On a class of permutation trinomials in characteristic 2

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form f(X)=X+aXq(q−1)+1+bX2(q−1)+1∈Fq2[X]$f(X)=X+aX^{q(q-1)+ 1}+bX^{2(q-1)+ 1}\in \mathbb {F}_{q^{2}}[X]$, where q is even and a,b∈Fq2∗$a,b\in \mathbb {F}_{q^{2}}^{*}$. They found sufficient conditions on a, b for f to be a permutation polynomial (PP) of Fq2$\mathbb {F}_{q^{2}}$ 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.