Abstract
Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form \(f(X)=X+aX^{q(q-1)+ 1}+bX^{2(q-1)+ 1}\in \mathbb {F}_{q^{2}}[X]\), where q is even and \(a,b\in \mathbb {F}_{q^{2}}^{*}\). 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.
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Hou, Xd. On a class of permutation trinomials in characteristic 2. Cryptogr. Commun. 11, 1199–1210 (2019). https://doi.org/10.1007/s12095-018-0342-1
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DOI: https://doi.org/10.1007/s12095-018-0342-1