Determination of a Class of Permutation Trinomials in Characteristic Three

Abstract

Let $f(X)=X(1+a{X}^{q(q-1)}+b{X}^{2(q-1)})\in {\mathbb{F}}_{q^2}[X]$, where $a,b\in {\mathbb{F}}_{q^2}^{\ast }$. 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 ${\mathbb{F}}_{q^2}$ 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 $\text{char}{\mathbb{F}}_q=3$, $f$ is a PP of ${\mathbb{F}}_{q^2}$ if and only if $(ab{)}^q=a(b^{q+1}-a^{q+1})$ and $1-(b/a{)}^{q+1}$ is a square in ${\mathbb{F}}_q^{\ast }$.

31 pages