Affine Cipher

How it is calculated

Each letter is turned into a number x (A = 0 … Z = 25). Encoding applies y = (a·x + b) mod 26. Decoding applies x = a⁻¹·(y − b) mod 26, where a⁻¹ is the modular inverse of a, the number for which a·a⁻¹ ≡ 1 (mod 26). The number is turned back into a letter.

The key a must share no factor with 26 (gcd(a, 26) = 1), otherwise the tool returns nothing, since decoding would be ambiguous. Upper and lower case are kept, and digits, spaces, punctuation and accented characters are copied through unchanged. Only the 26 letters A to Z are shifted. The mode field selects encode or decode.

Common mistakes

• Choosing an even a or a multiple of 13 (such as 2, 4 or 13). Those share a factor with 26, so the tool gives no result; use 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 or 25.
• Decoding with the wrong mode or different keys than were used to encode. You must set mode to decode and enter the same a and b.
• Using a = 1 and expecting something stronger than a Caesar shift. With a = 1 the cipher is just a shift by b.
• Expecting digits or accented letters to be encrypted. They pass through unchanged, which can reveal parts of the message.
• Treating the cipher as secure. With only 12 valid values of a and 26 of b, there are just 312 key pairs, which are easy to try all.