METHODS FOR CONSTRUCTING PERMUTATIONS OF AN ARBITRARY FINITE FIELD AND THEIR LINEAR CHARACTERISTICS

Abstract

Permutations in a finite field (bijective transformations) are actively studied in many applications, including in information security theory. Permutations are often used as elements for building information processing nodes. In the middle of the 20th century, K. Shannon theoretically justified the basic requirements for mapping performed on such nodes. Currently, when constructing bijective transformations, these requirements are provided by the composition of nonlinear representations given by the table in the field GF(2). The presented paper summarizes the results of work [1] on methods for increasing the dimension of stationary functional systems. Namely, in this paper, approaches to the construction of new permutations from the initial permutations in the finite field are investigated. The linear characteristic is calculated for the constructed permutations. The problem of constructing permutations given by coordinate functions is difficult. The relevance of the topic of the paper is determined by the need to search for new theoretically sound methods for constructing s permutation classes in multidimensional spaces with the required combinatorial-algebraic properties. The paper considers several methods for constructing substitutions of finite fields from initial permutations acting on vectors of smaller dimension. In the binary case, this allows us to find the nonlinearity of the substitutions under consideration, characterizing the proximity of linear combinations of coordinate functions of permutations to the entire class of affine functions. The results of the presented work somewhat expand the possibilities of constructing permutations for an arbitrary finite field.