Connected components of the graph generated by power maps in prime finite fields

TL;DR

This paper investigates the structure of the graph formed by the power map in prime finite fields, providing estimates for the number of cycles in the associated functional graph, which has implications for pseudorandom number generation.

Contribution

It offers new bounds and estimates for the number of cycles in the power map graph over prime finite fields, advancing understanding of its combinatorial properties.

Findings

Derived bounds for the maximum number of cycles.

Provided average cycle count estimates.

Analyzed the graph structure for prime finite fields.

Abstract

Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$. That is, for some integer $e\ge2$, one considers the sequence $u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb F}_q^\times$. This sequence is eventually periodic. One can consider the number of cycles that exist as the seed $u$ varies over ${\mathbb F}_q^\times$. This is the same as the number of cycles in the functional graph of the map $x\mapsto x^e$ in ${\mathbb F}_q^\times$. We prove some estimates for the maximal and average number of cycles in the case of prime finite fields.