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


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
Number of pages12
ISBN (Print)9783662476710
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)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference42nd International Colloquium on Automata, Languages and Programming, ICALP 2015


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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