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Ramp Hyper-invertible Matrices and Their Applications to MPC Protocols

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14438))

Abstract

Beerliová-Trubíniová and Hirt introduced hyper-invertible matrix technique to construct the first perfectly secure MPC protocol in the presence of maximal malicious corruptions \(\lfloor \frac{n-1}{3} \rfloor \) with linear communication complexity per multiplication gate [5]. This matrix allows MPC protocol to generate correct shares of uniformly random secrets in the presence of malicious adversary. Moreover, the amortized communication complexity of generating each sharing is linear. Due to this prominent feature, the hyper-invertible matrix plays an important role in the construction of MPC protocol and zero-knowledge proof protocol where the randomness needs to be jointly generated. However, the downside of this matrix is that the size of its base field is linear in the size of its matrix. This means if we construct an n-party MPC protocol over \(\mathbb {F}_q\) via hyper-invertible matrix, q is at least 2n.

In this paper, we propose the ramp hyper-invertible matrix which can be seen as the generalization of hyper-invertible matrix. Our ramp hyper-invertible matrix can be defined over constant-size field regardless of the size of this matrix. Similar to the arithmetic secret sharing scheme, to apply our ramp hyper-invertible matrix to perfectly secure MPC protocol, the maximum number of corruptions has to be compromised to \(\frac{(1-\epsilon )n}{3}\). As a consequence, we present the first perfectly secure MPC protocol in the presence of \(\frac{(1-\epsilon )n}{3}\) malicious corruptions with constant communication complexity. Besides presenting the variant of hyper-invertible matrix, we overcome several obstacles in the construction of this MPC protocol. Our arithmetic secret sharing scheme over constant-size field is compatible with the player elimination technique, i.e., it supports the dynamic changes of party number and corrupted party number. Moreover, we rewrite the public reconstruction protocol to support the sharings over constant-size field. Putting these together leads to the constant-size field variant of celebrated MPC protocol in [5].

We note that although it was widely acknowledged that there exists an MPC protocol with constant communication complexity by replacing Shamir secret sharing scheme with arithmetic secret sharing scheme, there is no reference seriously describing such protocol in detail. Our work fills the missing detail by providing MPC primitive for any applications relying on MPC protocol of constant communication complexity. As an application of our perfectly secure MPC protocol which implies perfect robustness in the MPC-in-the-Head framework, we present the constant-rate zero-knowledge proof with 3 communication rounds. The previous work achieves constant-rate with 5 communication rounds [32] due to the statistical robustness of their MPC protocol. Another application of our ramp hyper-invertible matrix is the information-theoretic multi-verifier zero-knowledge for circuit satisfiability [44]. We manage to remove the dependence of the size of circuit and security parameter from the share size.

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Notes

  1. 1.

    The perfectly secure MPC protocol in this paper is assumed to have guaranteed output delivery since \(t<n/3\).

  2. 2.

    We refer the reader to Sect. 4.1 for formal definition.

  3. 3.

    In fact, we are only concerned about the reconstruction of \(\varSigma _{2t}\).

  4. 4.

    We use 0 to represent the index of the secret and [n] to represent n indices of the shares.

  5. 5.

    In [15], a t-strongly multiplicative LSSS on n players for \(\mathbb {F}_q^k\) over \(\mathbb {F}_q\) is also called an (n, t, 2, t)-arithmetic secret sharing scheme with secret space \(\mathbb {F}_q^k\) and share space \(\mathbb {F}_q\).

  6. 6.

    In the Shamir SSS, one can identify this privacy d as the degree of polynomials.

  7. 7.

    Invoking Triples once can generate \(T=n'(1-\epsilon )-2t'-1=\varOmega (n)\) triples. Each triple contains \(\frac{\epsilon n}{6}\) secrets.

  8. 8.

    Since such triple consists of packed secret sharing scheme, we can further reduce the amortized communication complexity to constant if we evaluates \(\varOmega (n)\) instances of the same circuit in the online phase.

  9. 9.

    If we consider random oracle model, then statistically binding commitment scheme needs only one round [38]. We emphasize that regardless the model, our new MPCitH protocol saves two rounds of communication compared to [32].

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Acknowledgments

We thank the anonymous reviewers from ASIACRYPT 2023 for their insightful comments. The work of Chaoping Xing was supported in part by the National Key Research and Development Project under Grant 2022YFA1004900, in part by the National Natural Science Foundation of China under Grant 12031011. The work of Chen Yuan was supported in part by the National Natural Science Foundation of China under Grant 12101403.

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Authors and Affiliations

  1. Shanghai Jiao Tong University, Shanghai, China

    Hongqing Liu, Chaoping Xing & Chen Yuan

  2. Huawei International, Singapore, Singapore

    Yanjiang Yang

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  1. Hongqing Liu
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Correspondence to Chen Yuan .

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Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

A Player Elimination

A Player Elimination

Player elimination was first proposed in [31] to transform a non-robust (but detectable) protocol into a robust protocol at essentially no additional costs. This protocol cuts the preprocessing phase into many segments. At the beginning of each segment, all parties are happy. If some party detects the inconsistency, he becomes unhappy in this segment. At the end of this segment, if there is some party unhappy, the protocol enters into fault localization and removes a pair of parties from the rest of the computation. Then, the player elimination protocol repeats this segment. For completeness, we present the player elimination protocol in [5].

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Liu, H., Xing, C., Yang, Y., Yuan, C. (2023). Ramp Hyper-invertible Matrices and Their Applications to MPC Protocols. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14438. Springer, Singapore. https://doi.org/10.1007/978-981-99-8721-4_7

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