Permutation polynomials: iteration of shift and inversion maps over finite fields

TL;DR

This paper explores the structure and properties of permutation polynomials over finite fields, focusing on how shift and inversion maps generate permutation groups and their randomness characteristics.

Contribution

It demonstrates that all permutations can be generated by affine unicritical polynomials and analyzes the cycle structure using group theory.

Findings

Permutations in $S_n$ can be generated by affine unicritical polynomials.

Cycle structures of permutations with low Carlitz rank are characterized.

The tree structure of shift and inversion maps informs permutation randomness.

Abstract

We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group generated by shift and inversion maps is used to study the randomness properties of permutation polynomials.