About permutations on the sets of tuples from elements of the finite field

Abstract

© 2019, Kazan Federal University. All rights reserved. The following problem was considered: let S = S1×S2×…×Sm be the Cartesian product of subsets Si that are subgroups of the multiplicative group of a finite field Fq of q elements or their extensions by adding a zero element; a map f: S → S of S into itself can be specified by a system of polynomials f1,..., fm ε Fq[x1,..., xm]. Necessary and sufficient conditions, for which the map f = 〈f1,..., fm〉 is bijective, were obtained. Then this problem was generalized to the case when the subsets Si are any subsets of Fq. The obtained results can be used to construct S -boxes and P -boxes in block ciphers and to calculate automorphism groups of error-correcting codes.