TY - JOUR

T1 - An Efficient Reduction from Two-Source to Nonmalleable Extractors

T2 - Achieving Near-Logarithmic Min-Entropy

AU - Ben-Aroya, Avraham

AU - Doron, Dean

AU - Ta-Shma, Amnon

N1 - Funding Information:
\ast Received by the editors June 6, 2017; accepted for publication (in revised form) May 6, 2019; published electronically November 5, 2019. A preliminary version of this paper appeared in Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC'17), Montreal, Canada, 2017, pp. 1185--1194. This work was done while the second author was at Tel-Aviv University. https://doi.org/10.1137/17M1133245 Funding: This work was supported by Israel Science Foundation grant 994/14, and by United States-Israel Binational Science Foundation grant 2010120. \dagger The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel (avraham.benaroya@gmail.com, amnon@tau.ac.il). \ddagger Department of Computer Science, University of Texas at Austin, Austin, TX 78712 (deandoron@ utexas.edu).
Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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 2k 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.

AB - 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 2k 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.

KW - condensers

KW - nonmalleable extractors

KW - Ramsey graphs

KW - two-source extractors

UR - http://www.scopus.com/inward/record.url?scp=85129462999&partnerID=8YFLogxK

U2 - 10.1137/17M1133245

DO - 10.1137/17M1133245

M3 - Article

SN - 0097-5397

VL - 51

SP - 31

EP - 49

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 2

ER -