Construction of Substitution Box using Affine Transformation over a Galois Field

Abstract

In symmetric-key encryption systems, S-boxes are fundamental com-ponents that provide confusion, prevent statistical and algebraic attacks, and they are integrated into the design of the encryption algorithm. This paper describes the detailed construction of S-boxes that are cryptograph-ically secure, in particular ones created using affine mappings. The con-struction operates on the finite field, GF (28), in the field of cryptography. The process begins by computing the multiplicative inverse of the input byte with respect to a chosen irreducible polynomial. This byte inversion step alone is sufficient to guarantee a high degree of non-linearity and also and gives a one-to-one mapping of input and output values. The val-ues that have been inverse mapped are then processed through an affine transformation in GF (28) that is defined by an affine vector and an invert-ible matrix. This particular affine transformation is chosen to optimize the diffusion of the bits, eliminate fixed points and provide an additional layer of protection from potential attacks. The processed S-Box under-goes a preliminary evaluation using various possible established techniques in cryptography: evaluation of its differential uniformity, linearity, strict avalanche criteria (SAC), bit independence criteria (BIC), and linear ap-proximation. Experimental results reveal that the constructed S-Box has a balanced output distribution and exhibits proper output uncertainty, with low volumes of differential and linear cryptanalysis. Because the case requires an affine transformation, the inverse S-Box will also exist, which serves the proposed design for both encryption and decryption. Addition-ally, the proposed design retains low computational complexity suitable for efficient implementations in both hardware and software. These at-tributes confirm that the suggested affine transformation-based S-Box will serve efficiently in modern block ciphers, especially in low-resource and lightweight cryptographic applications.