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Publicly Verifiable Secret Sharing Over Class Groups and Applications to DKG and YOSO

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Advances in Cryptology – EUROCRYPT 2024 (EUROCRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14655))

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Abstract

Publicly Verifiable Secret Sharing (PVSS) allows a dealer to publish encrypted shares of a secret so that parties holding the corresponding decryption keys may later reconstruct it. Both dealing and reconstruction are non-interactive and any verifier can check their validity. PVSS finds applications in randomness beacons, distributed key generation (DKG) and in YOSO MPC (Gentry et al. CRYPTO’21), when endowed with suitable publicly verifiable re-sharing as in YOLO YOSO (Cascudo et al. ASIACRYPT’22).

We introduce a PVSS scheme over class groups that achieves similar efficiency to state-of-the art schemes that only allow for reconstructing a function of the secret, while our scheme allows the reconstruction of the original secret. Our construction generalizes the DDH-based scheme of YOLO YOSO to operate over class groups, which poses technical challenges in adapting the necessary NIZKs in face of the unknown group order and the fact that efficient NIZKs of knowledge are not as simple to construct in this setting.

Building on our PVSS scheme’s ability to recover the original secret, we propose two DKG protocols for discrete logarithm key pairs: a biasable 1-round protocol, which improves on the concrete communication/computational complexities of previous works; and a 2-round unbiasable protocol, which improves on the round complexity of previous works. We also add publicly verifiable resharing towards anonymous committees to our PVSS, so that it can be used to efficiently transfer state among committees in the YOSO setting. Together with a recent construction of MPC in the YOSO model based on class groups (Braun et al. CRYPTO’23), this results in the most efficient full realization (i.e. without assuming receiver anonymous channels) of YOSO MPC based on the CDN framework with transparent setup.

Ignacio Cascudo was partially supported by the Spanish Government under the project SecuRing (ref. PID2019-110873RJ-I00) and the PRODIGY Project (TED2021-132464B-I00), both funded by MCIN/AEI/10.13039/501100011033/. PRODIGY is also funded by the European Union NextGenerationEU/PRTR. He was also partially supported by the European Union under GA 101096435 (CONFIDENTIAL-6G). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

Bernardo David was supported by the Independent Research Fund Denmark (IRFD) grant number 0165-00079B.

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Notes

  1. 1.

    Up to a constant due to the time for group operations and size for group elements in class groups being higher than those for DDH-hard groups based on elliptic curves.

  2. 2.

    In coding-theoretic this set is the dual code to the Reed-Solomon code formed by the evaluations of polynomials of degree \(\le d\).

  3. 3.

    As well as for using the ZK proof protocol from [6] which we show in next section.

  4. 4.

    The proofs were in fact introduced for slightly more involved relations, but for simplicity we adapt them for just proving knowledge of discrete logarithm.

  5. 5.

    Notation: To avoid confusion with the group \(G^q\) of q-th powers of elements from G, we denote the direct product of m copies of G, for \(m\in \mathbb {N}\), as \((G)^m\).

  6. 6.

    Recall, that by definition of \(\textrm{Lag}_{}\), \(L_i(X)=\prod _{j\in \mathcal {T}'\setminus \{i\}}\frac{X-\alpha _j}{\alpha _i-\alpha _j}\).

  7. 7.

    In practice we consider \(\kappa =40\) is reasonable.

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Authors and Affiliations

  1. IMDEA Software Institute, Madrid, Spain

    Ignacio Cascudo

  2. IT University of Copenhagen, Copenhagen, Denmark

    Bernardo David

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Correspondence to Ignacio Cascudo .

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  1. Zama, Paris, France

    Marc Joye

  2. Ruhr University Bochum, Bochum, Germany

    Gregor Leander

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Cascudo, I., David, B. (2024). Publicly Verifiable Secret Sharing Over Class Groups and Applications to DKG and YOSO. In: Joye, M., Leander, G. (eds) Advances in Cryptology – EUROCRYPT 2024. EUROCRYPT 2024. Lecture Notes in Computer Science, vol 14655. Springer, Cham. https://doi.org/10.1007/978-3-031-58740-5_8

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