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)

Abstract

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.

References

  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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 http://www.daimi.au.dk/~ivan/MPCbook.pdf
  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Goldreich, O.: Foundations of Cryptography. Basic Applications, vol. 2. Cambridge University Press, New York (2004)CrossRefMATHGoogle Scholar
  24. 24.
    Goldreich, O.: Valiant’s polynomial-size monotone formula for majority (2011), http://www.wisdom.weizmann.ac.il/~oded/PDF/mono-maj.pdf
  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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), http://www.cs.illinois.edu/~mmp/research.htmlCrossRefGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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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