Affine Cipher

Affine cipher is a monoalphabetic substitution method where each letter of the plaintext is replaced by another letter according to an affine function of the form $ f(x) = A \times x + B \mod 26 $.

$ A $ and $ B $ are two integers that form the encryption key, and $ 26 $ corresponds to the length of the standard Latin alphabet.

To encode a message using affine cipher:

1/ Assign each letter of the plaintext its numerical position $ x $ in the alphabet. By default, $ A=0, B=1, \dots, Z=25 $. It is possible (but not recommended) to use $ A=1, \dots, Y=25, Z=0 $ by using the alphabet ZABCDEFGHIJKLMNOPQRSTUVWXY.

2/ Apply the affine function $ y = (A \times x + B) \mod 26 $ to obtain the position $ y $ of the ciphertext letter.

3/ Replace each letter of the plaintext with the letter corresponding to $ y $ in the alphabet.

To decode a message encrypted using Affine:

1/ Calculate the modular inverse $ A' $ of $ A $ modulo $ 26 $

2/ For each ciphertext letter at position $ y $, calculate $ x = A' \times (y - B) \mod 26 $

3/ Replace each ciphertext letter with the letter corresponding to $ x $ in the alphabet.

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.

Without knowing the coefficients $ A $ and $ B $, a brute-force attack can be used. Assuming the alphabet has 26 characters, then the coefficient $ A $ can only take 12 values (numbers coprime to 26) and the coefficient $ B $ can only take 16 values, for a total of 312 combinations.

Knowing some letters from the plaintext, it is possible to plot the plaintext/ciphertext letters on a cartesian plane and deduce $ A $ (the slope) and $ B $ (the y-intercept).

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:

Bézout's theorem states that $ A' $ exists only if $ A $ and $ 26 $ (the length of the alphabet) are coprime. This restricts $ A $ to the values $ 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25 $.

Yes, but this makes decryption non-unique: the same ciphered letter can correspond to several plaintext letters.

Yes, but every negative value of $ A $ has a positive equivalent modulo 26.

No, $ B $ can be any integer, but all values of $ B $ modulo 26 (the length of the alphabet) are equivalent. Thus, if $ B $ is negative, there exists 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.

An affine function is a linear function because its graph is a straight line.

Date and author are unknown, the affine cipher.