Determination of a Type of Permutation Binomials over Finite Fields

Abstract

Let $f=a\x+\x^{3q-2}\in\Bbb F_{q^2}[\x]$, where $a\in {\mathbb{F}}_{q^2}^{\ast }$. We prove that $f$ is a permutation polynomial of ${\mathbb{F}}_{q^2}$ if and only if one of the following occurs: (i) $q=2^e$, $e$ odd, and $a^{\frac{q+1}{3}}$ is a primitive $3$rd root of unity. (ii) $(q,a)$ belongs to a finite set which is determined in the paper.