The Hill cipher is a polygraphic substitution cipher invented by the American mathematician Lester S. Hill in 1929. Unlike simple substitution ciphers, the Hill cipher encrypts blocks of letters using linear algebra, making it one of the first ciphers to apply matrix operations to cryptography. Each block of plaintext letters is represented as a vector of numbers (with A=0, B=1, …, Z=25) and multiplied by an invertible key matrix modulo 26. The resulting vector is then converted back into letters to form the ciphertext. For example, using the key matrix [[3,3],[2,5]] and plaintext “HELLO” (split into digraphs HE, LL, O padded with X), each pair is transformed: “HE” → H=7, E=4 → vector [7,4], multiplied by the key matrix modulo 26 gives [25,2], which converts back to letters ZC. Repeating for remaining digraphs yields the full ciphertext. Decryption requires the inverse of the key matrix modulo 26, applied to the ciphertext vectors. The Hill cipher is symmetric, fully reversible, and more secure than monoalphabetic substitution ciphers because it obscures letter frequencies across blocks. Its strength depends on the invertibility of the key matrix and proper block size, while small matrices may still be vulnerable to known-plaintext attacks. The cipher illustrates the application of mathematical concepts to encryption, bridging linear algebra and classical cryptography, and remains an important historical and educational tool in understanding polygraphic substitution systems.
Hill Cipher
