Atomic Secure Multi-party Multiplication with Low Communication

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)


We consider the standard secure multi-party multiplication protocol due to M. Rabin. This protocol is based on Shamir’s secret sharing scheme and it can be viewed as a practical variation on one of the central techniques in the foundational results of Ben-Or, Goldwasser, and Wigderson and Chaum, Crépeau, and Damgaard on secure multi-party computation. Rabin’s idea is a key ingredient to virtually all practical protocols in threshold cryptography.

Given a passive t-adversary in the secure channels model with synchronous communication, for example, secure multiplication of two secret-shared elements from a finite field K based on this idea uses one communication round and has the network exchange O(n 2) field elements, if t = Θ(n) and t < n/2 and if n is the number of players. This is because each of O(n) players must perform Shamir secret sharing as part of the protocol. This paper demonstrates that under a few restrictions much more efficient protocols are possible; even at the level of a single multiplication.

We demonstrate a twist on Rabin’s idea that enables one-round secure multiplication with just O(n) bandwidth in certain settings, thus reducing it from quadratic to linear. The ideas involved can additionally be employed in the evaluation of arithmetic circuits, where under appropriate circumstances similar efficiency gains can be obtained.


  1. 1.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: Proceedings of STOC 1988, pp. 1–10. ACM Press, New York (1988)Google Scholar
  2. 2.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proceedings Proceedings of National Computer Conference ’79. AFIPS Proceedings, vol. 48, pp. 313–317 (1979)Google Scholar
  3. 3.
    Chaum, D., Crépeau, C., Damgaard, I.: Multi-party unconditionally secure protocols. In: Proceedings of STOC 1988, pp. 11–19. ACM Press, New York (1988)Google Scholar
  4. 4.
    Chen, H., Cramer, R.J.F.: Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 521–536. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Cramer, R.J.F., Damgård, I.B.: Secure Distributed Linear Algebra in a Constant Number of Rounds. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 119–136. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Cramer, R., Kiltz, E., Padró, C.: A Note on Secure Computation of the Moore-Penrose and Its Application to Secure Linear Algebra. Manuscript (2006)Google Scholar
  7. 7.
    Franklin, M., Yung, M.: Communication complexity of secure computation. In: Proceedings of STOC 1992, pp. 699–710. ACM Press, New York (1992)Google Scholar
  8. 8.
    Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The Round Complexity of Verifiable Secret Sharing and Secure Multicast. In: Proceedings of STOC 2001, pp. 580–589. ACM Press, New York (2001)Google Scholar
  9. 9.
    Gennaro, R., Rabin, M.O., Rabin, T.: Simplified VSS and fasttrack multiparty computations with applications to threshold cryptography. In: Proceedings of PODC 1997, pp. 101–111 (1998)Google 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.
    Kruskal, C.P., Rudolph, L., Snir, M.: A complexity theory of efficient parallel algorithms. Theoretical Computer Science 71(1), 95–132 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Shamir, A.: How to share a secret. Communications of the ACM 22(11), 612–613 (1979)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Mathematical InstituteLeiden UniversityThe Netherlands
  2. 2.Comp. Sc. Dept.Aarhus University & BRICSDenmark
  3. 3.CWIAmsterdamThe Netherlands

Personalised recommendations