Extracting randomness via repeated condensing

On an input probability distribution with some (min-)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). We construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of Nisan and Zuckerman (1996), the observation made by Srinivasan and Zuckerman (1994) and Nisan and Ta-Schma (1999) that a failure of the block extraction scheme is also useful, the recursive "win-win" case analysis of Impagliazzo et al. (1999, 2000), and the error correction of random sources used by Trevisan (1999). As a natural byproduct, (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 main two parameters-seed length and output length. Specifically, our extractors can make any of these two parameters optimal (up to a constant factor), only at a poly-logarithmic 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) to "strong " ones (which are good for mast seeds), with essentially the same parameters.