Zero-fixing extractors for sub-logarithmic entropy

Gil Cohen, Igor Shinkar*

*Corresponding author for this work

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

Abstract

An (n, k)-bit-fixing source is a distribution on n bit strings, that is fixed on n − k of the coordinates, and jointly uniform on the remaining k bits. Explicit constructions of bit-fixing extractors by Gabizon, Raz and Shaltiel [SICOMP 2006] and Rao [CCC 2009], extract (1 − o(1)) ・ k bits for k = poly log n, almost matching the probabilistic argument. Intriguingly, unlike other well-studied sources of randomness, a result of Kamp and Zuckerman [SICOMP 2006] shows that, for any k, some small portion of the entropy in an (n, k)-bit-fixing source can be extracted. Although the extractor does not extract all the entropy, it does extract log(k)/2 bits. In this paper we prove that when the entropy k is small enough compared to n, this exponential entropy-loss is unavoidable. More precisely, we show that forn > Tower(k2) one cannot extract more than log(k)/2+O(1) bits from (n, k)-bit-fixing sources. The remaining entropy is inaccessible, information theoretically. By the Kamp-Zuckerman construction, this negative result is tight. For small enough k, this strengthens a result by Reshef and Vadhan [RSA 2013], who proved a similar bound for extractors computable by space-bounded streaming algorithms. Our impossibility result also holds for what we call zero-fixing sources. These are bit-fixing sources where the fixed bits are set to 0. We complement our negative result, by giving an explicit construction of an (n, k)-zero-fixing extractor that outputs Ω(k) bits for k ≥ poly log log n. Finally, we give a construction of an (n, k)-bit-fixing extractor, that outputs k − O(1) bits, for entropy k = (1 + o(1)) ・ log log n, with running-time nO((log log n)2). This answers an open problem by Reshef and Vadhan [RSA 2013].

Original languageEnglish
Title of host publicationAutomata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings
EditorsMagnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama
PublisherSpringer Verlag
Pages343-354
Number of pages12
ISBN (Print)9783662476710
DOIs
StatePublished - 2015
Externally publishedYes
Event42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan
Duration: 6 Jul 201510 Jul 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9134
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference42nd International Colloquium on Automata, Languages and Programming, ICALP 2015
Country/TerritoryJapan
CityKyoto
Period6/07/1510/07/15

Funding

FundersFunder number
National Science Foundation0832795, CCF 1422159, 1061938
Israel Science Foundation
Planning and Budgeting Committee of the Council for Higher Education of Israel

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