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Two-Round MPC Without Round Collapsing Revisited – Towards Efficient Malicious Protocols

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

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Abstract

Recent works have made exciting progress on the construction of round optimal, two-round, Multi-Party Computation (MPC) protocols. However, most proposals so far are still complex and inefficient. In this work, we improve the simplicity and efficiency of two-round MPC in the setting with dishonest majority and malicious security. Our protocols make use of the Random Oracle (\({\textsf{RO}}\)) and a generalization of the Oblivious Linear Evaluation (\(\textsf{OLE}\)) correlated randomness, called tensor \(\textsf{OLE}\), over a finite field \(\mathbb {F}\), and achieve the following:

  • MPC for Boolean Circuits: Our two-round, maliciously secure MPC protocols for computing Boolean circuits, has overall (asymptotic) computational cost \(O(S\cdot n^3 \cdot \log |\mathbb {F}|)\), where S is the size of the circuit computed, n the number of parties, and \(\mathbb {F}\) a field of characteristic two. The protocols also make black-box calls to a Pseudo-Random Function (PRF).

  • MPC for Arithmetic Branching Programs (ABPs): Our two-round, information theoretically and maliciously secure protocols for computing ABPs over a general field \(\mathbb {F}\) has overall computational cost \(O(S^{1.5}\cdot n^3\cdot \log |\mathbb {F}|)\), where S is the size of ABP computed.

Both protocols achieve security levels inverse proportional to the size of the field \(|\mathbb {F}|\).

Our construction is built upon the simple two-round MPC protocols of [Lin-Liu-Wee TCC’20], which are only semi-honest secure. Our main technical contribution lies in ensuring malicious security using simple and lightweight checks, which incur only a constant overhead over the complexity of the protocols by Lin, Liu, and Wee. In particular, in the case of computing Boolean circuits, our malicious MPC protocols have the same complexity (up to a constant overhead) as (insecurely) computing Yao’s garbled circuits in a distributed fashion.

Finally, as an additional contribution, we show how to efficiently generate tensor \(\textsf{OLE}\) correlation in fields of characteristic two using OT.

The work was partially done when Liu was a postdoctoral researcher at University of Washington.

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Notes

  1. 1.

    These are protocols secure against corrupted parties who follow the protocol specification but may choose its input and randomness arbitrarily.

  2. 2.

    When the field is \(\textsf{GF}(2)\), \(\textsf{OLE}\) correlation coincides with the \(\textsf{OT}\) correlation.

  3. 3.

    An equivalent definition of semi-honest MPRE can be found in [ABT18], in which it is just called “MPRE”. In [ABT19], malicious MPRE is called “non-interactive reduction” and the canonical protocol of a MPRE is called “\(\hat{f}\)-oracle-aided protocol”. Both [ABT18] and [ABT19] consider the honest majority setting, so they only require the canonical protocol to be secure against a bounded number of corruptions.

  4. 4.

    Section 2.2 outlines how to canonicalize \(\hat{f}\). Formally, canonical form allows some coordinates to be linear instead of \({\textsf {2MultPlus}}\). The linear coordinates are easier to handle. We ignore them in the overview.

  5. 5.

    Note the transpose of \(\textbf{B}_2\). This makes the equation remains unchanged upon exchanging subscripts.

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Acknowledgement

We thank Antigoni Polychroniadou for being our shepherd and her help, and the anonymous Crypto reviewers for their helpful comments and suggestions. We thank Hoeteck Wee for being part of the initial discussion and his suggestion of directions. Finally, we thank Stefano Tessaro for his comments.

Huijia Lin and Tianren Liu were supported by NSF grants CNS-1936825 (CAREER), CNS-2026774, a JP Morgan AI research Award, a Cisco research award, and a Simons Collaboration on the Theory of Algorithmic Fairness. In addition, Tianren Liu was supported by NSFC excellent young scientists fund program.

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  1. University of Washington, Seattle, USA

    Huijia Lin

  2. CFCS, Peking University, Beijing, China

    Tianren Liu

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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Lin, H., Liu, T. (2022). Two-Round MPC Without Round Collapsing Revisited – Towards Efficient Malicious Protocols. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13507. Springer, Cham. https://doi.org/10.1007/978-3-031-15802-5_13

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