A proof of a conjecture on permutation trinomials

Abstract

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1}+\hat{I}\pm X^{pq}+X^{q+p-1}$, on finite fields ${\mathbb{F}}_{q^2},q=p^k$, from [A. Rai, R. Gupta, {\it Further results on a class of permutation trinomials}, Cryptogr. Commun. 15 (2023), 811--820].