Upper Bounds on the Communication Complexity of Optimally Resilient Cryptographic Multiparty Computation

  • Martin Hirt
  • Jesper Buus Nielsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3788)


We give improved upper bounds on the communication complexity of optimally-resilient secure multiparty computation in the cryptographic model. We consider evaluating an n-party randomized function and show that if f can be computed by a circuit of size c, then \(\mathcal{O}(cn^2\kappa)\) is an upper bound for active security with optimal resilience t < n/2 and security parameter κ. This improves on the communication complexity of previous protocols by a factor of at least n. This improvement comes from the fact that in the new protocol, only \(\mathcal{O}(n)\) messages (of size \(\mathcal{O}(\kappa)\) each) are broadcast during the whole protocol execution, in contrast to previous protocols which require at least \(\mathcal{O}(n)\) broadcasts per gate.

Furthermore, we improve the upper bound on the communication complexity of passive secure multiparty computation with resilience t<n from \(\mathcal{O}(cn^2\kappa)\) to \(\mathcal{O}(cn\kappa)\). This improvement is mainly due to a simple observation.


Communication Complexity Setup Phase Multiplication Gate Input Gate Output Gate 
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. [BB89]
    Bar-Ilan, J., Beaver, D.: Non-cryptographic fault-tolerant computing in constant number of rounds of interaction. In: PODC 1989, pp. 201–209 (1989)Google Scholar
  2. [BCG93]
    Ben-Or, M., Canetti, R., Goldreich, O.: Asynchronous secure computation (extended abstract). In: 25th STOC, pp. 52–61 (1993)Google Scholar
  3. [BDPR98]
    Bellare, M., Desai, A., Pointcheval, D., Rogaway, P.: Relations among notions of security for publickey encryption schemes. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 26–45. Springer, Heidelberg (1998)Google Scholar
  4. [Bea91a]
    Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992)Google Scholar
  5. [Bea91b]
    Beaver, D.: Secure multi-party protocols and zero-knowledge proof systems tolerating a faulty minority. Journal of Cryptology 4(2), 75–122 (1991)zbMATHCrossRefGoogle Scholar
  6. [BFKR90]
    Beaver, D., Feigenbaum, J., Kilian, J., Rogaway, P.: Security with low communication overhead (extended abstract). In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 62–76. Springer, Heidelberg (1991)Google Scholar
  7. [BGP92]
    Berman, P., Garay, J.A., Perry, K.J.: Optimal early stopping in distributed consensus. In: Proceedings of the sixth International Workshop on Distributed Algorithms, pp. 221–237 (1992)Google Scholar
  8. [BGW88]
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: 20th STOC, pp. 1–10 (1988)Google Scholar
  9. [BMR90]
    Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: 22nd STOC, pp. 503–513 (1990)Google Scholar
  10. [Can00]
    Canetti, R.: Security and composition of multiparty cryptographic protocols. Journal of Cryptology 13(1), 143–202 (winter 2000)Google Scholar
  11. [Can01]
    Canetti, R.: Universally composable security: A new paradigm for cryptographic protocols. In: 42nd FOCS (2001)Google Scholar
  12. [CCD88]
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: 20th STOC, pp. 11–19 (1988)Google Scholar
  13. [CDD+99]
    Cramer, R., Damgård, I., Dziembowski, S., Hirt, M., Rabin, T.: Efficient multiparty computations secure against an adaptive adversary. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 311–326. Springer, Heidelberg (1999)Google Scholar
  14. [CDD00]
    Cramer, R., Damgård, I., Dziembowski, S.: On the complexity of verifiable secret sharing and multiparty computation. In: 22nd STOC, pp. 325–334 (2000)Google Scholar
  15. [CDG87]
    Chaum, D., Damgård, I., van de Graaf, J.: Multiparty computations ensuring privacy of each party’s input and correctness of the result. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 87–119. Springer, Heidelberg (1988)Google Scholar
  16. [CDM00]
    Cramer, R., Damgård, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. [CDN01]
    Cramer, R., Damgaard, I., Nielsen, J.B.: Multiparty computation from threshold homomorphic encryption. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 280–300. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. [Cra96]
    Cramer, R.: Modular Design of Secure yet Practical Cryptographic Protocols. PhD thesis, CWI and University of Amsterdam (1996)Google Scholar
  19. [CW92]
    Coan, B.A., Welch, J.L.: Modular construction of a byzantine agreement protocol with optimal message complexity. Information and Computation 97(1), 61–85 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Dam00]
