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Fair and Sound Secret Sharing from Homomorphic Time-Lock Puzzles

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Provable and Practical Security (ProvSec 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12505))

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Abstract

Achieving fairness and soundness in non-simultaneous rational secret sharing schemes has proved to be challenging. On the one hand, soundness can be ensured by providing side information related to the secret as a check, but on the other, this can be used by deviant players to compromise fairness. To overcome this, the idea of incorporating a time delay was suggested in the literature: in particular, time-delay encryption based on memory-bound functions has been put forth as a solution. In this paper, we propose a different approach to achieve such delay, namely using homomorphic time-lock puzzles (HTLPs), introduced at CRYPTO 2019, and construct a fair and sound rational secret sharing scheme in the non-simultaneous setting from HTLPs.

HTLPs are used to embed sub-shares of the secret for a predetermined time. This allows to restore fairness of the secret reconstruction phase, despite players having access to information related to the secret which is required to ensure the soundness of the scheme. Key to our construction is the fact that the time-lock puzzles are homomorphic so that players can compactly evaluate sub-shares. Without this efficiency improvement, players would have to independently solve each puzzle sent from the other players to obtain a share of the secret, which would be computationally inefficient. We argue that achieving both fairness and soundness in a non-simultaneous scheme using a time delay based on CPU-bound functions rather than memory-bound functions is more cost-effective and realistic in relation to the implementation of the construction.

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Notes

  1. 1.

    See Appendices D.1, and D.2 for further discussion on payoff functions and equilibrium concepts.

  2. 2.

    A CMBF is a family of deterministic algorithms such that an efficiently generated key can decrypt the encrypted input, with a lower-bound on the number of memory-access steps to do so.

  3. 3.

    Note that [30] works under the assumption that players prefer everyone to obtain the correct output over misleading others, therefore soundness is not an issue that needs to be addressed.

  4. 4.

    Privacy and authentication of the distribution of shares is a standard cryptographic assumption in secret sharing schemes [34].

  5. 5.

    Whilst not explicit in the definition, there is an upper bound on how long players can communicate their shares for. Therefore, at the end of their communication, if a player \(P_i\) has not obtained a sufficient number of shares, then they output \(\bot \) at the end of the reconstruction phase.

  6. 6.

    What \(\varPsi \) is depends on the application the HTLP is being used for. It could be addition, multiplication or XOR for example.

  7. 7.

    We call the HTLP encryption of the sub-shares sub-puzzles for ease of understanding. They are simply time-lock puzzles that can be homomorphically evaluated to obtain a puzzle of the share which corresponds to the homomorphic evaluation of the given sub-shares.

  8. 8.

    At least one round of communication is required before players can start processing.

  9. 9.

    In a standard TLP scheme, the computational complexity of the puzzle-solving algorithm P.Solve is the same as HP.Solve.

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Authors and Affiliations

  1. Information Security Group, Royal Holloway, University of London, London, UK

    Jodie Knapp & Elizabeth A. Quaglia

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  1. Jodie Knapp
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Correspondence to Jodie Knapp or Elizabeth A. Quaglia .

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Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Khoa Nguyen

  2. Chinese Academy of Sciences, Beijing, China

    Wenling Wu

  3. Nanyang Technological University, Singapore, Singapore

    Kwok Yan Lam

  4. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Knapp, J., Quaglia, E.A. (2020). Fair and Sound Secret Sharing from Homomorphic Time-Lock Puzzles. In: Nguyen, K., Wu, W., Lam, K.Y., Wang, H. (eds) Provable and Practical Security. ProvSec 2020. Lecture Notes in Computer Science(), vol 12505. Springer, Cham. https://doi.org/10.1007/978-3-030-62576-4_17

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  • DOI: https://doi.org/10.1007/978-3-030-62576-4_17

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