Permutation polynomials of the form cx + Tr q l / q ( x a ) $cx+\text {Tr}_{q^{l}/ q}(x^{a})$ and permutation trinomials over finite fields with even characteristic

Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. Particularly, permutation polynomials with few terms are more popular for their simple algebraic form and additional extraordinary properties. Very recently, G. Kyureghyan and M.E. Zieve (2016) studied permutation polynomials over \(\mathbb {F}_{q^{n}}\) of the form \(x+\gamma \text {Tr}_{q^{n}/q}(x^{k})\), where q is odd, and nine classes of permutation polynomials were constructed. In this paper, we present fifteen new classes of permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) over finite fields with even characteristic, which explain most of the examples with q = 2^k, k> 1, kl< 14 and \(c\in \mathbb {F}_{q^{l}}^{*}\). Furthermore, we also construct four classes of permutation trinomials.