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One-Way Functions Imply Secure Computation in a Quantum World

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Advances in Cryptology – CRYPTO 2021 (CRYPTO 2021)

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Abstract

We prove that quantum-hard one-way functions imply simulation-secure quantum oblivious transfer (QOT), which is known to suffice for secure computation of arbitrary quantum functionalities. Furthermore, our construction only makes black-box use of the quantum-hard one-way function.

Our primary technical contribution is a construction of extractable and equivocal quantum bit commitments based on the black-box use of quantum-hard one-way functions in the standard model. Instantiating the Crépeau-Kilian (FOCS 1988) framework with these commitments yields simulation-secure QOT.

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Notes

  1. 1.

    It is important to note that extraction and equivocation are only made possible in an ideal world where a simulator has access to the adversary’s state. Participants in the real protocol cannot access each others’ state, which prevents them from extracting or equivocating.

  2. 2.

    Naive approaches to removing one direction of quantum communication appear to require the honest parties to be entangled and subsequently perform quantum teleportation.

  3. 3.

    [19] point out that the conclusions of Steps 1 and 2 together had also been established in prior works of [8, 13, 33].

  4. 4.

    We refer the reader to Sect. 6.1 for a formal definition of simulation-based QOT.

  5. 5.

    In slightly more detail, Naor’s commitment scheme makes black-box use of any pseudo-random generator (PRG). It is straightforward to verify that if the PRG is post-quantum secure, the commitment satisfies computational hiding against quantum attackers. A black-box construction of pseudo-random generators from one-way functions is due to [21]; Aaronson [1] and Zhandry [38] observed that [21] applies to non-uniform quantum attackers with classical advice. This can be extended to handle non-uniform quantum advice by giving the one-way function attacker constructed in the [21] reduction many copies of the PRG attacker’s non-uniform quantum advice (which only requires some polynomial upper bound on the number of times the reduction invokes the PRG attacker).

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Acknowledgments

The authors are grateful to the Simons Institute programs on Lattices: Algorithms, Complexity and Cryptography, and The Quantum Wave in Computing for fostering this collaboration. Thanks also to Alex Grilo, Huijia Lin, Fang Song, and Vinod Vaikuntanathan for discussions about similarities and differences with [19]. A.C. is supported by DoE under grant DOD ONR Federal. This material is based on work supported in part by DARPA under Contract No. HR001120C0024 (for DK). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

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Authors and Affiliations

  1. UC Berkeley, Berkeley, USA

    James Bartusek & Andrea Coladangelo

  2. UIUC, Champaign, USA

    Dakshita Khurana

  3. Princeton University and NTT Research, Princeton, USA

    Fermi Ma

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  1. Columbia University, New York City, NY, USA

    Tal Malkin

  2. University of Michigan, Ann Arbor, MI, USA

    Chris Peikert

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Bartusek, J., Coladangelo, A., Khurana, D., Ma, F. (2021). One-Way Functions Imply Secure Computation in a Quantum World. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12825. Springer, Cham. https://doi.org/10.1007/978-3-030-84242-0_17

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