A General Construction of Permutation Polynomials of $\Bbb F_{q^2}$

TL;DR

This paper presents a comprehensive algorithmic method to generate all permutation polynomials over the finite field b F_{q^2} with specific properties, expanding known classes and including new examples like binomials and trinomials.

Contribution

The authors introduce a general construction algorithm that produces all permutation polynomials of b F_{q^2} with certain subgroup-induced properties, surpassing previous ad hoc methods.

Findings

Algorithm generates all such permutation polynomials.

Includes many previously unknown permutation polynomials.

Explicit examples of binomials and trinomials are provided.

Abstract

Let $r$ be a positive integer, $h(X)\in\Bbb F_{q^2}[X]$, and $\mu_{q+1}$ be the subgroup of order $q+1$ of $\Bbb F_{q^2}^*$. It is well known that $X^rh(X^{q-1})$ permutes $\Bbb F_{q^2}$ if and only if $\text{gcd}(r,q-1)=1$ and $X^rh(X)^{q-1}$ permutes $\mu_{q+1}$. There are many ad hoc constructions of permutation polynomials of $\Bbb F_{q^2}$ of this type such that $h(X)^{q-1}$ induces monomial functions on the cosets of a subgroup of $\mu_{q+1}$. We give a general construction that can generate, through an algorithm, {\em all} permutation polynomials of $\Bbb F_{q^2}$ with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.