Paper 2022/485

Two new classes of permutation trinomials over ${\mathbb{F}}_{q^3}$ with odd characteristic

Xi Xie, Nian Li, Linjie Xu, Xiangyong Zeng, and Xiaohu Tang

Abstract

Let $q$ be an odd prime power and ${\mathbb{F}}_{q^3}$ be the finite field with $q^3$ elements. In this paper, we propose two classes of permutation trinomials of ${\mathbb{F}}_{q^3}$ for an arbitrary odd characteristic based on the multivariate method and some subtle manipulation of solving equations with low degrees over finite fields. Moreover, we demonstrate that these two classes of permutation trinomials are QM inequivalent to all known permutation polynomials over ${\mathbb{F}}_{q^3}$. To the best of our knowledge, this paper is the first to study the construction of nonlinearized permutation trinomials of ${\mathbb{F}}_{q^3}$ with at least one coefficient lying in ${\mathbb{F}}_{q^3}\mathrm{\setminus }{\mathbb{F}}_q$.