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

Source 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 $a\in\mathbb F_{q^e}^*$. 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 $(-a)^{(q^e-1)/(q-1)}\ne1, \gcd(r,q-1)=1$, and there exists $1\le h<e$ with $\gcd(h,e)=1$ such that $r(q^h-1)\equiv q-1\pmod{q^e-1}.$