Hill cipher

How the Hill cipher works

Turn the letters into numbers, A=0 to Z=25, and take them in blocks: pairs for a 2×2 key, triples for a 3×3. Each block is a column of numbers, and the key is a square grid of numbers, the key matrix. Multiply the matrix by the column and take each result mod 26 to get the ciphertext block.

With the key GYBNQKURP (6 24 1 / 13 16 10 / 20 17 15) the block ACT (0, 2, 19) becomes:

• 6·0 + 24·2 + 1·19 = 67 → 67 mod 26 = 15 → P
• 13·0 + 16·2 + 10·19 = 222 → 14 → O
• 20·0 + 17·2 + 15·19 = 319 → 7 → H

so ACT encrypts to POH. Decryption multiplies by the inverse matrix instead. A key only works if that inverse exists mod 26, which means its determinant must be odd and not divisible by 13. The tool checks this for you.

Why it mattered

Lester S. Hill published it in 1929. It was the first cipher built on algebra rather than tables, and the first practical one to encrypt blocks of more than three letters at once (Playfair, from 1854, works on pairs). Because every letter in a block affects every other, a single letter's frequency gives almost nothing away.

Breaking a 2×2 key

The Hill cipher falls quickly to a known stretch of plaintext, because the equations can simply be solved. Without one, a 2×2 key is still weak. Each row of the decryption matrix produces every other plaintext letter on its own, so each row's 676 possibilities can be scored separately against English letter frequencies, then the best rows combined. “Find the key” does exactly that. A 3×3 key has too many possibilities for that trick to work in a browser.