Permutation polynomials over finite fields by the local criterion

TL;DR

This paper explores permutation polynomials over finite fields using a local criterion, introduces new classes and inverses, and examines their properties relative to linear transformations.

Contribution

It introduces a new class of permutation polynomials and their inverses over finite fields, expanding understanding of local permutation properties.

Findings

Linearized polynomials are local permutation polynomials under all linear transformations.

Every permutation polynomial is a local permutation polynomial with respect to certain mappings.

The paper provides a new criterion for identifying permutation polynomials over finite fields.

Abstract

In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over $ \mathbb{F}_{q^2}$. Additionally, we demonstrate that linearized polynomial over $\mathbb{F}_{q^n}$ is a local permutation polynomial with respect to all linear transformations from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q ,$ and that every permutation polynomial is a local permutation polynomial with respect to certain mappings.