Some permutation pentanomials over finite fields of even characteristic

TL;DR

This paper explains why 14 out of 17 recently constructed permutation pentanomials over finite fields of even characteristic are permutations and extends these families into three broader classes.

Contribution

Provides a simple explanation for the permutation property of most of the recent families and extends them into three general classes.

Findings

14 families of permutation pentanomials are validated.

Extended these families into three broader classes.

Simplified understanding of permutation properties over finite fields.

Abstract

In a recent paper Zhang et al. constructed 17 families of permutation pentanomials of the form $x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}$ over $\mathbb{F}_{q^2}$ where $q=2^m$. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over $\mathbb{F}_{q^2}$.