On a conjecture about a class of permutation trinomials

Abstract

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{\alpha ,\beta }(x)=x+\alpha x^{q(q-1)+1}+\beta x^{2(q-1)+1}\in {\mathbb{F}}_{q^2}[x]$, $\alpha \beta \ne 0$, $q$ even, characterizing all the pairs $(\alpha ,\beta )\in {\mathbb{F}}_{q^2}^2$ for which $f_{\alpha ,\beta }(x)$ is a permutation of ${\mathbb{F}}_{q^2}$.