From Schreier graphs to secure s-boxes using a group-theoretic design framework.

Secure substitution boxes (S-boxes) are a fundamental design element of modern symmetric cryptography because they provide the necessary nonlinearity to achieve confusion. This paper presents a new [Formula: see text] S-box construction framework, which combines group theory, graph theory and cryptography. The construction starts from the action of the modular transformation group on the projective line over [Formula: see text], which produces Schreier graphs whose cycle structure is used to form an initial S-box. The design is then refined through a non-abelian permutation group acting on S-box positions. The proposed S-box is rigorously verified against conventional cryptographic criteria, including nonlinearity, differential uniformity, strict avalanche criterion, bit independence criterion, and linear approximation probability, which demonstrate that it is highly resistant to classical attacks. Besides, texture analysis is performed using the S-box in the image encryption setting that completely relies on substitution. The findings indicate enhanced randomness and reduced pixel correlation, as measured by gray-level co-occurrence matrix characteristics, such as contrast, correlation, energy, homogeneity, and entropy. The proposed methodology provides a strong theoretical foundation and demonstrates effectiveness for secure cryptographic and image processing applications.