Synthesis of highly nonlinear S-boxes satisfying higher order propagation criterion

Abstract

The S-box is the main component of modern block cryptographic algorithms and hash functions. One of the main criteria for the cryptographic quality of an S-boxes is the criterion of a high nonlinearity distance, as well as a propagation criterion of order m. One of the effective methods for constructing S-boxes is the method of their construction on the basis of a set of Boolean functions with a given level of cryptographic quality. This paper discusses the spectral classes (obtained on the basis of the classification of the Walsh-Hadamard transform coefficient vectors) of Boolean functions of 1 … 5 variables, for each of which the numbers of Boolean functions that satisfy the propagation criterion of order m were found. Using the constructive method among the full set of Boolean functions of 5 variables, a subclass of 12 balanced maximally nonlinear Boolean functions satisfying the propagation criterion PC(4) were found. It is shown that on the basis of this set of Boolean functions, bijective cryptographic S-boxes can be synthesized, which are the best among the entire set of S-boxes of length N = 32, from the point of view of the criterion of maximum nonlinearity distance and propagation criterion.