Some new results on permutation polynomials over finite fields

TL;DR

This paper introduces new classes of permutation polynomials over finite fields, confirming a conjecture and advancing understanding of their differential properties, with potential applications in cryptography and coding theory.

Contribution

It presents four classes of monomial complete permutation polynomials, one class of trinomial complete permutation polynomials, and two classes of trinomial permutation polynomials, confirming a conjecture and making progress on another.

Findings

Confirmed a conjecture by Wu et al. on permutation polynomials.

Constructed new classes of permutation polynomials.

Made progress on a conjecture about differential uniformity.

Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int. J. Inf. Coding Theory, 2010, 1, pp. 149-170).