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Annual International Cryptology Conference

CRYPTO 2019: Advances in Cryptology – CRYPTO 2019 pp 480–509Cite as

Leakage Resilient Secret Sharing and Applications

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11693))

Abstract

A secret sharing scheme allows a dealer to share a secret among a set of n parties such that any authorized subset of the parties can recover the secret, while any unauthorized subset learns no information about the secret. A leakage-resilient secret sharing scheme (introduced in independent works by Goyal and Kumar, STOC ’18 and Benhamouda, Degwekar, Ishai and Rabin, CRYPTO ’18) additionally requires the secrecy to hold against every unauthorized set of parties even if they obtain some bounded leakage from every other share. The leakage is said to be local if it is computed independently for each share. So far, the only known constructions of local leakage resilient secret sharing schemes are for threshold access structures for very low (O(1)) or very high (\(n -o(\log n)\)) thresholds.

In this work, we give a compiler that takes a secret sharing scheme for any monotone access structure and produces a local leakage resilient secret sharing scheme for the same access structure, with only a constant-factor asymptotic blow-up in the sizes of the shares. Furthermore, the resultant secret sharing scheme has optimal leakage-resilience rate, i.e., the ratio between the leakage tolerated and the size of each share can be made arbitrarily close to 1. Using this secret sharing scheme as the main building block, we obtain the following results:

  • Rate Preserving Non-Malleable Secret Sharing. We give a compiler that takes any secret sharing scheme for a 4-monotone access structure (A 4-monotone access structure has the property that any authorized set has size at least 4.) with rate R and converts it into a non-malleable secret sharing scheme for the same access structure with rate \(\varOmega (R)\). The previous such non-zero rate construction (Badrinarayanan and Srinivasan, EUROCRYPT ’19) achieved a rate of \(\varTheta (R/{t_{\max }\log ^2 n})\), where \(t_{\max }\) is the maximum size of any minimal set in the access structure. As a special case, for any threshold \(t \ge 4\) and an arbitrary \(n \ge t\), we get the first constant-rate construction of t-out-of-n non-malleable secret sharing.

  • Leakage-Tolerant Multiparty Computation for General Interaction Patterns. For any function f, we give a reduction from constructing a leakage-tolerant secure multi-party computation protocol for computing f that obeys any given interaction pattern to constructing a secure (but not necessarily leakage-tolerant) protocol for a related function that obeys the star interaction pattern. Together with the known results for the star interaction pattern, this gives leakage tolerant MPC for any interaction pattern with statistical/computational security. This improves upon the result of (Halevi et al., ITCS 2016), who presented such a reduction in a leak-free environment.

Research supported in part from DARPA/ARL SAFEWARE Award W911NF15C0210, AFOSR Award FA9550-15-1-0274, AFOSR YIP Award, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for LongTerm Cybersecurity (CLTC, UC Berkeley).

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Notes

  1. 1.

    The access structure of a secret sharing scheme is what we call the set of authorized subsets of parties.

  2. 2.

    As the extractor is a strong seeded extractor, Ext\((s,w_i)\) is statistically close to uniform even given the seed.

  3. 3.

    For the security of this modification to go through, we need the adversary to specify all the secrets and leakage functions upfront – it cannot adaptively choose the secrets and leakage functions depending on the previous leakage.

  4. 4.

    k-monotone means that all authorized sets in the access structure are of size at least k.

  5. 5.

    The case of multiple output parties reduces to the case of single output party by considering each output party computing a specific function of the other parties input.

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Authors and Affiliations

  1. University of California, Berkeley, USA

    Akshayaram Srinivasan & Prashant Nalini Vasudevan

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  1. Akshayaram Srinivasan
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  2. Prashant Nalini Vasudevan
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Correspondence to Akshayaram Srinivasan or Prashant Nalini Vasudevan .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Srinivasan, A., Vasudevan, P.N. (2019). Leakage Resilient Secret Sharing and Applications. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_17

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