The compositional inverses of three classes of permutation polynomials over finite fields

TL;DR

This paper proves that three classes of permutation trinomials over finite fields are indeed permutation polynomials and explicitly constructs their compositional inverses, enhancing understanding of permutation polynomial structures.

Contribution

The paper introduces a local method to verify permutation properties and explicitly derives compositional inverses for three classes of permutation trinomials over finite fields.

Findings

Confirmed permutation properties of the three classes of trinomials

Derived explicit compositional inverses for each class

Enhanced methods for analyzing permutation polynomials

Abstract

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over $\mathbb{F}_{q^3}$ in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses.