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

The following problem was considered: let $S=S_1\times S_2\times \dots \times S_m$ be the Cartesian product of subsets $S_i$ that are subgroups of the multiplicative group of a finite field ${\mathbb F}_q$ of $q$ elements or their extensions by adding a zero element; a map $f: S\rightarrow S$ of $S$ into itself can be specified by a system of polynomials $f_1, \dots, f_m\in {\mathbb F}_q [x_1, \dots, x_m ]$. Necessary and sufficient conditions, for which the map $f=\langle f_1, \dots ,f_m\rangle$ is bijective, were obtained. Then this problem was generalized to the case when the subsets $S_i$ are any subsets of ${\mathbb F}_q$. 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.