Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors

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 -source if X assigns probability at most 2-n to any string of length n. For every > 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 -sources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2n . A corollary is that the function D is a 2-coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N by N is not monochromatic. 3-source extractor: E : ({0, 1}n )3 {0, 1} such that for any three independent -sources X1, X2, X3 we have that E(X1, X2, X3) is o(1)-close to being an unbiased random bit. No previous explicit construction was known for either of these for any < 1/2, and these results constitute significan...

• Type: Conference
• Year: 2005
• Where: STOC
• Authors: Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, Avi Wigderson
• Keywords: Algorithms, Combinatorics General Term, Long-standing Open Problems, STOC 2005, Unbiased Random Bit