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Classical Binding for Quantum Commitments

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Theory of Cryptography (TCC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13042))

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Abstract

In classical commitments, statistical binding means that for almost any commitment transcript there is at most one possible opening. While quantum commitments (for classical messages) sometimes have benefits over their classical counterparts (e.g. in terms of assumptions), they provide a weaker notion of binding. Essentially that the sender cannot open a given commitment to a random value with probability noticeably greater than 1/2.

We introduce a notion of classical binding for quantum commitments which provides guarantees analogous to the classical case. In our notion, the receiver performs a (partial) measurement of the quantum commitment string, and the outcome of this measurement determines a single value that the sender may open. We expect that our notion can replace classical commitments in various settings, leaving the security proof essentially unchanged. As an example we show a soundness proof for the GMW zero-knowledge proof system.

We construct a non-interactive quantum commitment scheme which is classically statistically-binding and has a classical opening, based on the existence of any post-quantum one-way function. Prior candidates had inherently quantum openings and were not classically binding. In contrast, we show that it is impossible to achieve classical binding for statistically hiding commitments, regardless of assumption or round complexity.

Our scheme is simply Naor’s commitment scheme (which classically requires a common random string, CRS), but executed in superposition over all possible values of the CRS, and repeated several times. We hope that this technique for using quantum communication to remove a CRS may find other uses.

N. Bitansky—member of the checkpoint institute of information security. Supported by ISF grants 18/484 and 19/2137, by Len Blavatnik and the Blavatnik Family Foundation, by the Blavatnik Interdisciplinary Cyber Research Center at Tel Aviv University, and by the European Union Horizon 2020 Research and Innovation Program via ERC Project REACT (Grant 756482).

Z. Brakerski—supported by the Binational Science Foundation (Grant No. 2016726), and by the European Union Horizon 2020 Research and Innovation Program via ERC Project REACT (Grant 756482) and via Project PROMETHEUS (Grant 780701).

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Notes

  1. 1.

    Alternatively, it suffices to perform an operation that is equivalent to a measurement from the viewpoint of an adversary, such as copying (via CNOT operation) the “measured” value into a space that is inaccessible by the adversary.

  2. 2.

    The proof there considers classical distinguishers (and inverters), but is known to extend to the quantum setting.

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Acknowledgements

We thank the reviewers of TCC 2021 for their comments.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Nir Bitansky

  2. Weizmann Institute of Science, Rehovot, Israel

    Zvika Brakerski

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  1. Nir Bitansky
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Editor information

Editors and Affiliations

  1. Georgetown University, Washington, WA, USA

    Kobbi Nissim

  2. The University of Texas at Austin, Austin, TX, USA

    Brent Waters

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Bitansky, N., Brakerski, Z. (2021). Classical Binding for Quantum Commitments. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13042. Springer, Cham. https://doi.org/10.1007/978-3-030-90459-3_10

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  • DOI: https://doi.org/10.1007/978-3-030-90459-3_10

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