Efficient Multiparty Protocols via Log-Depth Threshold Formulae

(Extended Abstract)
  • Gil Cohen
  • Ivan Bjerre Damgård
  • Yuval Ishai
  • Jonas Kölker
  • Peter Bro Miltersen
  • Ran Raz
  • Ron D. Rothblum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8043)


We put forward a new approach for the design of efficient multiparty protocols:

  1. 1

    Design a protocol π for a small number of parties (say, 3 or 4) which achieves security against a single corrupted party. Such protocols are typically easy to construct, as they may employ techniques that do not scale well with the number of corrupted parties.

  2. 2

    Recursively compose π with itself to obtain an efficient n-party protocol which achieves security against a constant fraction of corrupted parties.


The second step of our approach combines the “player emulation” technique of Hirt and Maurer (J. Cryptology, 2000) with constructions of logarithmic-depth formulae which compute threshold functions using only constant fan-in threshold gates.

Using this approach, we simplify and improve on previous results in cryptography and distributed computing. In particular:

  • We provide conceptually simple constructions of efficient protocols for Secure Multiparty Computation (MPC) in the presence of an honest majority, as well as broadcast protocols from point-to-point channels and a 2-cast primitive.

  • We obtain new results on MPC over blackbox groups and other algebraic structures.

The above results rely on the following complexity-theoretic contributions, which may be of independent interest:

  • We show that for every j,k ∈ ℕ such that \(m \triangleq \frac{k-1}{j-1}\) is an integer, there is an explicit (poly(n)-time) construction of a logarithmic-depth formula which computes a good approximation of an (n/m)-out-of-n threshold function using only j-out-of-k threshold gates and no constants.

  • For the special case of n-bit majority from 3-bit majority gates, a non-explicit construction follows from the work of Valiant (J. Algorithms, 1984). For this special case, we provide an explicit construction with a better approximation than for the general threshold case, and also an exact explicit construction based on standard complexity-theoretic or cryptographic assumptions.


