Permutation Polynomials of $\mathbb{F}_{q^2}$ : A Linear Algebraic Approach

TL;DR

This paper introduces a linear algebraic method to characterize and enumerate a specific class of permutation polynomials over finite fields, providing conditions, inverses, and algorithms for their computation.

Contribution

It offers a new linear algebraic framework for identifying and counting permutation polynomials over _{q^2}, including novel subclasses and their compositional inverses.

Findings

Derived necessary and sufficient conditions for permutation polynomials.

Enumerated the number of such permutation polynomials.

Provided algorithms for computing compositional inverses.

Abstract

In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over $\mathbb{F}_{q^2}$, in the context of rank deficient and full rank linear maps over $\mathbb{F}_{q^2}$. We derive necessary and sufficient conditions under which the given class of polynomials are permutation polynomials. We further show that the number of such permutation polynomials can be easily enumerated. Only a subset of these permutation polynomials have been reported in literature earlier. It turns out that this class of permutation polynomials have compositional inverses of the same kind and we provide algorithms to evaluate the compositional inverses of most of these permutation polynomials.