Some permutation pentanomials over finite fields of even characteristic

TL;DR

This work analyzes permutation pentanomials over the even-characteristic field $\mathbb{F}_{q^{2}}$ by recasting the problem via the rational function $g_\bullet(x)$ on the $(q+1)$-st roots of unity $\mu_{q+1}$. It demonstrates that 14 of Zhang et al.'s 17 families permute $\mathbb{F}_{q^{2}}$ under simple gcd and root-avoidance conditions, and it introduces three general classes $f_A,f_B,f_C$ that unify these cases through linear equivalence into tractable monomial-type forms. The key methodological step is reducing permutation questions to the study of $H_\bullet(x)$, $N_\bullet(x)$, and their gcd with $Q(x)=x^2+x+1$, plus conjugations by degree-one maps to obtain explicit linear-equivalence reductions to easily verifiable monomial forms. The findings offer a simpler explanatory framework and extendable structure for permutation pentanomials in even characteristic, while leaving open questions about the remaining three families and odd characteristic cases.

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 $\mathbb{F}_{q^2}$.

Theorems & Definitions (18)

• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Definition 1
• Definition 2