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

V.S. Kugurakov* , A.F. Gainutdinova** , V.T. Dubrovin***
Kazan Federal University, Kazan, 420008 Russia
E-mail: *Vladimir.Kugurakov@kpfu.ru, **aida.ksu@gmail.com, ***yacheslav.Dubrovin@kpfu.ru
Received March 11, 2019

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DOI: 10.26907/2541-7746.2019.2.292-300

For citation: Kugurakov V.S., Gainutdinova A.F., Dubrovin V.T. About permutations on the sets of tuples from elements of the finite field. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 2, pp. 292–300. doi: 10.26907/2541-7746.2019.2.292-300. (In Russian)

Abstract

The following problem was considered: let S = S₁× S₂×…× S_m be the Cartesian product of subsets S_i that are subgroups of the multiplicative group of a finite field F_q 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,…,f_m є F_q[x₁,…,x _m]. Necessary and sufficient conditions, for which the map f =< f₁,…,f_m >  is bijective, were obtained. Then this problem was generalized to the case when the subsets S_i are any subsets of 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.

Keywords: cryptography, error-correcting codes, finite fields, permutation polynomials

References

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