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Your Reputation’s Safe with Me: Framing-Free Distributed Zero-Knowledge Proofs

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14369))

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Abstract

Distributed Zero-Knowledge (dZK) proofs, recently introduced by Boneh et al. (CRYPTO‘19), allow a prover \(\mathcal{P}\) to prove NP statements on an input x which is distributed between k verifiers \(\mathcal{V}_1,\ldots ,\mathcal{V}_k\), where each \(\mathcal{V}_i\) holds only a piece of x. As in standard ZK proofs, dZK proofs guarantee Completeness when all parties are honest; Soundness against a malicious prover colluding with t verifiers; and Zero Knowledge against a subset of t malicious verifiers, in the sense that they learn nothing about the NP witness and the input pieces of the honest verifiers.

Unfortunately, dZK proofs provide no correctness guarantee for an honest prover against a subset of maliciously corrupted verifiers. In particular, such verifiers might be able to “frame” the prover, causing honest verifiers to reject a true claim. This is a significant limitation, since such scenarios arise naturally in dZK applications, e.g., for proving honest behavior, and such attacks are indeed possible in existing dZKs (Boneh et al., CRYPTO‘19).

We put forth and study the notion of strong completeness for dZKs, guaranteeing that true claims are accepted even when t verifiers are maliciously corrupted. We then design strongly-complete dZK proofs using the “MPC-in-the-head” paradigm of Ishai et al. (STOC‘07), providing a novel analysis that exploits the unique properties of the distributed setting.

To demonstrate the usefulness of strong completeness, we present several applications in which it is instrumental in obtaining security. First, we construct a certifiable version of Verifiable Secret Sharing (VSS), which is a VSS in which the dealer additionally proves that the shared secret satisfies a given NP relation. Our construction withstands a constant fraction of corruptions, whereas a previous construction of Ishai et al. (TCC‘14) required \(k={\textsf{poly}}\left( t\right) \). We also design a reusable version of certifiable VSS that we introduce, in which the dealer can prove an unlimited number of predicates on the same shared secret.

Finally, we extend a compiler of Boneh et al. (CRYPTO‘19), who used dZKs to transform a class of “natural” semi-honest protocols in the honest-majority setting into maliciously secure ones with abort. Our compiler uses strongly-complete dZKs to obtain identifiable abort.

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Notes

  1. 1.

    Various works have considered other models, e.g., when security is only computational, or when the input statement is known in full to all verifiers. These models are discussed in Sect. 1.5, but similar to [BBC+19b] our focus is on information-theoretic security when the input statement is distributed between the verifiers.

  2. 2.

    More specifically, this paradigm applies to a class of natural protocols which guarantee, among other things, that privacy is preserved up to the final round even in the presence of malicious corruptions; see the full version for further details.

  3. 3.

    Roughly, this holds in their protocols because the verifiers do not have any private coins, and \(\mathcal{P}\) knows the entire input statement x.

  4. 4.

    Notice that the dZK proof is for input statements that are distributed between the verifiers using a robust encoding. [BBC+19b] make the same assumption. The reason to focus on such languages is because they show [BBC+19a, Sec. 6.3.2] limitations on the existence of dZK proofs for languages that are not robustly encoded.

  5. 5.

    See Sect. 1.5 for a comparison between our construction and other constructions using this technique in the two-party and in other distributed settings.

  6. 6.

    The messages sent from party i to party j appear explicitly in the view of party j, and the messages it sent to party i can be computed from its view.

  7. 7.

    [AKP22] also obtain a fully information-theoretically secure VRS assuming ideal non-interactive commitments, as well as a computationally sound and statistically ZK (statistically sound and computationally ZK, respectively) VRS based on computationally binding and statistically hiding (statistically binding and computationally hiding, respectively) non-interactive commitments [App22].

  8. 8.

    In the computational setting one can use standard tools such as commitments to help resolve disputes between parties, but in the information theoretic setting this seems to require a more sophisticated dispute-resolution sub-protocol.

  9. 9.

    Notice that if \(\textsf{Enc}\) is randomized then x might have several corresponding k-distributed inputs X.

  10. 10.

    We note that w may also be the empty string, e.g., if \(\widehat{\mathcal{R}}\) corresponds to a language in \(\textsf {P}\).

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Acknowledgment

We thank Benny Applebaum for helpful discussions and for pointing out to us the reduction from VRS to dZK. The first and third authors are supported by the BIU Center for Research in Applied Crypytography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office. The first author is supported by ISF grant No. 1316/18. The first and second authors are supported by DARPA under Contract No. HR001120C0087. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA. The first author is supported by the Algorand Centres of Excellence programme managed by Algorand Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Algorand Foundation.

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Authors and Affiliations

  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay & Mor Weiss

  2. Georgetown University and Ligero Inc., Washington, USA

    Muthuramakrishnan Venkitasubramaniam

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  1. Carmit Hazay
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  2. Muthuramakrishnan Venkitasubramaniam
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  3. Mor Weiss
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Correspondence to Mor Weiss .

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  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Hazay, C., Venkitasubramaniam, M., Weiss, M. (2023). Your Reputation’s Safe with Me: Framing-Free Distributed Zero-Knowledge Proofs. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14369. Springer, Cham. https://doi.org/10.1007/978-3-031-48615-9_2

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