A Class of Permutation Trinomials over Finite Fields

Abstract

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

18 pages