New Independent Source Extractors with Exponential Improvement

We study the problem of constructing extractors for independent weak random sources. The probabilistic method shows that such an extractor exists for two sources on n bits with min-entropy k >= 2 log n. On the other hand, explicit constructions are far from optimal. Previously the best known extractor for (n,k) sources requires O(log n/log k) independent sources [Rao06, Barak-Rao-Shaltiel-Wigderson06]. In this talk I will give a new extractor that uses only O(log (log n/log k))+O(1) independent sources. This improves the previous best result exponentially. The extractor is based on a new method for condensing somewhere random sources, which seems promising for further improvements.