Regular complete permutation polynomials over quadratic extension fields

TL;DR

This paper investigates conditions under which certain composed permutation polynomials over quadratic extension fields are regular complete permutation polynomials, expanding understanding of their algebraic structure and construction.

Contribution

It provides new criteria for constructing regular complete permutation polynomials over quadratic extension fields using permutation polynomials and linear maps.

Findings

Established conditions for regular complete permutation polynomials over quadratic fields.

Constructed explicit examples of such polynomials under certain algebraic conditions.

Extended the class of known permutation polynomials with regularity properties.

Abstract

Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $\tau_1, \tau_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $\sigma_M$ is an invertible linear map over $\mathbb{F}_{q^2}$ and $\sigma=\tau_1\circ\sigma_M\circ\tau_2$. In this paper, we prove that, for suitable $\tau_1, \tau_2$ and $\sigma_M$, the map $\sigma$ could be $r$-regular complete permutation polynomials over quadratic extension fields.