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

Area of Science:

• Cryptography
• Applied Mathematics
• Computer Science

Background:

• Secure substitution boxes (S-boxes) are crucial for nonlinearity in symmetric cryptography.
• Existing S-box constructions require robust theoretical underpinnings and rigorous security validation.

Purpose of the Study:

• To propose a new S-box construction framework combining group theory, graph theory, and cryptography.
• To evaluate the cryptographic strength and image encryption capabilities of the proposed S-boxes.

Main Methods:

• Utilizing the action of the modular transformation group on the projective line over a finite field to generate Schreier graphs.
• Forming an initial S-box from the cycle structure of Schreier graphs.
• Refining the S-box design using a non-abelian permutation group.

Main Results:

• The proposed S-box meets key cryptographic criteria: nonlinearity, differential uniformity, strict avalanche criterion, bit independence, and low linear approximation probability.
• Texture analysis in image encryption shows enhanced randomness and reduced pixel correlation (contrast, correlation, energy, homogeneity, entropy).

Conclusions:

• The novel S-box construction framework provides a strong theoretical foundation for cryptographic applications.
• The proposed S-boxes demonstrate high resistance to classical attacks and effectiveness in image encryption.