Overdrive2k: Efficient Secure MPC over \(\mathbb {Z}_{2^k}\) from Somewhat Homomorphic Encryption

  • Emmanuela Orsini
  • Nigel P. SmartEmail author
  • Frederik Vercauteren
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12006)


Recently, Cramer et al. (CRYPTO 2018) presented a protocol, \(\mathsf {SPDZ2k}\), for actively secure multiparty computation for dishonest majority in the pre-processing model over the ring \(\mathbb {Z}_{2^k}\), instead of over a prime field \(\mathbb {F}_p\). Their technique used oblivious transfer for the pre-processing phase, more specifically the MASCOT protocol (Keller et al. CCS 2016). In this paper we describe a more efficient technique for secure multiparty computation over \(\mathbb {Z}_{2^k}\) based on somewhat homomorphic encryption. In particular we adapt the Overdrive approach (Keller et al. EUROCRYPT 2018) to obtain a protocol which is more like the original SPDZ protocol (Damgård et al. CRYPTO 2012). To accomplish this we introduce a special packing technique for the BGV encryption scheme operating on the plaintext space defined by the \(\mathsf {SPDZ2k}\) protocol, extending the ciphertext packing method used in SPDZ to the case of \(\mathbb {Z}_{2^k}\). We also present a more complete pre-processing phase for secure computation modulo \(2^k\) by adding a new technique to produce shared random bits.



We thank Cyprien Delpech de Saint Guilhem for many helpful discussions. This work has been supported in part by ERC Advanced Grant ERC-2015-AdG-IMPaCT, by the Defense Advanced Research Projects Agency (DARPA) and Space and Naval Warfare Systems Center, Pacific (SSC Pacific) under contract No. N66001-15-C-4070, and by the FWO under an Odysseus project GOH9718N.


  1. 1.
    Aly, A., et al.: SCALE-MAMBA v1.6: Documentation (2019).
  2. 2.
    Baum, C., Cozzo, D., Smart, N.P.: Using TopGear in Overdrive: a more efficient ZKPoK for SPDZ. Cryptology ePrint Archive, Report 2019/035 (2019).
  3. 3.
    Beaver, D.: Foundations of secure interactive computing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 377–391. Springer, Heidelberg (1992). Scholar
  4. 4.
    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). Scholar
  5. 5.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, pp. 309–325. ACM, New York (2012)Google Scholar
  6. 6.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky, R. (ed.) 52nd FOCS, pp. 97–106. IEEE Computer Society Press, October 2011Google Scholar
  7. 7.
    Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: 42nd FOCS, pp. 136–145. IEEE Computer Society Press, October 2001Google Scholar
  8. 8.
    Cassels, J.W.: Local Fields. Cambridge University Press, Cambridge (1986)CrossRefGoogle Scholar
  9. 9.
    Cramer, R., Damgård, I.: On the amortized complexity of zero-knowledge protocols. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 177–191. Springer, Heidelberg (2009). Scholar
  10. 10.
    Cramer, R., Damgård, I., Escudero, D., Scholl, P., Xing, C.: SPD\(\mathbb{Z}_{2^k}\): efficient MPC mod \(2^k\) for dishonest majority. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 769–798. Springer, Cham (2018). Scholar
  11. 11.
    Damgård, I., Escudero, D., Frederiksen, T.K., Keller, M., Scholl, P., Volgushev, N.: New primitives for actively-secure MPC over rings with applications to private machine learning. In: 2019 IEEE Symposium on Security and Privacy, pp. 1102–1120. IEEE Computer Society Press, May 2019Google Scholar
  12. 12.
    Damgård, I., Keller, M., Larraia, E., Pastro, V., Scholl, P., Smart, N.P.: Practical covertly secure MPC for dishonest majority – or: breaking the SPDZ limits. In: Crampton, J., Jajodia, S., Mayes, K. (eds.) ESORICS 2013. LNCS, vol. 8134, pp. 1–18. Springer, Heidelberg (2013). Scholar
  13. 13.
    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). Scholar
  14. 14.
    Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 1–16. Springer, Heidelberg (2012). Scholar
  15. 15.
    Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 465–482. Springer, Heidelberg (2012). Scholar
  16. 16.
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012). Scholar
  17. 17.
    Halevi, S., Shoup, V.: Algorithms in HElib. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 554–571. Springer, Heidelberg (2014). Scholar
  18. 18.
    Keller, M., Orsini, E., Scholl, P.: MASCOT: faster malicious arithmetic secure computation with oblivious transfer. In: Weippl, E.R., Katzenbeisser, S., Kruegel, C., Myers, A.C., Halevi, S. (eds.) ACM CCS 2016, pp. 830–842. ACM Press, New York (2016)Google Scholar
  19. 19.
    Keller, M., Pastro, V., Rotaru, D.: Overdrive: making SPDZ great again. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part III. LNCS, vol. 10822, pp. 158–189. Springer, Cham (2018). Scholar
  20. 20.
    Nielsen, J.B., Nordholt, P.S., Orlandi, C., Burra, S.S.: A new approach to practical active-secure two-party computation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 681–700. Springer, Heidelberg (2012). Scholar
  21. 21.
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Crypt. 71(1), 57–81 (2014)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.imec-COSICKU LeuvenLeuvenBelgium
  2. 2.University of BristolBristolUK

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