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On Homomorphic Secret Sharing from Polynomial-Modulus LWE

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

Homomorphic secret sharing (HSS) is a form of secret sharing that supports the local evaluation of functions on the shares, with applications to multi-server private information retrieval, secure computation, and more.

Insisting on additive reconstruction, all known instantiations of HSS from “Learning with Error (LWE)”-type assumptions either have to rely on LWE with superpolynomial modulus, come with non-negligible error probability, and/or have to perform expensive ciphertext multiplications, resulting in bad concrete efficiency.

In this work, we present a new 2-party local share conversion procedure, which allows to locally convert noise encoded shares to non-noise plaintext shares such that the parties can detect whenever a (potential) error occurs and in that case resort to an alternative conversion procedure.

Building on this technique, we present the first HSS for branching programs from (Ring-)LWE with polynomial input share size which can make use of the efficient multiplication procedure of Boyle et al. (Eurocrypt 2019) and has no correctness error. Our construction comes at the cost of a – on expectation – slightly increased output share size (which is insignificant compared to the input share size) and a more involved reconstruction procedure.

More concretely, we show that in the setting of 2-server private information retrieval we can choose ciphertext sizes of only a quarter of the size of the scheme of Boyle et al. at essentially no extra cost.

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Notes

  1. 1.

    Here, we consider \(\mathbb {Z}_q\) to be represented as integers in the interval \(\left( -\frac{q}{2},\frac{q}{2}\right] \). For \(y\in \left\{ -\frac{q}{4},\frac{q}{4}\right\} \), by \(\left[ y\pm |e|\right] \) we denote the interval containing all \(z\in \mathbb {Z}_q\) having at most distance |e| from y (considered as integer).

  2. 2.

    We assume that for every instruction \((\textsf{add},\textsf{id},u,v,w)\) such that u (resp. v) is the output wire of a previous instruction with id \(\textsf{id}_u\) (resp. \(\textsf{id}_v\)) we have \(\textsf{id}_u<\textsf{id}_v\). This ensures that shares corresponding to u are computed before shares corresponding to v in our evaluation algorithm.

  3. 3.

    We assume here that \(\beta \) divides q, so that shares mod q are also shares mod \(\beta \). If we wish to avoid this assumption, we can simply perform a lifting step to obtain shares over \(\mathbb {Z}\) before reducing them mod \(\beta \).

  4. 4.

    Here we again consider the case \(\mathcal {R}=\mathbb {Z}\) for simplicity. For \(\mathcal {R}\) of dimension N, the equation applies to each coordinate of y.

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Acknowledgments

Thomas Attema was supported by the Vraaggestuurd Programma Cyber Security & Resilience, part of the Dutch Top Sector High Tech Systems and Materials program. Pedro Capitão and Lisa Kohl have been supported by the NWO Gravitation project QSC.

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Authors and Affiliations

  1. Cryptology Group, CWI, Amsterdam, The Netherlands

    Thomas Attema, Pedro Capitão & Lisa Kohl

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands

    Thomas Attema & Pedro Capitão

  3. Cyber Security and Robustness, TNO, The Hague, The Netherlands

    Thomas Attema

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Correspondence to Pedro Capitão .

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

    Alexandra Boldyreva

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

    Vladimir Kolesnikov

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Attema, T., Capitão, P., Kohl, L. (2023). On Homomorphic Secret Sharing from Polynomial-Modulus LWE. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_1

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