Some new classes of permutation polynomials and their compositional inverses

TL;DR

This paper introduces six new classes of permutation polynomials over finite fields, providing explicit inverse formulas and demonstrating their novelty compared to existing classes.

Contribution

The paper presents six new classes of permutation polynomials of a specific form over finite fields, with explicit inverse formulas and proof of their inequivalence to known classes.

Findings

Six new classes of permutation polynomials are identified.

Explicit compositional inverses are derived for each class.

The new classes are proven to be inequivalent to existing ones.

Abstract

We focus on the permutation polynomials of the form $L(X)+\Tr_{m}^{3m}(X)^{s}$ over $\F_{q^3}$, where $\F_q$ is the finite field with $q=p^m$ elements, $p$ is a prime number, $m$ is a positive integer, $\Tr_{m}^{3m}$ is the relative trace function from $\F_{p^{3m}}$ to $\F_{p^{m}}$, $L(X)$ is a linearized polynomial over $\F_{q^{3}}$, and $s>1$ is a positive integer. More precisely, we present six new classes of permutation polynomials over $\F_{q^3}$ of the aforementioned form: one class over finite fields of even characteristic, three classes over finite fields of odd characteristic, and the remaining two over finite fields of arbitrary characteristic. Furthermore, we show that these classes of permutation polynomials are inequivalent to the known ones of the same form. We also provide the explicit expressions for the compositional inverses of each of these classes of permutation polynomials.