Permutation binomials over finite fields

TL;DR

This paper investigates conditions under which binomials of the form x^m + c*x^n permute finite fields, establishing bounds on gcd(m-n, p-1) for prime fields and existence criteria for larger fields.

Contribution

It provides new bounds and criteria for permutation binomials over finite fields, linking gcd conditions to permutation properties.

Findings

gcd(m-n,p-1) > sqrt{p} - 1 for permutation binomials over GF(p)

Existence of permutation binomials over GF(q) depends on gcd(m,n,q-1) = 1

Conditions relate gcd bounds to field size and permutation properties.

Abstract

We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then there exist permutation binomials over GF(q) of the form x^m + c*x^n if and only if gcd(m,n,q-1) = 1.