Pseudorandom Generators with Long Stretch and Low Locality from Random Local One-Way Functions
Benny Applebaum · SIAM Journal on Computing · 2012
We continue the study of locally-computable pseudorandom generators (PRG) G:{0,1}n -> {0,1}m that each of their outputs depend on a small number of d input bits. While it is known that such generators are likely to exist for the case of small sub-linear stretch m=n+n1-δ, it is less clear whether achieving larger stretch such as m=n+Ω(n), or even m=n1+δ 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 (ECCC 2000). Specifically, we construct collections of PRGs with the following parameters: 1. Linear stretch m=n+Ω(n) and constant locality d=O(1). 2. Polynomial stretch m=n1+δ and any (arbitrarily slowly growing) super-constant locality d=ω(1), e.g., log*n. 3. Polynomial stretch m=n1+δ, constant locality d=O(1), and inverse polynomial distinguishing advantage (as opposed to the standard case of n-ω(1)).