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Efficient Protocols for Oblivious Linear Function Evaluation from Ring-LWE

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Security and Cryptography for Networks (SCN 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12238))

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Abstract

An oblivious linear function evaluation protocol, or OLE, is a two-party protocol for the function \(f(x) = ax + b\), where a sender inputs the field elements a, b, and a receiver inputs x and learns f(x). OLE can be used to build secret-shared multiplication, and is an essential component of many secure computation applications including general-purpose multi-party computation, private set intersection and more.

In this work, we present several efficient OLE protocols from the ring learning with errors (RLWE) assumption. Technically, we build two new passively secure protocols, which build upon recent advances in homomorphic secret sharing from (R)LWE (Boyle et al., Eurocrypt 2019), with optimizations tailored to the setting of OLE. We upgrade these to active security using efficient amortized zero-knowledge techniques for lattice relations (Baum et al., Crypto 2018), and design new variants of zero-knowledge arguments that are necessary for some of our constructions.

Our protocols offer several advantages over existing constructions. Firstly, they have the lowest communication complexity amongst previous, practical protocols from RLWE and other assumptions; secondly, they are conceptually very simple, and have just one round of interaction for the case of OLE where b is randomly chosen. We demonstrate this with an implementation of one of our passively secure protocols, which can perform more than 1 million OLEs per second over the ring \(\mathbb {Z}_m\), for a 120-bit modulus m, on standard hardware.

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Notes

  1. 1.

    Our protocols do not directly use the decryption algorithm, but our simulator in the proof of Theorem 2 does.

  2. 2.

    This optimization is possible since we skip the “modulus lifting” step from  [13], which is only needed when doing several repeated multiplications.

  3. 3.

    This restriction can be easily overcome by modifying the definition of security against passive adversaries, allowing the adversary to choose its output. However, we prefer to stick to more standard security definitions.

  4. 4.

    That is, the bounds in one of the two relations have an extra factor of \(\tau \). This corresponds to what can be guaranteed for a corrupt party via the zero knowledge argument.

  5. 5.

    As in public-key protocol from Sect. 3.2, \({\mathcal {P}_{\mathsf {Alice}}}\) does not need to prove the bound on her input \({\varvec{v}}\).

  6. 6.

    We have used the LWE security estimator of Albrecht et al.  [2] (available online in https://bitbucket.org/malb/lwe-estimator.) to give bit-security estimates.

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Acknowledgements

We thank the anonymous reviewers for comments which helped to improve the paper. This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 669255 (MPCPRO), the Danish Independent Research Council under Grant-ID DFF–6108-00169 (FoCC), an Aarhus University Research Foundation starting grant, the Xunta de Galicia & ERDF under projects ED431G2019/08 and Grupo de Referencia ED431C2017/53, and by the grant #2017-201 (DPPH) of the Strategic Focal Area “Personalized Health and Related Technologies (PHRT)” of the ETH Domain.

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Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Carsten Baum, Daniel Escudero & Peter Scholl

  2. University of Vigo, Vigo, Galicia, Spain

    Alberto Pedrouzo-Ulloa

  3. EPFL, Lausanne, Switzerland

    Juan Ramón Troncoso-Pastoriza

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  1. Carsten Baum
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Correspondence to Peter Scholl .

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  1. Università degli Studi di Salerno, Fisciano, Italy

    Clemente Galdi

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Baum, C., Escudero, D., Pedrouzo-Ulloa, A., Scholl, P., Troncoso-Pastoriza, J.R. (2020). Efficient Protocols for Oblivious Linear Function Evaluation from Ring-LWE. In: Galdi, C., Kolesnikov, V. (eds) Security and Cryptography for Networks. SCN 2020. Lecture Notes in Computer Science(), vol 12238. Springer, Cham. https://doi.org/10.1007/978-3-030-57990-6_7

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  • DOI: https://doi.org/10.1007/978-3-030-57990-6_7

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