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Perfect MPC over Layered Graphs

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14081))

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Abstract

The classical “BGW protocol” (Ben-Or, Goldwasser, and Wigderson, STOC 1988) shows that secure multiparty computation (MPC) among n parties can be realized with perfect full security if \(t < n/3\) parties are corrupted. This holds against malicious adversaries in the “standard” model for MPC, where a fixed set of n parties is involved in the full execution of the protocol. However, the picture is less clear in the mobile adversary setting of Ostrovsky and Yung (PODC 1991), where the adversary may periodically “move” by uncorrupting parties and corrupting a new set of t parties. In this setting, it is unclear if full security can be achieved against an adversary that is maximally mobile, i.e., moves after every round. The question is further motivated by the “You Only Speak Once” (YOSO) setting of Gentry et al. (Crypto 2021), where not only the adversary is mobile but also each round is executed by a disjoint set of parties. Previous positive results in this model do not achieve perfect security, and either assume probabilistic corruption and a nonstandard communication model, or only realize the weaker goal of security-with-abort. The question of matching the BGW result in these settings remained open.

In this work, we tackle the above two challenges simultaneously. We consider a layered MPC model, a simplified variant of the fluid MPC model of Choudhuri et al. (Crypto 2021). Layered MPC is an instance of standard MPC where the interaction pattern is defined by a layered graph of width n, allowing each party to send secret messages and broadcast messages only to parties in the next layer. We require perfect security against a malicious adversary who may corrupt at most t parties in each layer. Our main result is a perfect, fully secure layered MPC protocol with an optimal corruption threshold of \(t < n/3\), thus extending the BGW feasibility result to the layered setting. This implies perfectly secure MPC protocols against a maximally mobile adversary.

This paper is a merged version of the papers [19, 20].

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Notes

  1. 1.

    A term from [30] for protocols where a new set of parties executes each round.

  2. 2.

    In CNF-based secret sharing, the secret is first split into \({n\atopwithdelims ()t}\) additive shares–a share \(r_T\) for each set \(T\subset [n]\) of size t–and party i receives all shares \(r_T\) such that \(i\not \in T\).

  3. 3.

    The inherent issue with state complexity originates from a common misconception (see fx [18]) that any general arithmetic circuit can be transformed into a layered circuit with same depth and only linear overhead in width.

  4. 4.

    In the coming sections our security analysis is with respect to non-adaptive adversaries for simplicity. In Sect. 2.2 we justify this leap appealing to the work of [10].

  5. 5.

    While we can meaningfully argue that the final protocol for computing general functionalities is UC-secure, we do not treat individual components of this protocol in a UC manner. This would require a significant modelling effort of communication and synchronization for layered MPC and would be counterproductive in our effort to present layered MPC as a simple special case of secure MPC as in [8, 31].

  6. 6.

    The instance of Future Messaging with honest sender in \(\mathcal {L}_0\) and honest receiver in \(\mathcal {L}_2\) is equivalent to perfect 1-way SMT.

  7. 7.

    Here, we refer to the primitive in the setting of layered MPC that ensures termination, validity and agreement among all parties located in some layer \(d>1\). Not Future Broadcast as defined in [28].

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Acknowledgement

We thank the anonymous reviewers for helpful comments. B. David was supported by the Independent Research Fund Denmark (IRFD) grants number 9040-00399B (TrA2C), 9131-00075B (PUMA) and 0165-00079B. A. Konring was supported by IRFD (TrA2C) and by the Otto Mønsted Foundation in a joint program with Innovation Center Denmark - Israel. Y. Ishai, E. Kushilevitz, and V. Narayanan were supported by ISF grant 2774/20 and BSF grant 2018393. Y. Ishai and V. Narayanan were also supported by ERC Project NTSC (742754).

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Authors and Affiliations

  1. IT University of Copenhagen, Copenhagen, Denmark

    Bernardo David & Anders Konring

  2. ETH Zurich, Zürich, Switzerland

    Giovanni Deligios

  3. NTT Research, Sunnyvale, USA

    Aarushi Goel & Chen-Da Liu-Zhang

  4. Technion - Israel Institute of Technology, Haifa, Israel

    Yuval Ishai, Eyal Kushilevitz & Varun Narayanan

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  1. Bernardo David
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Correspondence to Bernardo David .

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  1. Rambus Inc., San Jose, CA, USA

    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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David, B. et al. (2023). Perfect MPC over Layered Graphs. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14081. Springer, Cham. https://doi.org/10.1007/978-3-031-38557-5_12

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