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Crypto Dark Matter on the Torus

Oblivious PRFs from Shallow PRFs and TFHE

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Advances in Cryptology – EUROCRYPT 2024 (EUROCRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14656))

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Abstract

Partially Oblivious Pseudorandom Functions (POPRFs) are 2-party protocols that allow a client to learn pseudorandom function (PRF) evaluations on inputs of its choice from a server. The client submits two inputs, one public and one private. The security properties ensure that the server cannot learn the private input, and the client cannot learn more than one evaluation per POPRF query. POPRFs have many applications including password-based key exchange and privacy-preserving authentication mechanisms. However, most constructions are based on classical assumptions, and those with post-quantum security suffer from large efficiency drawbacks.

In this work, we construct a novel POPRF from lattice assumptions and the “Crypto Dark Matter” PRF candidate (TCC’18) in the random oracle model. At a conceptual level, our scheme exploits the alignment of this family of PRF candidates, relying on mixed modulus computations, and programmable bootstrapping in the torus fully homomorphic encryption scheme (TFHE). We show that our construction achieves malicious client security based on circuit-private FHE, and client privacy from the semantic security of the FHE scheme. We further explore a heuristic approach to extend our scheme to support verifiability, based on the difficulty of computing cheating circuits in low depth. This would yield a verifiable (P)OPRF. We provide a proof-of-concept implementation and preliminary benchmarks of our construction. For the core online OPRF functionality, we require amortised 10.0 KB communication per evaluation and a one-time per-client setup communication of 2.5 MB.

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Notes

  1. 1.

    The security of the PRF candidate in [11] rests on the absence of any low-degree polynomial interpolating it, ruling out efficient implementations using FHE schemes that only provide additions and multiplications.

  2. 2.

    These are: plaintext moduli that are not powers of two, circuit privacy, ciphertext and bootstrapping-key compression.

  3. 3.

    We note that while the large sizes for achieving malicious security in [2] can be avoided using improved NIZKs, the semi-honest base size of 2 MB per query stems from requiring \(q\approx 2^{256}\) for statistical correctness and security arguments.

  4. 4.

    We note that quantum algorithms offer only marginal, i.e. less than square-root, speedups here [3].

  5. 5.

    This allows to restrict the number of sequential bootstrappings that can be performed.

  6. 6.

    As it stands, this strong PRF candidate maps zero to zero and is thus trivially distinguished from a random function, we will address this below.

  7. 7.

    If the reader’s PDF viewer does not support PDF attachments (e.g. Preview on MacOS does not), then e.g. pdfdetach can be used to extract these files.

  8. 8.

    Note that we may pick \(q \ne 3\) but require \(p=2\).

  9. 9.

    All of these figures assume that we apply public-key compression.

  10. 10.

    \(F_\textsf{poprf}.\textsf{Request}\) runs in \(28.9\) ms and \(F_\textsf{poprf}.\textsf{Finalise}\) in \(0.2\) ms in our benchmarks. The former does not include the time to prove well-formedness, but – as below – we do not expect this to radically change this picture.

  11. 11.

    Table 3 of [40] reports an overhead of 5x to 10x for circuit privacy.

  12. 12.

    On a single core, server blind evaluation took \(7.1\) s.

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Acknowledgements

We thank Christian Weinert for discussing MPC approaches with us; Ilaria Chillotti, Ben Curtis and Jean-Baptiste Orfila for answering questions about TFHE and tfhe-rs; Nicolas Gama for answering questions about TFHE. We thank Ward Beullens and Gregor Seiler for answering questions about LaBRADOR and sharing their size estimation Pari/GP script.

The research of Martin Albrecht was supported by UKRI grants EP/S02-0330/1, EP/-S02087X/1, EP/Y02432X/1 and by the European Union Horizon 2020 Research and Innovation Program Grant 780701. The research of Alex Davidson was supported by NOVA LINCS (UIDB/04516/2020), with the financial support of FCT.IP. Part of this work was done while Martin Albrecht was at Royal Holloway, University of London, while Alex Davidson was at Brave Software, and while Amit Deo was at Crypto Quantique.

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Authors and Affiliations

  1. King’s College London, London, UK

    Martin R. Albrecht

  2. SandboxAQ, New York, USA

    Martin R. Albrecht

  3. NOVA LINCS & DI, FCT, Universidade NOVA de Lisboa, Lisbon, Portugal

    Alex Davidson

  4. Zama, Paris, France

    Amit Deo

  5. University of Surrey, Guildford, UK

    Daniel Gardham

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  1. Martin R. Albrecht
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  2. Alex Davidson
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Correspondence to Martin R. Albrecht .

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  1. Zama, Paris, France

    Marc Joye

  2. Ruhr University Bochum, Bochum, Germany

    Gregor Leander

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Albrecht, M.R., Davidson, A., Deo, A., Gardham, D. (2024). Crypto Dark Matter on the Torus. In: Joye, M., Leander, G. (eds) Advances in Cryptology – EUROCRYPT 2024. EUROCRYPT 2024. Lecture Notes in Computer Science, vol 14656. Springer, Cham. https://doi.org/10.1007/978-3-031-58751-1_16

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