A quasi quadratic time algorithm for hyperelliptic curve point counting

Ramanujan Journal

Volumn 12, Issue 3, 2006, Pages 399-423

  Lercier, Reynald ^a   Lubicz, David ^a  

^a CELAR   (France)

Author keywords

AGM; Cryptography; Finite fields; Hyperelliptic curves; Point counting algorithms

Indexed keywords

EID: 33751552694     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-006-0151-6     Document Type: Article

References (38)

1. Bosma, W., Cannon, J., Playoust, C.: The magma algebra system I: the user language. J. Symbolic Comp. 24(3), 235-265 (1997)
2. Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48(177), 95-101 (1987)
3. Carls, R.: Generalized AGM sequences and approximation of canonical lifts (2003). Available at http://www.math.leidenuniv.nl/~carls
4. Chabaud, F., Lercier, R.: ZEN, User Manual (1996)
5. Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptology 10(4), 233-260 (1997)
6. Denef, J., Vercauteren F.: An extension of Kedlaya's algorithm to Artin-Schreier curves in characteristic 2. In: Fieker C. Kohel D.R. (eds.) Algorithmic Number Theory, 5th International Symposium, ANTS-V, pp 369-384. Springer (2002)
7. Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, vol. 352. Springer-Verlag, Berlin (1973)
8. Gaudry, P.: Algorithmique des courbes hyperelliptiques et applications à la cryptologie. PhD thesis, École Polytechnique (2000)
9. 24 + 41). Email at the Number Theory List, (September, 2002)
10. Free Software Foundation GNU. GMP library (2002). Available at http://www.swox.com/gmp/
11. Griffiths, P., Harris, J.: Principles of algebraic geometry. Wiley Classics Library. Reprint of the 1978 original. John Wiley & Sons Inc., New York (1994)
12. Harley, R.: Asymptotically optimal p-adic point-counting. E-mail to the NMBRTHRY mailing list (December 2002)
13. Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323-328 (2001)
14. Kim, H.Y., Park, J.Y., Cheon, J.H., Park, J.H., Kim, J.H., Hahn. S.G.: Fast elliptic curve point counting using gaussian normal basis. In: Fieker C., Kohel D.R. (eds.) Algorithmic Number Theory, 5th International Symposium, ANTS-V, pp. 292-307. Springer Verlag, Berlin (2002)
15. Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptology 1(3), 139-150 (1989)
16. Lang, S.: Algebra. 3rd revised ed. Graduate Texts in Mathematics. 211. Springer, New York, NY (2002)
17. Lauder, A.G.B., Wan, D.: Computing Zeta functions of Artin-Schreier curves over finite fields. LMS J. Comput. Math. 5, 34-55 (electronic) (2002)
18. Lenstra, A.K., Lenstra, H.W., Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 513-534 (1982)
19. Lercier, R., Lubicz, D.: Counting points on elliptic curves over finite fields of small characteristic in quasi quadratic time. In: Biham E (ed.) Advances in Cryptology - EUROCRYPT 2003, Lecture Notes in Computer Science. Springer-Verlag (2003)
20. Lubin, J., Serre, J.-P., Tate, J.: Elliptic curves and formal groups. Available at http:/-/ma.utexas.edu/users/voloch/lst.html (1964)
21. Menezes, A.J., Blake, I.F., Gao, X., Mullin, R.C., Vanstone, S.A., Yaghoobian, T.: xi, 218, Applications of finite fields. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, p. (1993)
22. Mestre, J.-F.: Lettre à Gaudry et Harley (2001). Available at http://www.math.jussieu.fr/~mestre
23. Mestre, J.-F.: Notes of a talk given at the seminar of cryptography of Rennes (2002). Available at http://www.math.univ-rennes1.fr/crypto/2001-02/ mestre.ps
24. Mumford, D.: On the equations defining abelian varieties. I. Invent. Math. 1, 287-354 (1966)
25. Mumford, D.: On the equations defining abelian varieties. II. Invent. Math. 3, 75-135 (1967)
26. Mumford, D.: Tata lectures on theta I, volume 28 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (1983). With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman
27. Mumford, D.: Tata lectures on theta II, volume 43 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (1984). Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura
28. Neukirch, J.: Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder. Springer-Verlag, Berlin (1999)
29. Pila, J.: Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp. 55(192), 745-763 (1990)
30. Ritzenthaler, C.: Problèmes arithmétiques relatifs à certaines familles de courbes sur les corps finis. PhD thesis, Université Paris 7-Denis Diderot (2003)
31. Satoh, T.: The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc. 15(4), 247-270 (2000)
32. Satoh, T., Skjernaa, B., Taguchi, Y.: Fast computation of canonical lifts of elliptic curves and its application to point counting. Finite Fields and Their Applications 9(1), 89-101 (2003)
33. Schoof, R.: Counting points on elliptic curves over finite fields. J. Théorie des nombres de Bordeaux 7, 483-494 (1998)
34. Serre, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble 6, 1-42 (1955-1956)
35. Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions. Princ. Univ. Press, NJ (1998)
36. Silverman, J.H.: The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Corrected reprint of the 1986 original. Springer-Verlag, New York (1986)
37. Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134-144 (1966)
38. Vercauteren, F.: Computing zeta functions of curves over finite fields. PhD thesis, Katholieke Universiteit Leuven (2003). Preprint