Deterministic extractors for independent-symbol sources

Chia Jung Lee*, Chi Jen Lu, Shi-Chun Tsai

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations

Abstract

In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0, 1}d. The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which can extract Ω(log k -log d-log log(1/ε)) bits with error ε, for any n, d, k ∈ ℕ and ε ∈ (0, 1). For sources with a larger min-entropy, we can extract even more randomness, When k ≥ n 1/2+γ, for any constant γ ∈ (0, 1/2), we can extract m = k - O(d log (1/ε)) bits with any error ε ≥ 2 Ωnγ, When k ≥ logc n, for some constant c > 0, we can extract m = k -d(1/ε)O(1) bits with any error ε ≥ k-Ω(1). Our results generalize those of Kamp & Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d = 1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a non-explicit deterministic extractor which can extract m = k-O(log(1/ε)) bits whenever k = w(d+log(n/ε)), Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k -m = ω(log(1/ε)). This generalizes a lower bound of Radhakrishnan & Ta-Shma with respect to general sources.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings
PublisherSpringer Verlag
Pages84-95
Number of pages12
ISBN (Print)3540359044, 9783540359043
DOIs
StatePublished - 1 Jan 2006
Event33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 - Venice, Italy
Duration: 10 Jul 200614 Jul 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4051 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference33rd International Colloquium on Automata, Languages and Programming, ICALP 2006
CountryItaly
CityVenice
Period10/07/0614/07/06

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    Lee, C. J., Lu, C. J., & Tsai, S-C. (2006). Deterministic extractors for independent-symbol sources. In Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings (pp. 84-95). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4051 LNCS). Springer Verlag. https://doi.org/10.1007/11786986_9