Oblivious Transfer Broadcast Protocol Majority Gate Threshold Gate Byzantine Agreement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ajtai, M., Komlós, J., Szemerédi, E.: An o(n log n) sorting network. In: STOC, pp. 1–9 (1983)Google Scholar
  2. 2.
    Akers, S., Robbins, T.: Logical design with three-input majority gates. Computer Design 45(3), 12–27 (1963)Google Scholar
  3. 3.
    Barkol, O., Ishai, Y., Weinreb, E.: On locally decodable codes, self-correctable codes, and t-private PIR. Algorithmica 58(4), 831–859 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: STOC, pp. 1–10 (1988)Google Scholar
  5. 5.
    Bracha, G.: An O(log n) expected rounds randomized byzantine generals protocol. J. ACM 34(4), 910–920 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bro Miltersen, P.: Lecutre notes. Available from author (1992)Google Scholar
  7. 7.
    Canetti, R.: Security and composition of multiparty cryptographic protocols. J. Cryptology 13(1), 143–202 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Canetti, R.: Universally composable security: A new paradigm for cryptographic protocols. In: FOCS, pp. 136–145 (2001)Google Scholar
  9. 9.
    Chaum, D.: The spymasters double-agent problem: Multiparty computations secure unconditionally from minorities and cryptograhically from majorities. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 591–602. Springer, Heidelberg (1990)Google Scholar
  10. 10.
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: STOC, pp. 11–19 (1988)Google Scholar
  11. 11.
    Considine, J., Fitzi, M., Franklin, M.K., Levin, L.A., Maurer, U.M., Metcalf, D.: Byzantine agreement given partial broadcast. J. Cryptology 18(3), 191–217 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cramer, R., Damgård, I., Nielsen, J.B.: Secure Multiparty Computation and Secret Sharing - An Information Theoretic Appoach (2012), Book draft, available at
  13. 13.
    Cramer, R., Fehr, S., Ishai, Y., Kushilevitz, E.: Efficient multi-party computation over rings. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 596–613. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Damgård, I., Ishai, Y., Krøigaard, M., Nielsen, J.B., Smith, A.: Scalable multiparty computation with nearly optimal work and resilience. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 241–261. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Desmedt, Y., Pieprzyk, J., Steinfeld, R.: Active security in multiparty computation over black-box groups. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 503–521. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Desmedt, Y., Pieprzyk, J., Steinfeld, R., Sun, X., Tartary, C., Wang, H., Yao, A.C.-C.: Graph coloring applied to secure computation in non-abelian groups. J. Cryptology 25(4), 557–600 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Desmedt, Y., Pieprzyk, J., Steinfeld, R., Wang, H.: On secure multi-party computation in black-box groups. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 591–612. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Dolev, D.: The byzantine generals strike again. J. Algorithms 3(1), 14–30 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: STOC, pp. 554–563 (1994)Google Scholar
  20. 20.
    Fitzi, M., Maurer, U.M.: Efficient byzantine agreement secure against general adversaries. In: Kutten, S. (ed.) DISC 1998. LNCS, vol. 1499, pp. 134–148. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Fitzi, M., Maurer, U.M.: From partial consistency to global broadcast. In: STOC, pp. 494–503 (2000)Google Scholar
  22. 22.
    Garay, J.A., Moses, Y.: Fully polynomial byzantine agreement for n > 3t processors in t + 1 rounds. SIAM J. Comput. 27(1), 247–290 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goldreich, O.: Foundations of Cryptography. Basic Applications, vol. 2. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Goldreich, O.: Valiant’s polynomial-size monotone formula for majority (2011),
  25. 25.
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or A completeness theorem for protocols with honest majority. In: STOC, pp. 218–229. ACM (1987)Google Scholar
  26. 26.
    Gupta, A., Mahajan, S.: Using amplification to compute majority with small majority gates. Computational Complexity 6(1), 46–63 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Harnik, D., Ishai, Y., Kushilevitz, E., Nielsen, J.B.: OT-combiners via secure computation. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 393–411. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  28. 28.
    Hirt, M., Maurer, U.M.: Player simulation and general adversary structures in perfect multiparty computation. J. Cryptology 13(1), 31–60 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hoory, S., Magen, A., Pitassi, T.: Monotone circuits for the majority function. In: APPROX-RANDOM, pp. 410–425 (2006)Google Scholar
  30. 30.
    Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Zero-knowledge proofs from secure multiparty computation. SIAM J. Comput. 39(3), 1121–1152 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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), Scholar
  32. 32.
    Kilian, J.: Founding cryptography on oblivious transfer. In: STOC, pp. 20–31 (1988)Google Scholar
  33. 33.
    Kushilevitz, E., Lindell, Y., Rabin, T.: Information-theoretically secure protocols and security under composition. SIAM J. Comput. 39(5), 2090–2112 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lindell, Y., Oxman, E., Pinkas, B.: The IPS compiler: Optimizations, variants and concrete efficiency. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 259–276. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  35. 35.
    Lucas, C., Raub, D., Maurer, U.M.: Hybrid-secure mpc: trading information-theoretic robustness for computational privacy. In: PODC, pp. 219–228 (2010)Google Scholar
  36. 36.
    Maurer, U.M.: Secure multi-party computation made simple. Discrete Applied Mathematics 154(2), 370–381 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pease, M.C., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority (extended abstract). In: Johnson, D.S. (ed.) STOC, pp. 73–85. ACM (1989)Google Scholar
  39. 39.
    Sun, X., Yao, A.C.-C., Tartary, C.: Graph design for secure multiparty computation over non-abelian groups. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 37–53. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  40. 40.
    Valiant, L.G.: Short monotone formulae for the majority function. J. Algorithms 5(3), 363–366 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yao, A.C.-C.: Protocols for secure computations (extended abstract). In: FOCS, pp. 160–164 (1982)Google Scholar
  42. 42.

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Gil Cohen
    • 1
  • Ivan Bjerre Damgård
    • 2
  • Yuval Ishai
    • 3
  • Jonas Kölker
    • 2
  • Peter Bro Miltersen
    • 2
  • Ran Raz
    • 1
  • Ron D. Rothblum
    • 1
  1. 1.Weizmann InstituteRehovotIsrael
  2. 2.Aarhus UniversityAarhusDenmark
  3. 3.TechnionHaifaIsrael

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