A family of permutation trinomials in $\mathbb{F}_{q^2}$

TL;DR

This paper characterizes permutation polynomials of a specific form over finite fields, establishing necessary and sufficient conditions by linking algebraic curves and finite field theory.

Contribution

It provides a complete characterization of permutation trinomials in $_{q^2}$, confirming known conditions are also necessary through algebraic curve analysis.

Findings

Necessary and sufficient conditions for permutation polynomials are established.

Connections with algebraic curves over finite fields are used to prove the results.

The characterization applies to polynomials of the form $f_{a,b}(X)$ in $_{q^2}$.

Abstract

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary.