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

Abstract

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.

Keywords

• Action of modular group
• cryptographic properties’ analyses
• finite field
• majority logic criterion
• S-Box