Statistical zero knowledge protocols to prove modular polynomial relations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1294)


This paper proposes a bit commitment scheme, BC(·), and efficient statistical zero knowledge (in short, SZK) protocols in which, for any given multi-variable polynomial f(X 1,..,X t) and any given modulus n, prover P gives (I 1,..,I t) to verifier V and can convince V that V knows (x 1,..,x t) satisfying f(x 1,..,x t) = 0 (mod n) and I i = BC(x i), (i = l,..,t). The proposed protocols are O(n) times more efficient than the corresponding previous ones [Dam93, Dam95, Oka95]. The (knowledge) soundness of our protocols holds under a computational assumption, the intractability of a modified RSA problem (see Def.3), while the (statistical) zero-knowledgeness of the protocols needs no computational assumption. The protocols can be employed to construct various practical cryptographic protocols, such as fair exchange, untraceable electronic cash and verifiable secret snaring protocols.


Success Probability Secret Sharing Cryptographic Protocol Commitment Scheme Basic Protocol 
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. [BCC86]
    G.Brassard, D.Chaum, and C.Crépeau, “Minimum Disclosure Proofs of Knowledge,” Journal of Computer and System Sciences, Vol.37, pp.156–189 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [BG92]
    Bellare, M. and Goldreich, O., “On Defining Proofs of Knowledge”, Proceedings of Crypto 92, pp.390–420 (1992).Google Scholar
  3. [Bra95]
    Brands, S., “Restrictive Blinding of Secret-Key Certificates”, Proceedings of Eurocrypt 95, pp.231–247 (1995).Google Scholar
  4. [CDS94]
    Cramer, R., Damgård, I. and Schoenmakers, B., “Proofs of Partial Knowledge and Simplified Design of Witness Hiding Protocols”, Proc. of Crypto'94, LNCS, Springer, pp.174–187 (1994)Google Scholar
  5. [CGMA85]
    Chor, B., Goldwasser, S., Micali, S. and Awerbuch, B., “Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults”, Proc. of FOCS, pp.383–395 (1985).Google Scholar
  6. [Dam93]
    Damgård, I., “Practical and Provably Secure Release of a Secret and Exchange of Signatures,” Proceedings of Eurocrypt 93 (1993).Google Scholar
  7. [Dam95]
    Damgård, I., “Practical and Provably Secure Release of a Secret and Exchange of Signatures,” vol. 8 pp.201–222, Journal of CRYPTOLOGY(1995).zbMATHCrossRefGoogle Scholar
  8. [FFS88]
    U.Feige, A.Fiat and A.Shamir, “Zero Knowledge Proofs of Identity,” Journal of Cryptology, Vol. 1, pp.77–94 (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  9. [FS90]
    U.Feige, and A.Shamir, “Witness Indistinguishable and Witness Hiding Protocols,” Proc. of STOC90.Google Scholar
  10. [GMRa89]
    Goldwasser, S., Micali, S., and Rackoff, C., “The knowledge complexity of interactive proof systems”, SIAM J. Comput., vol.18, pp.186–208 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  11. [GMW86]
    O.Goldreich, S.Micali, and A.Wigderson, “Proofs that Yield Nothing But their Validity and a Methodology of Cryptographic Protocol Design,” Proc. FOCS, pp.174–187 (1986)Google Scholar
  12. [Mil76]
    Miller, G.L., “Riemann's Hypothesis and Tests for Primality”, Journal of Computer and System Sciences 13, 300–317 (1976).zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Oka95]
    Okamoto, T., “An Efficient Divisible Electronic Cash Scheme”, Proceedings of Crypto 95, pp.438–451 (1995).Google Scholar
  14. [Ped91]
    Pedersen, T. P., “Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing”, Proceedings of Crypto 91, pp. 129–140 (1992).Google Scholar
  15. [Sta96]
    Stadler, M., “Publicly Verifiable Secret Sharing”, Proc. of Eurocrypt'96, LNCS 1070, Springer, pp.190–199 (1996)Google Scholar
  16. [TW87]
    Tompa, M., and Woll, H., “Random Self-Reducibility and Zero-Knowledge Interactive-Proofs of Possession of Information”, Proc. FOCS, pp 472–482 (1987).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  1. 1.NTT LaboratoriesYokosuka-shiJapan

Personalised recommendations