Generating k-independent variables in constant time

TL;DR

This paper introduces a novel generator for k-independent pseudorandom variables over finite fields that operates in constant time per output, significantly improving efficiency in randomized algorithms.

Contribution

It presents the first construction of a k-independent generator that outputs each value in constant time, independent of k, with near-optimal space complexity.

Findings

Generator operates in constant time per output

Period length is |inite| poly log k

Uses near-optimal space complexity

Abstract

The generation of pseudorandom elements over finite fields is fundamental to the time, space and randomness complexity of randomized algorithms and data structures. We consider the problem of generating $k$-independent random values over a finite field $\mathbb{F}$ in a word RAM model equipped with constant time addition and multiplication in $\mathbb{F}$, and present the first nontrivial construction of a generator that outputs each value in constant time, not dependent on $k$. Our generator has period length $|\mathbb{F}|\,\mbox{poly} \log k$ and uses $k\,\mbox{poly}(\log k) \log |\mathbb{F}|$ bits of space, which is optimal up to a $\mbox{poly} \log k$ factor. We are able to bypass Siegel's lower bound on the time-space tradeoff for $k$-independent functions by a restriction to sequential evaluation.