Four new classes of permutation trinomials and their compositional inverses

TL;DR

This paper introduces four new classes of permutation trinomials over finite fields of even characteristic, providing explicit compositional inverses and extending previous work with a specific example.

Contribution

The paper constructs four new classes of permutation trinomials over finite fields and explicitly derives their compositional inverses, including an extension of a recent permutation trinomial.

Findings

Four new classes of permutation trinomials are constructed.

Explicit polynomial forms of the compositional inverses are provided.

The inverse of a recent permutation trinomial is derived.

Abstract

We construct four new classes of permutation trinomials over the cubic extension of a finite field with even characteristic. Additionally, we explicitly provide the compositional inverse of each class of permutation trinomials in polynomial form. Furthermore, we derive the compositional inverse of the permutation trinomial $\alpha X^{q(q^2 - q + 1)} + \beta X^{q^2 - q + 1} + 2X$ for $\alpha = 1$ and $\beta = 1$, originally proposed by Xie, Li, Xu, Zeng, and Tang (2023).