Permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields

Abstract

This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\in {\mathbb{F}}_{q^e}[X]$ with $a\in {\mathbb{F}}_{q^e}^{\ast }$. We present a complete characterization for $F_{r,a}$ to be a permutation of ${\mathbb{F}}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda--Rubio--Santiago \cite{masuda2022permutation}. More precisely, we show that \(F_{r,a}\) permutes \(\mathbb F_{q^e}\) if and only if $(-a{)}^{(q^e-1)/(q-1)}\ne 1,\gcd (r,q-1)=1$, and there exists \(1\le h<e\) with \(\gcd(h,e)=1\) such that $r(q^h-1)\equiv q-1(\mathrm{mod}{q}^e-1).$

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