TR05-109 Authors: Ariel Gabizon, Ran Raz, Ronen Shaltiel

Publication: 29th September 2005 19:21

Downloads: 3548

Keywords:

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.