On Permutation Binomials over Finite Fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field of characteristic $p$ containing $q=p^r$ elements and $f(x)=a{x}^n+x^m$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x)=a{x}^n+x^m$ permutes ${\mathbb{F}}_p$, where $n>m>0$ and $a\in {{\mathbb{F}}_p}^{\ast }$, then $p-1\le (d-1)d$, where $d=gcd(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of ${\mathbb{F}}_q$ from a permutation binomial over ${\mathbb{F}}_q$.

Cited by in corpus (1)

• Constructing permutation polynomials over finite fields