Non-interactive Proofs for Integer Multiplication

  • Ivan Damgård
  • Rune Thorbek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)


We present two universally composable and practical protocols by which a dealer can, verifiably and non-interactively, secret-share an integer among a set of players. Moreover, at small extra cost and using a distributed verifier proof, it can be shown in zero-knowledge that three shared integers a,b,c satisfy ab = c. This implies by known reductions non-interactive zero-knowledge proofs that a shared integer is in a given interval, or that one secret integer is larger than another. Such primitives are useful, e.g., for supplying inputs to a multiparty computation protocol, such as an auction or an election. The protocols use various set-up assumptions, but do not require the random oracle model.


Random Oracle Commitment Scheme Random Oracle Model Adversary Structure Common Reference String 
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.
    Abe, M., Cramer, R.J.F., Fehr, S.: Non-interactive Distributed-Verifier Proofs and Proving Relations among Commitments. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 206–223. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for Non-Cryptographic Fault-Tolerant Distributed Computation. In: Proc. ACM STOC ’88, pp. 1–10 (1988)Google Scholar
  3. 3.
    Boneh, D., Franklin, M.K.: Identity-Based Encryption from the Weil Pairing. SIAM J. Comput. 32(3), 586–615 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bogetoft, P., Damgård, I.B., Jakobsen, T., Nielsen, K., Pagter, J., Toft, T.: A Practical Implementation of Secure Auctions Based on Multiparty Integer Computation. In: Di Crescenzo, G., Rubin, A. (eds.) FC 2006. LNCS, vol. 4107, pp. 142–147. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Boudot, F.: Efficient Proofs that a Committed Number Lies in an Interval. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 431–444. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Canetti, R.: Universally Composable Security: A New Paradigm for Cryptographic Protocols. In: Proc. of FOCS 2001, pp. 136–145 (2001), See also updated version on the Eprint archive,
  7. 7.
    Canetti, R., Halevi, S., Katz, J.: Chosen-Ciphertext Security from Identity-Based Encryption. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 207–222. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Chaum, D., Crépeau, C., Damgård, I.: Multi-Party Unconditionally Secure Protocols. In: Proc. of ACM STOC ’88, pp. 11–19 (1988)Google Scholar
  9. 9.
    Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos (1985)Google Scholar
  10. 10.
    Cramer, R., Damgård, I., Ishai, Y.: Share Conversion, Pseudorandom Secret-Sharing and Applications to Secure Computation. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 342–362. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Cramer, R.J.F., Fehr, S., Stam, M.: Black-Box Secret Sharing from Primitive Sets in Algebraic Number Fields. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 344–360. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Damgård, I., Fitzi, M., Kiltz, E., Nielsen, J.B., Toft, T.: Unconditionally Secure Constant-Rounds Multi-party Computation for Equality, Comparison, Bits and Exponentiation. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 285–304. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Damgård, I.B., Thorbek, R.: Linear Integer Secret Sharing and Distributed Exponentiation. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 75–90. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Damgård, I., Thorbek, R.: Non-Interactive Proofs for Integer Multiplication (full version). The Eprint archive (,
  15. 15.
    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
  16. 16.
    Fujisaki, E., Okamoto, T.: Statistical Zero Knowledge Protocols to Prove Modular Polynomial Relations. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997)Google Scholar
  17. 17.
    Gennaro, R., Rabin, M., Rabin, T.: Simplified VSS and Fast-Track Multiparty Computations with Applications to Threshold Cryptography. In: Proc. of ACM PODC’98 (1998)Google Scholar
  18. 18.
    Goldreich, O., Micali, S., Wigderson, A.: How to Play Any Mental Game or a Completeness Theorem for Protocols with Honest Majority. In: Proc. of ACM STOC ’87, pp. 218–229 (1987)Google Scholar
  19. 19.
    Hirt, M., Maurer, U.: Player Simulation and General Adversary Structures in Perfect Multiparty Computation. Journal of Cryptology: the journal of the International Association for Cryptologic Research 13, 31–60 (2000)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structures. In: Proc. IEEE Global Telecommunication Conf., Globecom 87, pp. 99–102 (1987)Google Scholar
  21. 21.
    Karchmer, M., Wigderson, A.: On Span Programs. In: Proc. of 8th IEEE Structure in Complexity Theory, pp. 102–111 (1993)Google Scholar
  22. 22.
    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
  23. 23.
    Shamir, A.: How to share a secret. Communication of the Association for Computing Machinery 22(11) (1979)Google Scholar
  24. 24.
    Waters, B.: Efficient Identity-Based Encryption Without Random Oracles. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ivan Damgård
    • 1
  • Rune Thorbek
    • 1
  1. 1.BRICS, Dept. of Computer ScienceUniversity of Aarhus 

Personalised recommendations