A New Approach for Constructing Low-Error, Two-Source Extractors

Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework [CZ16], and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient in [CZ16] and follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck. The parameters we require from t-non-malleable extractors for our reduction to work hold (quite comfortably) in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. 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. †Cornell University and IAS. Email: [email protected]. Supported by NSF grants CCF-1526952, CCF-1412958 and the Simons Foundation. Part of this work was done when the author was a graduate student in UT Austin and while visiting the Simons Institute for the Theory of Computing at UC Berkeley. ‡The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel. Email: [email protected]. 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: [email protected]. Supported by NSF Grant CCF-1617713. ¶The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel. Email: [email protected]. 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.