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Homomorphic Secret Sharing from Lattices Without FHE

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11477))

Abstract

Homomorphic secret sharing (HSS) is an analog of somewhat- or fully homomorphic encryption (S/FHE) to the setting of secret sharing, with applications including succinct secure computation, private manipulation of remote databases, and more. While HSS can be viewed as a relaxation of S/FHE, the only constructions from lattice-based assumptions to date build atop specific forms of threshold or multi-key S/FHE. In this work, we present new techniques directly yielding efficient 2-party HSS for polynomial-size branching programs from a range of lattice-based encryption schemes, without S/FHE. More concretely, we avoid the costly key-switching and modulus-reduction steps used in S/FHE ciphertext multiplication, replacing them with a new distributed decryption procedure for performing “restricted” multiplications of an input with a partial computation value. Doing so requires new methods for handling the blowup of “noise” in ciphertexts in a distributed setting, and leverages several properties of lattice-based encryption schemes together with new tricks in share conversion.

The resulting schemes support a superpolynomial-size plaintext space and negligible correctness error, with share sizes comparable to SHE ciphertexts, but cost of homomorphic multiplication roughly one order of magnitude faster. Over certain rings, our HSS can further support some level of packed SIMD homomorphic operations. We demonstrate the practical efficiency of our schemes within two application settings, where we compare favorably with current best approaches: 2-server private database pattern-match queries, and secure 2-party computation of low-degree polynomials.

E. Boyle—Supported in part by ISF grant 1861/16, AFOSR Award FA9550-17-1-0069, and ERC grant 742754 (project NTSC).

L. Kohl—Supported by ERC Project PREP-CRYPTO (724307), by DFG grant HO 4534/2-2 and by a DAAD scholarship. This work was done in part while visiting the FACT Center at IDC Herzliya, Israel.

P. Scholl—Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 731583 (SODA), and the Danish Independent Research Council under Grant-ID DFF-6108-00169 (FoCC).

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Notes

  1. 1.

    Namely, solving the Discrete Logarithm in a Interval problem with interval length R in time \(o(\sqrt{R})\).

  2. 2.

    Although the cost of bootstrapping has fallen dramatically in recent years [16, 17, 23, 32], the efficiency is still orders of magnitude worse than low-depth somewhat homomorphic encryption using SIMD operations.

  3. 3.

    So-called “third generation” SHE schemes based on GSW [28] have simpler homomorphic multiplication, but much larger ciphertexts that grow with \(\varOmega (N \log ^2 q)\) instead of \(O(N \log q)\), for (R)LWE dimension N and modulus q.

  4. 4.

    Note that nearly linear decryption generically implies existence of a public-key encryption procedure.

  5. 5.

    This can be decreased to \((d-1)\) \(R_q\)-elements communicated, as \(s_1=1 \in R_q\).

  6. 6.

    To simplify the analysis, we restrict the definition to 2-power cyclotomic rings. However, our construction can be generalized to arbitrary cyclotomics.

  7. 7.

    Choosing a sparse secret like this does incur a small loss in security, and only gives us a small gain in parameters for the HSS. The main reason we choose s like this is to allow a fair comparison with SHE schemes, which typically have to use sparse secrets to obtain reasonable parameters.

  8. 8.

    Using S/FHE alone instead of HSS allows for the stronger setting of single-server PIR. However, a major advantage of HSS with additive reconstruction is that shares across many rows can easily be combined, allowing more expressive queries with simpler computation.

  9. 9.

    Actually, these works use function secret-sharing [8] for point functions, which in this case is equivalent to HSS for the same class of functions.

  10. 10.

    Other queries such as returning the record identifier, or min/max and range queries can easily be supported with similar techniques, as previously shown in [6, 40].

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Acknowledgements

We would like to thank the anonymous reviewers of Eurocrypt 2019 for their thorough and generous comments.

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Authors and Affiliations

  1. IDC Herzliya, Herzliya, Israel

    Elette Boyle

  2. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Lisa Kohl

  3. Aarhus University, Aarhus, Denmark

    Peter Scholl

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  1. Elette Boyle
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Correspondence to Elette Boyle .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Boyle, E., Kohl, L., Scholl, P. (2019). Homomorphic Secret Sharing from Lattices Without FHE. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11477. Springer, Cham. https://doi.org/10.1007/978-3-030-17656-3_1

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