On a Class of Permutation Trinomials in Characteristic 2

Abstract

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.

References in corpus (2)

• Several Classes of Permutation Trinomials From Niho Exponents
• Permutation polynomials on F_q induced from bijective Redei functions on subgroups of the multiplicative group of F_q

Cited by in corpus (1)

• Determination of a Class of Permutation Trinomials in Characteristic Three