How to get more mileage from randomness extractors

• Authors: Ronen Shaltiel
• Venue: Random Structures & Algorithms
• Year: 2008

Let 𝒞 be a class of distributions over Bn. A deterministic randomness extractor for 𝒞 is a function E : BnarBm such that for any X in 𝒞 the distribution E(X) is statistically close to the uniform distribution. A long line of research deals with explicit constructions of such extractors for various classes 𝒞 while trying to maximize m. In this paper we give a general transformation that transforms a deterministic extractor E which extracts “few” bits into an extractor E′ that extracts “almost all the bits present in the source distribution.” More precisely, we prove a general theorem saying that if E and 𝒞 satisfy certain properties, then we can transform E into an extractor E′. Our methods build on (and generalize) a technique of Gabizon et al. [FOCS (2004) 394–403] that presents such a transformation for the very restricted class 𝒞 of “oblivious bit-fixing sources.” The high level idea is to find properties of E and 𝒞 which allow “recycling” the output of E so that it can be “reused” to operate on the source distribution. An obvious obstacle is that the output of E is correlated with the source distribution. Using our transformation we give an explicit construction of a two-source extractor E : Bn × BnarBm such that for every two independent distributions X1 and X2 over Bn with min-entropy at least k = (1-2 + δ)n and eps ≤ 2- log 4n, E(X1,X2) is eps-close to the uniform distribution on m = 2k - Cδ log(1-eps) bits. This result is optimal except for the precise constant Cδ and improves previous results by Chor and Goldreich [SICOMP 17 (1988) 230–261], Vazirani [Combinatorica 7 (1987) 375–392], and Dodis et al. [RANDOM (2004) 334–344]. We also give explicit constructions of extractors for samplable distributions that extract many bits even out of “low-entropy” samplable distributions. These constructions rely on average case hardness assumptions and extend some previous results by Trevisan and Vadhan [FOCS (2000) 32–42] which give such extractors only for distributions with “high entropy.” © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008