Construction of Nonlinear Component of Block Cipher by Action of Modular Group PSL(2, Z) on Projective Line PL(GF(28))
Wei Dong Gao, Bazgha Idrees, Sohail Zafar, Tabasam Rashid · IEEE Access · 2020
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.