Advertisement

Communication Efficient Perfectly Secure VSS and MPC in Asynchronous Networks with Optimal Resilience

  • Arpita Patra
  • Ashish Choudhury
  • C. Pandu Rangan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6055)

Abstract

Verifiable Secret Sharing (VSS) is a fundamental primitive used in many distributed cryptographic tasks, such as Multiparty Computation (MPC) and Byzantine Agreement (BA). It is a two phase (sharing, reconstruction) protocol. The VSS and MPC protocols are carried out among n parties, where t out of n parties can be under the influence of a Byzantine (active) adversary, having unbounded computing power. It is well known that protocols for perfectly secure VSS and perfectly secure MPC exist in an asynchronous network iff n ≥ 4t + 1. Hence, we call any perfectly secure VSS (MPC) protocol designed over an asynchronous network with n = 4t + 1 as optimally resilient VSS (MPC) protocol.

A secret is d-shared among the parties if there exists a random degree-d polynomial whose constant term is the secret and each honest party possesses a distinct point on the degree-d polynomial. Typically VSS is used as a primary tool to generate t-sharing of secret(s). In this paper, we present an optimally resilient, perfectly secure Asynchronous VSS (AVSS) protocol that can generate d-sharing of a secret for any d, where t ≤ d ≤ 2t. This is the first optimally resilient, perfectly secure AVSS of its kind in the literature. Specifically, our AVSS can generate d-sharing of ℓ ≥ 1 secrets from \({\mathbb F}\) concurrently, with a communication cost of \({\cal O}(\ell n^2 \log{|{\mathbb F}|})\) bits, where \({\mathbb F}\) is a finite field. Communication complexity wise, the best known optimally resilient, perfectly secure AVSS is reported in [2]. The protocol of [2] can generate t-sharing of ℓ secrets concurrently, with the same communication complexity as our AVSS. However, the AVSS of [2] and [4] (the only known optimally resilient perfectly secure AVSS, other than [2]) does not generate d-sharing, for any d > t.

Interpreting in a different way, we may also say that our AVSS shares ℓ(d + 1 − t) secrets simultaneously with a communication cost of \({\cal O}(\ell n^2 \log{|{\mathbb F}|})\) bits. Putting d = 2t (the maximum value of d), we notice that the amortized cost of sharing a single secret using our AVSS is only \({\cal O}(n \log{|{\mathbb F}|})\) bits. This is a clear improvement over the AVSS of [2] whose amortized cost of sharing a single secret is \({\cal O}(n^2 \log{|{\mathbb F}|})\) bits.

As an interesting application of our AVSS, we propose a new optimally resilient, perfectly secure Asynchronous Multiparty Computation (AMPC) protocol that communicates \({\cal O}(n^2 \log|{\mathbb F}|)\) bits per multiplication gate. The best known optimally resilient perfectly secure AMPC is due to [2], which communicates \({\cal O}(n^3 \log|{\mathbb F}|)\) bits per multiplication gate. Thus our AMPC improves the communication complexity of the best known AMPC of [2] by a factor of Ω(n).

Keywords

Verifiable Secret Sharing Multiparty Computation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beerliová-Trubíniová, Z., Hirt, M.: Efficient multi-party computation with dispute control. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 305–328. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Beerliová-Trubíniová, Z., Hirt, M.: Simple and efficient perfectly-secure asynchronous MPC. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 376–392. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Beerliová-Trubíniová, Z., Hirt, M.: Perfectly-secure MPC with linear communication complexity. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 213–230. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Ben-Or, M., Canetti, R., Goldreich, O.: Asynchronous secure computation. In: STOC, pp. 52–61 (1993)Google Scholar
  5. 5.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: STOC, pp. 1–10 (1988)Google Scholar
  6. 6.
    BenOr, M., Kelmer, B., Rabin, T.: Asynchronous secure computations with optimal resilience. In: PODC, pp. 183–192 (1994)Google Scholar
  7. 7.
    Bracha, G.: An asynchronous \(\lfloor (n - 1) / 3 \rfloor\)-resilient consensus protocol. In: PODC, pp. 154–162 (1984)Google Scholar
  8. 8.
    Canetti, R.: Studies in Secure Multiparty Computation and Applications. PhD thesis, Weizmann Institute, Israel (1995)Google Scholar
  9. 9.
    Canetti, R., Rabin, T.: Fast asynchronous Byzantine Agreement with optimal resilience. In: STOC, pp. 42–51 (1993)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.
    Damgård, I., Nielsen, J.B.: Scalable and unconditionally secure multiparty computation. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 572–590. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Franklin, M.K., Yung, M.: Communication complexity of secure computation. In: STOC, pp. 699–710 (1992)Google Scholar
  13. 13.
    Feldman, P., Micali, S.: An optimal algorithm for synchronous Byzantine Agreemet. In: STOC, pp. 639–648 (1988)Google Scholar
  14. 14.
    Fitzi, M., Garay, J., Gollakota, S., Pandu Rangan, C., Srinathan, K.: Round-optimal and efficient verifiable secret sharing. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 329–342. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: STOC, pp. 580–589 (2001)Google Scholar
  16. 16.
    Katz, J., Koo, C., Kumaresan, R.: Improving the round complexity of VSS in point-to-point networks. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 499–510. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Patra, A., Choudhary, A., Rabin, T., Pandu Rangan, C.: The round complexity of verifiable secret sharing revisited. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 487–504. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Patra, A., Choudhary, A., Pandu Rangan, C.: Efficient asynchronous multiparty computation with optimal resilience. Cryptology ePrint Archive, Report 2008/425 (2008)Google Scholar
  19. 19.
    Patra, A., Choudhary, A., Pandu Rangan, C.: Efficient asynchronous Byzantine Agreement with optimal resilience. In: PODC, pp. 92–101 (2009)Google Scholar
  20. 20.
    Patra, A., Choudhary, A., Pandu Rangan, C.: Unconditionally secure asynchronous multiparty computation with quadratic communication per multiplication gate. Cryptology ePrint Archive, Report 2009/087 (2009)Google Scholar
  21. 21.
    Patra, A., Choudhary, A., Pandu Rangan, C.: Communication Efficient Perfectly Secure VSS and MPC in Asynchronous Networks with Optimal Resilience Cryptology ePrint Archive, Report 2010/007 (2010)Google Scholar
  22. 22.
    Prabhu, B., Srinathan, K., Pandu Rangan, C.: Trading players for efficiency in unconditional multiparty computation. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 342–353. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority. In: STOC, pp. 73–85 (1989)Google Scholar
  24. 24.
    Srinathan, K., Pandu Rangan, C.: Efficient asynchronous secure multiparty distributed computation. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 117–129. Springer, Heidelberg (2000)Google Scholar
  25. 25.
    Yao, A.C.: Protocols for secure computations. In: FOCS, pp. 160–164 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Arpita Patra
    • 1
  • Ashish Choudhury
    • 1
  • C. Pandu Rangan
    • 1
  1. 1.Dept of Computer Science and EngineeringIIT MadrasChennaiIndia

Personalised recommendations