Permutation polynomials of finite fields of even characteristic from character sums

TL;DR

This paper characterizes permutation polynomials over finite fields of even characteristic using character sums, focusing on a specific polynomial form involving trace and linear polynomials, and offers new constructions.

Contribution

It introduces a novel characterization of permutation polynomials of a specific form over finite fields of characteristic two using character sums and provides new construction methods.

Findings

Characterization of permutation polynomials via character sums

New constructions of permutation polynomials in the specified form

Conditions for permutation property over finite fields of even characteristic

Abstract

In this paper, we investigate permutation polynomials over the finite field $\mathbb F_{q^n}$ with $q=2^m$, focusing on those in the form $\mathrm{Tr}(Ax^{q+1})+L(x)$, where $A\in\mathbb F_{q^n}^*$ and $L$ is a $2$-linear polynomial over $\mathbb F_{q^n}$. By calculating certain character sums, we characterize these permutation polynomials and provide additional constructions.