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Towards a Unified Approach to Black-Box Constructions of Zero-Knowledge Proofs

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Advances in Cryptology – CRYPTO 2021 (CRYPTO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12828))

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Abstract

General-purpose zero-knowledge proofs for all \(\mathsf {NP} \) languages greatly simplify secure protocol design. However, they inherently require the code of the underlying relation. If the relation contains black-box calls to a cryptographic function, the code of that function must be known to use the ZK proof, even if both the relation and the proof require only black-box access to the function. Rosulek (Crypto’12) shows that non-trivial proofs for even simple statements, such as membership in the range of a one-way function, require non-black-box access.

We propose an alternative approach to bypass Rosulek’s impossibility result. Instead of asking for a ZK proof directly for the given one-way function f, we seek to construct a new one-way function F given only black-box access to f, and an associated ZK protocol for proving non-trivial statements, such as range membership, over its output. We say that F, along with its proof system, is a proof-based one-way function. We similarly define proof-based versions of other primitives, specifically pseudo-random generators and collision-resistant hash functions.

We show how to construct proof-based versions of each of the primitives mentioned above from their ordinary counterparts under mild but necessary restrictions over the input. More specifically,

  • We first show that if the prover entirely chooses the input, then proof-based pseudo-random generators cannot be constructed from ordinary ones in a black-box manner, thus establishing that some restrictions over the input are necessary.

  • We next present black-box constructions handling inputs of the form (x, r) where r is chosen uniformly by the verifier. This is similar to the restrictions in the widely used Goldreich-Levin theorem. The associated ZK proofs support range membership over the output as well as arbitrary predicates over prefixes of the input.

Our results open up the possibility that general-purpose ZK proofs for relations that require black-box access to the primitives above may be possible in the future without violating their black-box nature by instantiating them using proof-based primitives instead of ordinary ones.

This material is based upon work supported in part by DARPA SIEVE Award HR00112020026, NSF grants 1907908 and 2028920, and a Cisco Research Award. 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, DARPA, NSF, or Cisco.

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Notes

  1. 1.

    Note that the protocol is allowed to depend on \(G^{(\cdot )}\) but not on the oracle which may be arbitrarily chosen later. The same holds for the relation \(R^f_G\) introduced next.

  2. 2.

    For now, we only focus on range-membership proofs. The definitional approach is consistent with the commit-and-prove literature, although there are important differences since we are dealing with deterministic primitives.

  3. 3.

    We use Prover/Verifier and Sender/Receiver interchangeably since our ZKP is captured by considering a secure computation style definition for two parties.

  4. 4.

    This holds if the size of range of h is exponentially larger than its image space. It is also worth noting that \(\tau \) does not need to be efficiently computable, because our soundness proof (or the editing technique) is only an existential argument.

  5. 5.

    Note that these values already determine the sets \(\ddot{Q}\), \(\ddot{Q}^\mathsf {Easy} \), and \(\ddot{Q}^\mathsf {Hard} \) as defined above.

  6. 6.

    The symbol “\(*\)” denotes the wildcard that matches any answer to q.

  7. 7.

    Technically, elements in \(Q^{\mathsf {Hard}}_n\) are query-answer pairs. From here on, we override the notation “\(\in \)” such that \(q \in Q^{\mathsf {Hard}}_n\) means that there exists a pair \((q, *)\) in \(Q^{\mathsf {Hard}}_n\).

  8. 8.

    Note that these values also fix the corresponding \(\{\ddot{Y}_n\} _{n\in \mathbb {N}}\), \(\{\ddot{Q}^\mathsf {Easy} _n\} _{n\in \mathbb {N}}\) and \(\{\ddot{Q}^\mathsf {Hard} _n\} _{n\in \mathbb {N}}\) as in the above.

  9. 9.

    Note that the input to \(h_i\) should have length 2m. But \(|\varLambda | > 2m\). This can be handled using domain-extension techniques, e.g., the Merkle-Damgård transformation [5, 37].

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, NY, 11790, USA

    Xiao Liang & Omkant Pandey

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  1. Xiao Liang
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Correspondence to Xiao Liang .

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

    Tal Malkin

  2. University of Michigan, Ann Arbor, MI, USA

    Chris Peikert

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Liang, X., Pandey, O. (2021). Towards a Unified Approach to Black-Box Constructions of Zero-Knowledge Proofs. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12828. Springer, Cham. https://doi.org/10.1007/978-3-030-84259-8_2

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