The Group Diffie-Hellman Problems

  • Emmanuel Bresson
  • Olivier Chevassut
  • David Pointcheval
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2595)


In this paper we study generalizations of the Diffie-Hellman problems recently used to construct cryptographic schemes for practical purposes. The Group Computational and the Group Decisional Diffie- Hellman assumptions not only enable one to construct efficient pseudorandom functions but also to naturally extend the Diffie-Hellman protocol to allow more than two parties to agree on a secret key. In this paper we provide results that add to our confidence in the GCDH problem. We reach this aim by showing exact relations among the GCDH, GDDH, CDH and DDH problems.


Group Computational Basic Trigon Cyclic Multiplicative Group Hellman Assumption Probabilistic Turing Machine 
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.
    E. Biham, D. Boneh, and O. Reingold. Breaking generalized Diffie-Hellman modulo a composite is no easier than factoring. In Information Processing Letters (IPL), volume 70(2), pages 83–87. Elsevier Science, April 1999.MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Blake-Wilson, D. Johnson, and A. Menezes. Key agreement protocols and their security analysis. In M. Darnell, editor, Proc. of 6th IMA International Conference on Crypotography and Coding, volume 1355 of LNCS, pages 30–45. Springer-Verlag, 1997.zbMATHGoogle Scholar
  3. 3.
    S. Blake-Wilson and A. Menezes. Authenticated Diffie-Hellman key agreement protocols. In H. Meijer and S. Tavares, editors, Proc. of Selected Areas in Cryptography SAC’ 98, volume 1556 of LNCS, pages 339–361. Springer-Verlag, August 1998.zbMATHGoogle Scholar
  4. 4.
    Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In Proc. of ACM CCS’ 93, pages 62–73. ACM Press, November 1993.Google Scholar
  5. 5.
    D. Boneh. The decision Diffie-Hellman problem. In J. P. Buhler, editor, Proc. of the 3 rdANTS Symposium, volume 1423 of LNCS, pages 48–63, Portland, OR, USA, June 1998. Springer-Verlag.zbMATHGoogle Scholar
  6. 6.
    E. Bresson, O. Chevassut, D. Pointcheval, and J.-J. Quisquater. Provably authenticated group Diffie-Hellman key exchange. In P. Samarati, editor, Proc. of ACM CCS’ 01, pages 255–264. ACM Press, November 2001.Google Scholar
  7. 7.
    E. Bresson, O. Chevassut, and D. Pointcheval. Provably authenticated group Diffie-Hellman key exchange-the dynamic case. In C. Boyd, editor, Proc. of Asiacrypt’ 01, volume 2248 of LNCS, pages 290–309. Springer-Verlag, December 2001. Full Version available at Scholar
  8. 8.
    E. Bresson, O. Chevassut, and D. Pointcheval. Dynamic group Diffie-Hellman key exchange under standard assumptions. In L. R. Knudsen, editor, Proc. of Eurocrypt’ 02, volume 2332 of LNCS, pages 321–336. Springer-Verlag, May 2002. Full Version available at Scholar
  9. 9.
    E. Bresson, O. Chevassut, and D. Pointcheval. Group diffie-hellman key exchange secure against dictionary attacks. In Y. Zheng, editor, Proc. of Asiacrypt’ 2002. Springer, December 2002. Full Version available at Scholar
  10. 10.
    R. Cramer and V. Shoup. A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In H. Krawczyk, editor, Proc. of Crypto’ 98, volume 1462 of LNCS, pages 13–25. Springer-Verlag, August 1998.zbMATHGoogle Scholar
  11. 11.
    W. Diffie and M. E. Hellman. New directions in cryptography. Transactions on Information Theory, IT-22(6):644–654, November 1976.MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. In Proc. of Crypto’ 84, LNCS 196, pp. 10–18.zbMATHGoogle Scholar
  13. 13.
    M. Naor and O. Reingold. Number-theoretic constructions of efficient pseudorandom functions. In Proc. of FOCS’ 97, pages 458–467. IEEE Computer Society Press, October 1997.Google Scholar
  14. 14.
    R. L. Rivest, A. Shamir, and L. M. Adleman. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2):120–126, February 1978.MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. Shoup. On formal models for secure key exchange. Technical Report RZ 3120, IBM Zürich Research Lab, November 1999.Google Scholar
  16. 16.
    M. Steiner, B. Pfitzmann, and M. Waidner. A formal model for multi-party group key agreement. PhD Thesis RZ 3383, IBM Research, April 2002.Google Scholar
  17. 17.
    M. Steiner, G. Tsudik, and M. Waidner. Diffie-Hellman key distribution extended to group communication. In Proc. of ACM CCS’ 96, pages 31–37. ACM Press, March 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Emmanuel Bresson
    • 1
  • Olivier Chevassut
    • 2
    • 3
  • David Pointcheval
    • 1
  1. 1.École normale supérieureParis Cedex 05France
  2. 2.Lawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.Université Catholique de LouvainLouvain-la-NeuveBelgium

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