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Deterministic Extractors for BitFixing Sources by Obtaining an Independent Seed

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shaltiel, Ronen Raz, Ran Gabizon, Ariel
Copyright Year	2006
Abstract	An $(n,k)$bitfixing source is a distribution X over $\{0,1\}^n$ such that there is a subset of k variables in $X_1,\ldots,X_n$ which are uniformly distributed and independent of each other, and the remaining $n-k$ variables are fixed. A deterministic bitfixing source extractor is a function $E:\{0,1\}^n \rightarrow \{0,1\}^m$ which on an arbitrary $(n,k)$bitfixing source outputs m bits that are statistically close to uniform. Recently, Kamp and Zuckerman Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 92101 gave a construction of a deterministic bitfixing source extractor that extracts $\Omega(k^2/n)$ bits and requires $k>\sqrt{n}$.In this paper we give constructions of deterministic bitfixing source extractors that extract $(1-o(1))k$ bits whenever $k>(\log n)^c$ for some universal constant $c>0$. Thus, our constructions extract almost all the randomness from bitfixing sources and work even when k is small. For $k \gg \sqrt{n}$ the extracted bits have statistical distance $2^{-n^{\Omega(1)}}$ from uniform, and for $k \le \sqrt{n}$ the extracted bits have statistical distance $k^{-\Omega(1)}$ from uniform.Our technique gives a general method to transform deterministic bitfixing source extractors that extract few bits into extractors which extract almost all the bits.
Starting Page	1072
Ending Page	1094
Page Count	23
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539705447049
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4 (Special Issue on Randomness and Complexity)
Volume Number	36
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-12-15
Access Restriction	Subscribed
Subject Keyword	seed obtainers deterministic extractors seeded extractors derandomization Combinatorics bitfixing sources
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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