A note regarding permutation binomials over finite fields

TL;DR

This paper investigates specific permutation polynomials over finite fields, providing explicit conditions on parameters for when these polynomials permute the field elements.

Contribution

It offers explicit criteria for permutation behavior of certain binomials over finite fields, focusing on cases where r is 5 or 7.

Findings

Derived explicit conditions for permutation polynomials with r=5 or 7.

Identified parameter ranges (q, a) ensuring permutation properties.

Enhanced understanding of permutation binomials over finite fields.

Abstract

Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$