A Class of Permutation Trinomials over Finite Fields

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 $\Bbb F_{q^2}$ifandonlyifoneofthefollowingoccurs:(i)$q$isevenand$$\text{Tr}$_{q/2}(\frac 1t)=0$;(ii)$q\equiv 1\pmod 8$and$t^2=-2$.