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Secret-Sharing Schemes for General and Uniform Access Structures

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11478))

Abstract

A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size \(2^{n-o(n)}\) and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to \(O(2^{0.994n})\). Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).

Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:

  • A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is \(O(k^2)\) times the size of the secret.

  • A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is \(\tilde{O}(2^{h(k/n)n/2})\) (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).

  • A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is \(2^{\tilde{O}(\sqrt{k \log n})}\).

Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.

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Notes

  1. 1.

    Formally, such a statement implicitly refers to an infinite sequence of (collections of) access structures that is parameterized by the number of participants n.

  2. 2.

    The notation \(\mathbf {M}\) stands for “middle”.

  3. 3.

    The notation \(\mathbf {X}\) stands for eXternal slices.

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Acknowledgement

The first and fourth authors are supported by the European Union’s Horizon 2020 Programme (ERC-StG-2014-2020) under grant agreement no. 639813 ERC-CLC, and the Check Point Institute for Information Security. Part of this work was done while the second author was visiting Georgetown university, supported by NSF grant no. 1565387, TWC: Large: Collaborative: Computing Over Distributed Sensitive Data. The second author is also supported by ISF grant 152/17 and by a grant from the Cyber Security Research Center at Ben-Gurion University of the Negev. The third author is supported by the Spanish Government through TIN2014-57364-C2-1-R and by the Government of Catalonia through Grant 2017 SGR 705. The fifth author is supported by ISF grant 152/17, by a grant from the Cyber Security Research Center at Ben-Gurion University of the Negev, and by the Frankel center for computer science.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Benny Applebaum & Oded Nir

  2. Ben-Gurion University of the Negev, Be’er-Sheva, Israel

    Amos Beimel & Naty Peter

  3. Universitat Rovira i Virgili, Tarragona, Catalonia, Spain

    Oriol Farràs

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  1. Benny Applebaum
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Correspondence to Benny Applebaum .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Applebaum, B., Beimel, A., Farràs, O., Nir, O., Peter, N. (2019). Secret-Sharing Schemes for General and Uniform Access Structures. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_15

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