Determination of a Class of Permutation Trinomials in Characteristic Three

TL;DR

This paper characterizes when a specific class of permutation trinomials over finite fields of characteristic three are permutations, confirming that certain algebraic conditions are both necessary and sufficient.

Contribution

It establishes the exact necessary and sufficient conditions for permutation trinomials of a particular form over finite fields of characteristic three.

Findings

Permutation polynomial characterization in characteristic 3

Necessary and sufficient conditions proven

Conditions involve algebraic relations and quadratic residues

Abstract

Let $f(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})\in\Bbb F_{q^2}[X]$, where $a,b\in\Bbb F_{q^2}^*$. In a series of recent papers by several authors, sufficient conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial (PP) of $\Bbb F_{q^2}$ and, in characteristic $2$, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when $\text{char}\,\Bbb F_q=3$, $f$ is a PP of $\Bbb F_{q^2}$ if and only if $(ab)^q=a(b^{q+1}-a^{q+1})$ and $1-(b/a)^{q+1}$ is a square in $\Bbb F_q^*$.