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Beyond MPC-in-the-Head: Black-Box Constructions of Short 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

In their seminal work, Ishai, Kushilevitz, Ostrovsky, and Sahai (STOC‘07) presented the MPC-in-the-Head paradigm, which shows how to design Zero-Knowledge Proofs (ZKPs) from secure Multi-Party Computation (MPC) protocols. This paradigm has since then revolutionized and modularized the design of efficient ZKP systems, with far-reaching applications beyond ZKPs. However, to the best of our knowledge, all previous instantiations relied on fully-secure MPC protocols and have not been able to leverage the fact that the paradigm only imposes relatively weak privacy and correctness requirements on the underlying MPC.

In this work, we extend the MPC-in-the-Head paradigm to game-based cryptographic primitives supporting homomorphic computations (e.g., fully-homomorphic encryption, functional encryption, randomized encodings, homomorphic secret sharing, and more). Specifically, we present a simple yet generic compiler from these primitives to ZKPs which use the underlying primitive as a black box. We also generalize our paradigm to capture commit-and-prove protocols, and use it to devise tight black-box compilers from Interactive (Oracle) Proofs to ZKPs, assuming One-Way Functions (OWFs).

We use our paradigm to obtain several new ZKP constructions:

1. The first ZKPs for \(\textsf {NP}\) relations \(\mathcal{R}\) computable in (polynomial-time uniform) \(\textsf{NC}^1\), whose round complexity is bounded by a fixed constant (independent of the depth of \(\mathcal{R}\)’s verification circuit), with communication approaching witness length (specifically, \(n\cdot {\textsf{poly}}\left( \kappa \right) \), where n is the witness length, and \(\kappa \) is a security parameter), assuming DCR. Alternatively, if we allow the round complexity to scale with the depth of the verification circuit, our ZKPs can make black-box use of OWFs.

2. Constant-round ZKPs for NP relations computable in bounded polynomial space, with \(O\left( n\right) +o\left( m\right) \cdot {\textsf{poly}}\left( \kappa \right) \) communication assuming OWFs, where m is the instance length. This gives a black-box alternative to a recent non-black-box construction of Nassar and Ron (CRYPTO‘22).

3. ZKPs for NP relations computable by a logspace-uniform family of depth-\(d\left( m\right) \) circuits, with \(n\cdot {\textsf{poly}}\left( \kappa ,d\left( m\right) \right) \) communication assuming OWFs. This gives a black-box alternative to a result of Goldwasser, Kalai and Rothblum (JACM).

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Notes

  1. 1.

    This dependence on the randomness can be removed by generating the randomness using a PRG whose output is indistinguishable from random, against non-uniform distinguishers. This causes only a negligible increase in the soundness error.

  2. 2.

    By polynomial-time uniform \(\textsf{NC}^1\) we mean that there exist a polynomial p(n) and a Turing machine that on input \(1^n\) runs in time p(n) and outputs the circuit (in \(\textsf{NC}^1\)) for input length n.

  3. 3.

    In a public-coin IP, the verifier’s messages are simply random bits.

  4. 4.

    This is reminiscent of the [IKOS07] construction from passively-secure MPC protocols, in which the witness is secret-shared between the parties participating in the execution “in-the-head”. We note, however, that our use of secret sharing is conceptually different: in our case, there is no underlying two- or multi-party computation. Instead, one of the shares is hard-wired into the computed function, making its identity secret, whereas [IKOS07] compute a public function by emulating multiple parties “in-the-head”.

  5. 5.

    We note that a similar construction could be obtained from the paradigm of [IKOS07] by instantiating an appropriate 2-party protocol from FHE.

  6. 6.

    See Sect. 4 for a generalization to imperfect correctness; e.g., in the HSS-based construction of Theorem 2.

  7. 7.

    The reason the protocol requires logspace-uniformity is to provide an efficient way for the verifier to evaluate a point on the low-degree extension of the circuit wiring predicate. If the circuit class was just polynomial-time uniform, the verifier would need time that is quasi-linear in the size of the predicate.

  8. 8.

    [GR20] provide a constant-round protocol for sufficiently uniform (i.e., adjacency predicate) circuits in \(\textsf{NC}^1\). However, following the observation made on the protocol of [GKR15], the protocol of [GR20] also yields a constant-round protocol for polynomial-time uniform \(\textsf{NC}^1\) with short communication.

  9. 9.

    We will assume the multi-tape formulation to capture sub-linear space computations.

  10. 10.

    A safe prime is a prime number of the form \(2p + 1\), where p is also a prime.

  11. 11.

    We say that \(t\in \mathbb {Z}^*_{N^2}\) is a perfect power of N if there exists \(r\in \mathbb {Z}_N^*\) such that \(t=r^N\bmod ~\mathbb {Z}^*_{N^2}\).

  12. 12.

    We note that \(\mathcal{D}'\) does not need to generate the commitments - these do not contribute to distinguishability because the commitments are ideal.

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Acknowledgments

We thank Shweta Agarwal, Elette Boyle, Yuval Ishai, Justin Thaler, and Daniel Wichs for several discussions on the various cryptographic primitives. We also thank Guy Rothblum and Ron Rothblum for substantial discussions on the state-of-the-art for succinct proofs. We thank the anonymous TCC reviewers for their insightful comments and suggestions. Distribution Statement “A” (Approved for Public Release, Distribution Unlimited). 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.

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  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay & Mor Weiss

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

    Carmit Hazay & Muthuramakrishnan Venkitasubramaniam

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  1. Carmit Hazay
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Hazay, C., Venkitasubramaniam, M., Weiss, M. (2023). Beyond MPC-in-the-Head: Black-Box Constructions of Short 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_1

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