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The Tate-Lichtenbaum Pairing on a Hyperelliptic Curve via Hyperelliptic Nets

  • Yukihiro Uchida
  • Shigenori Uchiyama
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7708)

Abstract

Recently, Stange proposed a new algorithm to compute the Tate pairing on an elliptic curve. Her algorithm is based on elliptic nets, which are also defined by Stange as a generalization of elliptic divisibility sequences. In this paper, we define hyperelliptic nets as a generalization of elliptic nets to hyperelliptic curves. We also give an expression for the Tate-Lichtenbaum pairing on a hyperelliptic curve in terms of hyperelliptic nets. Using this expression, we give an algorithm to compute the Tate-Lichtenbaum pairing on a hyperelliptic curve of genus 2.

Keywords

Tate-Lichtenbaum pairing hyperelliptic curve hyperelliptic net 

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References

  1. 1.
    Arledge, J., Grant, D.: An explicit theorem of the square for hyperelliptic Jacobians. Michigan Math. J. 49, 485–492 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 21. Springer, Berlin (1990)zbMATHCrossRefGoogle Scholar
  3. 3.
    Buchstaber, V.M., Enolskii, V.Z., Leykin, D.V.: Kleinian functions, hyperelliptic Jacobians and applications. Rev. Math. Math. Phys. 10, 1–125 (1997)Google Scholar
  4. 4.
    Buchstaber, V.M., Enolskii, V.Z., Leykin, D.V.: A recursive family of differential polynomials generated by the Sylvester identity and addition theorems for hyperelliptic Kleinian functions. Funct. Anal. Appl. 31, 240–251 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A.: Fundamental Algebraic Geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs, vol. 123. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hone, A.N.W.: Analytic solutions and integrability for bilinear recurrences of order six. Appl. Anal. 89, 473–492 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kanayama, N.: Division polynomials and multiplication formulae of Jacobian varieties of dimension 2. Math. Proc. Cambridge Philos. Soc. 139, 399–409 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kanayama, N.: Corrections to “Division polynomials and multiplication formulae in dimension 2”. Math. Proc. Cambridge Philos. Soc. 149, 189–192 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lichtenbaum, S.: Duality theorems for curves over p-adic fields. Invent. Math. 7, 120–136 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Maxima.sourceforge.net: Maxima, a Computer Algebra System. Version 5.25.1 (2011), http://maxima.sourceforge.net/
  12. 12.
    Miller, V.S.: Short programs for functions on curves (1986) (unpublished manuscript), http://crypto.stanford.edu/miller/
  13. 13.
    Miller, V.S.: The Weil pairing, and its efficient calculation. J. Cryptology 17, 235–261 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Mumford, D.: Tata Lectures on Theta I. Progress in Mathematics, vol. 28. Birkhäuser, Boston (1983)zbMATHGoogle Scholar
  15. 15.
    Mumford, D.: Tata Lectures on Theta II. Progress in Mathematics, vol. 43. Birkhäuser, Boston (1984)zbMATHCrossRefGoogle Scholar
  16. 16.
    Ônishi, Y.: Determinant expressions for hyperelliptic functions (with an appendix by Shigeki Matsutani). Proc. Edinb. Math. Soc. 48, 705–742 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Stange, K.E.: The Tate Pairing Via Elliptic Nets. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 329–348. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Stange, K.E.: Elliptic nets and elliptic curves. Algebra Number Theory 5, 197–229 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Uchida, Y.: Division polynomials and canonical local heights on hyperelliptic Jacobians. Manuscripta Math. 134, 273–308 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    PARI/GP, version 2.3.4, Bordeaux (2008), http://pari.math.u-bordeaux.fr/

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Yukihiro Uchida
    • 1
  • Shigenori Uchiyama
    • 1
  1. 1.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachiojiJapan

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