A Novel Scheme for Designing Secure Substitution Boxes (S-Boxes) Based on Mobius Group and Finite Field

Data encryption with substitution-box (S-box) is now widely regarded as the most secure method of communicating in order to prevent unauthorized access to sensitive and confidential information. An S-box is the basic component in a cryptosystem, which provides confusion in data. An S-box that holds a high degree of nonlinearity, as well as a low level of linear and differential approximation probability, is considered cryptographically secure. In this regard, cryptography plays an important role in the encryption of private data and information. The field of graph theory is important in a variety of domains. Graph theory, in particular, is commonly utilized as an encryption technique, as graphs can be transformed into matrices. In this study, we use graph theory and adjacency matrices to generate highly nonlinear S-boxes. We use an adjacency matrix corresponding to the coset diagram for the action of the Mobius group M = ⟨x,y:x^2 = y^6 = 1⟩ on PL( F_7) . We then use this matrix to the Galois field GF( 2^8) to obtain new S-boxes. The proposed S-boxes are then analyzed using algebraic and statistical methods. The analyses and findings are encouraging, demonstrating that the proposed S-boxes are well protected. Also, as a means of confusion, the proposed S-boxes have a high level of security and enhance resistance against the potential attacks.