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- W2938082011 abstract "A random variable $X$ is an $(n,k)$-zero-fixing source if for some subset $Vsubseteq[n]$, $X$ is the uniform distribution on the strings ${0,1}^n$ that are zero on every coordinate outside of $V$. An $epsilon$-extractor for $(n,k)$-zero-fixing sources is a mapping $F:{0,1}^nto{0,1}^m$, for some $m$, such that $F(X)$ is $epsilon$-close in statistical distance to the uniform distribution on ${0,1}^m$ for every $(n,k)$-zero-fixing source $X$. Zero-fixing sources were introduced by Cohen and Shinkar in [10] in connection with the previously studied extractors for bit-fixing sources. They constructed, for every $mu>0$, an efficiently computable extractor that extracts a positive fraction of entropy, i.e., $Omega(k)$ bits, from $(n,k)$-zero-fixing sources where $kgeq(loglog n)^{2+mu}$. In this paper we present two different constructions of extractors for zero-fixing sources that are able to extract a positive fraction of entropy for $k$ essentially smaller than $loglog n$. The first extractor works for $kgeq Clogloglog n$, for some constant $C$. The second extractor extracts a positive fraction of entropy for $kgeq log^{(i)}n$ for any fixed $iin mathbb{N}$, where $log^{(i)}$ denotes $i$-times iterated logarithm. The fraction of extracted entropy decreases with $i$. The first extractor is a function computable in polynomial time in~$n$ (for $epsilon=o(1)$, but not too small); the second one is computable in polynomial time when $kleqalphaloglog n/logloglog n$, where $alpha$ is a positive constant. The subject studied in this paper is closely related to Ramsey theory. We use methods developed in Ramsey theory and our results can also be interpreted as a contribution to this field." @default.
- W2938082011 created "2019-04-25" @default.
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- W2938082011 date "2019-04-16" @default.
- W2938082011 modified "2023-10-17" @default.
- W2938082011 title "Extractors for small zero-fixing sources" @default.
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