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On Black-Box Knowledge-Sound Commit-And-Prove SNARKs

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14439))

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Abstract

Gentry and Wichs proved that adaptively sound SNARGs for hard languages need non-falsifiable assumptions. Lipmaa and Pavlyk claimed Gentry-Wichs is tight by constructing a non-adaptively sound zk-SNARG FANA for \(\textsf{NP}\) from falsifiable assumptions. We show that FANA is flawed. We define and construct a fully algebraic F-position-binding vector commitment scheme \(\textrm{VCF}\). We construct a concretely efficient commit-and-prove zk-SNARK Punic, a version of FANA with an additional \(\textrm{VCF}\) commitment to the witness. Punic satisfies semi-adaptive black-box \(G\)-knowledge-soundness, a new natural knowledge-soundness notion for commit-and-prove SNARKs. We use a new proof technique to achieve global consistency using a functional somewhere-extractable commitment scheme to extract vector commitment’s local proofs.

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Notes

  1. 1.

    [53] noted that FANA is insecure (and referred to a private conversation with the authors of [46]), but they did not explain why. We will provide full details.

  2. 2.

    The initial QA-NIZK constructions were for linear subspaces [36, 40]. They (and the bilateral linear subspace QA-SNARG, used in [16, 17] and the current paper) have a language parameter that is not a commitment key. We use the acronym QA-SNARG since it fits our framework better.

  3. 3.

    We use the standard additive bracket notation for pairings. For example, for \(s \in \mathbb {Z}_{p}\), \([s]_{1} = s [1]_{1} \in \mathbb {G}_{1}\). See Sect. 3.

  4. 4.

    Defined as functional somewhere statistically binding (SSB) commitment in [17]; generalizes SSB hashes [32, 48]. In SSB hashes, \(\mathcal {F}\) is the family of point functions, and q is always equal to one. On the other hand, we do not need the local opening property, thus obtaining better efficiency. Since extractability is essential, we call them functional \(\textrm{SE}\). \(\textrm{DGPRS}\) and \(\textrm{FLPS}\) predate [13]. \(\textrm{SE}\) commitments have been used to build SNARGs for \(\textsf{P}\) and batch-arguments for \(\textsf{NP}\) [13, 27].

  5. 5.

    \((\boldsymbol{U}, \boldsymbol{V}, \boldsymbol{W})\) is a part of the instance and thus our SNARKs are non-universal. The most efficient known universal SNARKs [30] in the standard model (without random oracles) have quadratic size CRS and are thus too inefficient for practice.

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Acknowledgment and History

The author became aware of the error in FANA in December 2021; the error was (partially) caused by the fact that he was severely sick when submitting [46] and its camera-ready version. We thank Daniel Wichs and anonymous reviewers for helpful comments.

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    Helger Lipmaa

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    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

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Lipmaa, H. (2023). On Black-Box Knowledge-Sound Commit-And-Prove SNARKs. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14439. Springer, Singapore. https://doi.org/10.1007/978-981-99-8724-5_2

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