A Note on Permutation Binomials and Trinomials over Finite Fields

TL;DR

This paper characterizes permutation binomials and trinomials derived from reversed Dickson polynomials over finite fields, specifically for certain exponents, providing a complete classification in this case.

Contribution

It offers a complete explanation of permutation binomials and trinomials from reversed Dickson polynomials over finite fields for the case when n=p^l+2.

Findings

Classification of permutation binomials and trinomials for n=p^l+2

Explicit conditions for permutation properties over finite fields

Extension of known results on Dickson polynomials

Abstract

Let $p$ be an odd prime and $e$ be a positive integer. We completely explain the permutation binomials and trinomials arising from the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over $\mathbb{F}_{p^e}$ when $n=p^l+2$, where $l\in \mathbb{N}$.