Affine Cipher

Affine encryption is the name given to a substitution cipher whose correspondence is given by an affine function endowed with 2 coefficients A and B.

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.

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 $, resulting of the affine function $ y = A \times x + B \mod 26 $ (with $ 26 $ the alphabet size). For each value $ y $, corresponds 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 corresponds the value of its position in the alphabet.

For each letter of value $ y $ of the message, corresponds a value $ x $, result of the inverse function $ x = A' \times (y-B) \mod 26 $ (with $ 26 $ the alphabet size)

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 $, corresponds a 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 $).

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

Pour an affine encryption with the function $ y = A x + B $, then the reciproqual 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 it exists a positive corresponding value, a value of A = -1 is equals 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.

No date nor known author for affine cipher.