Counting Method for Multi-party Computation over Non-abelian Groups

  • Youming Qiao
  • Christophe Tartary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5339)


In the Crypto’07 paper [5], Desmedt et al. studied the problem of achieving secure n-party computation over non-Abelian groups. The function to be computed is f G (x 1,...,x n ) : = x 1 ·...·x n where each participant P i holds an input x i from the non-commutative group G. The settings of their study are the passive adversary model, information-theoretic security and black-box group operations over G.

They presented three results. The first one is that honest majority is needed to ensure security when computing f G . Second, when the number of adversary \(t\leq\lceil\frac{n}{2}\rceil-1\), they reduced building such a secure protocol to a graph coloring problem and they showed that there exists a deterministic secure protocol computing f G using exponential communication complexity. Finally, Desmedt et al. turned to analyze random coloring of a graph to show the existence of a probabilistic protocol with polynomial complexity when t < n/μ, in which μ is a constant less than 2.948.

We call their analysis method of random coloring the counting method as it is based on the counting of the number of a specific type of random walks. This method is inspiring because, as far as we know, it is the first instance in which the theory of self-avoiding walk appears in multiparty computation.

In this paper, we first give an altered exposition of their proof. This modification will allow us to adapt this method to a different lattice and reduce the communication complexity by 1/3, which is an important saving for practical implementations of the protocols. We also show the limitation of the counting method by presenting a lower bound for this technique. In particular, we will deduce that this approach would not achieve the optimal collusion resistance \(\lceil \frac{n}{2} \rceil - 1\).


Multiparty Computation Passive Adversary Non-Abelian Groups Graph Coloring Neighbor-Avoiding Walk Random Walk 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: 20th Annual ACM Symposium on Theory of Computing, Chicago, USA, May 1988, pp. 1–10. ACM Press, New York (1988)Google Scholar
  2. 2.
    Bogetoft, P., Christensen, D.L., Damgård, I.B., Geisler, M., Jakobsen, T., Krøigaard, M., Nielsen, J.D., Nielsen, J.B., Nielsen, K., Pagter, J., Schwartzbach, M., Toft, T.: Multiparty computation goes lives. Cryptology ePrint Archive, Report 2008/068 (January 2008),
  3. 3.
    Cramer, R., Damgård, I.B., 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
  4. 4.
    Damgård, I.B., Ishai, Y.: Scalable secure multiparty computation. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 501–520. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Goldreich, O.: Foundations of Cryptography. Basic Applications, vol. II. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    Goldwasser, S.: Multi-party computations: Past and present. In: 16th annual ACM symposium on Principles of Distributed Computing, Santa Barbara, USA, August 1997, pp. 1–6. ACM Press, New York (1997)Google Scholar
  9. 9.
    Guttmann, A.J., Parviainen, R., Rechnitzer, A.: Self-avoiding walks and trails on the 3.12 lattice. Journal of Physics A: Mathematical and General 38, 543–554 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    Hirt, M., Nielsen, J.B.: Robust multiparty computation with linear communication complexity. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 463–482. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Lang, S.: Algebra (Revised Third Edition). Springer, Heidelberg (2002)Google Scholar
  14. 14.
    Lin, K.-Y., Hsaio, Y.C.: Self-avoiding walks and related problems. Chinese Journal of Physics 31(6-I), 695–708 (1993)Google Scholar
  15. 15.
    Madras, N., Slade, G.: The Self-Avoiding Walk. Probability and Its Applications. Birkhäuser, Basel (1996)CrossRefzbMATHGoogle Scholar
  16. 16.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26(5), 1484–1509 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yao, A.C.-C.: Protocols for secure computations. In: 23rd Annual IEEE Symposium on Foundations of Computer Science, Chicago, USA, November 1982, pp. 80–91. IEEE Press, Los Alamitos (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Youming Qiao
    • 1
  • Christophe Tartary
    • 1
    • 2
  1. 1.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Division of Mathematical Sciences School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore

Personalised recommendations