On a Class of Permutation Trinomials in Characteristic 2

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form $f(X)=X+a{X}^{q(q-1)+1}+b{X}^{2(q-1)+1}\in {\mathbb{F}}_{q^2}[X]$, where $q$ is even and $a,b\in {\mathbb{F}}_{q^2}^{\ast }$. They found sufficient conditions on $a,b$ for $f$ to be a permutation polynomial (PP) of ${\mathbb{F}}_{q^2}$ and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach.