A Class of Permutation Binomials over Finite Fields

TL;DR

This paper proves a conjecture characterizing when a specific class of permutation binomials over finite fields are permutations, based on parameters $t$ and the prime power $q$, confirming the precise conditions.

Contribution

The paper establishes necessary and sufficient conditions for a class of permutation binomials over finite fields, confirming a recent conjecture.

Findings

Confirmed the conjecture for permutation binomials over finite fields.

Identified exact conditions on parameters $t$ and $q$ for permutation behavior.

Provided a complete characterization of the permutation property for the class.

Abstract

Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv 1\pmod 4$; (ii) $t=-3$, $q\equiv \pm1\pmod{12}$; (iii) $t=3$, $q\equiv -1\pmod 6$. We confirm this conjecture in the present paper.