Аннотация:
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.
