David Zuckerman: Explicit Two-Source Extractors and Resilient Functions

Tuesday, April 12, 2016 - 4:00pm to 5:00pm
Light Refreshments at 3:50pm
Patil/Kiva G449
David Zuckerman

Abstract: 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 -

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.