Construction Of Condensers Research Articles

Simulating independence

We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. We say that a distribution X on binary strings of length n is a δ-source if X assigns probability at most 2 −δ n to any string of length n . For every δ>0, we construct the following poly( n )-time computable functions: 2-source disperser: D:({0, 1} n ) 2 → {0, 1} such that for any two independent δ-sources X 1 , X 2 we have that the support of D ( X 1 , X 2 ) is {0, 1}. Bipartite Ramsey graph: Let N =2 n . A corollary is that the function D is a 2-coloring of the edges of K N,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3-source extractor: E :({0, 1} n ) 3 → {0, 1} such that for any three independent δ-sources X 1 , X 2 , X 3 we have that E ( X 1 , X 2 , X 3 ) is o (1)-close to being an unbiased random bit. No previous explicit construction was known for either of these for any δ<1/2, and these results constitute significant progress to long-standing open problems. A component in these results is a new construction of condensers that may be of independent interest: This is a function C :{0, 1} n → ({0, 1} n/c ) d (where c and d are constants that depend only on δ) such that for every δ-source X one of the output blocks of C(X) is (exponentially close to) a 0.9-source. (This result was obtained independently by Ran Raz.) The constructions are quite involved and use as building blocks other new and known objects. A recurring theme in these constructions is that objects that were designed to work with independent inputs, sometimes perform well enough with correlated, high entropy inputs. The construction of the disperser is based on a new technique which we call “the challenge-response mechanism” that (in some sense) allows “identifying high entropy regions” in a given pair of sources using only one sample from the two sources.

Extractors for a Constant Number of Polynomially Small Min-Entropy Independent Sources

We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have min-entropy $n^\gamma$, for an arbitrarily small constant $\gamma>0$. Our extractor is obtained by composing seeded extractors in simple ways. We introduce a new technique to condense independent somewhere-random sources which looks like a useful way to manipulate independent sources. Our techniques are different from those used in recent work [B. Barak, R. Impagliazzo, and A. Wigderson, SIAM J. Comput., 36 (2006), pp. 1095–1118; B. Barak, G. Kindler, R. Shaltiel, B. Sudakov, and A. Wigderson, Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 1–10; R. Raz, Extractors with weak random seeds, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 11–20; J. Bourgain, Int. J. Number Theory, 1 (2005), pp. 1–32] for this problem in the sense that they do not rely on any results from arithmetic combinatorics. Using an extractor of Bourgain's [Int. J. Number Theory, 1 (2005), pp. 1–32] as a black box, we obtain a new extractor for two independent block sources with few blocks, even when the min-entropy is as small as $\operatorname{polylog}(n)$. We also show how to modify the 2 source disperser for linear min-entropy of Barak et al. [Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 1–10] and the three source extractor of Raz [Extractors with weak random seeds, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 11–20] to get dispersers/extractors with exponentially small error and linear output length where previously both were constant.