LINEAR ALGEBRA IN CRYPTOGRAPHY AND SECURE COMMUINCATION

Abstract

This study examines the role of linear algebra in the development and analysis of cryptographic systems, motivated by the growing need for mathematically robust security frameworks in an era of expanding digital infrastructure. The research analyzes both classical and modern encryption techniques — including the Hill cipher, affine transformations, RSA, and AES — to trace how algebraic structures directly shape cryptographic design. Core concepts such as matrix invertibility, modular arithmetic, and finite field operations were studied in detail, with particular attention to how these properties determine the security and reversibility of encryption schemes. The Hill cipher served as a foundational case, illustrating how the invertibility of a key matrix under modular arithmetic is not merely a mathematical convenience but the actual mechanism of decryption. AES extended this further, where MixColumns operates as a matrix multiplication over GF(2⁸), selected for measurable diffusion strength rather than arbitrary construction. Decomposition methods — LDU, QR, and SVD — were also examined for their computational relevance, particularly in efficient matrix operations and their emerging role in lattice-based cryptographic frameworks that aim to resist quantum attacks. The findings confirm that linear algebra is central to cryptographic design, governing both security guarantees and implementation efficiency across the systems studied. Overall, this study establishes that linear algebra provides not just theoretical grounding for cryptography but the actual design logic that makes modern encryption systems function — and holds up under adversarial conditions.