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Improved Computational Extractors and Their Applications

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Advances in Cryptology – CRYPTO 2021 (CRYPTO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12827))

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Abstract

Recent exciting breakthroughs have achieved the first two-source extractors that operate in the low min-entropy regime. Unfortunately, these constructions suffer from non-negligible error, and reducing the error to negligible remains an important open problem. In recent work, Garg, Kalai, and Khurana (GKK, Eurocrypt 2020) investigated a meaningful relaxation of this problem to the computational setting, in the presence of a common random string (CRS). In this relaxed model, their work built explicit two-source extractors for a restricted class of unbalanced sources with min-entropy \(n^{\gamma }\) (for some constant \(\gamma \)) and negligible error, under the sub-exponential DDH assumption.

In this work, we investigate whether computational extractors in the CRS model be applied to more challenging environments. Specifically, we study network extractor protocols (Kalai et al., FOCS 2008) and extractors for adversarial sources (Chattopadhyay et al., STOC 2020) in the CRS model. We observe that these settings require extractors that work well for balanced sources, making the GKK results inapplicable.

We remedy this situation by obtaining the following results, all of which are in the CRS model and assume the sub-exponential hardness of DDH.

  • We obtain “optimal” computational two-source and non-malleable extractors for balanced sources: requiring both sources to have only poly-logarithmic min-entropy, and achieving negligible error. To obtain this result, we perform a tighter and arguably simpler analysis of the GKK extractor.

  • We obtain a single-round network extractor protocol for poly-logarithmic min-entropy sources that tolerates an optimal number of adversarial corruptions. Prior work in the information-theoretic setting required sources with high min-entropy rates, and in the computational setting had round complexity that grew with the number of parties, required sources with linear min-entropy, and relied on exponential hardness (albeit without a CRS).

  • We obtain an “optimal” adversarial source extractor for poly-logarithmic min-entropy sources, where the number of honest sources is only 2 and each corrupted source can depend on either one of the honest sources. Prior work in the information-theoretic setting had to assume a large number of honest sources.

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Notes

  1. 1.

    The quality of an independent source extractor is determined by three parameters, (i) the number of independent sources, (ii) the min-entropy of these sources, and (iii) the error which is the statistical distance between the output of the extractor and the uniform distribution.

  2. 2.

    In a work that is concurrent and independent to Chattopadhyay et al., Aggarwal et al. [AOR+20b] studied another model of adversarial sources called as SHELA sources. They showed that it is impossible to extract uniform random bits from SHELA sources and gave constructions of extractors whose output is somewhere random. In another work, Dodis et al. [DVW20] studied a notion of extractor dependent sources which arise in the setting where the source sampler could depend on the output of the previous invocations of the extractor using the same seed.

  3. 3.

    There are many other subtleties involved, most importantly, a circularity: the CRS must be programmed according to h(y), but y is sampled as a function of the CRS. The work of [GKK20] develops techniques to avoid these subtleties, but we do not discuss them here as they are less relevant to the current approach.

  4. 4.

    The reason why two-source extractor is sufficient in this case but non-malleable extractor was needed in the previous case is that the parties here can be thought of as following the protocol whereas in the previous case, they could deviate arbitrarily from the protocol specification.

  5. 5.

    This is often denoted by \(\widetilde{H}_\infty (X|Y)\) in the literature.

  6. 6.

    This condition follows from the way X and Y are sampled, and like [GKK20], we add it only for the sake of being explicit.

  7. 7.

    This condition follows from the way X and Y are sampled, and we add it only for the sake of being explicit.

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Author information

Authors and Affiliations

  1. University of Illinois Urbana-Champaign, Champaign, USA

    Dakshita Khurana

  2. Tata Institute of Fundamental Research, Bengaluru, India

    Akshayaram Srinivasan

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  1. Dakshita Khurana
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Editors and Affiliations

  1. Columbia University, New York City, NY, USA

    Tal Malkin

  2. University of Michigan, Ann Arbor, MI, USA

    Chris Peikert

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Khurana, D., Srinivasan, A. (2021). Improved Computational Extractors and Their Applications. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12827. Springer, Cham. https://doi.org/10.1007/978-3-030-84252-9_19

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  • DOI: https://doi.org/10.1007/978-3-030-84252-9_19

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