Time-Space Tradeoffs for Element Distinctness and Set Intersection via Pseudorandomness

TL;DR

This paper simplifies the analysis of a pseudorandom hash family used for element distinctness, enabling a clear time-space tradeoff without relying on a random oracle, and extends the approach to set intersection.

Contribution

It provides a simplified analysis of pseudorandom hash families for element distinctness, establishing a time-space tradeoff without random oracle assumptions.

Findings

Achieves a time-space tradeoff for element distinctness without random oracle.

Extends the approach to the set intersection problem with tight bounds.

Improves the seed length for pseudorandom hash function sampling.

Abstract

In the Element Distinctness problem, one is given an array $a_1,\dots, a_n$ of integers from $[poly(n)]$ and is tasked to decide if $\{a_i\}$ are mutually distinct. Beame, Clifford and Machmouchi (FOCS 2013) gave a low-space algorithm for this problem running in space $S(n)$ and time $T(n)$ where $T(n) \le \widetilde{O}(n^{3/2}/S(n)^{1/2})$, assuming a random oracle (i.e., random access to polynomially many random bits). A recent breakthrough by Chen, Jin, Williams and Wu (SODA 2022) showed how to remove the random oracle assumption in the regime $S(n) = polylog(n)$ and $T(n) = \widetilde{O}(n^{3/2})$. They designed the first truly $polylog(n)$-space, $\widetilde{O}(n^{3/2})$-time algorithm by constructing a small family of hash functions $\mathcal{H} \subseteq \{h | h:[poly(n)]\to [n]\}$ with a certain pseudorandom property. In this paper, we give a significantly simplified analysis of the pseudorandom hash family by Chen et al. Our analysis clearly identifies the key pseudorandom property required to fool the BCM algorithm, allowing us to explore the full potential of this construction. As our main result, we show a time-space tradeoff for Element Distinctness without random oracle. Namely, for every $S(n),T(n)$ such that $T\approx \widetilde{O}(n^{3/2}/S(n)^{1/2})$, our algorithm can solve the problem in space $S(n)$ and time $T(n)$. Our algorithm also works for a related problem Set Intersection, for which this tradeoff is tight due to a matching lower bound by Dinur (Eurocrypt 2020). As two additional contributions, we show a more general pseudorandom property of the hash family, and slightly improve the seed length to sample the pseudorandom hash function.