Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Construction of CM Picard curves

Authors: Kenji Koike and Annegret Weng
Journal: Math. Comp. 74 (2005), 499-518
MSC (2000): Primary 14H45, 11G15; Secondary 14G50, 14K22
Published electronically: May 21, 2004
MathSciNet review: 2085904
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.

References [Enhancements On Off] (What's this?)

  • 1. S. Arita.
    Construction of secure $C\sb {ab}$ curves using modular curves.
    Algorithmic number theory (Leiden, 2000), LNCS 1838, pages 113-126, 2000. MR 2002f:11067
  • 2. A.O.L. Atkin.
    The number of points on an elliptic curve modulo a prime.
    Unpublished manuscript, 1991.
  • 3. A.O.L. Atkin and F. Morain.
    Elliptic curves and primality proving.
    Math. Comp., 61:29-68, 1993. MR 93m:11136
  • 4. M. Bauer.
    The arithmetic of certain cubic function fields.
    Math. Comp., 73:387-413, 2003.
  • 5. C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier.
    User's guide to pari-gp.
  • 6. I. Dolgachev and D. Ortland.
    Point sets in projective spaces and theta functions.
    Asterisque, 165, 1985. MR 90i:14009
  • 7. S. Galbraith, S.M. Paulus, and N. Smart.
    Arithmetic on superelliptic curves.
    Math. Comp., 71(237):393-405, 2002. MR 2002h:14102
  • 8. P. Gaudry.
    An algorithm for solving the discrete logarithm problem on hyperelliptic curves.
    Eurocrypt 2000, LNCS 1807, Springer, pages 19-34, 2000.
  • 9. P. Gaudry and N. Gurel.
    An extension of Kedlaya's algorithm to superelliptic curves.
    Asiacrypt 2001, LNCS 2248, Springer, pages 480-494, 2001. MR 2003h:11159
  • 10. E. Gottschling.
    Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades.
    Math. Ann., 138:103-124, 1959. MR 21:5748
  • 11. R. Hartshorne.
    Algebraic geometry.
    Springer, 1977. MR 57:3116
  • 12. R.-P. Holzapfel.
    The ball and some Hilbert problems.
    Birkhäuser, 1995. MR 97g:11059
  • 13. T. Honda.
    Isogeny classes of abelian varieties over finite fields. J.
    Math. Soc. Japan, 20:83-95, 1968. MR 37:5216
  • 14. N. Koblitz.
    Elliptic curve cryptosystems.
    Math. Comp., 48(177):203-209, 1987. MR 88b:94017
  • 15. N. Koblitz.
    Hyperelliptic cryptosystems.
    J. Cryptology, 1:139-150, 1989. MR 90k:11165
  • 16. S. Lang.
    Introduction to Algebraic and Abelian Functions.
    Springer, 1982. MR 84m:14032
  • 17. S. Lang.
    Complex Multiplication.
    Springer, 1983. MR 85f:11042
  • 18. H. Lange and Ch. Birkenhake.
    Complex Abelian varieties.
    Springer, 1982.
  • 19. MAGMA.
    University of Sydney, 2002.
  • 20. V. S. Miller.
    The use of elliptic curves in cryptography.
    Advances in cryptology--CRYPTO '85 (Santa Barbara, Calif., 1985), Springer, Berlin LNCS, 218:417-426, 1986. MR 88b:68040
  • 21. J.-S. Milne.
    Jacobian varieties.
    In Cornell G. and J.H. Silverman, editors, Arithmetic Geometry, pages 167-212. Springer, 1986.
  • 22. D. Mumford.
    Tata Lectures on Theta I.
    Birkhäuser, 1983. MR 85h:14026
  • 23. M. Newman.
    Integral matrices.
    Pure and applied mathematics, Vol. 45, Academic Press, New York-London,
    1972. MR 49:5038
  • 24. F. Oort and K. Ueno.
    Principally polarized abelian varieties of dimension two or three are Jacobian varieties.
    J. Fac. Sci. Univ. Tokyo Sect. IA Math, 20:377-381, 1973. MR 51:520
  • 25. E. Picard.
    Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires.
    Acta math., 2:114-135, 1883.
  • 26. E. Picard.
    Sur les formes quadratiques ternaire indéfinies et sur les fonctions hyperfuchsiennes,
    Acta math., 5:121-182, 1884.
  • 27. S. Pohlig and M. Hellmann.
    An improved algorithm for computing logarithms over ${GF}(p)$ and its cryptographic significance.
    IEEE Trans. Inform. Theory, IT-24:106-110, 1978. MR 58:4617
  • 28. E. Pohst and H. Zassenhaus.
    Algorithmic Number Theory.
    Cambridge University Press, 1989. MR 92b:11074
  • 29. E. Reinaldo-Barreiro, J. Estrada-Sarlabois, and J.P. Cherdieu.
    Efficient reduction on the Jacobian variety of Picard curves.
    Coding theory, cryptography and related areas (Guanajuato, 1998), Springer, pages 13-28, 2000. MR 2001f:14057
  • 30. H. Shiga.
    On the representation of the Picard modular function by $\theta$ constants i-ii.
    Publ. RIMS, Kyoto Univ., 24(3):311-360, 1988. MR 89k:11036
  • 31. G. Shimura.
    Abelian varieties with complex multiplication and modular functions.
    Princeton University Press, revised edition, 1998. MR 99e:11076
  • 32. G. Shimura.
    Introduction to the Arithmetic Theory of Automorphic Functions.
    Princeton University Press, 1971. MR 47:3318
  • 33. C.L. Siegel.
    Topics in Complex Function Theory. Vol. II.
    John Wiley and Sons, 1972
  • 34. A.-M. Spallek.
    Kurven vom Geschlecht 2 und ihre Anwendung in Public-Key-Kryptosystemen.
    PhD thesis, Institut für Experimentelle Mathematik, Universität GH Essen, 1994.
  • 35. J. Tate.
    Classes d'isogénie des variétès abéliennes sur un corps fini (d'après T. Honda).
    Seminaire Bourbaki, Soc. Math. France., 352, 95-110. 1968
  • 36. P. van Wamelen.
    Examples of genus two CM curves defined over the rationals.
    Math. Comp., 68, 1999. MR 99c:11079
  • 37. A. Weng.
    Constructing hyperelliptic curves of genus 2 suitable for cryptography. Math. Comp., 72:435-458, 2003. MR 2003i:14029
  • 38. A. Weng.
    A class of hyperelliptic CM-curves of genus three.
    Journal of the Ramanujan Mathematical Society 16, 4:339-372, 2001. MR 2002k:11099
  • 39. K. Yamamura.
    On unramified Galois extensions of real quadratic number fields.
    Osaka J. Math. 23, 471-486, 1986. MR 88a:11112

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 14H45, 11G15, 14G50, 14K22

Retrieve articles in all journals with MSC (2000): 14H45, 11G15, 14G50, 14K22

Additional Information

Kenji Koike
Affiliation: Institut für Algebra und Geometrie, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60054 Frankfurt am Main, Germany

Annegret Weng
Affiliation: Institut für Algebra und Geometrie, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60054 Frankfurt am Main, Germany

Received by editor(s): February 3, 2003
Received by editor(s) in revised form: July 14, 2003
Published electronically: May 21, 2004
Additional Notes: The first author was supported by the Alexander von Humboldt Stiftung. The second author was supported by the Maria Sibylla Merian program of the university of Essen
Dedicated: Dedicated to the 60th birthday of Professor Rolf Peter Holzapfel
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society