Almost Chor-Goldreich Sources and Adversarial Random Walks

Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

A Chor-Goldreich (CG) source is a sequence of random variables X = X1 . Xt, where each Xi ∼{0,1}d and Xi has δd min-entropy conditioned on any fixing of X1 . Xi-1. The parameter 0<-≤ 1 is the entropy rate of the source. We typically think of d as constant and t as growing. We extend this notion in several ways, defining almost CG sources. Most notably, we allow each Xi to only have conditional Shannon entropy δd. We achieve pseudorandomness results for almost CG sources which were not known to hold even for standard CG sources, and even for the weaker model of Santha-Vazirani sources: We construct a deterministic condenser that on input X, outputs a distribution which is close to having constant entropy gap, namely a distribution Z ∼{0,1}m for m ≈ δdt with min-entropy m-O(1). Therefore, we can simulate any randomized algorithm with small failure probability using almost CG sources with no multiplicative slowdown. This result extends to randomized protocols as well, and any setting in which we cannot simply cycle over all seeds, and a "one-shot"simulation is needed. Moreover, our construction works in an online manner, since it is based on random walks on expanders. Our main technical contribution is a novel analysis of random walks, which should be of independent interest. We analyze walks with adversarially correlated steps, each step being entropy-deficient, on good enough lossless expanders. We prove that such walks (or certain interleaved walks on two expanders), starting from a fixed vertex and walking according to X1 . Xt, accumulate most of the entropy in X.

Original languageEnglish
Title of host publicationSTOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
EditorsBarna Saha, Rocco A. Servedio
PublisherAssociation for Computing Machinery
Pages1-9
Number of pages9
ISBN (Electronic)9781450399135
DOIs
StatePublished - 2 Jun 2023
Event55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: 20 Jun 202323 Jun 2023

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/TerritoryUnited States
CityOrlando
Period20/06/2323/06/23

Keywords

  • Santha-Vazirani sources
  • condensers
  • expander Graphs
  • extractors
  • random Walks
  • randomized algorithm

ASJC Scopus subject areas

  • Software

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