Loss-less condensers, unbalanced expanders, and extractors

An extractor is a procedure which extracts randomness from a detective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important derandomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits required is suboptimal when the input source has low-min-entropy. Significant progress toward overcoming this bottleneck has been made, but so far has required complicated recursive techniques that lose the simplicity of Trevisan's construction. We give a clean method for overcoming this bottleneck by constructing {\em loss-less condensers}. which compress the n-bit input source without losing any min-entropy, using O(\log n) additional random bits. Our condensers are built using a simple modification of Trevisan's construction, and yield the best extractor constructions to date. Loss-less condensers also produce unbalanced bipartite expander graphs with small (polylogarithmic) degree D and very strong expansion of (1-\epilon)D. We give other applications of our construction, including dispersers with entropy loss O(\log n), depth two super-concentrators whose size is within a polylog of optimal, and an improved hardness of approximation result.