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Adaptively Secure MPC with Sublinear Communication Complexity

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Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

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Abstract

A central challenge in the study of MPC is to balance between security guarantees, hardness assumptions, and resources required for the protocol. In this work, we study the cost of tolerating adaptive corruptions in MPC protocols under various corruption thresholds. In the strongest setting, we consider adaptive corruptions of an arbitrary number of parties (potentially all) and achieve the following results:

  • A two-round secure function evaluation (SFE) protocol in the CRS model, assuming LWE and indistinguishability obfuscation (iO). The communication, the CRS size, and the online-computation are sublinear in the size of the function. The iO assumption can be replaced by secure erasures. Previous results required either the communication or the CRS size to be polynomial in the function size.

  • Under the same assumptions, we construct a “Bob-optimized” 2PC (where Alice talks first, Bob second, and Alice learns the output). That is, the communication complexity and total computation of Bob are sublinear in the function size and in Alice’s input size. We prove impossibility of “Alice-optimized” protocols.

  • Assuming LWE, we bootstrap adaptively secure NIZK arguments to achieve proof size sublinear in the circuit size of the NP-relation.

On a technical level, our results are based on laconic function evaluation (LFE) (Quach, Wee, and Wichs, FOCS’18) and shed light on an interesting duality between LFE and FHE.

Next, we analyze adaptive corruptions of all-but-one of the parties and show a two-round SFE protocol in the threshold PKI model (where keys of a threshold FHE scheme are pre-shared among the parties) with communication complexity sublinear in the circuit size, assuming LWE and NIZK. Finally, we consider the honest-majority setting, and show a two-round SFE protocol with guaranteed output delivery under the same constraints.

R. Cohen—Research supported by the Northeastern University Cybersecurity and Privacy Institute Post-doctoral fellowship, NSF grant TWC-1664445, NSF grant 1422965, and by the NSF MACS project.

A. Shelat—Research supported by NSF grant TWC-1664445 and a Google Faculty fellowship.

D. Wichs—Research supported by NSF grants CNS-1314722, CNS-1413964, CNS-1750795 and the Alfred P. Sloan Research Fellowship.

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Notes

  1. 1.

    We note that in certain cases it is reasonable to erase the random coins, e.g., when encrypting a message it is normally fine not to store the encryption randomness; however, is some cases one cannot erase all of its random tape, e.g., when sending a public encryption key it is normally essential to store the decryption key. We refer the reader to [18, 20] for further discussion on secure erasures.

  2. 2.

    In the common random string model, all parties receive a uniformly random string generated in a trusted setup phase. In the common reference string model, the common string is sampled according to some pre-defined distribution.

  3. 3.

    The protocols in [24, 38] use the CLOS compiler [21] to get malicious security. Since the communication of previously known adaptively secure ZK protocols depends on the NP relation (see [44, 58, 70] and references therein), the communication of the maliciously secure protocols depended on the CRS. Our short NIZK (Theorem 3) can be used to reduce the communication of [24, 38] in the malicious setting as well.

  4. 4.

    The basic construction in [75] holds under the standard LWE assumption; however, for the purpose of (semi-)malicious MPC, in which the inputs to the protocol can be chosen adaptively, after the CRS is published, we require the stronger variant.

  5. 5.

    Another approach for compact MPC is using function secret sharing (FSS) [15, 16]. This approach does not seem to support adaptive corruptions.

  6. 6.

    In the semi-malicious setting, the adversary follows the protocol as in the semi-honest case, but he can choose arbitrary random coins for corrupted parties.

  7. 7.

    We emphasize that the lower bounds hold given a public-coin setup, where all parties get the same information, and does not hold given a private-coin setup such as threshold PKI.

  8. 8.

    Other properties such as privacy and independence of inputs are always required to hold.

  9. 9.

    We note that the same problem arises also in the threshold FHE scheme for more general access structures [14, Def. 5.5], where the simulation is defined only for maximal invalid party sets.

  10. 10.

    Recently, Boneh et al. [14] showed that this problem can be overcome in a different way, by using a special secret sharing scheme that ensures the Lagrange coefficients are binary values.

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Correspondence to Ran Cohen or Abhi Shelat .

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© 2019 International Association for Cryptologic Research

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Cohen, R., Shelat, A., Wichs, D. (2019). Adaptively Secure MPC with Sublinear Communication Complexity. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_2

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  • DOI: https://doi.org/10.1007/978-3-030-26951-7_2

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