Cryptography

Affine Cipher

History and Description

Since a shift cipher can produce only 25 different distinct transformations for the text, it is not a very secure encryption method. The affine cipher is a generalization of the shift cipher that provides a little bit more security. The affine cipher applies multiplication and addition to each character using the function:
y = (ax + b) MOD m
where x is the numerical value of the letter in the plaintext, m is the number of letters in the plaintext alphabet, a and b are the secret numbers, and y is the result of transformation. y can be decrypted back to x by using the formula x = inverse(a) (y – b) MOD m, inverse(a) is a value such that if it is multiplied with a MOD m the result will be 1, i.e. (a * inverse(a)) MOD m = 1.

Using the encryption function y = 11x + 4 MOD 26, letter E and S will be encoded to W and U as shown in example below. Since the computation involves modulo 26 arithmetic, several letters may fail to be uniquely decoded if the multiplier  has a common divisor with 26. Therefore, the greatest common divisor of a and m must be 1.

Example

Encipher
Assume the message is encrypted by the function y = (11x + 4) MOD 26
To encrypt the plaintext MONEY, we first convert each letter in plaintext into a numerical value between 0 and 25 according to following list

Thus, the numerical values corresponding to the plaintext MONEY are 12, 14, 13, 4, and 24.
Applying the given function for each numerical value, we have

The corresponding letters are GCRWI, which is the ciphertext.

Decipher
To decipher, we transform the function y as:
x = inverse (a) (y – b) MOD m
Then we have, x = inverse(11) (y – 4) MOD 26
Inverse(11) MOD 26 = 19, and the decryption function will be x = 19 (y – 4) MOD 26

We now decipher the ciphertext GCRWI by applying the decryption function. We have:

The corresponding plaintext letters are MONEY.

Analysis

Since we know that each letter in plaintext is enciphered in the function of y = (ax + b) MOD m, we can break the affine cipher by solving two linear equations with two examples of x and y. Once we obtain the values of a and b, we can decipher the entire  ciphertext.

For example,

Assume that “IF” is enciphered as “PQ”.
I -> P: 8a + b = 15 MOD 26
F -> Q: 5a + b = 16 MOD 26
By solving these equations, we have the keys a = 17, b = 9