Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes
Journal of the ACM (JACM)
Variations on Muchnik's Conditional Complexity Theorem
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Simulating independence: New constructions of condensers, ramsey graphs, dispersers, and extractors
Journal of the ACM (JACM)
Simpler constant-seed condensers
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Kakeya Sets, New Mergers, and Old Extractors
SIAM Journal on Computing
Randomness condensers for efficiently samplable, seed-dependent sources
TCC'12 Proceedings of the 9th international conference on Theory of Cryptography
TCC'13 Proceedings of the 10th theory of cryptography conference on Theory of Cryptography
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Extractors (as defined by Nisan and Zuckerman) are procedures that use a small number of truly random bits (called the seed) to extract many (almost) truly random bits from arbitrary distributions as long as distributions have sufficient (min)-entropy. A natural weakening of an extractor is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). An extractor can be viewed as an ultimate condenser because it outputs a distribution with the maximal entropy rate.In this paper we construct explicit condensers with short seed length. The condenser constructions combine (variants of or more efficient versions of) ideas from several works, including the block extraction scheme of [N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43-52], the observation made in [A. Srinivasan and D. Zuckerman, SIAM J. Comput., 28 (1999), pp. 1433-1459; N. Nisan and A. Ta-Shma, J. Comput. System Sci., 58 (1999), pp. 148-173] that a failure of the block extraction scheme is also useful, the recursive "win-win" case analysis of [R. Impagliazzo, R. Shaltiel, and A. Wigderson, Near-optimal conversion of hardness into pseudo-randomness, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1999, pp. 181-190; R. Impagliazzo, R. Shaltiel, and A. Wigderson, Extractors and pseudo-random generators with optimal seed length, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, ACM, New York, 2000, pp. 1-10], and the error correction of random sources used in [L. Trevisan, J. ACM, 48 (2001), pp. 860-879]. As a by-product (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary min-entropy; they are nearly optimal simultaneously in the two main parameters of seed length and output length. Specifically, our extractors can make any one of these two parameters optimal (up to a constant factor) only at a polylogarithmic loss in the other. Previous constructions require polynomial loss in both cases for general sources.We also give a simple reduction converting "standard" extractors (which are good for an average seed) into "strong" ones (which are good for most seeds), with essentially the same parameters. With this reduction, all the above improvements apply to strong extractors as well.