Modular equations for hyperelliptic curves

Mathematics of Computation

Volumn 74, Issue 249, 2005, Pages 429-454

  Gaudry, P ^a   Schost, E ^b  

^a ÉCOLE POLYTECHNIQUE   (France)

^b ECOLE POLYTECHNIQUE   (France)

Author keywords

Hyperelliptic curves; Modular equations; Schoof elkies atkin algorithm

Indexed keywords

EID: 11244288438     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-04-01682-5     Document Type: Article

References (32)

1. Magma, http://www.maths.usyd.edu.au:8000/u/magma/.
2. Medicis, http://www.medicis.polytechnique.fr/.
3. Tera, http://tera.medicis.polytechnique.fr/.
4. L. Adleman and M.-D. Huang, Counting points on curves and abelian varieties over finite fields, J. Symbolic Comput. 32 (2001), 171-189. MR2002j:14027
5. A. O. L. Atkin, The number of points on an elliptic curve modulo a prime, Series of e-mails to the NMBRTHRY mailing list, 1992.
6. I. Blake, G. Seroussi, and N. Smart, Elliptic curves in cryptography, London Math. Soc. Lecture Note Ser., vol. 265, Cambridge University Press, 1999. MR2001i:94048
7. J. Boxall and D. Grant, Examples of torsion points on genus two curves, Trans. Amer. Math. Soc. 352 (2000), 4533-4555. MR2001b:11057
8. B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, Chapter 6, in N. K. Bose et al., Multidimensional Systems Theory, Reidel, Dordrecht, 1985, pp. 374-383. MR87m:93017
9. D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), 95401. MR88f:11118
10. _, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. 447 (1994), 91-145. MR94m:11071
11. L. S. Charlap, R. Coley, and D. P. Robbins, Enumeration of rational points on elliptic curves over finite fields, Draft, 1991.
12. N. Elkies, Elliptic and modular curves over finite fields and related computational issues, Computational Perspectives on Number Theory (D.A. Buell and J.T. Teitelbaum, eds.), AMS/International Press, 1998, Proceedings of a Conference in Honor of A.O.L. Atkin, pp. 21-76. MR99a:11078
13. P. Gaudry and R. Harley, Counting points on hyperelliptic curves over finite fields, ANTS-IV (W. Bosma, ed.), Lecture Notes in Comput. Sci., vol. 1838, Springer-Verlag, 2000, pp. 313-332. MR2002f: 11072
14. M. Giusti, J. Heintz, J. E. Morais, J. Morgenstern, and L. M. Pardo, Straight-line programs in geometric elimination theory, J. of Pure and App. Algebra 124 (1998), 101-146. MR99d:68128
15. M. Giusti, G. Lecerf, and B. Salvy, A Gröbner free alternative for polynomial system solving, J. Complexity 17 (2001), 154-211. MR2002b:68123
16. J. Heintz, T. Krick, S. Puddu, J. Sabia, and A. Waissbein, Deformation techniques for efficient polynomial equation solving, J. Complexity 16 (2000), 70-109. MR2001c:65067
17. M. Hindry and J. Silverman, Diophantine geometry, an introduction, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, 2000. MR2001e:11058
18. M.-D. Huang and D. Ierardi, Counting points on curves over finite fields, J. Symbolic Comput. 25 (1998), 1-21. MR98i:11040
19. C. Dicrescenzo J. Della Dora and D. Duval, About a new method for computing in algebraic number fields, Proceedings Eurocal '85 Vol. 2, Lecture Notes in Comput. Sci., vol. 204, 1985, pp. 289-290
20. E. Kaltofen and V. Shoup, Subguadratic-time factoring of polynomials over finite fields, Math. Comp., 67 (1998), 1179-1197. MR99m:68097
21. W. Kampkötter, Explizite Gleichungen für Jacobische Varietäten hyperelliptischer Kurven, Ph.D. thesis, Universität Gesamthochschule Essen, August 1991
22. J. S. Milne, Abelian varieties, Arithmetic Geometry (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, 1986, pp. 103-150. MR89b: 14029
23. _, Jacobian varieties, Arithmetic Geometry (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, 1986, pp. 167-212. MR89b:14029
24. F. Morain, Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques, J. Théor. Nombres Bordeaux 7 (1995), 255-282. MR97i:11069
25. D. Mumford, Tata lectures on theta I, Progr. Math., vol. 28, Birkhäuser, 1983. MR85h:14026
26. J. Pila, frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp. 55 (1990), 745-763. MR91a:11071
27. R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985), 483-494. MR86e:11122
28. _, Counting points on elliptic curves over finite fields, J. Théor. Nombres Bordeaux 7 (1995), 219-254. MR97i:11070
29. É. Schost, Complexity results for triangular sets, J. Symbolic Comput. 36 (2003), 555-594.
30. J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, 1986. MR87g:11070
31. J. von zur Gathen and J. Gerhard, Modern computer algebra, Cambridge University Press, 1999. MR2000j:68205
32. D. Y. Y. Yun, On square-free decompositions algorithms, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, ACM Press, 1976, pp. 26-35.