About permutations on the sets of tuples from elements of the finite field - V.S. Kugurakov; A.F. Gainutdinova; V.T. Dubrovin (Учёные записки Казанского университета: Серия Физико-математические науки)
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 є Fq[x1,…,x m]. 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.