Short Cycles in Repeated Exponentiation Modulo a Prime

TL;DR

This paper investigates the behavior of repeated exponentiation modulo a prime, providing bounds on fixed points and short cycles, which are relevant for cryptographic pseudorandom generators.

Contribution

It introduces new upper bounds on fixed points and short cycles in the dynamical system generated by repeated exponentiation modulo a prime.

Findings

Nontrivial upper bounds on fixed points

Bounds on the number of short cycles

Implications for cryptographic pseudorandom generators

Abstract

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.