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Efficient Indifferentiable Hashing into Ordinary Elliptic Curves

  • Eric Brier
  • Jean-Sébastien Coron
  • Thomas Icart
  • David Madore
  • Hugues Randriam
  • Mehdi Tibouchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6223)

Abstract

We provide the first construction of a hash function into ordinary elliptic curves that is indifferentiable from a random oracle, based on Icart’s deterministic encoding from Crypto 2009. While almost as efficient as Icart’s encoding, this hash function can be plugged into any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle model.

We also describe a more general (but less efficient) construction that works for a large class of encodings into elliptic curves, for example the Shallue-Woestijne-Ulas (SWU) algorithm. Finally we describe the first deterministic encoding algorithm into elliptic curves in characteristic 3.

References

  1. 1.
    Baek, J., Zheng, Y.: Identity-based threshold decryption. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 262–276. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Bellare, M., Rogaway, P.: Random oracles are practical: A paradigm for designing efficient protocols. In: ACM Conference on Computer and Communications Security, pp. 62–73 (1993)Google Scholar
  3. 3.
    Boldyreva, A.: Threshold signatures, multisignatures and blind signatures based on the gap-diffie-hellman-group signature scheme. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 31–46. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Boneh, D., Franklin, M.K.: Identity-based encryption from the weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Boneh, D., Gentry, C., Lynn, B., Shacham, H.: Aggregate and verifiably encrypted signatures from bilinear maps. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 416–432. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Boneh, D., Lynn, B., Shacham, H.: Short signatures from the weil pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Boyd, C., Montague, P., Nguyen, K.Q.: Elliptic curve based password authenticated key exchange protocols. In: Varadharajan, V., Mu, Y. (eds.) ACISP 2001. LNCS, vol. 2119, pp. 487–501. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Boyen, X.: Multipurpose identity-based signcryption (a swiss army knife for identity-based cryptography). In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 383–399. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Boyko, V., MacKenzie, P.D., Patel, S.: Provably secure password-authenticated key exchange using diffie-hellman. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 156–171. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Brier, E., Coron, J.-S., Icart, T., Madore, D., Randriam, H., Tibouchi, M.: Efficient indifferentiable hashing into ordinary elliptic curves. Cryptology ePrint Archive, Report 2009/340 (2009) (full version of this paper), http://eprint.iacr.org/
  11. 11.
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. J. ACM 51(4), 557–594 (2004)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Cha, J.C., Cheon, J.H.: An identity-based signature from gap diffie-hellman groups. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 18–30. Springer, Heidelberg (2002)Google Scholar
  13. 13.
    Chevallier-Mames, B.: An efficient cdh-based signature scheme with a tight security reduction. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 511–526. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Coron, J.-S., Dodis, Y., Malinaud, C., Puniya, P.: Merkle-damgård revisited: How to construct a hash function. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 430–448. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Farashahi, R.R., Shparlinski, I.E., Voloch, J.F.: On hashing into elliptic curves (2010) (preprint), http://www.ma.utexas.edu/users/voloch/preprint.html
  16. 16.
    Fouque, P.-A., Tibouchi, M.: Estimating the size of the image of deterministic hash functions to elliptic curves. Cryptology ePrint Archive, Report 2010/037 (2010), http://eprint.iacr.org/
  17. 17.
    Gentry, C., Silverberg, A.: Hierarchical id-based cryptography. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 548–566. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Horwitz, J., Lynn, B.: Toward hierarchical identity-based encryption. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 466–481. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Icart, T.: How to hash into elliptic curves. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 303–316. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Jablon, D.P.: Strong password-only authenticated key exchange. SIGCOMM Comput. Commun. Rev. 26(5), 5–26 (1996)CrossRefGoogle Scholar
  21. 21.
    Libert, B., Quisquater, J.-J.: Efficient signcryption with key privacy from gap diffie-hellman groups. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 187–200. Springer, Heidelberg (2004)Google Scholar
  22. 22.
    Maurer, U.M., Renner, R., Holenstein, C.: Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 21–39. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Menezes, A., Okamoto, T., Vanstone, S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory 39(5), 1639–1646 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mestre, J.-F.: Rang de courbe elliptiques d’invariant donné. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 314(12), 297–319 (1992)MathSciNetGoogle Scholar
  25. 25.
    Shallue, A., van de Woestijne, C.E.: Construction of rational points on elliptic curves over finite fields. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 510–524. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Ulas, M.: Rational points on certain hyperelliptic curves over finite fields. Bull. Polish Acad. Sci. Math. 55(2), 97–104 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Zhang, F., Kim, K.: Id-based blind signature and ring signature from pairings. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 533–547. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Eric Brier
    • 1
  • Jean-Sébastien Coron
    • 2
  • Thomas Icart
    • 2
  • David Madore
    • 3
  • Hugues Randriam
    • 3
  • Mehdi Tibouchi
    • 2
    • 4
  1. 1.Ingenico 
  2. 2.Université du Luxembourg 
  3. 3.TELECOM-ParisTech 
  4. 4.École normale supérieure 

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