Application of Linear Algebra in the Hill Cipher for Encryption and Decryption

Abstract

Cryptography is the science of securing information through mathematical transformations. Hill Cipher, developed by Lester S. Hill in 1929, is a classical cryptographic algorithm that directly applies matrix operations within modular arithmetic. This study analyzes the linear algebra foundations of Hill Cipher, covering plaintext representation as column vectors, encryption via n×n key matrix multiplication in Z26, and decryption using modular matrix inverses. The method employed is a systematic literature review (2021–2025) and computational simulation using a 2×2 key matrix. Results show that the invertibility condition for key K is gcd(det(K), 26) = 1; a simulation of plaintext “HELP” with key K = [[3,3],[2,5]] produced ciphertext “HIAT”, which was successfully verified through decryption. This research affirms the pedagogical value of Hill Cipher as a bridge between linear algebra theory and applied cryptography, while also identifying its vulnerability to known-plaintext attack.