Loss-less Condensers, Unbalanced Expanders, and Extractors

Abstract

An extractor is a procedure which extracts randomness from a defective 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 lossless condensers, which compress the n-bit input source without losing any minentropy, 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