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.

Example: Encrypt DCODE with the keys A=5, B=3 and the English/latin alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ.

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

Example: By default, A=0, B=1, …, Z=25, but it is possible (but not recommended) to use A=1, …, Y=25, Z=0 using the alphabet ZABCDEFGHIJKLMNOPQRSTUVWXY.

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.

Example: DCODE is crypted SNVSX

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

Example: Decrypt the ciphered message SNVSX with keys A=5 and B=3

For each letter of the alphabet corresponds the value of its position in the alphabet.

Example: The alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ, starting at 0 gives A=0, B=1, …, Z=25.

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.

Example: A coefficient $ A' $ for $ A = 5 $ with an alphabet size of $ 26 $ is $ 21 $ because $ 5 \times 21 = 105 \equiv 1 \mod 26 $.
For S ( $ y = 18 $ ), $ x = A' \times (18-B) = 21 \times (18-3) \equiv 315 \mod 26 = 3 $

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.

Example: For S ( $ x = 3 $ ) corresponds the letter at position 3: D, etc. The original plain text is DCODE.

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

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

If the alphabet is 26 characters long, then A coefficient has only 12 possible values, and B has 26 values, so there are only 312 test to try.

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.

Example: 'B = -1' is equivalent to 'B = 25' (modulo 26)

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.