Galois Finite Fields and the Advanced Encryption Standard (AES)

This explains how the bytes used in the Advanced Encryption Standard are elements of the Galois field GF(2^8), and how that algebra determines the core AES operations. Bytes are treated as polynomials with coefficients in GF(2), so bit patterns map directly to polynomial terms; addition in the field is bitwise XOR (polynomial addition with no carries). Multiplication is polynomial multiplication followed by reduction modulo an irreducible “reducing” polynomial (for AES, x^8 + x^4 + x^3 + x + 1, 0x11b), which keeps results within one byte. The write-up gives concrete examples and multiplication tables (including the common multiply-by-2 table) and notes practical techniques used in implementations: precomputed inverse or log/antilog tables and simple bitwise algorithms.

That finite-field structure is exactly what AES uses for its round functions: the S-box is the multiplicative inverse in GF(2^8) followed by an affine transform; ShiftRows permutes bytes by row shifts; MixColumns applies an invertible linear transformation built from finite-field multiplications so that each output byte depends on all input bytes in a column; and round keys are XORed (field addition). Using GF(2^8) keeps values one byte long, guarantees inverses for decryption, and provides well-understood, noncommuting operations (inverses, transposes, matrix mixes) that combine to produce strong nonlinearity and diffusion.