Large classes of permutation polynomials over $\mathbb{F}_{q^2}$

TL;DR

This paper develops a general framework for constructing permutation polynomials over finite fields using the AGW criterion, unifying and extending previous classes of permutation polynomials with new conditions.

Contribution

The paper introduces a general form of permutation polynomials over _{q^2} and employs the AGW criterion twice to derive simple conditions for permutation behavior, unifying and extending prior results.

Findings

Derived simple conditions for permutation polynomials over _{q^2}

Unified and generalized known classes of permutation polynomials

Constructed a framework using the AGW criterion for broader classes

Abstract

Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{\frac{q^2 -1}{d}+1} -bx$ over $\mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form \[ f(x)=(ax^{q} +bx +c)^r \phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~\text{over $\mathbb{F}_{q^2}$}, \] where $a,b,c,u,v \in \mathbb{F}_{q^2}$, $r \in \mathbb{Z}^{+}$, $\phi(x)\in \mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $\mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $\mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $\mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.