The compositional inverses of permutation polynomials of the form $\sum_{i=1}^kb_i(x^{p^m}+x+\delta)^{s_i}-x$ over $\mathbb{F}_{p^{2m}}$

TL;DR

This paper derives explicit compositional inverses for a class of permutation polynomials over finite fields, expanding understanding of their algebraic structure and potential applications in cryptography.

Contribution

The paper provides explicit formulas for the compositional inverses of permutation polynomials of a specific form over finite fields, a novel contribution to permutation polynomial theory.

Findings

Explicit inverse formulas for the permutation polynomials derived.

Enhanced understanding of the algebraic structure of these polynomials.

Potential applications in cryptography and coding theory.

Abstract

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i(x^{p^m}+x+\delta)^{s_i}-x$ over $\mathbb{F}_{p^{2m}}$, where for $1\leq i \leq k,$ $s_i, m$ are positive integers, $b_i, \delta \in \mathbb{F}_{p^{2m}},$ and $p$ is prime.