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VSS from Distributed ZK Proofs and Applications

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14438))

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Abstract

Non-Interactive Verifiable Secret Sharing (NI-VSS) is a technique for distributing a secret among a group of individuals in a verifiable manner, such that shareholders can verify the validity of their received share and only a specific number of them can access the secret. VSS is a fundamental tool in cryptography and distributed computing. In this paper, we present an extremely efficient NI-VSS scheme using Zero-Knowledge (ZK) proofs on secret shared data. While prior VSS schemes have implicitly used ZK proofs on secret shared data, we specifically use their formal definition recently provided by Boneh et al. in CRYPTO 2019. The proposed NI-VSS scheme uses a quantum random oracle and a quantum computationally hiding commitment scheme in a black-box manner, which ensures its ease of use, especially in post-quantum threshold protocols. Implementation results further solidify its practicality and superiority over current constructions. With the new VSS scheme, for parameter sets \((n, t)=(128, 63)\) and (2048, 1023), a dealer can share a secret in less than 0.02 and 2.0 s, respectively, and shareholders can verify their shares in less than 0.4 and 5.0 ms. Compared to the well-established Pedersen VSS scheme, for the same parameter sets, at the cost of \(2.5\times \) higher communication, the new scheme is respectively \(22.5\times \) and \(3.25\times \) faster in the sharing phase, and notably needs \(271\times \) and \(479\times \) less time in the verification. Leveraging the new NI-VSS scheme, we revisit several classic and PQ-secure threshold protocols and improve their efficiency. Our revisions led to more efficient versions of both the Pedersen DKG protocol and the GJKR threshold signature scheme. We show similar efficiency enhancements and improved resilience to malicious parties in isogeny-based DKG and threshold signature schemes. We think, due to its remarkable efficiency and ease of use, the new NI-VSS scheme can be a valuable tool for a wide range of threshold protocols.

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Notes

  1. 1.

    This choice is done for two main reasons. First, strong ZK might lead to misunderstandings, as the majority of verifiers is actually honest in our cases. Second, we risk confusion by using the abbreviation SZK, which usually stands for statistical zero-knowledge.

  2. 2.

    In general \(\mathbb {Z}_N\) will constitute a ring. In later applications, we sometimes choose N to be a prime, so that \(\mathbb {Z}_N\) becomes a field.

  3. 3.

    Such a hash function can easily be implemented by hashing into a set \(\{1,\dots ,k\}\) and then using the output value \(i\in \{1,\dots ,k\}\) as an index in \(\varXi _k\), i.e. \(c_i\in \varXi _k\).

  4. 4.

    For \(|I|\le t\), there always exists a witness that satisfies the relation.

  5. 5.

    Read component-wise, e.g. \(\forall i\in I:\ \textsf{C}'_i=\mathcal {C}(x_i,y'_i)\).

  6. 6.

    Actually, the authors implicitly prove TZK, but do not call it as such.

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Acknowledgments

We would like to express our gratitude to the anonymous reviewers for their valuable insights and suggestions. We also extend our thanks to Navid Ghaedi Bardeh for his valuable assistance during the implementation of VSS schemes.

This work has been supported in part by the Defense Advanced Research Projects Agency (DARPA) under contract No. HR001120C0085, by the FWO under an Odysseus project GOH9718N, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101020788 - Adv-ERC-ISOCRYPT), by CyberSecurity Research Flanders with reference number VR20192203, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under project PICOCRYPT (grant agreement No. 101001283), by the Spanish Government under projects PRODIGY (TED2021-132464B-I00) and ESPADA (PID2022-142290OB-I00). The last two projects are co-funded by European Union EIE, and NextGenerationEU/PRTR funds.

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Authors and Affiliations

  1. COSIC, KU Leuven, Leuven, Belgium

    Shahla Atapoor, Karim Baghery, Daniele Cozzo & Robi Pedersen

  2. IMDEA Software Institute, Madrid, Spain

    Daniele Cozzo

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  1. Shahla Atapoor
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  2. Karim Baghery
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  3. Daniele Cozzo
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Correspondence to Karim Baghery .

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Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

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Atapoor, S., Baghery, K., Cozzo, D., Pedersen, R. (2023). VSS from Distributed ZK Proofs and Applications. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14438. Springer, Singapore. https://doi.org/10.1007/978-981-99-8721-4_13

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