Pseudorandom Generators with Long Stretch and Low Locality from Random Local One-Way Functions

Abstract

We continue the study of locally computable pseudorandom generators (PRGs) $G: \{0,1\}^n \rightarrow \{0,1\}^m$ such that each of their outputs depends on a small number $d$ of input bits. While it is known that such generators are likely to exist for the case of small sublinear stretch $m=n+n^{1-\delta}$, it is less clear whether achieving larger stretch such as $m=n+\Omega(n)$ or even $m=n^{1+\delta}$ is possible. The existence of such PRGs, which was posed as an open question in previous works, has recently gained an additional motivation due to several interesting applications. We make progress towards resolving this question by obtaining several local constructions based on the one-wayness of “random” local functions---a variant of an assumption made by Goldreich [Candidate one-way functions based on exporter graphs, 7 (ECCC 2000), TRCO-090]. Specifically, we construct collections of PRGs with the following parameters: (1) linear stretch $m=n+\Omega(n)$ and constant locality $d=O(1)$; (2) polynomial stretch $m=n^{1+\delta}$ and any (arbitrarily slowly growing) super-constant locality $d=\omega(1)$, e.g., $\log^{\star} n$; (3) polynomial stretch $m=n^{1+\delta}$, constant locality $d=O(1)$, and inverse polynomial distinguishing advantage (as opposed to the standard case of $n^{-\omega(1)}$). Our constructions match the parameters achieved by previous “ad hoc” candidates and are the first to do this under a one-wayness assumption. At the core of our results lies a new search-to-decision reduction for random local functions. This reduction also shows that some of the previous PRG candidates can be based on one-wayness assumptions. Altogether, our results fortify the existence of local PRGs of long stretch. As an additional contribution, we show that our constructions give rise to strong inapproximability results for the densest-subgraph problem in $d$-uniform hypergraphs for constant $d$. This allows us to improve the previous bounds of Feige [Relations between average case complexity and approximation complexity, in Proceedings of STOC 2002, pp. 534--543] and Khot [Ruling out PTAS for graph min-bisection-densest subgraph and bipartate clique, in Proceedings of FOCS 2004, pp. 136--145] from constant inapproximability factor to $n^{\varepsilon}$-inapproximability, at the expense of relying on stronger assumptions.