Extracting randomness: A survey and new constructions

Noam Nisan*, Amnon Ta-Shma

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

138 Scopus citations

Abstract

Extractors are Boolean functions that allow, in some precise sense, extraction of randomness from somewhat random distributions, using only a small amount of truly random bits. Extractors, and the closely related "dispersers," exhibit some of the most "random-like" properties of explicitly constructed combinatorial structures. In this paper we do two things. First, we survey extractors and dispersers: what they are, how they can be designed, and some of their applications. The work described in the survey is due to a long list of research papers by various authors - most notably by David Zuckerman. Then, we present a new tool for constructing explicit extractors and give two new constructions that greatly improve upon previous results. The new tool we devise, a "merger," is a function that accepts d strings, one of which is uniformly distributed and outputs a single string that is guaranteed to be uniformly distributed. We show how to build good explicit mergers, and how mergers can be used to build better extractors. Using this, we present two new constructions. The first construction succeeds in extracting all of the randomness from any somewhat random source. This improves upon previous extractors that extract only some of the randomness from somewhat random sources with "enough" randomness. The amount of truly random bits used by this extractor, however, is not optimal. The second extractor we build extracts only some of the randomness and works only for sources with enough randomness, but uses a nearoptimal amount of truly random bits. Extractors and dispersers have many applications in "removing randomness" in various settings and in making randomized constructions explicit. We survey some of these applications and note whenever our new constructions yield better results, e.g., plugging our new extractors into a previous construction we achieve the first explicit N-superconcentrators of linear size and polyloglog(N) depth.

Original languageAmerican English
Pages (from-to)148-173
Number of pages26
JournalJournal of Computer and System Sciences
Volume58
Issue number1
DOIs
StatePublished - Feb 1999

Bibliographical note

Funding Information:
* This paper is a combination of the paper ‘‘On Extracting Randomness from Weak Random Sources’’ [Ta-96] and the paper ‘‘Refining Randomness: Why and How’’ [Nis96]. This work was supported by BSF Grant 92-0043 and by a Wolfeson award administered by the Israeli Academy of Sciences.

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