NONLINEAR NYBERG CONSTRUCTION TRANSFORMS OVER ISOMORPHIC REPRESENTATIONS OF FIELDS GALOIS

Abstract

Further development of cryptographic algorithms based on the principles of many-valued logic requires more accurate research of non-binary cryptographic primitives – S-boxes. One of the most promising constructions for the synthesis of S-boxes is the Nyberg construction, which ensures high quality of the designed S-boxes in the binary case. The disadvantage of the Nyberg construction is the small cardinality of the classes of the constructed S-boxes. Nevertheless, this disadvantage can be overcome by considering all the isomorphic representations of the main field, substantially expanding the choice of available high-quality S-boxes. The research carried out in this paper has shown that the advantages of the Nyberg construction can be easily transferred to a many-valued case. Thus, we construct complete sets of S-boxes of the Nyberg construction over all isomorphic representations of fields GF(pᵏ), р = 3,5, and research their nonlinear characteristics. As a criterion of nonlinearity, we measure the distances from the component many-valued functions to the set of Vilenkin–Chrestenson functions that are considered to be the most linear. The correlation coefficients of the output and input vectors of the obtained S-boxes are calculated. The researches performed have shown the high quality of the constructed cryptographic primitives and allow recommendation of them for use in cryptoalgorithms based on the principles of many-valued logic.

Keywords

• s-box
• nyberg construction
• many-valued logic