# Permutations from an arithmetic setting

## TL;DR

This paper introduces a special class of permutations on finite sets with structured cycle properties, explores their cycle decompositions, and connects them to permutation polynomials over finite fields.

## Contribution

It defines a new class of piecewise-affine permutations with specific modular cycle properties and provides their explicit cycle decompositions and applications to finite field permutations.

## Key findings

- Cycle decomposition of the introduced permutations is explicitly characterized.
- Permutations give rise to permutation polynomials over finite fields.
- Explicit classes of permutation polynomials with known cycle structures are constructed.

## Abstract

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$ consecutive elements in any of its cycles is, up to a cyclic shift, a fixed permutation of $[1, m]$. Our main result provides the cycle decomposition of such permutations. We further show that such permutations give rise to permutations of finite fields. In particular, we explicitly obtain classes of permutation polynomials of finite fields whose cycle decomposition and its inverse are explicitly given.