet, associate the value of its position in the alphabet.

Each letter of value $ y $ of the message corresponds to a value $ x $, result of the inverse function $ x = A' \times (y-B) \mod 26 $ (with $ 26 $ the alphabet size). If the value obtained is negative, add 26.

The value $ A' $ is an integer such as $ A \times A' = 1 \mod 26 $ (with $ 26 $ the alphabet size). To find $ A' $, calculate its modular inverse.

For each value $ x $, associate the letter with the same position in the alphabet: the coded letter. The plain text is the replacement of all characters with calculated new letters.

A message encrypted by Affine has a coincidence index close to the plain text language's one.

Any reference to an affine function (in a straight line), a graph, an abscissa or an ordinate is a clue (the function $ f(x) = ax + b $ can be represented in an orthonormal coordinate system like a classical affine function, it is therefore possible from a graph to find the slope coefficient $ a $ and the y-intercept $ b $).

The affine ciphers in fact group together several ciphers which are special cases:

— The multiplicative cipher is a special case of the Affine cipher where B is 0.

— The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offest.

The affine cipher is itself a special case of the Hill cipher, which uses an invertible matrix, rather than a straight-line equation, to generate the substitution alphabet.

To crack Affine, it is possible to bruteforce/test all values for A and B coefficients. Use the Brute-force attack button.

For an affine encryption with the function $ y = A x + B $, then the reciprocal/inverse decryption function is expressed $ y' = A' x + B $

Calculate the modular inverse of A, modulo the length of the alphabet (see below for pre-calculated values).

B' has the same value as B, for this reason, this variable should not be called B' but B.

The value of A' depends on A but also on the alphabet's length, if it is a classic one, it is 26 characters long. The values of A' are then:

The Bezout's theorem indicates that A' only exists if A and 26 (alphabet length) are coprime. This limits A values to 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 (if the alphabet is 26 characters long)

Yes, but an automatic decryption process becomes impossible, a single ciphered letter will have multiple plain letters possible.

Yes, but there exists a positive corresponding value, a value of A = -1 is equal to a value of A = 25 (because 25 = -1 mod 26).

No, B can take any value.

All the values of B modulo 26 (length of the alphabet) are equivalent. So if B is negative, there is an equivalent positive value of B.

In mathematics, an affine function is defined by addition and multiplication of the variable (often $ x $) and written $ f(x) = ax + b $. The affine cipher is similar to the $ f $ function as it uses the values $ a $ and $ b $ as a coefficient and the variable $ x $ is the letter to be encrypted.

Date and author are unknown, the affine cipher.