We study the problem of constructing explicit extractors for independent general weak random sources. Given weak sources on n bits, the probabilistic method shows that there exists a deterministic extractor for two independent sources with min-entropy as small as log n+O(1). However, even to extract from a constant number of independent sources, previously the best known extractors require the min-entropy to be at least nδ for any constant δ>0 Rao06, BarakRSW06, Li12d. For sources on n bits with min-entropy k ≥ q polylog(n), previously the best known extractor needs to use O(log(log n/log k))+O(1) independent sources Li12d. In this paper, we construct the first explicit extractor for a constant number of independent sources on n bits with min-entropy k ≥ q polylog(n). Thus, for the first time we get extractors for independent sources that are close to optimal. Our extractor is obtained by improving the condenser for structured somewhere random sources in Li12d, which is based on a connection between the problem of condensing somewhere random sources and the problem of leader election in distributed computing.
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机译:我们研究为独立的一般弱随机源构造显式提取器的问题。给定n位上的弱信号源,概率方法表明存在两个最小熵最小为log n + O(1)的独立信号源的确定性提取器。然而,即使是从恒定数量的独立来源中提取,对于任何恒定δ> 0 Rao06,BarakRSW06,Li12d,以前最著名的提取器都要求最小熵至少为nδ。对于最小熵k≥q polylog(n)的n位源,以前最著名的提取器需要使用O(log(log n / log k))+ O(1)独立源Li12d。在本文中,我们构造了第一个显式提取器,用于在最小熵k≥q polylog(n)的n位上为恒定数量的独立源。因此,我们首次获得了接近最优的独立来源提取器。我们的提取器是通过改进Li12d中某处结构化随机源的冷凝器而获得的,该冷凝器基于将某处随机源浓缩的问题与分布式计算中的领导者选举问题之间的联系。
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