New Constructions of Permutation Polynomials of the Form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$

TL;DR

This paper introduces new methods using the AGW Criterion to construct permutation polynomials of the form $x^rh(x^{q-1})$ over $F_{q^2}$, generalizing previous results and explaining many known permutation trinomials.

Contribution

The paper develops a novel approach applying both multiplicative and additive AGW Criterion to characterize and construct permutation polynomials over $F_{q^2}$ with arbitrary $h(x)$ over $F_q$, extending prior work.

Findings

Constructed many new explicit permutation polynomials over $F_{q^2}$.

Unified explanation for most known permutation trinomials in even characteristic.

Reduced permutation polynomial problem to smaller subsets of subfields.

Abstract

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing the corresponding fractional polynomials permute a smaller set $\mu_{q+1}$, where $\mu_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. Motivated by these results, we characterize the permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$ such that $h(x)\in\mathbb{F}_q[x]$ is arbitrary and $q$ is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing permutations over a small subset $S$ of a proper subfield $\mathbb{F}_{q}$, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic.