New independent source extractors with exponential improvement

Abstract

We study the problem of constructing explicit extractors for independent general weak random sources. For weak sources on n bits with min-entropy k, perviously the best known extractor needs to use at least log n/log k independent sources [22, 3]. In this paper we give a new extractor that only uses O(log(log n/log k))+O(1) independent sources. Thus, our result improves the previous best result exponentially. We then use our new extractor to give improved network extractor protocols, as defined in [14]. The network extractor protocols also give new results in distributed computing with general weak random sources, which dramatically improve previous results. For example, we can tolerate a nearly optimal fraction of faulty players in synchronous Byzantine agreement and leader election, even if the players only have access to independent n-bit weak random sources with min-entropy as small as k=polylog(n). Our extractor for independent sources is based on a new condenser for somewhere random sources with a special structure. We believe our techniques are interesting in their own right and are promising for further improvement.