Permutation polynomials over finite fields by the local criterion

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.