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The Superlinearity Problem in Post-quantum Blockchains

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Financial Cryptography and Data Security (FC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13950))

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Abstract

The proof of work mechanism by which many blockchain-based protocols achieve consensus may be undermined by the use of quantum computing in mining—even when all cryptographic primitives are replaced with post-quantum secure alternatives. First, we offer an impossibility result: we prove that quantum (Grover) speedups in solving a large, natural class of proof-of-work puzzles cause an inevitable incentive incompatibility in mining, by distorting the reward structure of mining in proof-of-work-based protocols such as Bitcoin. We refer to such distortion as the Superlinearity Problem. Our impossibility result suggests that for robust post-quantum proof-of-work-based consensus, we may need to look beyond standard cryptographic models. We thus propose a proof-of-work design in a random-beacon model, which is tailored to bypass the earlier impossibility. We conclude with a discussion of open problems, and of the challenges of integrating our new proof-of-work scheme into decentralised consensus protocols under realistic conditions.

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Notes

  1. 1.

    Classical means computing without quantum computers.

  2. 2.

    Classical network takeover attacks are also possible with the collusion of much less than half of mining power [24]. The Quantum Superlinearity Problem worsens those attacks too: basically, an attack that requires a certain fraction of classical mining power may require a much smaller fraction of quantum mining power.

  3. 3.

    E.g., verifiable random functions and verifiable delay functions.

  4. 4.

    This is not strictly true: the left hand side is bounded by 1 whereas t grows without bound. A refined definition of proportionality (Definition 4) handles this issue.

  5. 5.

    This rules out deterministic proofs of work (whose reward functions are 0–1).

  6. 6.

    The upper bound is necessary since probabilities are upper-bounded by 1.

  7. 7.

    Jakobsson and Juels proposed a definition that includes interactive protocols [29].

  8. 8.

    Stebila et al. proposed a definition where verification is keyed [49].

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Acknowledgments

We are grateful to Thaddeus Dryja for the conversation that sparked this research, and to Chris Peikert for a helpful discussion at early stages of the work.

SP’s work on this project was supported by a 2021–22 Computing Innovation Fellowship, funded by the National Science Foundation under Grant #2127309 to the Computing Research Association, by Cornell Tech’s Digital Life Initiative, and by the MIT Media Lab’s Digital Currency Initiative.

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Authors and Affiliations

  1. Columbia University, New York, USA

    Sunoo Park

  2. University of Warwick, Coventry, UK

    Nicholas Spooner

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Correspondence to Sunoo Park or Nicholas Spooner .

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  1. George Mason University, Fairfax, VA, USA

    Foteini Baldimtsi

  2. University of Bern, Bern, Switzerland

    Christian Cachin

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Park, S., Spooner, N. (2024). The Superlinearity Problem in Post-quantum Blockchains. In: Baldimtsi, F., Cachin, C. (eds) Financial Cryptography and Data Security. FC 2023. Lecture Notes in Computer Science, vol 13950. Springer, Cham. https://doi.org/10.1007/978-3-031-47754-6_12

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