Rédei permutations with cycles of the same length

Abstract

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

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• Rédei permutations with the same cycle structure