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Constrained Pseudorandom Functions from Homomorphic Secret Sharing

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14006))

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Abstract

We propose and analyze a simple strategy for constructing 1-key constrained pseudorandom functions (CPRFs) from homomorphic secret sharing. In the process, we obtain the following contributions: first, we identify desirable properties for the underlying HSS scheme for our strategy to work. Second, we show that (most of) recent existing HSS schemes satisfy these properties, leading to instantiations of CPRFs for various constraints and from various assumptions. Notably, we obtain the first (1-key selectively secure, private) CPRFs for inner-product and (1-key selectively secure) CPRFs for \(\textsf{NC}^1\) from the DCR assumption, and more. Last, we revisit two applications of HSS equipped with these additional properties to secure computation: we obtain secure computation in the silent preprocessing model with one party being able to precompute its whole preprocessing material before even knowing the other party, and we construct one-sided statistically secure computation with sublinear communication for restricted forms of computation.

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Notes

  1. 1.

    The inverse condition is often used (pseudorandomness if \(C(x) = 0\) and partial evaluation if \(C(x) = 1\)). Our choice slightly simplifies our constructions.

  2. 2.

    It is not too hard to see that having both parties execute the bulk of the computation prior to interacting (while keeping a non-cryptographic online phase) is impossible.

  3. 3.

    If the key could depend on C, one could just generate two independent PRF keys \(k_0,k_1\) and define the evaluation as \(F_{k_{C(x)}}(x)\). Revealing \(k_0\) then allows to compute the evaluation on any x such that \(C(x) = 0\) and reveals nothing about the key \(k_1\) used when \(C(x) = 1\).

  4. 4.

    This scheme does not yield CPRFs as it does not achieve statistical correctness, but staged-HSS is easily illustrated with it.

  5. 5.

    Actually of x and \(x \cdot s_i\)’s for each bit \(s_i\) of s.

  6. 6.

    s is encrypted bit-by-bit in the actual construction.

  7. 7.

    An arithmetic circuit is layered if its nodes can be partitioned into layers, such that any wire connects adjacent layers.

  8. 8.

    In the remaining of the paper, we drop the \(\lambda \) subscript when it is clear from context.

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Acknowledgments

We thank the anonymous reviewers of Eurocrypt 2023. Geoffroy Couteau was supported by the French ANR SCENE (ANR-20-CE39-0001) and the PEPR Cyber France 2030 programme (ANR-22-PECY-0003). Pierre Meyer was supported by ERC Project HSS (852952). Alain Passelègue and Mahshid Riahinia were supported by the French ANR RAGE project (ANR-20-CE48-0011) and the PEPR Cyber France 2030 programme (ANR-22-PECY-0003).

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Authors and Affiliations

  1. Université Paris Cité, CNRS, IRIF, Paris, France

    Geoffroy Couteau & Pierre Meyer

  2. Reichman University, Herzliya, Israel

    Pierre Meyer

  3. Inria, Paris, France

    Alain Passelègue

  4. ENS de Lyon, Laboratoire LIP (U. Lyon, CNRS, ENSL, Inria, UCBL), Paris, France

    Alain Passelègue & Mahshid Riahinia

Authors
  1. Geoffroy Couteau
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Correspondence to Pierre Meyer .

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Editors and Affiliations

  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

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Couteau, G., Meyer, P., Passelègue, A., Riahinia, M. (2023). Constrained Pseudorandom Functions from Homomorphic Secret Sharing. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_7

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  • DOI: https://doi.org/10.1007/978-3-031-30620-4_7

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