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

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 \[ f(x)=(ax^{q} +bx +c)^r ϕ((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~\text{over ${\mathbb{F}}_{q^2}$}, \] where $a,b,c,u,v\in {\mathbb{F}}_{q^2}$, $r\in {\mathbb{Z}}^{+}$, $\varphi (x)\in {\mathbb{F}}_{q^2}[x]$ 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.

Cited by in corpus (4)

• On Inverses of Permutation Polynomials of Small Degree over Finite Fields
• Finding compositional inverses of permutations from the AGW criterion
• On inverses of some permutation polynomials over finite fields of characteristic three
• Some new classes of permutation polynomials and their compositional inverses