Algebraic Foundations of AES (Advanced Encryption Standard): Group Theory and Finite Field Applications in Symmetric Cryptography - Vikas Kumar; Jitendra Singh (Recent Trends in Mathematics, 2025)
This paper presents a mathematical study of symmetric cryptographic algorithms, with a particular emphasis on the Advanced Encryption Standard (AES), which is one of the most widely used encryption schemes in modern security applications. The study highlights how abstract mathematical frameworks such as group theory, finite fields, and vector space concepts provide the foundation for the design, implementation, and analysis of AES. By approaching the algorithm from a mathematical perspective, we aim to shed light on the theoretical principles that underlie its security and efficiency. The discussion begins with an overview of finite field arithmetic, specifically the structure of the Galois Field GF(28), which forms the backbone of operations in AES. The construction of substitution boxes (S-Boxes) is examined in detail, emphasizing their role in introducing non-linearity and resistance against linear and differential cryptanalysis. The algebraic properties of these S-Boxes, including their design based on multiplicative inverses in finite fields combined with affine transformations, are analyzed to show how they enhance cryptographic robustness. Attention is also given to the MixColumns transformation, which is modeled as a linear transformation over a vector space. This operation diffuses input data across multiple bytes, thereby strengthening the avalanche effect and improving resistance against statistical attacks. We further examine how the cyclic structure of AES rounds, consisting of repeated applications of SubBytes, ShiftRows, MixColumns, and AddRoundKey, reflects principles of group operations, ensuring both complexity and uniformity throughout the encryption process. By situating AES within a broader mathematical framework, this study demonstrates how abstract algebraic concepts are translated into practical mechanisms that safeguard digital communication. The results highlight the deep interplay between pure mathematics and applied cryptography, providing insight into why AES remains a cornerstone of secure information systems.
Keywords: AES; Cryptographic Primitives; Finite Fields; Group Theory; Linear Algebra; MixColumns; S-Box