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Lattice-Based SNARKs: Publicly Verifiable, Preprocessing, and Recursively Composable

(Extended Abstract)

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13508))

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Abstract

A succinct non-interactive argument of knowledge (SNARK) allows a prover to produce a short proof that certifies the veracity of a certain NP-statement. In the last decade, a large body of work has studied candidate constructions that are secure against quantum attackers. Unfortunately, no known candidate matches the efficiency and desirable features of (pre-quantum) constructions based on bilinear pairings.

In this work, we make progress on this question. We propose the first lattice-based SNARK that simultaneously satisfies many desirable properties: It (i) is tentatively post-quantum secure, (ii) is publicly-verifiable, (iii) has a logarithmic-time verifier and (iv) has a purely algebraic structure making it amenable to efficient recursive composition. Our construction stems from a general technical toolkit that we develop to translate pairing-based schemes to lattice-based ones. At the heart of our SNARK is a new lattice-based vector commitment (VC) scheme supporting openings to constant-degree multivariate polynomial maps, which is a candidate solution for the open problem of constructing VC schemes with openings to beyond linear functions. However, the security of our constructions is based on a new family of lattice-based computational assumptions which naturally generalises the standard Short Integer Solution (SIS) assumption.

M. R. Albrecht—The research of MA was supported by EPSRC grants EP/S020330/1, EP/S02087X/1 and by the European Union Horizon 2020 Research and Innovation Program Grant 780701.

M. R. Albrecht, V. Cini, R. W. F. Lai, G. Malavolta, S. A. Thyagarajan—This work was supported by Protocol Labs under PL-RGP1-2021-050.

V. Cini—This work was in part done while visiting Max Planck Institute for Security and Privacy. The research of VC was in part funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 830929 (CyberSec4Europe), No. 871473 (KRAKEN), and by the Austrian Science Fund (FWF) and netidee SCIENCE grant P31621-N38 (PROFET).

R. W. F. Lai—This work was done at Friedrich-Alexander-Universität Erlangen-Nürnberg.

G. Malavolta—This work has been partially supported by the German Federal Ministry of Education and Research BMBF (grant 16K15K042, project 6GEM).

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Notes

  1. 1.

    It can be succinctly verified that SNARKs, like sharks, are creatures of the sea.

  2. 2.

    Though we mention that there is recent progress [5, 40] in crafting hash functions that are friendlier to multiparty computation and argument systems.

  3. 3.

    A Laurent monomial is a monomial where negative powers are allowed. Generally, one could consider \(k\text {-}R\text {-}\textsf{ISIS} \) problems for rational functions.

  4. 4.

    This generalises position binding.

  5. 5.

    We stress that this does not contradict any of the claims made in [63], but rather exemplifies the difference between their approach and ours.

  6. 6.

    The literature routinely simplifies the first expression to \(\approx \delta ^{d} \cdot {\det (\varLambda )}^{1/d}\).

  7. 7.

    Concretely, let \(\mathcal {T}\) be the set of all \(\mathcal {R}_q\) elements t where half of the components of t in the Chinese remainder theorem (CRT) representation are zero and the other half are non-zero. Note that this is well-defined only when \(\langle q \rangle \) is not a prime ideal in \(\mathcal {R}\).

  8. 8.

    In practice the gap may be smaller or larger and when picking parameters we optimise over these gaps.

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Authors and Affiliations

  1. Royal Holloway, University of London, Egham, UK

    Martin R. Albrecht

  2. AIT Austrian Institute of Technology, Seibersdorf, Austria

    Valerio Cini

  3. Aalto University, Espoo, Finland

    Russell W. F. Lai

  4. Max Planck Institute for Security and Privacy, Bochum, Germany

    Giulio Malavolta

  5. Carnegie Mellon University, Pittsburgh, USA

    Sri AravindaKrishnan Thyagarajan

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  1. Martin R. Albrecht
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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Albrecht, M.R., Cini, V., Lai, R.W.F., Malavolta, G., Thyagarajan, S.A. (2022). Lattice-Based SNARKs: Publicly Verifiable, Preprocessing, and Recursively Composable. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13508. Springer, Cham. https://doi.org/10.1007/978-3-031-15979-4_4

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  • DOI: https://doi.org/10.1007/978-3-031-15979-4_4

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