Public Verifiability from Pairings in Secret Sharing Schemes

  • Somayeh Heidarvand
  • Jorge L. Villar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5381)


In this paper we propose a new publicly verifiable secret sharing scheme using pairings with close relations to Shoenmakers’ scheme. This scheme is efficient, multiplicatively homomorphic and with unconditional verifiability in the standard model. We formalize the notion of Indistinguishability of Secrets and prove that out scheme achieves it under the Decisional Bilinear Square (DBS) Assumption that is a natural variant of the Decisional Bilinear Diffie Hellman Assumption. Moreover, our scheme tolerates active and adaptive adversaries.


Public Verification Secret Sharing Pairings Semantic Security Threshold Cryptography 


  1. 1.
    Boldyreva, A.: Threshold signatures, multisignatures and blind signatures based on the gap-diffie-hellman-group signature scheme. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 31–46. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults. In: Proc. 26th IEEE Symp. on Found. of Comp. Sci., pp. 383–395 (1985)Google Scholar
  3. 3.
    Feldman, P.: A Practical Scheme for Non-interactive Verifiable Secret Sharing. In: Proceedings 28th IEEE Symp. on Found. of Comp. Sci., pp. 427–437 (1987)Google Scholar
  4. 4.
    Fiat, A., Shamir, A.: How to prove yourself: Practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  5. 5.
    Fujisaki, E., Okamoto, T.: A practical and provably secure scheme for publicly verifiable secret sharing and its applications. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 32–46. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Goldwasser, S., Tauman, Y.: On the (In)security of the Fiat-Shamir Paradigm. In: Proc. 44th Annual IEEE Symp. on Found. of Comp. Sci., pp. 102–115 (2003)Google Scholar
  7. 7.
    Groth, J., Ostrovsky, R., Sahai, A.: Perfect non-interactive zero knowledge for NP. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 339–358. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Joux, A.: A One Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Libert, B., Vergnaud, D.: Unidirectional chosen-ciphertext secure proxy re-encryption. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 360–379. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)Google Scholar
  12. 12.
    Ruiz, A., Villar, J.L.: Publicly Verifiable Secret Sharing from Paillier’s Cryptosystem. In: WEWoRC 2005. LNI P-74, pp. 98–108 (2005)Google Scholar
  13. 13.
    Sadeghi, A.-R., Steiner, M.: Assumptions related to discrete logarithms: Why subtleties make a real difference. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 243–260. Springer, Heidelberg (2001)Google Scholar
  14. 14.
    Schoenmakers, B.: A simple publicly verifiable secret sharing scheme and its application to electronic voting. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 148–164. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  15. 15.
    Shamir, A.: How to share a secret. Commun. of the ACM 22, 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stadler, M.A.: Publicly verifiable secret sharing. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 190–199. Springer, Heidelberg (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Somayeh Heidarvand
    • 1
  • Jorge L. Villar
    • 1
  1. 1.Universitat Politècnica de CatalunyaSpain

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