AN EFFICIENT REDUCTION FROM TWO-SOURCE TO NONMALLEABLE EXTRACTORS: ACHIEVING NEAR-LOGARITHMIC MIN-ENTROPY

The breakthrough result of Chattopadhyay and Zuckerman [Explicit two-source extractors and resilient functions, in Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), ACM, 2016, pp. 670-683] gives a reduction from the construction of explicit two-source extractors to the construction of explicit nonmalleable extractors. However, even assuming the existence of optimal explicit nonmalleable extractors, we only obtain a two-source extractor for poly(log n) entropy, rather than the optimal O(log n). In this paper we modify the construction to solve the above barrier. Using the currently best explicit nonmalleable extractors, we get explicit bipartite Ramsey graphs for sets of size 2(k) for k = O(log n log log n/log log log n). Any further improvement in the construction of nonmalleable extractors would immediately yield a corresponding two-source extractor. Intuitively, Chattopadhyay and Zuckerman use an extractor as a sampler, and we observe that we could use a weaker object-a somewhere-random condenser with a small entropy gap and a very short seed. We also show how to explicitly construct this weaker object using the error reduction technique of Raz, Reingold, and Vadhan [Error reduction for extractors, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE, 1999, pp. 191-201], and the constant-degree dispersers of Zuckerman [Linear degree extractors and the inapproximability of max clique and chromatic number, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (STOC), ACM, 2006, pp. 681-690] that also work against extremely small tests.