Permutation Polynomials of the form $L(X)+\gamma Tr_q^{q^3}(h(X))$ over finite fields with even characteristic

TL;DR

This paper characterizes a new class of permutation polynomials over finite fields of even characteristic, transforming the problem into multivariate permutation construction and generalizing recent results.

Contribution

It introduces a novel approach to construct permutation polynomials by converting the univariate problem into multivariate permutations over finite fields.

Findings

Complete characterization of a class of permutation polynomials over _{q^3}

Transformation of univariate permutation problems into multivariate permutations

Generalization of recent results by Jiang, Li, and Qu

Abstract

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+\gamma Tr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,\gamma\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026).