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Efficient Protocols for Oblivious Linear Function Evaluation from Ring-LWE

Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 12238)

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 ab, 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.

Notes

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.

References

  1. 1.
    Lattigo 1.3.1 (2020). http://github.com/ldsec/lattigo. EPFL-LDS
  2. 2.
    Albrecht, M.R., Player, R., Scott, S.: On the concrete hardness of learning with errors. J. Math. Cryptol. 9(3), 169–203 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Applebaum, B., Damgård, I., Ishai, Y., Nielsen, M., Zichron, L.: Secure arithmetic computation with constant computational overhead. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 223–254. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_8CrossRefGoogle Scholar
  4. 4.
    Baum, C., Bootle, J., Cerulli, A., del Pino, R., Groth, J., Lyubashevsky, V.: Sub-linear lattice-based zero-knowledge arguments for arithmetic circuits. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 669–699. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96881-0_23CrossRefGoogle Scholar
  5. 5.
    Baum, C., Damgård, I., Larsen, K.G., Nielsen, M.: How to prove knowledge of small secrets. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 478–498. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53015-3_17CrossRefGoogle Scholar
  6. 6.
    Baum, C., Damgård, I., Lyubashevsky, V., Oechsner, S., Peikert, C.: More efficient commitments from structured lattice assumptions. In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 368–385. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-98113-0_20CrossRefGoogle Scholar
  7. 7.
    Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-46766-1_34CrossRefGoogle Scholar
  8. 8.
    Bendlin, R., Damgård, I., Orlandi, C., Zakarias, S.: Semi-homomorphic encryption and multiparty computation. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 169–188. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-20465-4_11CrossRefGoogle Scholar
  9. 9.
    Boneh, D., Ishai, Y., Sahai, A., Wu, D.J.: Lattice-based SNARGs and their application to more efficient obfuscation. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 247–277. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_9CrossRefGoogle Scholar
  10. 10.
    Boyle, E., Couteau, G., Gilboa, N., Ishai, Y.: Compressing vector OLE. In: Lie, D., Mannan, M., Backes, M., Wang, X. (eds.) ACM CCS 2018, pp. 896–912. ACM Press (2018)Google Scholar
  11. 11.
    Boyle, E., et al.: Efficient two-round OT extension and silent non-interactive secure computation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019, pp. 291–308. ACM Press (2019)Google Scholar
  12. 12.
    Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 509–539. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53018-4_19CrossRefGoogle Scholar
  13. 13.
    Boyle, E., Kohl, L., Scholl, P.: Homomorphic secret sharing from lattices without FHE. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 3–33. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-17656-3_1CrossRefGoogle Scholar
  14. 14.
    Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_29CrossRefGoogle Scholar
  15. 15.
    Cramer, R., Damgård, I., Xing, C., Yuan, C.: Amortized complexity of zero-knowledge proofs revisited: achieving linear soundness slack. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 479–500. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_17CrossRefGoogle Scholar
  16. 16.
    Damgård, I., Pastro, V., Smart, N., Zakarias, S.: Multiparty computation from somewhat homomorphic encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 643–662. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_38CrossRefGoogle Scholar
  17. 17.
    Damgård, I.B., Pedersen, T.P., Pfitzmann, B.: On the existence of statistically hiding bit commitment schemes and fail-stop signatures. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 250–265. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-48329-2_22CrossRefGoogle Scholar
  18. 18.
    Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky encryption and its applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53015-3_4CrossRefGoogle Scholar
  19. 19.
    Döttling, N., Ghosh, S., Nielsen, J.B., Nilges, T., Trifiletti, R.: TinyOLE: efficient actively secure two-party computation from oblivious linear function evaluation. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 2263–2276. ACM Press (2017)Google Scholar
  20. 20.
    Genkin, D., Ishai, Y., Prabhakaran, M., Sahai, A., Tromer, E.: Circuits resilient to additive attacks with applications to secure computation. In: Shmoys, D.B. (ed.) 46th ACM STOC, pp. 495–504. ACM Press (2014)Google Scholar
  21. 21.
    Ghosh, S., Nielsen, J.B., Nilges, T.: Maliciously secure oblivious linear function evaluation with constant overhead. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 629–659. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70694-8_22CrossRefGoogle Scholar
  22. 22.
    Ghosh, S., Nilges, T.: An algebraic approach to maliciously secure private set intersection. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 154–185. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-17659-4_6CrossRefGoogle Scholar
  23. 23.
    Ishai, Y., Prabhakaran, M., Sahai, A.: Secure arithmetic computation with no honest majority. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 294–314. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-00457-5_18CrossRefGoogle Scholar
  24. 24.
    Keller, M., Pastro, V., Rotaru, D.: Overdrive: making SPDZ great again. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 158–189. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78372-7_6CrossRefGoogle Scholar
  25. 25.
    Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_43CrossRefGoogle Scholar
  26. 26.
    Lyubashevsky, V., Peikert, C., Regev, O.: A toolkit for ring-LWE cryptography. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 35–54. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_3CrossRefGoogle Scholar
  27. 27.
    Mohassel, P., Zhang, Y.: SecureML: a system for scalable privacy-preserving machine learning. In: 2017 IEEE Symposium on Security and Privacy, pp. 19–38. IEEE Computer Society Press (2017)Google Scholar
  28. 28.
    Naor, M., Pinkas, B.: Oblivious transfer and polynomial evaluation. In: 31st ACM STOC, pp. 245–254. ACM Press (1999)Google Scholar
  29. 29.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th ACM STOC, pp. 84–93. ACM Press (2005)Google Scholar
  30. 30.
    Schoppmann, P., Gascón, A., Reichert, L., Raykova, M.: Distributed vector-OLE: improved constructions and implementation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019, pp. 1055–1072. ACM Press (2019)Google Scholar
  31. 31.
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Cryptogr. 71(1), 57–81 (2012).  https://doi.org/10.1007/s10623-012-9720-4CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Aarhus UniversityAarhusDenmark
  2. 2.University of VigoVigoSpain
  3. 3.EPFLLausanneSwitzerland

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