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ECLIPSE: Enhanced Compiling Method for Pedersen-Committed zkSNARK Engines

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Public-Key Cryptography – PKC 2022 (PKC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13177))

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Abstract

We advance the state-of-the art for zero-knowledge commit-and-prove SNARKs (CP-SNARKs). CP-SNARKs are an important class of SNARKs which, using commitments as “glue”, allow to efficiently combine proof systems—e.g., general-purpose SNARKs (an efficient way to prove statements about circuits) and \(\varSigma \)-protocols (an efficient way to prove statements about group operations). Thus, CP-SNARKs allow to efficiently provide zero-knowledge proofs for composite statements such as \(h=H(g^{x})\) for some hash-function H.

Our main contribution is providing the first construction of CP-SNARKs where the proof size is succinct in the number of commitments.

We achieve our result by providing a general technique to compile Algebraic Holographic Proofs (AHP) (an underlying abstraction used in many modern SNARKs) with special “decomposition” properties into an efficient CP-SNARK. We then show that some of the most efficient AHP constructions—Marlin, PLONK, and Sonic—satisfy our compilation requirements.

Our resulting SNARKs achieve universal and updatable reference strings, which are highly desirable features as they greatly reduce the trust needed in the SNARK setup phase.

Full version available at [3].

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Notes

  1. 1.

    The service can afford to wait for \(\ell \) orders depending on the application, and the expected throughput and time-to-service of the application. As an example, the ID-Layer in Concordium [28] orders may be even serviced each epoch.

  2. 2.

    The reason why we apply our compiler to all three proof systems is that Marlin, PLONK and Sonic are a sort of rock-paper-scissor for AHPs (the first can outperform the second, which can outperform the third, which can in turn outperform the first). This is because they use different models of computations, and therefore it may be possible to prove some statements more efficiently with one system rather than the others.

  3. 3.

    While there are approaches that mitigate this problem [1, 22, 43], they are curve-dependent—hindering generality and interoperability—and still relatively expensive (at 4–6 constraints per curve operation).

  4. 4.

    While it is also necessary to prove \(w_\mathsf {com}(X)\) only maps to \((w_i)_{i\in I_\mathsf {com}}\), this is trivially achieved by knowledge soundness of the \(\varSigma \)-protocol.

  5. 5.

    We use proof and argument as synonymous in this paper, as we are only interested in computational soundness.

  6. 6.

    More formally, if the underlying AHP is \((\mathsf {b}+1,\mathsf {C})\)-zero knowledge, where \(\mathsf {b}\) is the maximum number of queries made by the verifier to polynomials, one can retain ZK of the resulting SNARK by compiling AHP via PCS with somewhat hiding security, a weaker notion of hiding [20]. Because the deterministic KZG is already somewhat hiding and every WCP in \(\mathsf {AHP}_\textsf {PLONK} \) is queried once, it suffices to add \(v_\mathbb {H}\) multiplied by a masking polynomial of degree 1 to tolerate 2 openings (i.e., one evaluation and one commitment).

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Acknowledgments

The authors are grateful for Sean Bowe, Ben Fisch, Ariel Gabizon, and Mary Maller for clarifying the details of their works. We thank the anonymous reviewers of PKC 2022 for helpful comments. This work has been supported by: the Concordium Blockchain Research Center, Aarhus University, Denmark; the Carlsberg Foundation under the Semper Ardens Research Project CF18-112 (BCM); the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No 803096 (SPEC).

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Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Diego F. Aranha, Matteo Campanelli, Claudio Orlandi & Akira Takahashi

  2. Concordium, Aarhus, Denmark

    Emil Madsen Bennedsen

  3. Indian Institute of Science, Bangalore, India

    Chaya Ganesh

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  1. Diego F. Aranha
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Correspondence to Diego F. Aranha .

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Editors and Affiliations

  1. National Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan

    Goichiro Hanaoka

  2. Yokohama National University, Yokohama, Japan

    Junji Shikata

  3. The University of Electro-Communications, Tokyo, Japan

    Yohei Watanabe

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Aranha, D.F., Bennedsen, E.M., Campanelli, M., Ganesh, C., Orlandi, C., Takahashi, A. (2022). ECLIPSE: Enhanced Compiling Method for Pedersen-Committed zkSNARK Engines. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds) Public-Key Cryptography – PKC 2022. PKC 2022. Lecture Notes in Computer Science(), vol 13177. Springer, Cham. https://doi.org/10.1007/978-3-030-97121-2_21

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