R\'edei permutations with the same cycle structure

TL;DR

This paper characterizes when Rédéi permutations over finite fields share the same cycle structure, identifies isolated permutations, and explores their relationships and explicit families, with implications for related bijections.

Contribution

It provides a complete characterization of pairs of Rédéi permutations with identical cycle structures and identifies the unique isolated Rédéi involutions.

Findings

Characterization of all pairs (m,n) with same cycle structure when a and b have same quadratic character.

Identification that only Rédéi involutions can be isolated permutations.

Explicit families of Rédéi permutations sharing cycle structures.

Abstract

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathbb P^1(\mathbb{F}_q) = \mathbb F_q\cup \{\infty\}$. Write $(x+\sqrt y)^m$ as $N(x,y)+D(x,y)\sqrt{y}$. For $m\in\mathbb N$ and $a \in \mathbb{F}_q$, the R\'edei function $R_{m,a}\colon \mathbb P^1(\mathbb F_q) \to \mathbb P^1(\mathbb F_q)$ is defined by $N(x,a)/D(x,a)$ if $D(x,a)\neq 0$ and $x\neq\infty$, and $\infty$, otherwise. In this paper we give a complete characterization of all pairs $(m,n)\in\mathbb N^2$ such that the R\'edei permutations $R_{m,a}$ and $R_{n,b}$ have the same cycle structure when $a$ and $b$ have the same quadratic character and $q$ is odd. We explore some relationships between such pairs $(m,n)$, and provide explicit families of R\'edei permutations with the same cycle structure. When a R\'edei permutation has a unique cycle structure that is not shared by any other R\'edei permutation, we call it isolated. We show that the only isolated R\'edei permutations are the isolated R\'edei involutions. Moreover, all our results can be transferred to bijections of the form $mx$ and $x^m$ on certain domains.