Pseudorandom Numbers and Hash Functions from Iterations of Multivariate Polynomials

TL;DR

This paper explores the use of polynomial iterations with slow degree growth to generate pseudorandom vectors and hash functions, providing new theoretical bounds and constructions for their mixing properties.

Contribution

It introduces new methods and extends previous results to more general polynomial orbits, leading to novel hash functions with proven mixing properties.

Findings

Good estimates of exponential sums along polynomial orbits

Stronger bounds on discrepancy for pseudorandom vectors

Design of new hash functions with mixing properties

Abstract

Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather stronger bounds on the discrepancy for pseudorandom vectors generated by these iterations. Here we add new arguments to our original approach and also extend some of our recent constructions and results to more general orbits of polynomial iterations which may involve distinct polynomials as well. Using this construction we design a new class of hash functions from iterations of polynomials and use our estimates to motivate their "mixing" properties.