Generalised Jacobians in Cryptography and Coding Theory

  • Florian Hess
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7369)

Abstract

The use of generalised Jacobians in discrete logarithm based cryptosystems has so far been rather limited since they offer no advantage over traditional discrete logarithm based systems. In this paper we continue the search for possible applications in two directions.

Firstly, we investigate pairings on generalised Jacobians and show that these are insecure. Secondly, generalising and extending prior work, we show how the discrete logarithm problem in generalised Jacobians can be reduced to the minimal non zero weight word and maximum likelihood decoding problems in generalised algebraic geometric codes.

Keywords

Elliptic Curve Cayley Graph Discrete Logarithm Code Theory Discrete Logarithm Problem 
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.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Augot, D., Morain, F.: Discrete logarithm computations over finite fields using Reed-Solomon codes (2012), http://hal.inria.fr/hal-00672050
  2. 2.
    Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. In: Melles, C.G., et al. (eds.) Topics in Algebraic and Noncommutative Geometry; Proceedings in Memory of Ruth Michler, Luminy, France, Annapolis, MD, USA, July 20-22, October 25-28. American Mathematical Society (AMS), Providence (2001); Contemp. Math. 324, 71–90 (2003)Google Scholar
  3. 3.
    Cheng, Q.: Hard problems of algebraic geometry codes. IEEE Trans. Inform. Theory 54(1), 402–406 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cheng, Q., Wan, D.: On the list and bounded distance decodability of Reed-Solomon codes. SIAM J. Comput. 37(1), 195–209 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cheng, Q., Wan, D.: Complexity of Decoding Positive-Rate Reed-Solomon Codes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 283–293. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Déchène, I.: Arithmetic of Generalized Jacobians. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 421–435. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Déchène, I.: On the security of generalized Jacobian cryptosystems. Adv. Math. Commun. 1(4), 413–426 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Erdős, P., Rényi, A.: Probabilistic methods in group theory. J. Analyse Math. 14, 127–138 (1965)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Frey, G., Rück, H.-G.: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp. 62, 865–874 (1994)MathSciNetMATHGoogle Scholar
  10. 10.
    Galbraith, S.D., Smith, B.: Discrete Logarithms in Generalized Jacobians (2006), http://hal.inria.fr/inria-00537887
  11. 11.
    Galbraith, S.D., Hess, F., Vercauteren, F.: Aspects of pairing inversion. IEEE Trans. Inf. Theory 54(12), 5719–5728 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hasse, H.: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172, 37–54 (1934)Google Scholar
  13. 13.
    Hess, F.: Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symbolic Comp. 33(4), 425–445 (2002)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hess, F.: A note on the Tate pairing of curves over finite fields. Arch. Math. 82, 28–32 (2004)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hess, F., Pauli, S., Pohst, M.E.: Computing the multiplicative group of residue class rings. Math. Comp. 72(243), 1531–1548 (2003) (electronic)Google Scholar
  16. 16.
    Huang, M.-D., Raskind, W.: A multilinear generalization of the Tate pairing. In: McGuire, G., et al. (eds.) Finite Fields. Theory and Applications. Proceedings of the 9th International Conference on Finite Fields and Applications, Dublin, Ireland, July 13-17, American Mathematical Society (AMS), Providence (2009); Contemporary Mathematics 518, 255–263 (2010)Google Scholar
  17. 17.
    Jao, D., Miller, S.D., Venkatesan, R.: Expander graphs based on GRH with an application to elliptic curve cryptography. J. Number Theory 129(6), 1491–1504 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kohel, D.: Constructive and destructive facets of torus-based cryptography (2004), http://echidna.maths.usyd.edu.au/kohel/pub/torus.ps
  19. 19.
    Papamanthou, C., Tamassia, R., Triandopoulos, N.: Optimal Authenticated Data Structures with Multilinear Forms. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 246–264. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Serre, J.-P.: Algebraic groups and class fields, Transl. of the French edn. Graduate Texts in Mathematics, vol. 117, ix, 207 p. Springer, New York (1988)Google Scholar
  21. 21.
    Stichtenoth, H.: Algebraic function fields and codes, 2nd edn. Graduate Texts in Mathematics, vol. 254, xiii, 355 p. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Florian Hess
    • 1
  1. 1.Carl-von-Ossietzky Universität OldenburgGermany

Personalised recommendations