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Explicit Two-Source Extractors and Resilient Functions

Banff International Research Station for Mathematical Innovation and Discovery

We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least log^C n for a large enough constant C. Our extractor outputs one bit and has error n^{-\Omega(1)}. The best previous extractor, by Bourgain, required each source to have min-entropy .499n. A key ingredient in our construction is an explicit construction of a monotone, almost-balanced boolean function on n bits that is resilient to coalitions of size n^{1-delta}, for any delta>0. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown n-q bits are chosen almost polylog(n)-wise independently, and the remaining q=n^{1-\delta} bits are chosen by an adversary as an arbitrary function of the n-q bits. The best previous construction, by Viola, achieved q=n^{1/2 - \delta}. Our explicit two-source extractor directly implies an explicit construction of a 2^{(log log N)^{O(1)}}-Ramsey graph over N vertices, improving bounds obtained by Barak et al. and matching independent work by Cohen. Joint work with Eshan Chattopadhyay.

Mathematics; Computer science; Theoretical computer science

video/mp4

Author affiliation: University of Texas, Austin

2017-03-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60799

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/