Graph Design for Secure Multiparty Computation over Non-Abelian Groups

  • Xiaoming Sun
  • Andrew Chi-Chih Yao
  • Christophe Tartary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5350)


Recently, Desmedt et al. studied the problem of achieving secure n-party computation over non-Abelian groups. They considered the passive adversary model and they assumed that the parties were only allowed to perform black-box operations over the finite group G. They showed three results for the n-product function fG(x1,...,xn) : = x1 ·x2 ·...·xn, where the input of party Pi is xi ∈ G for i ∈ {1,...,n}. First, if \(t \geq \lceil \tfrac{n}{2} \rceil\) then it is impossible to have a t-private protocol computing fG. Second, they demonstrated that one could t-privately compute fG for any \(t \leq \lceil \tfrac{n}{2} \rceil - 1\) in exponential communication cost. Third, they constructed a randomized algorithm with O(nt2) communication complexity for any \(t < \tfrac{n}{2.948}\).

In this paper, we extend these results in two directions. First, we use percolation theory to show that for any fixed ε> 0, one can design a randomized algorithm for any \(t\leq \frac{n}{2+\epsilon}\) using O(n3) communication complexity, thus nearly matching the known upper bound \(\lceil \tfrac{n}{2} \rceil - 1\). This is the first time that percolation theory is used for multiparty computation. Second, we exhibit a deterministic construction having polynomial communication cost for any t = O(n1 − ε) (again for any fixed ε> 0). Our results extend to the more general function \(\widetilde{f}_{G}(x_{1},\ldots,x_{m}) := x_{1} \cdot x_{2} \cdot \ldots \cdot x_{m}\) where m ≥ n and each of the n parties holds one or more input values.


Multiparty Computation Passive Adversary Non-Abelian Groups Graph Coloring Percolation Theory 


  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, pp. 1–10. ACM Press, New York (1988)Google Scholar
  2. 2.
    Bollobàs, B., Riordan, O.: Percolation. Cambridge University Press, Cambridge (September 2006)CrossRefMATHGoogle Scholar
  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.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    El Gamal, T.: A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory 31(4), 469–472 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Hammersley, J.M.: Percolation processes: Lower bounds for the critical probability. The Annals of Mathematical Statistics 28(3), 790–795 (1957)MathSciNetCrossRefMATHGoogle 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.
    Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser, Basel (November 1982)CrossRefMATHGoogle Scholar
  14. 14.
    Lang, S.: Algebra (Revised Third Edition). Springer, Heidelberg (November 2002)Google Scholar
  15. 15.
    Magliveras, S.S., Stinson, D.R., van Trung, T.: New approaches to designing public key cryptosystems using one-way functions and trapdoors in finite groups. Journal of Cryptology 15(4), 285–297 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Paeng, S.-H., Ha, K.-C., Kim, J.H., Chee, S., Park, C.: New public key cryptosystem using finite non Abelian groups. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 470–485. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public key cryptosystems. Communication of the ACM 21(2), 120–126 (1978)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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

  • Xiaoming Sun
    • 1
  • Andrew Chi-Chih Yao
    • 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