Extractors for small zero-fixing sources
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A random variable X is an (n,k)-zero-fixing source if for some subset V⊆[n], X is the uniform distribution on the strings {0,1}^n that are zero on every coordinate outside of V. An ϵ-extractor for (n,k)-zero-fixing sources is a mapping F:{0,1}^n→{0,1}^m, for some m, such that F(X) is ϵ-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 μ>0, an efficiently computable extractor that extracts a positive fraction of entropy, i.e., Ω(k) bits, from (n,k)-zero-fixing sources where k≥( n)^2+μ. 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 n. The first extractor works for k≥ C n, for some constant C. The second extractor extracts a positive fraction of entropy for k≥^(i)n for any fixed i∈N, where ^(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 ϵ=o(1), but not too small); the second one is computable in polynomial time when k≤α n/ n, where α is a positive constant.
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