Construction of Nonlinear Component of Block Cipher by Action of Modular Group PSL(2, Z) on Projective Line PL(GF(2⁸))

Substitution box (S-Box) has a prominent significance being the fundamental nonlinear component of block cipher which fulfils confusion, one of the properties proposed by Claude Shannon in 1949. In this paper, we proposed an S-Box by using the action of modular group PSL(2,Z )on projective line PL(F₂₅₇ )over Galois field GF(2⁸ ). In the first step we obtained elements of GF(2⁸ )by using powers ofα, whereαis the primitive root of irreducible polynomialp (x )of order 8 over fieldZ₂, then applied the generators of PSL(2,Z )and followed steps to get rid of infinity from output. In the final step of proposed scheme, one of the permutations ofS₁₆is applied which enhanced the possible number of S-Boxes obtained by any single specific irreducible polynomial$p(x)$over fieldZ₂of order 8. We analyzed performance of the proposed8× 8S-Box under cryptographic properties such as strict avalanche criterion, bit independence criterion, nonlinearity, differential approximation probability, linear approximation probability; and compared obtained results with a number of renowned S-Boxes. Lastly, we performed statistical analysis (which comprises of contrast analysis, homogeneity analysis, energy analysis, correlation analysis, entropy analysis and mean of absolute deviation analysis) on our proposed S-Box and obtained results have been compared with adequate number of S-Boxes.