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A Note on the Communication Complexity of Multiparty Computation in the Correlated Randomness Model

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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

Secure multiparty computation (\(\mathsf {MPC}\)) addresses the challenge of evaluating functions on secret inputs without compromising their privacy. A central question in multiparty computation is to understand the amount of communication needed to securely evaluate a circuit of size s. In this work, we revisit this fundamental question in the setting of information-theoretically secure \(\mathsf {MPC}\) in the correlated randomness model, where a trusted dealer distributes correlated random coins, independent of the inputs, to all parties before the start of the protocol. This setting is of strong theoretical interest, and has led to the most practically efficient \(\mathsf {MPC}\) protocols known to date.

While it is known that protocols with optimal communication (proportional to input plus output size) can be obtained from the LWE assumption, and that protocols with sublinear communication o(s) can be obtained from the DDH assumption, the question of constructing protocols with o(s) communication remains wide open for the important case of information-theoretic \(\mathsf {MPC}\) in the correlated randomness model; all known protocols in this model require O(s) communication in the online phase.

In this work, we exhibit the first generic multiparty computation protocol in the correlated randomness model with communication sublinear in the circuit size, for a large class of circuits. More precisely, we show the following: any size-s layered circuit (whose nodes can be partitioned into layers so that any edge connects adjacent layers) can be evaluated with \(O(s/\log \log s)\) communication. Our results holds for both boolean and arithmetic circuits, in the honest-but-curious setting, and do not assume honest majority. For boolean circuits, we extend our results to handle malicious corruption.

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Notes

  1. 1.

    The work of [BGI16] considered, as we will do in this work, boolean circuits which can be divided into layers such as any edge connects adjacent layers. Such circuits are called layered boolean circuits.

  2. 2.

    We assume \(w\cdot d = O(s)\) in this high level explanation for simplicity only, this is not a necessary condition in the actual construction.

  3. 3.

    More precisely, the protocol needs to assume the circular security of an LWE-based encryption scheme; alternatively, it can be based on the LWE assumption only, but the communication will grow with the depth of the circuit.

  4. 4.

    A technique to amortize this overhead, using a linear MAC scheme, is described in [DNNR17]; it applies to our setting as well, and allows to remove this factor \(\kappa \) overhead in the storage complexity, but we focus on the more naive approach in this work for simplicity.

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Acknowledgements

We thank Yuval Ishai for helpful comments and pointers. Work supported by ERC grant 724307 (project PREP-CRYPTO).

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  1. KIT, Karlsruhe, Germany

    Geoffroy Couteau

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  1. Geoffroy Couteau
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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Couteau, G. (2019). A Note on the Communication Complexity of Multiparty Computation in the Correlated Randomness Model. 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_17

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