Affine Cipher Calculator

What is the affine cipher?

The affine cipher is a substitution cipher that generalises the Caesar cipher. Each letter is first converted to a number (A = 0, …, Z = 25), then multiplied by a key number a, shifted by a second key number b, and reduced modulo 26: E(x) = (a·x + b) mod 26. The Caesar cipher is the special case a = 1.

The key is the pair (a, b). Since b can be any of the 26 values and a can only be one of 12 values, the cipher has 12 × 26 = 312 possible keys.

Why a must be coprime to 26

Encryption must be reversible, which means different letters have to produce different cipher letters. If a shares a factor with 26, that fails. Take a = 2: the letters A (0) and N (13) give 2·0 + b = b and 2·13 + b = 26 + b, which are both b modulo 26, so two letters collide and the receiver cannot tell which one was sent.

The condition is gcd(a, 26) = 1, so a must be odd and not a multiple of 13. The twelve valid values are 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25. Their count is Euler's totient φ(26) = 12.

Decryption and the modular inverse

To undo the encryption, solve y = a·x + b for x: D(y) = a⁻¹·(y − b) mod 26, where a⁻¹ is the number with a·a⁻¹ ≡ 1 (mod 26). It is found with the extended Euclidean algorithm. For a = 5 the algorithm gives 5·(−5) + 26·1 = 1, so a⁻¹ = −5 mod 26 = 21; check: 5·21 = 105 = 4·26 + 1.

The inverses of the twelve valid values are: 1 → 1, 3 → 9, 5 → 21, 7 → 15, 9 → 3, 11 → 19, 15 → 7, 17 → 23, 19 → 11, 21 → 5, 23 → 17 and 25 → 25.

Special cases and security

With a = 25 and b = 25 the formula becomes 25·x + 25 ≡ 25 − x (mod 26), which reverses the alphabet: this is the Atbash cipher. The affine cipher is easy to break: there are only 312 keys, letter frequencies are preserved, and just two known plaintext letters are enough to solve for a and b.