A reduction from efficient non-malleable extractors to low-error two-source extractors with arbitrary constant rate

We show a reduction from the existence of explicit t-non-malleable extractors with a small seed length, to the construction of explicit two-source extractors with small error for sources with arbitrarily small constant rate. Previously, such a reduction was known either when one source had entropy rate above half [Raz05] or for general entropy rates but only for large error [CZ16]. As in previous reductions we start with the Chattopadhyay and Zuckerman approach [CZ16], where an extractor is applied on one source to create a table, and the second source is used to sample a sub-table. In previous work, a resilient function was applied on the sub-table and the use of resilient functions immediately implied large error. In this work we replace the resilient function with the parity function (that is not resilient). We prove correctness by showing that doing the sampling properly, the sample size can be made so small that it is smaller then the non-malleability parameter t of the first extractor, which is enough for the correctness. The parameters we require from the non-malleable construction hold (quite comfortably) in a non-explicit construction, but currently it is not known how to explicitly construct such graphs. As a result we do not give an unconditional construction of an explicit low-error twosource extractor. However, the reduction shows a new connection between non-malleable and two-source extractors, and also shows resilient functions do not play a crucial role in the twosource construction framework suggested in [CZ16]. Furthermore, the reduction highlights a barrier in constructing non-malleable extractors, and reveals its importance. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors. ∗The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel. Supported by the Israel science Foundation grant no. 994/14 and by the United States – Israel Binational Science Foundation grant no. 2010120. †Simons Institute for the Theory of Computing, UC Berkeley, Berkeley, CA 94720, USA. Email: eshanc@ias.edu. Part of this work was done when the author was a graduate student in UT Austin under NSF grant CCF-1526952 ‡The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel. Email: deandoron@mail.tau.ac.il. Supported by the Israel science Foundation grant no. 994/14 and by the United States – Israel Binational Science Foundation grant no. 2010120. This work was done in part while visiting the Simons Institute for the Theory of Computing at UC Berkeley. §Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218, USA. Email: lixints@cs.jhu.edu. Supported by NSF Grant CCF-1617713. ¶The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel. Email: amnon@tau.ac.il. Supported by the Israel science Foundation grant no. 994/14 and by the United States – Israel Binational Science Foundation grant no. 2010120. This work was done in part while visiting the Simons Institute for the Theory of Computing at UC Berkeley. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 27 (2017)

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