Abstract
The last ten years have seen a tremendous growth in the interest and practicality of secure multiparty computation (MPC) and its possible applications. Secure MPC is indeed a very hot research topic and recent advances in the field have already been translated into commercial products world-wide. A major pillar in this advance has been in the case of active security with a dishonest majority, mainly due to the SPDZ-line of work protocols. This survey gives an overview of these protocols, with a focus of the original SPDZ paper (Damgård et al. CRYPTO 2012) and its subsequent optimizations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
OT is a fundamental cryptographic primitive [63, 69]. In its classical formulation, a (one-out-of-two) oblivious transfer is a two-party protocol between a sender \(P_S\) and a receiver \(P_R\): \(P_S\) inputs two messages \(x_0,x_1\), \(P_R\) inputs a bit b, and the goal is for the receiver to learn \(x_b\) and nothing more, whilst the sender learns no information about b.
- 2.
Given a set S, a positive polynomial on S is such that \(p(x)>0\) for every \(x \in S\).
- 3.
Roughly, the LPN assumption says that given a random linear code C, a noisy random codeword of C is pseudo-random.
- 4.
This generalisation of oblivious transfer is also referred to as oblivious linear function evaluation (OLE) [57].
References
Aly, A., et al.: Scale - mamba v1.9: documentation
Asharov, G., Jain, A., López-Alt, A., Tromer, E., Vaikuntanathan, V., Wichs, D.: Multiparty computation with low communication, computation and interaction via threshold FHE. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 483–501. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_29
Asharov, G., Lindell, Y., Schneider, T., Zohner, M.: More efficient oblivious transfer and extensions for faster secure computation. In: Sadeghi, A.R., Gligor, V.D., Yung, M. (eds.) ACM CCS 2013, pp. 535–548. ACM Press, November 2013. https://doi.org/10.1145/2508859.2516738
Asharov, G., Lindell, Y., Schneider, T., Zohner, M.: More efficient oblivious transfer extensions with security for malicious adversaries. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 673–701. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_26
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_23
Baum, C., Cozzo, D., Smart, N.P.: Using TopGear in overdrive: a more efficient ZKPoK for SPDZ. In: Paterson, K.G., Stebila, D. (eds.) SAC 2019. LNCS, vol. 11959, pp. 274–302. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38471-5_12
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_17
Baum, C., Damgård, I., Toft, T., Zakarias, R.: Better preprocessing for secure multiparty computation. In: Manulis, M., Sadeghi, A.-R., Schneider, S. (eds.) ACNS 2016. LNCS, vol. 9696, pp. 327–345. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-39555-5_18
Baum, C., Orsini, E., Scholl, P.: Efficient secure multiparty computation with identifiable abort. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9985, pp. 461–490. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53641-4_18
Baum, C., Orsini, E., Scholl, P., Soria-Vazquez, E.: Efficient constant-round MPC with identifiable abort and public verifiability. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 562–592. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_20
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_34
Beaver, D.: Precomputing oblivious transfer. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 97–109. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-44750-4_8
Beaver, D.: Correlated pseudorandomness and the complexity of private computations. In: 28th ACM STOC, pp. 479–488. ACM Press, May 1996. https://doi.org/10.1145/237814.237996
Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: 22nd ACM STOC, pp. 503–513. ACM Press, May 1990. https://doi.org/10.1145/100216.100287
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: 20th ACM STOC, pp. 1–10. ACM Press, May 1988. https://doi.org/10.1145/62212.62213
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_11
Boneh, D., et al.: Threshold cryptosystems from threshold fully homomorphic encryption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 565–596. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_19
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, November 2019. https://doi.org/10.1145/3319535.3354255
Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Efficient pseudorandom correlation generators: silent OT extension and more. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 489–518. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_16
Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Efficient pseudorandom correlation generators from ring-LPN. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 387–416. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_14
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, pp. 309–325. ACM, January 2012. https://doi.org/10.1145/2090236.2090262
Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: 42nd FOCS, pp. 136–145. IEEE Computer Society Press, October 2001. https://doi.org/10.1109/SFCS.2001.959888
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: 20th ACM STOC, pp. 11–19. ACM Press, May 1988. https://doi.org/10.1145/62212.62214
Cleve, R.: Limits on the security of coin flips when half the processors are faulty (extended abstract). In: 18th ACM STOC, pp. 364–369. ACM Press, May 1986. https://doi.org/10.1145/12130.12168
Cohen, R., Lindell, Y.: Fairness versus guaranteed output delivery in secure multiparty computation. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8874, pp. 466–485. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45608-8_25
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). https://doi.org/10.1007/978-3-642-03356-8_11
Cunningham, R., Fuller, B., Yakoubov, S.: Catching MPC cheaters: identification and openability. In: Shikata, J. (ed.) ICITS 2017. LNCS, vol. 10681, pp. 110–134. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72089-0_7
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). https://doi.org/10.1007/978-3-642-40203-6_1
Damgård, I., Orlandi, C.: Multiparty computation for dishonest majority: from passive to active security at low cost. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 558–576. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_30
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_38
del Pino, R., Lyubashevsky, V.: Amortization with fewer equations for proving knowledge of small secrets. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10403, pp. 365–394. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63697-9_13
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, October/November 2017. https://doi.org/10.1145/3133956.3134024
Even, S., Goldreich, O., Lempel, A.: A randomized protocol for signing contracts. Commun. ACM 28(6), 637–647 (1985)
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, May/Jun 2014. https://doi.org/10.1145/2591796.2591861
Genkin, D., Ishai, Y., Weiss, M.: Binary AMD circuits from secure multiparty computation. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9985, pp. 336–366. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53641-4_14
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_22
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or A completeness theorem for protocols with honest majority. In: Aho, A. (ed.) 19th ACM STOC, pp. 218–229. ACM Press, May 1987. https://doi.org/10.1145/28395.28420
Goldwasser, S., Lindell, Y.: Secure multi-party computation without agreement. J. Cryptol. 18(3), 247–287 (2005). https://doi.org/10.1007/s00145-005-0319-z
Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)
Hastings, M., Hemenway, B., Noble, D., Zdancewic, S.: SoK: general purpose compilers for secure multi-party computation. In: 2019 IEEE Symposium on Security and Privacy, pp. 1220–1237. IEEE Computer Society Press, May 2019. https://doi.org/10.1109/SP.2019.00028
Hazay, C., Venkitasubramaniam, M., Weiss, M.: The price of active security in cryptographic protocols. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12106, pp. 184–215. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_7
Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: 21st ACM STOC, pp. 44–61. ACM Press, May 1989. https://doi.org/10.1145/73007.73012
Ishai, Y., Kilian, J., Nissim, K., Petrank, E.: Extending oblivious transfers efficiently. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 145–161. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_9
Ishai, Y., Ostrovsky, R., Zikas, V.: Secure multi-party computation with identifiable abort. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 369–386. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_21
Ishai, Y., Prabhakaran, M., Sahai, A.: Founding cryptography on oblivious transfer – efficiently. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 572–591. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_32
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_18
Keller, M.: MP-SPDZ: a versatile framework for multi-party computation. IACR Cryptology ePrint Archive 2020, 521 (2020)
