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 \left ({2,\mathbb {Z} }\right) on projective line PL \left ({F_{257} }\right) over Galois field GF \left ({2^{8} }\right) . In the first step we obtained elements of GF \left ({2^{8} }\right) by using powers of \alpha , where \alpha is the primitive root of irreducible polynomial p\left ({x }\right) of order 8 over field \mathbb {Z}_{2} , then applied the generators of PSL \left ({2,\mathbb {Z} }\right) and followed steps to get rid of infinity from output. In the final step of proposed scheme, one of the permutations of S_{16} is applied which enhanced the possible number of S-Boxes obtained by any single specific irreducible polynomial p(x) over field \mathbb {Z}_{2} of order 8. We analyzed performance of the proposed 8\times 8 S-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