Degree 2 is Complete for the Round-Complexity of Malicious MPC

  • Benny Applebaum
  • Zvika BrakerskiEmail author
  • Rotem Tsabary
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11477)


We show, via a non-interactive reduction, that the existence of a secure multi-party computation (MPC) protocol for degree-2 functions implies the existence of a protocol with the same round complexity for general functions. Thus showing that when considering the round complexity of MPC, it is sufficient to consider very simple functions.

Our completeness theorem applies in various settings: information theoretic and computational, fully malicious and malicious with various types of aborts. In fact, we give a master theorem from which all individual settings follow as direct corollaries. Our basic transformation does not require any additional assumptions and incurs communication and computation blow-up which is polynomial in the number of players and in \(S,2^D\), where SD are the circuit size and depth of the function to be computed. Using one-way functions as an additional assumption, the exponential dependence on the depth can be removed.

As a consequence, we are able to push the envelope on the state of the art in various settings of MPC, including the following cases.

  • 3-round perfectly-secure protocol (with guaranteed output delivery) against an active adversary that corrupts less than 1/4 of the parties.

  • 2-round statistically-secure protocol that achieves security with “selective abort” against an active adversary that corrupts less than half of the parties.

  • Assuming one-way functions, 2-round computationally-secure protocol that achieves security with (standard) abort against an active adversary that corrupts less than half of the parties. This gives a new and conceptually simpler proof to the recent result of Ananth et al. (Crypto 2018).

Technically, our non-interactive reduction draws from the encoding method of Applebaum, Brakerski and Tsabary (TCC 2018). We extend these methods to ones that can be meaningfully analyzed even in the presence of malicious adversaries.



We thank Yuval Ishai for helpful discussions, for providing us several useful pointers, and for sharing with us the full version of [11].


  1. 1.
    Ananth, P., Choudhuri, A.R., Goel, A., Jain, A.: Round-optimal secure multiparty computation with honest majority. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 395–424. Springer, Cham (2018). Scholar
  2. 2.
    Ananth, P., Choudhuri, A.R., Goel, A., Jain, A.: Two round information-theoretic MPC with malicious security. Cryptology ePrint Archive, Report 2018/1078 (2018).
  3. 3.
    Applebaum, B., Brakerski, Z., Tsabary, R.: Perfect secure computation in two rounds. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018, Part I. LNCS, vol. 11239, pp. 152–174. Springer, Cham (2018). Scholar
  4. 4.
    Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: Ortiz, H. (ed.) Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, Baltimore, Maryland, USA, 13–17 May 1990, pp. 503–513. ACM (1990)Google Scholar
  5. 5.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 1–10. ACM (1988)Google Scholar
  6. 6.
    Benhamouda, F., Lin, H.: \(k\)-round multiparty computation from \(k\)-round oblivious transfer via garbled interactive circuits. In: Nielsen and Rijmen [18], pp. 500–532 (2018)Google Scholar
  7. 7.
    Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21–23 October 1985, pp. 383–395. IEEE Computer Society (1985)Google Scholar
  8. 8.
    Damgård, I., Ishai, Y.: Constant-round multiparty computation using a black-box pseudorandom generator. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 378–394. Springer, Heidelberg (2005). Scholar
  9. 9.
    Garg, S., Srinivasan, A.: Garbled protocols and two-round MPC from bilinear maps. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, 15–17 October 2017, pp. 588–599. IEEE Computer Society (2017)Google Scholar
  10. 10.
    Garg, S., Srinivasan, A.: Two-round multiparty secure computation from minimal assumptions. In: Nielsen and Rijmen [18], pp. 468–499 (2018)Google Scholar
  11. 11.
    Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: Vitter, J.S., Spirakis, P.G., Yannakakis, M. (eds.) Proceedings on 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Crete, Greece, 6–8 July 2001, pp. 580–589. ACM (2001)Google Scholar
  12. 12.
    Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: On 2-round secure multiparty computation. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 178–193. Springer, Heidelberg (2002). Scholar
  13. 13.
    Goldreich, O.: The Foundations of Cryptography - Volume 2, Basic Applications. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  14. 14.
    Goldwasser, S., Lindell, Y.: Secure computation without agreement. In: Malkhi, D. (ed.) DISC 2002. LNCS, vol. 2508, pp. 17–32. Springer, Heidelberg (2002). Scholar
  15. 15.
    Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, Redondo Beach, California, USA, 12–14 November 2000, pp. 294–304. IEEE Computer Society (2000)Google Scholar
  16. 16.
    Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002). Scholar
  17. 17.
    Ishai, Y., Kushilevitz, E., Paskin, A.: Secure multiparty computation with minimal interaction. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 577–594. Springer, Heidelberg (2010). Scholar
  18. 18.
    Nielsen, J.B., Rijmen, V. (eds.): EUROCRYPT 2018, Part II. LNCS, vol. 10821. Springer, Cham (2018). Scholar
  19. 19.
    Paskin-Cherniavsky, A.: Secure computation with minimal interaction. Ph.D. thesis, Technion – Israel Institute of Technology (2012)Google Scholar
  20. 20.
    Patra, A., Ravi, D.: On the exact round complexity of secure three-party computation. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 425–458. Springer, Cham (2018). Scholar
  21. 21.
    Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority (extended abstract). In: Johnson, D.S. (ed.) Proceedings of the 21st Annual ACM Symposium on Theory of Computing, Seattle, Washigton, USA, 14–17 May 1989, pp. 73–85. ACM (1989)Google Scholar
  22. 22.
    Rogaway, P.: The round-complexity of secure protocols. Ph.D. thesis, MIT (1991)Google Scholar
  23. 23.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: FOCS, pp. 162–167 (1986)Google Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  • Benny Applebaum
    • 1
  • Zvika Brakerski
    • 2
    Email author
  • Rotem Tsabary
    • 2
  1. 1.Tel-Aviv UniversityTel AvivIsrael
  2. 2.Weizmann Institute of ScienceRehovotIsrael

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