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Impossibilities in Succinct Arguments: Black-Box Extraction and More

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Progress in Cryptology - AFRICACRYPT 2023 (AFRICACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14064))

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Abstract

The celebrated result by Gentry and Wichs established a theoretical barrier for succinct non-interactive arguments (SNARGs), showing that for (expressive enough) hard-on-average languages, we must assume non-falsifiable assumptions. We further investigate those barriers by showing new negative and positive results related to the proof size.

  1. 1.

    We start by formalizing a folklore lower bound for the proof size of black-box extractable arguments based on the hardness of the language. This separates knowledge-sound SNARGs (SNARKs) in the random oracle model (that can have black-box extraction) and those in the standard model.

  2. 2.

    We find a positive result in the non-adaptive setting. Under the existence of non-adaptively sound SNARGs (without extractability) and from standard assumptions, it is possible to build SNARKs with black-box extractability for a non-trivial subset of \({\textbf {NP}}\).

  3. 3.

    On the other hand, we show that (under some mild assumptions) all \({\textbf {NP}}\) languages cannot have SNARKs with black-box extractability even in the non-adaptive setting.

  4. 4.

    The Gentry-Wichs result does not account for the preprocessing model, under which fall several efficient constructions. We show that also, in the preprocessing model, it is impossible to construct SNARGs that rely on falsifiable assumptions in a black-box way.

Along the way, we identify a class of non-trivial languages, which we dub “trapdoor languages”, that can bypass these impossibility results.

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Notes

  1. 1.

    There exist efficient SNARKs (SNARGs of knowledge) in idealized models like ROM (random oracle model) [12] and GGM (generic group model) [28].

  2. 2.

    For example, DLOG is a falsifiable assumption since the challenger can efficiently test if the adversary has found the correct discrete logarithm.

  3. 3.

    It is an open problem to obtain non-adaptively secure SNARGs from polynomial-time secure falsifiable assumptions. The SNARG in [39], was shown to be flawed [46].

  4. 4.

    For example, in pairing-based constructions such as [23] it consists of at least one group element per wire in the circuit to be proven.

  5. 5.

    Contrast this with the adaptive case, where the prover is stateless and rewinding is not useful.

  6. 6.

    This class should include FHE encryption and CRHF and should be closed under conjunction. In our theorem statement, we simply require a SNARG for \({\textbf {NP}}\).

  7. 7.

    More generally, if poly-to-one one-way functions exist then \({\textbf {P}}\ne {\textbf {FewP}}\) [2].

  8. 8.

    A slightly simpler construction for the case of \({\textbf {UP}}\) (\({\textbf {NP}}\) statements with a unique witness) can be seen in the full version.

  9. 9.

    The proof of [35] contains 1 group element and bypasses the  [28] impossibility as well. This is not contradictory because the [28] impossibility only applies to pairing-based NIZKs that are compiled from NILPs.

  10. 10.

    Here, \(\mathbb {G}_1\) is an additive pairing group and \([x]_{1}\) denotes a group element with a discrete logarithm x.

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Acknowledgement

We thank the reviewers of CRYPTO 2022 for constructive feedback, in particular, for pointing out the connection between black-box extractability and leakage-resilient cryptography. We thank Helger Lipmaa for comments on the paper.

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Authors and Affiliations

  1. Protocol Labs, San Francisco, USA

    Matteo Campanelli

  2. Indian Institute of Science, Bengaluru, India

    Chaya Ganesh

  3. Concordium, Aarhus, Denmark

    Hamidreza Khoshakhlagh

  4. Simula UiB, Bergen, Norway

    Janno Siim

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  1. Matteo Campanelli
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  2. Chaya Ganesh
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  3. Hamidreza Khoshakhlagh
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  4. Janno Siim
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Correspondence to Hamidreza Khoshakhlagh .

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Editors and Affiliations

  1. EMSE, Saint-Étienne, France

    Nadia El Mrabet

  2. IBM Research Europe, Rüschlikon, Switzerland

    Luca De Feo

  3. Université de Rennes 1, Rennes Cedex, France

    Sylvain Duquesne

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Campanelli, M., Ganesh, C., Khoshakhlagh, H., Siim, J. (2023). Impossibilities in Succinct Arguments: Black-Box Extraction and More. In: El Mrabet, N., De Feo, L., Duquesne, S. (eds) Progress in Cryptology - AFRICACRYPT 2023. AFRICACRYPT 2023. Lecture Notes in Computer Science, vol 14064. Springer, Cham. https://doi.org/10.1007/978-3-031-37679-5_20

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