Метод синтезу s-блоків на основі матриць gf-перетворення

Abstract

Cryptographic methods today are a crucial tool for constructing information security systems. At the same time, to solve the problem of encrypting large amounts of information, block or stream symmetric ciphers are mainly preferred because of their efficiency and proven cryptographic strength, including against perspective quantum cryptanalysis. The effectiveness of modern symmetric ciphers largely depends on the cryptographic S-boxes applied in their construction, the quality of which largely determines the degree of implementation of the concepts of diffusion and confusion by the cryptographic algorithm, while the presence of large sets of cryptographically high-quality S-boxes is also important, in the terms of their application as a long-term key. Today, the Nyberg construction is well-known and widely applied in ciphers, including widespread AES block symmetric cipher. This construction allows you to synthesize high-quality S-boxes that harmoniously satisfy the main criteria for cryptographic quality, however, the set of S-boxes synthesized using this construction is small, which makes the task of developing new methods for synthesizing large sets of cryptographically high-quality S-boxes highly relevant. At the same time, as research shows, the constructions of extended Galois fields are a promising raw material for solving this problem. In this paper, the Galois field transform matrices of order N=256 are constructed for all isomorphic representations of the extended Galois field GF(256) which are analogous to the Reed-Muller transform but for the case of many-valued logic functions. As part of the research, the isomorphism invariant row numbers of the Galois field transform matrices are identified, which allows to obtain bijective S-boxes, as well as bijective S-boxes that correspond to the main criteria for cryptographic quality of component Boolean functions such as algebraic degree of nonlinearity, distance of nonlinearity, error propagation criterion, and criterion of minimization of correlation of output and input vectors of the S-box. At the same time, the cardinality of the set of synthesized S-boxes is ~23 times higher than the cardinality of the set of S-boxes of the Nyberg construction, which allows them to be used as a long-term key. The proposed S-boxes can become the basis for improving the effectiveness of existing symmetric cryptographic algorithms and developing new ciphers.