R\'edei permutations with cycles of the same length

TL;DR

This paper investigates Rédéi permutations over finite fields that have cycle structures consisting only of fixed points and cycles of a fixed length j, providing conditions, characterizations, and explicit constructions for such permutations.

Contribution

It offers necessary and sufficient conditions for the existence of these permutations, characterizes their structure, and provides explicit formulas and construction procedures for specific cycle lengths.

Findings

Characterization of Rédéi permutations with only 1- and j-cycles.

Explicit formulas for Rédéi involutions based on fixed points.

Procedures to construct permutations with specified cycle structures.

Abstract

Let $\mathbb{F}_q$ be a finite field of odd characteristic. We study R\'edei functions that induce permutations over $\mathbb{P}^1(\mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $j\geq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a R\'edei permutation of this type to exist over $\mathbb{P}^1(\mathbb{F}_q)$, characterize R\'edei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for R\'edei involutions based on the number of fixed points, and procedures to construct R\'edei permutations with a prescribed number of fixed points and $j$-cycles for $j \in \{3,4,5\}$.