Keller, M., Orsini, E., Rotaru, D., Scholl, P., Soria-Vazquez, E., Vivek, S.: Faster secure multi-party computation of AES and DES using lookup tables. In: Gollmann, D., Miyaji, A., Kikuchi, H. (eds.) ACNS 2017. LNCS, vol. 10355, pp. 229–249. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61204-1_12
Keller, M., Orsini, E., Scholl, P.: Actively secure OT extension with optimal overhead. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 724–741. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_35
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, October 2016. https://doi.org/10.1145/2976749.2978357
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_6
Kilian, J.: Founding cryptography on oblivious transfer. In: 20th ACM STOC, pp. 20–31. ACM Press, May 1988. https://doi.org/10.1145/62212.62215
Kolesnikov, V., Kumaresan, R.: Improved OT extension for transferring short secrets. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 54–70. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_4
Larraia, E., Orsini, E., Smart, N.P.: Dishonest majority multi-party computation for binary circuits. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 495–512. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_28
Lindell, Y., Pinkas, B., Oxman, E.: The IPS compiler: optimizations, variants and concrete efficiency. Cryptology ePrint Archive, Report 2011/435 (2011). http://eprint.iacr.org/2011/435
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_3
Naor, M., Pinkas, B.: Oblivious transfer and polynomial evaluation. In: 31st ACM STOC, pp. 245–254. ACM Press, May 1999. https://doi.org/10.1145/301250.301312
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). https://doi.org/10.1007/978-3-642-32009-5_40
Orlandi, C.: Is multiparty computation any good in practice? In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011, Prague Congress Center, Prague, Czech Republic, 22–27 May 2011, pp. 5848–5851. IEEE (2011)
Orrù, M., Orsini, E., Scholl, P.: Actively secure 1-out-of-N OT extension with application to private set intersection. In: Handschuh, H. (ed.) CT-RSA 2017. LNCS, vol. 10159, pp. 381–396. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-52153-4_22
Orsini, E., Smart, N.P., Vercauteren, F.: Overdrive2k: efficient secure MPC over \(\mathbb{Z}_{2^k}\) from somewhat homomorphic encryption. In: Jarecki, S. (ed.) CT-RSA 2020. LNCS, vol. 12006, pp. 254–283. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40186-3_12
Pinkas, B., Schneider, T., Smart, N.P., Williams, S.C.: Secure two-party computation is practical. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 250–267. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_15
Rabin, M.O.: How to exchange secrets with oblivious transfer (1981)
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, May 2005. https://doi.org/10.1145/1060590.1060603
Rotaru, D., Smart, N.P., Tanguy, T., Vercauteren, F., Wood, T.: Actively secure setup for SPDZ. Cryptology ePrint Archive, Report 2019/1300 (2019). https://eprint.iacr.org/2019/1300
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, November 2019. https://doi.org/10.1145/3319535.3363228
Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Cryptogr. 71(1), 57–81 (2014)
Wang, X., Ranellucci, S., Katz, J.: Authenticated garbling and efficient maliciously secure two-party computation. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 21–37. ACM Press, October/November 2017). https://doi.org/10.1145/3133956.3134053
Wiesner, S.: Conjugate coding. ACM SIGACT News 15(1), 78–88 (1983)
Yang, K., Weng, C., Lan, X., Zhang, J., Wang, X.: Ferret: fast extension for correlated OT with small communication. IACR Cryptology ePrint Archive 2020, 924 (2020)
Yao, A.C.C.: How to generate and exchange secrets (extended abstract). In: 27th FOCS, pp. 162–167. IEEE Computer Society Press, October 1986. https://doi.org/10.1109/SFCS.1986.25
Acknowledgements
I would like to thank the organizers of WAIFI 2020 for inviting me to give a talk there. I am also grateful to Axel Mertens and Nigel Smart for helpful comments. This work has been supported in part by ERC Advanced Grant ERC-2015-AdG-IMPaCT and by the FWO under an Odysseus project GOH9718N.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Orsini, E. (2021). Efficient, Actively Secure MPC with a Dishonest Majority: A Survey. In: Bajard, J.C., Topuzoğlu, A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science(), vol 12542. Springer, Cham. https://doi.org/10.1007/978-3-030-68869-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-68869-1_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-68868-4
Online ISBN: 978-3-030-68869-1
eBook Packages: Computer ScienceComputer Science (R0)