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Turbospeedz: Double Your Online SPDZ! Improving SPDZ Using Function Dependent Preprocessing

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Applied Cryptography and Network Security (ACNS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11464))

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Abstract

Secure multiparty computation allows a set of mutually distrusting parties to securely compute a function of their private inputs, revealing only the output, even if some of the parties are corrupt. Recent years have seen an enormous amount of work that drastically improved the concrete efficiency of secure multiparty computation protocols. Many secure multiparty protocols work in an “offline-online” model. In this model, the computation is split into two main phases: a relatively slow “offline phase”, which the parties execute before they know their input, and a fast “online phase”, which the parties execute after receiving their input.

One of the most popular and efficient protocols for secure multiparty computation working in this model is the SPDZ protocol (Damgård et al., CRYPTO 2012). The SPDZ offline phase is function independent, i.e., does not require knowledge of the computed function at the offline phase. Thus, a natural question is: can the efficiency of the SPDZ protocol be improved if the function is known at the offline phase?

In this work, we answer the above question affirmatively. We show that by using a function dependent preprocessing protocol, the online communication of the SPDZ protocol can be brought down significantly, almost by a factor of 2, and the online computation is often also significantly reduced. In scenarios where communication is the bottleneck, such as strong computers on low bandwidth networks, this could potentially almost double the online throughput of the SPDZ protocol, when securely computing the same circuit many times in parallel (on different inputs).

We present two versions of our protocol: Our first version uses the SPDZ offline phase protocol as a black-box, which achieves the improved online communication at the cost of slightly increasing the offline communication. Our second version works by modifying the state-of-the-art SPDZ preprocessing protocol, Overdrive (Keller et al., Eurocrypt 2018). This version improves the overall communication over the state-of-the-art SPDZ.

A. Ben-Efraim and E. Omri—Research supported by ISF grant 152/17 and the Ariel Cyber Innovation Center.

M. Nielsen—Partially supported by the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No. 669255 (MPCPRO).

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Notes

  1. 1.

    Additional online communication includes squaring gates and communication in the MACCheck protocol, where we do not improve over [23]. However, this communication is relatively small, especially in large circuits. Therefore, our online communication is only slightly more than half the online communication of [23].

  2. 2.

    We note that our “aligning” method works even better with the SPDZ preprocessing of [23], but the overall improvement would still probably not surpass using Overdrive. In contrast, due to a randomization technique used in MASCOT [31] triple generation, it is not clear if this “alignment” can also be applied to MASCOT preprocessing.

  3. 3.

    To be more precise, these protocols perform best over small characteristic fields. However, they can be somewhat efficiently extended to arithmetic computations over the integers using the Chinese Remainder Theorem, e.g., [5, 10], and to extension fields with small characteristic using multiplication embedding.

  4. 4.

    An input masking \((r_i,[[ r_i ]])\) is a random \([[ \cdot ]]\)-shared element, where the value \(r_i\) is known to party i.

  5. 5.

    In [23, 24] they do not distinguish between the wire and its value – there \(v_{\omega }\) and \([[ v_{\omega } ]]\) are denoted \(\omega \) and \([[ \omega ]]\), respectively. Our notation is similar to notations used for multiparty garbled circuits, e.g., [9, 11].

  6. 6.

    It might be tempting to naïvely set \(\lambda _{z}=c\), but this would not be secure, because \(\lambda _{z}\) must be independently random. However, in Sect. 4 we show that by modifying Overdrive, this part can be optimized.

  7. 7.

    Note that due to the asymmetry in the multiplication, this is not possible if the value plays b in the other multiplication.

  8. 8.

    Of course a full proof would also require including the details of the zero-knowledge proofs, noise drowning, etc., as done in [33]. But these are beyond the scope of this paper and therefore left to the full version.

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Acknowledgements

We would like to thank Amos Beimel for helpful discussions. Special thanks to Ivan Damgård and Marcel Keller for helping us to understand SPDZ and Overdrive better.

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Authors and Affiliations

  1. Department of Computer Science, Ariel University, Ariel, Israel

    Aner Ben-Efraim & Eran Omri

  2. Uber, Aarhus, Denmark

    Michael Nielsen

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Correspondence to Eran Omri .

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    Robert H. Deng

  2. Universidad del Rosario, Bogota, Colombia

    Valérie Gauthier-Umaña

  3. Cyxtera Technologies, Bogota, Colombia

    Martín Ochoa

  4. Columbia University, New York, NY, USA

    Moti Yung

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Ben-Efraim, A., Nielsen, M., Omri, E. (2019). Turbospeedz: Double Your Online SPDZ! Improving SPDZ Using Function Dependent Preprocessing. In: Deng, R., Gauthier-Umaña, V., Ochoa, M., Yung, M. (eds) Applied Cryptography and Network Security. ACNS 2019. Lecture Notes in Computer Science(), vol 11464. Springer, Cham. https://doi.org/10.1007/978-3-030-21568-2_26

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