    Damgård, I.: Efficient concurrent zero-knowledge in the auxiliary string model. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 418–430. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. [DCIO98]
    Di Crescenzo, G., Ishai, Y., Ostrovsky, R.: Non-interactive and non-malleable commitment. In: 30th STOC, pp. 141–150 (1998)Google Scholar
  22. [DG02]
    Damgaard, I., Groth, J.: Non-interactive and reusable non-maleable commitment schemes. In: 34th STOC (2002)Google Scholar
  23. [DJ01]
    Damgård, I., Jurik, M.: A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 110–136. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. [DN03]
    Damgård, I., Nielsen, J.B.: Universally composable efficient multiparty computation from threshold homomorphic encryption. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 247–264. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  25. [FF00]
    Fischlin, M., Fischlin, R.: Efficient non-malleable commitment schemes. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 414–432. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  26. [FH96]
    Franklin, M., Haber, S.: Joint encryption and message-efficient secure computation. Journal of Cryptology 9(4), 217–232 (Autumn 1996)Google Scholar
  27. [FPS00]
    Fouque, P.-A., Poupard, G., Stern, J.: Sharing decryption in the context of voting or lotteries. In: Proceedings of Financial Crypto 2000 (2000)Google Scholar
  28. [GIKR02]
    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)CrossRefGoogle Scholar
  29. [CDI05]
    Cramer, R., Damgård, I., Ishai, Y.: Local conversion of secret-sharing schemes with applications to threshold cryptography. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 342–362. Springer, Heidelberg (2005)Google Scholar
  30. [GL90]
    Goldwasser, S., Levin, L.: Fair computation of general functions in presence of immoral majority. In: Menezes, A.J., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 77–93. Springer, Heidelberg (1991)Google Scholar
  31. [GMR85]
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof-systems (extended abstract). In: 17th STOC, pp. 291–304 (1985)Google Scholar
  32. [GMW87]
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: 19th STOC, pp. 218–229 (1987)Google Scholar
  33. [GRR98]
    Gennaro, R., Rabin, M., Rabin, T.: Simplified VSS and fast-track multi-party computations with applications to threshold cryptography. In: PODC 1998 (1998)Google Scholar
  34. [GV87]
    Goldreich, O., Vainish, R.: How to solve any protocol problem - an efficiency improvement. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 73–86. Springer, Heidelberg (1988)Google Scholar
  35. [HM00]
    Hirt, M., Maurer, U.: Player simulation and general adversary structures in perfect multiparty computation. Journal of Cryptology 13(1), 31–60 (winter 2000)Google Scholar
  36. [HM01]
    Hirt, M., Maurer, U.: Robustness for free in unconditional multi-party computation. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 101–118. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  37. [HMP00]
    Hirt, M., Maurer, U., Przydatek, B.: Efficient secure multi-party computation. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 143–161. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  38. [JJ00]
    Jakobsson, M., Juels, A.: Mix and match: Secure function evaluation via ciphertexts. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 162–177. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  39. [KY02]
    Katz, J., Yung, M.: Threshold cryptosystems based on factoring. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 192–205. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  40. [LSP82]
    Lamport, L., Shostak, R., Pease, M.: The Byzantine generals problem. ACM Transactions on Programming Languages and Systems 4(3), 381–401 (1982)CrossRefGoogle Scholar
  41. [MR91]
    Micali, S., Rogaway, P.: Secure computation. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 392–404. Springer, Heidelberg (1992)Google Scholar
  42. [Nie03]
    Nielsen, J.B.: On protocol security in the cryptographic model. Dissertation Series DS-03-8, BRICS, Department of Computer Science, University of Aarhus (August 2003)Google Scholar
  43. [Pai99]
    Paillier, P.: Public-key cryptosystems based on composite degree residue classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)Google Scholar
  44. [RB89]
    Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority. In: 21th STOC, pp. 73–85 (1989)Google Scholar
  45. [Yao82]
    Yao, A.C.-C.: Protocols for secure computations (extended abstract). In: 23rd FOCS, pp. 160–164 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Hirt
    • 1
  • Jesper Buus Nielsen
    • 2
  1. 1.Deptartment of Computer ScienceETH Zurich 
  2. 2.Department of Computer ScienceUniversity of Aarhus 

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