On the Security of the Threshold Scheme Based on the Chinese Remainder Theorem

  • Michaël Quisquater
  • Bart Preneel
  • Joos Vandewalle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2274)


Threshold schemes enable a group of users to share a secret by providing each user with a share. The scheme has a threshold t+1 if any subset with cardinality t + 1 of the shares enables the secret to be recovered.

In 1983, C. Asmuth and J. Bloom proposed such a scheme based on the Chinese remainder theorem. They derived a complex relation between the parameters of the scheme in order to satisfy some notion of security. However, at that time, the concept of security in cryptography had not yet been formalized.

In this paper, we revisit the security of this threshold scheme in the modern context of security. In particular, we prove that the scheme is asymptotically optimal both from an information theoretic and complexity theoretic viewpoint when the parameters satisfy a simplified relationship. We mainly present three theorems, the two first theorems strengthen the result of Asmuth and Bloom and place it in a precise context, while the latest theorem is an improvement of a result obtained by Goldreich et al.


Average Mutual Information Uniform Probability Threshold Scheme Chinese Remainder Theorem Ideal Scheme 
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. AB83.
    Asmuth, C., Bloom, J.: A modular approach to key safeguarding. IEEE Trans. inform. Theory, 1983, IT-29, pp. 208–210.CrossRefMathSciNetGoogle Scholar
  2. B79.
    Blakley, G.R.: Safeguarding cryptographic keys. AFIPS Conf. Proc., 1979, 48, pp. 313–317.Google Scholar
  3. DF94.
    Desmedt, Y., Frankel, Y.: Homomorphic zero-knowledge threshold schemes over any finite abelian group. SIAM J. discr. math., 1994, 7, pp. 667–679.zbMATHCrossRefMathSciNetGoogle Scholar
  4. G68.
    Gallager, R.G.: Information Theory and Reliable Communication. Willey, 1968.Google Scholar
  5. GB01.
    Goldwasser S., Bellare M.: Lectures Notes on Cryptography. 1996–2001.
  6. GRS00.
    Goldreich, O., Ron, D., Sudan, M.: Chinese remainder with errors. IEEE Trans. Inform. Theory, 2000, IT-46, pp. 1330–1338.CrossRefMathSciNetGoogle Scholar
  7. KGH83.
    Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. Inform. Theory, 1983, IT-29, pp. 35–41.CrossRefMathSciNetGoogle Scholar
  8. M82.
    Mignotte, M.: How to share a secret. Advances in Cryptology — Eurocrypt’82, LNCS, 1983, 149, Springer-Verlag, pp. 371–375.Google Scholar
  9. R88.
    Ribenboim, P.: The Book of Prime Number Records. Springer-Verlag, 1988.Google Scholar
  10. S79.
    Shamir, A.: How to share a secret. Commun. ACM 1979, 22, pp. 612–613.zbMATHCrossRefMathSciNetGoogle Scholar
  11. SV88.
    Stinson, D.R., Vanstone S.A. SIAM J. discr. math., 1988, 1, pp. 230–236.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michaël Quisquater
    • 1
  • Bart Preneel
    • 1
  • Joos Vandewalle
    • 1
  1. 1.Department Electrical Engineering-ESAT, COSICKatholieke Universiteit LeuvenHeverleeBelgium

Personalised recommendations