A note on inverses of cyclotomic mapping permutation polynomials over finite fields

TL;DR

This paper provides a simplified proof for the inverse formulas of generalized cyclotomic permutation polynomials over finite fields, characterizes involutions among them, and offers an efficient algorithm for generating these polynomials and their inverses.

Contribution

It introduces a shorter proof for inverse formulas, characterizes involutions, and presents a fast modular algorithm for generating generalized cyclotomic permutation polynomials and their inverses.

Findings

Simplified proof of inverse formulas for cyclotomic permutation polynomials

Characterization of involutions among these polynomials

Efficient modular algorithm for generation of polynomials and inverses

Abstract

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.