S-Boxes construction over Galois elds of order 256

Abstract

Although over the last 40 years, cryptography is weighed up as a developed branch of science nonetheless it is a new eld of study compare to other subjects and every day brings so many expansion. Symmetric cryptography is one of the signi cant branch by which two parties share secret information and keys by encryption and decryption procedures. Symmetric cryptography splits into two main branches; Block cipher and Stream cipher. S-box is the important component of block cipher algorithm, used in many famous cipher system such as data encryption standard (DES), International data encryption algorithm (IDEA), advanced encryption standard (AES) [1,2]. S-box is one of the nonlinear components of the block cipher, hence the security strength of block cipher depends on the quality of an S-box. As a result of this many researchers have shown their interest to design new and powerful S-boxes. Owing to their strong cryptographic features, S-boxes that are created on algebraic systems have much at- tention and which are robust against linear and di erential cryptanalysis. Thus a secure communication based on di erent types of S-boxes are always encouraged. Like AES S-box, the a ne power a ne (APA) S-box is proposed which upsurges the algebraic complexityalgebraic complexity though possession the anticipat avail- able encryption properties [3]. The action of the symmetric group S8 on the original S-box used in AES, the S8 AES S-box is o ered in [4]. On applying additional trans- form based on binary Gray codes on the original S-box of AES. The Gray S-box is i ii obtained [5]. The Gray S-box has a 255-term polynomial as compare to 8-term poly- nomial which carries all the properties and rises the security for AES. Similarly, Xyi S-box, Residue Prime S-box and Skipjack S-box are normally used S-boxes in the en- cryption and decryption techniques [6, 7]. Typically the algebraic strength of an S-box is measured by Nonlinearity Nonlinearity, strict avalanche criterion strict avalanche criterion (SAC), bit independence criterion bit independence criterion(BIC), linear approximation probability linear approximation probability (LP) and Di erential ap- proximation probability Di erential approximation probability (DP) [8]. It is evident by the study of novelty in algorithms for S-box construction that the alteration of the model and the selection of Boolean functions give small to the performance in- dices of an S-box. In this research, we suggest that the performance of an S-box is momentously link with the contextual Galois eld. The nite elds of the same order are isomorphic but the scaling e ect of a nonlinear Boolean function apply on two or more di erent elds of the same order may diverge. An S-box is a signi - cant component in a block cipher used to produce confusion in the data; it is valued take in that the confusion making ability is allied with the optimal of the irreducible polynomial used to form the contextual Galois eld. The main aim of this research is to understand the basic concept of cryptography, but mainly focused on the con- struction of S-boxes based on the group action of projective general linear group on the Galois eld GF(28). The algebraic analysis such as nonlinearity, strict avalanche criteria, linear approximation probability, bit independence criteria and di erential approximation probability on the newly generated S-box is performed to determine the strength of the S-boxes.