Affine Cipher Lab

The affine cipher is a substitution cipher: every letter is always replaced by the same other letter. Each letter becomes a number (A = 0, B = 1, and so on up to Z = 25), a formula changes the number, and the result becomes a letter again. The two numbers a and b in the formula are the key.

• Caesar and Atbash. The Caesar cipher is the special case a = 1, which shifts every letter b places along the alphabet. Julius Caesar is said to have shifted by 3, so b = 3. Atbash, which reverses the alphabet (A becomes Z, B becomes Y), is a = 25, b = 25.
• Not every a works. a must share no factor with 26, which is written gcd(a, 26) = 1. Otherwise two letters encode to the same letter and the message can't be decoded. With a = 2, for example, A (0) and N (13) both become A. That leaves 12 usable values: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25.
• Easy to break. 12 choices for a and 26 for b give only 12 × 26 = 312 keys, so anyone can break the cipher by trying every key, or by counting how often each letter appears (frequency analysis, a method described by the 9th-century scholar al-Kindi). The affine cipher is for learning, not for keeping secrets.
• Decoding needs an inverse. a⁻¹a inverse is the inverse of a mod 26: the number that gives a remainder of 1 when you multiply it by a and divide by 26. For a = 7 it is 15, because 7 × 15 = 105 = 4 × 26 + 1.