A Class of Permutation Trinomials over Finite Fields

TL;DR

This paper characterizes when a specific class of permutation trinomials over finite fields $F_{q^2}$ exists, based on conditions involving the field's characteristic and trace or specific element relations.

Contribution

It provides a complete characterization of permutation properties for a class of trinomials over finite fields, extending understanding of permutation polynomials.

Findings

Permutation polynomial conditions depend on field characteristic and trace

Permutation occurs if $q$ is even and trace condition holds

Permutation occurs if $q mod 8=1$ and $t^2=-2$

Abstract

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and $\text{Tr}_{q/2}(\frac 1t)=0$; (ii) $q\equiv 1\pmod 8$ and $t^2=-2$.