Further results on permutation pentanomials over ${\mathbb F}_{q^3}$ in characteristic two
Tongliang Zhang, Lijing Zheng, Hengtai Wang, Jie Peng, Yanjun Li
TL;DR
This work extends the study of permutation pentanomials over ${\mathbb F}_{q^{3}}$ in characteristic two by classifying new pentanomials of the form $\epsilon_{0}x^{d_{0}}+L(\epsilon_{1}x^{d_{1}}+\epsilon_{2}x^{d_{2}})$ with $d_{0}\in\{1,2,4\}$ and $L(x)=x+x^{q}$. Building on LiKK's multivariate framework, the authors derive several explicit permutation classes under precise congruence conditions on $m$ (where $q=2^{m}$), proving permutation property via the nonexistence of nontrivial solutions to $f(x+a)=f(x)$ for all $a\neq0$ in ${\mathbb F}_{q^{3}}$ and employing resultant, trace, and Magma-assisted computations. They also analyze quasi-multiplicative (QM) equivalence, demonstrating non-equivalence to known families and establishing inequivalence among their new classes, supported by a comparative table. The results expand the catalog of permutation pentanomials over ${\mathbb F}_{q^{3}}$ and provide robust techniques for identifying further families in characteristic two, with a conjecture identifying two parametric families governed by gcd conditions on $2^{k}+1$ and $q-1$ as a next step.
Abstract
Let $q=2^m.$ In a recent paper \cite{Zhang3}, Zhang and Zheng investigated several classes of permutation pentanomials of the form $ε_0x^{d_0}+L(ε_{1}x^{d_1}+ε_{2}x^{d_2})$ over ${\mathbb F}_{q^3}~(d_0=1,2,4)$ from some certain linearized polynomial $L(x)$ by using multivariate method and some techniques to determine the number of the solutions of some equations. They proposed an open problem that there are still some permutation pentanomials of that form have not been proven. In this paper, inspired by the idea of \cite{LiKK}, we further characterize the permutation property of the pentanomials with the above form over ${\mathbb F}_{q^3}~(d_0=1,2,4)$. The techniques in this paper would be useful to investigate more new classes of permutation pentanomials.