On the Heuristic of Approximating Polynomials over Finite Fields by Random Mappings

TL;DR

This paper investigates the heuristic that polynomial iterations over finite fields behave like random mappings, analyzing their indegree distributions and coalescence properties through theoretical proofs and experiments, especially for quartic and Chebyshev polynomials.

Contribution

It provides new proofs and improved results on indegree distributions of polynomials over finite fields, extending the heuristic to higher degrees and comparing behaviors of specific polynomial classes.

Findings

Indegree distributions of polynomials and mappings are asymptotically similar.

Polynomials generally behave like random mappings in terms of coalescence.

Chebyshev polynomials exhibit distinct behavior compared to other polynomial classes.

Abstract

The study of iterations of functions over a finite field and the corresponding functional graphs is a growing area of research with connections to cryptography. The behaviour of such iterations is frequently approximated by what is know as the Brent-Pollard heuristic, where one treats functions as random mappings. We aim at understanding this heuristic and focus on the expected rho length of a node of the functional graph of a polynomial over a finite field. Since the distribution of indegrees (preimage sizes) of a class of functions appears to play a central role in its average rho length, we survey the known results for polynomials over finite fields giving new proofs and improving one of the cases for quartic polynomials. We discuss the effectiveness of the heuristic for many classes of polynomials by comparing our experimental results with the known estimates for random mapping models defined by different restrictions on their distribution of indegrees. We prove that the distribution of indegrees of general polynomials and mappings have similar asymptotic properties, including the same asymptotic average coalescence. The combination of these results and our experiments suggests that these polynomials behave like random mappings, extending a heuristic that was known only for degree $2$. We show numerically that the behaviour of Chebyshev polynomials of degree $d \geq 2$ over finite fields present a sharp contrast when compared to other polynomials in their respective classes.