New permutation polynomials over $\mathbb{F}_{q^2}$

TL;DR

The paper addresses constructing permutation polynomials over $\mathbb{F}_{q^2}$ and proposes a novel multivariate reduction method that bypasses the AGW criterion. It applies this framework to polynomials of the form $(x^{q}-x+\delta)^s+\gamma L(x)$, deriving explicit necessary and sufficient conditions in odd and even characteristics and producing multiple new classes of PPs, complemented by an alternative construction for higher extensions. Central to the results are reductions to two-variable permutation tests over $\mathbb{F}_q$ via a chosen $\mathbb{F}_q$-basis, with conditions expressed through traces and norms like $Tr_q^{q^2}(\cdot)$ and $N_q^{q^2}(\cdot)$. The work broadens the catalog of PPs over $\mathbb{F}_{q^2}$ and offers templates for systematic construction beyond AGW, while also outlining extensions to $\mathbb{F}_{q^d}$ and connections to direction-set concepts.

Abstract

In this paper, we propose a new method to obtain new permutation polynomials over $\mathbb{F}_{q^2}$. Using this method, we extend many known permutation polynomials, which take the form $\sum_i(x^q-x+δ)^{s_i}+L(x)$, where $L(x)$ is a $q$-polynomial over $\mathbb{F}_q$ and $δ\in\mathbb{F}_{q^2}$. We also present an alternative approach for constructing permutation polynomials of the form $x+γTr_q^{q^d}(x^{q+1}+x^{2q+2})$ for the cases where $q=2^m$, $2\nmid d$ and $ Tr_q^{q^d}(x)=x+x^q+\dots+x^{q^{d-1}}$.

Theorems & Definitions (34)

• Proposition 2.1
• proof
• Proposition 2.2
• Example 2.1
• proof
• Lemma 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3