Iterations of Quadratic Polynomials over Finite Fields

TL;DR

This paper investigates the behavior of quadratic polynomial iterations over finite fields, providing heuristic and experimental evidence that their cycle lengths resemble a scaled Birthday Problem distribution.

Contribution

It introduces a heuristic model linking quadratic polynomial iterations to the Birthday Problem and offers experimental validation of the distribution convergence.

Findings

Set lengths behave like a scaled Birthday Problem distribution

Experimental evidence supports the heuristic model

Distribution convergence observed for primes up to x

Abstract

Given a map f:Z-->Z and an initial argument alpha, we can iterate the map to get a finite set of iterates modulo a prime p. In particular, for a quadratic map f(z)=z^2 +c, c constant, work by Pollard suggests that this set should have length on the order of p^(1/2). We give a heuristic argument that suggests that the statistical properties of this set might be very similar to the Birthday Problem random variable X_n, for an n=p day year, and offer considerable experimental evidence that the limiting distribution of these set lengths, divided by p^(1/2), for p\leq x as x goes to infinity, converges to the limiting distribution of X_n/n^(1/2), as n goes to infinity.