On Permutation Binomials over Finite Fields

TL;DR

This paper investigates conditions under which binomials over finite fields permute elements, establishing bounds on the prime characteristic and providing methods to derive permutation binomials over subfields.

Contribution

It proves bounds on the prime characteristic for permutation binomials and shows how to construct such binomials over subfields from those over larger fields.

Findings

If a binomial permutes $F_p$, then $p-1 leq (d-1)d$, with $d = gcd(n-m,p-1)$.

The bound on $p$ in terms of $d$ is sharp.

Methods are provided to obtain permutation binomials over subfields.

Abstract

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ 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 $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (d -1)d$, 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 $\mathbb{F}_{q}$.