Cycle types of complete mappings of finite fields

TL;DR

This paper investigates the cycle structures of complete mappings in finite fields, especially focusing on first-order cyclotomic mappings, and establishes conditions under which various cycle types can be realized, including new examples with specific permutation properties.

Contribution

It provides new existence results for cycle types of first-order cyclotomic permutations and complete mappings in finite fields, expanding understanding of their permutation behaviors.

Findings

All cycle types of certain cyclotomic permutations are achievable for large enough fields.

Complete mappings can permute nonzero elements in a single cycle under specific conditions.

New examples of complete mappings with orthomorphisms permuting elements in one cycle are constructed.

Abstract

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$ that fix $0$ and restrict to the multiplication $x\mapsto a_ix$ by a fixed element $a_i\in\mathbb{F}_q$ on each coset $C_i$ of a given subgroup $C$ of $\mathbb{F}_q^{\ast}$. The gist of two of our main results is that as long as $q$ is large enough relative to the index $|\mathbb{F}_q^{\ast}:C|$, all cycle types of first-order cyclotomic permutations with only long cycles on $\mathbb{F}_q^{\ast}$ can be achieved through a complete mapping, as can all permutations of the cosets of $C$. Our third main result provides new examples of complete mappings $f$ such that both $f$ and its associated orthomorphism $f+\operatorname{id}$ permute the nonzero field elements in one cycle.