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### Paper:

TR05-109 | 28th September 2005 00:00

#### Deterministic Extractors for Bit-fixing Sources by Obtaining an Independent Seed

TR05-109
Authors: Ariel Gabizon, Ran Raz, Ronen Shaltiel
Publication: 29th September 2005 19:21
Downloads: 3212
Keywords:

Abstract:

An $(n,k)$-bit-fixing source is a distribution $X$ over $\B^n$ 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 bit-fixing source extractor is a function $E:\B^n \ar \B^m$ which on an arbitrary $(n,k)$-bit-fixing source outputs $m$ bits that
are statistically-close to uniform. Recently, Kamp and Zuckerman [44th FOCS,
2003] gave a construction of a deterministic bit-fixing source extractor that
extracts $\Omega(k^2/n)$ bits and requires $k>\sqrt{n}$.

In this paper we give constructions of deterministic bit-fixing
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 bit-fixing 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
bit-fixing source extractors that extract few bits into extractors
which extract almost all the bits.

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