Linear Integer Secret Sharing and Distributed Exponentiation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3958)


We introduce the notion of Linear Integer Secret-Sharing (LISS) schemes, and show constructions of such schemes for any access structure. We show that any LISS scheme can be used to build a secure distributed protocol for exponentiation in any group. This implies, for instance, distributed RSA protocols for arbitrary access structures and with arbitrary public exponents.


Secret Sharing Access Structure Secret Sharing Scheme Random Oracle Model Monotone Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Benaloh, J.C., Leichter, J.: Generalized Secret Sharing and Monotone Functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, Heidelberg (1988)Google Scholar
  2. 2.
    Boppana, R.B.: Amplification of Probabilistic Boolean Formulas. Advances in Computing Research 5, 27–45 (1989)Google Scholar
  3. 3.
    Radhakrishnan, J.: Better Lower Bounds for Monotone Threshold Formulas. J. Comput. Syst. Sci. 54(2), 221–226 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Canetti, R.: Universally Composable Security: A New Paradigm for Cryptographic Protocols. In: FOCS 2001, pp. 136–145 (2001)Google Scholar
  5. 5.
    Chor, B., Kushilevitz, E.: Secret Sharing Over Infinite Domains. J. Cryptology 6(2), 87–95 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chor, B., Geréb-Graus, M., Kushilevitz, E.: Private Computations over the Integers. SIAM J. Comput. 24(2), 376–386 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cramer, R., Damgård, I.B.: Secret-Key Zero-Knowlegde and Non-interactive Verifiable Exponentiation. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 223–237. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Cramer, R., Fehr, S.: Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 272–287. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Cramer, R., Fehr, S., Stam, M.: Black-Box Secret Sharing from Primitve Sets in Algebraic Number Fields. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 344–360. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Damgård, I., Dupont, K.: Efficient threshold RSA signatures with general moduli and no extra assumptions. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 346–361. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Damgård, I.B., Fazio, N., Nicolosi, A.: Non-Interactive Zero-Knowledge Proofs from Homomorphic Encryption. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 41–59. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Damgård, I.B., Koprowski, M.: Practical Threshold RSA Signatures without a Trusted Dealer. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 152–165. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Desmedt, Y., Frankel, Y.: Perfect Homomorphic Zero-Knowledge Threshold Schemes over any Finite Abelian Group. SIAM J. Discrete Math. 7(4), 667–679 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Damgård, I., Thorbek, R.: Linear Integer Secret Sharing and Distributed Exponentiation (full version), the Eprint archive,
  15. 15.
    Frankel, Y., Gemmell, P., MacKenzie, P.D., Yung, M.: Optimal Resilience Proactive Public-Key Cryptosystems. In: FOCS 1997, pp. 384–393 (1997)Google Scholar
  16. 16.
    Gennaro, R., Rabin, T., Jarecki, S., Krawczyk, H.: Robust and Efficient Sharing of RSA Functions. J. Cryptology 13(2), 273–300 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that Yield Nothing But Their Validity or All Languages in NP Have Zero-Knowledge Proof Systems. J. ACM 38(3), 691–729 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rabin, T.: A Simplified Approach to Threshold and Proactive RSA. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 89–104. Springer, Heidelberg (1998)Google Scholar
  19. 19.
    Schnorr, C.-P.: Efficient Signature Generation by Smart Cards. J. Cryptology 4(3), 161–174 (1991)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shamir, A.: How to Share a Secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    De Santis, A., Desmedt, Y., Frankel, Y., Yung, M.: How to share a function securely. In: STOC 1994, pp. 522–533 (1994)Google Scholar
  22. 22.
    Shoup, V.: Practical Threshold Signatures. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 207–220. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  23. 23.
    Valiant, L.G.: Short Monotone Formulae for the Majority Function. J. Algorithms 5(3), 363–366 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.BRICS, Dept. of Computer ScienceUniversity of AarhusDenmark

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