Nonlinear properties of Rijndael S-boxes represented by the many-valued logic functions

Abstract

S-boxes of the Nyberg construction are one of the most important cryptographic primitives, which are used in the AES cryptographic algorithm and largely determines its effectiveness. Numerous researches have confirmed the high cryptographic quality of their component Boolean functions. Nevertheless, the cryptanalyst is not constrained in the methods used and can also use the mathematical apparatus of the functions of many-valued logic for cryptanalysis. This work is devoted to the research of the nonlinear properties of S-boxes of the Nyberg construction, presented in the form of component 4-functions and 16-functions. The paper proposes a method for calculating the nonlinearity value of 16-functions, for which the formula of the recursive construction of hexadecimal Vilenkin-Chrestenson matrices of arbitrary order is discovered. The performed researches made it possible to establish that the nonlinearity values of component 4-functions and 16-functions of S-boxes of the Nyberg construction is not stable and depends on the type of irreducible polynomial used to construct them. In the paper we present the irreducible polynomial for which the nonlinearity values of component 4-functions and 16-functions is evenly high. At the same time, it was established that the same polynomial also provides the uniform minimization of the correlation coefficients between output and input vectors of the S-box. The specified polynomial can be recommended for the practical use.