An extension of Kedlaya's algorithm to hyperelliptic curves in characteristic 2

Journal of Cryptology

Volumn 19, Issue 1, 2006, Pages 1-25

  Denef, Jan ^a   Vercauteren, Frederik ^a,b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Cryptography; Hyperelliptic curves; Kedlaya's algorithm; Monsky Washnitzer cohomology

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; COMPUTER APPLICATIONS; COMPUTER GRAPHICS; COMPUTER SCIENCE; INFORMATION TECHNOLOGY;

CRYPTOLOGIC RESEARCH; HYPERELLIPTIC CURVES; KEDLAYA'S ALGORITHM; MONSKY-WASHNITZER COHOMOLOGY;

CRYPTOGRAPHY;

EID: 29644434493     PISSN: 09332790     EISSN: 14321378     Source Type: Journal    
DOI: 10.1007/s00145-004-0231-y     Document Type: Article

References (42)

1. L. M. Adleman and M.-D. Huang. Counting rational points on curves and abelian varieties over finite fields. In H. Cohen, editor, Algorithmic Number Theory. 2nd International Symposium. ANTS-II, pages 1-16. Volume 1122 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1996.
2. ab curves and their application to discrete-log-based public key cryptosystems. In Proceedings of Conference on the Mathematics of Public Key Cryptography, pages 165-175, 1999.
3. 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.
4. I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series, 265. Cambridge University Press, Cambridge, 1999.
5. S. Bosch. A rigid analytic version of M. Artin's theorem on analytic equations. Math. Ann., 255:395-404, 1981.
6. H. Cohen. A Course in Computational Algebraic Number Theory. Volume 138 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1993.
7. J. Denef and F. Vercauteren. An extension of Kedlaya's algorithm to Artin-Schreier curves in characteristic 2. In C. Fieker and D.R. Kohel, editors, Algorithmic Number Theory. 5th International Symposium. ANTS-V, pages 308-323. Volume 2369 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2002.
8. ab curves using Monsky-Washnitzer cohomology. Available at http://www.cs.bris.ac.uk/ ~frederik/, 2003.
9. N. Elkies. Elliptic and modular curves over finite fields and related computational issues. In D. A. Buell and J. T. Teitelbaum, editors, Computational Perspectives on Number Theory, pages 21-76. American Mathematical Society/International Press, Providence, RI/Somerville, MA, 1998.
10. R. Elkik. Solutions d'équations a coefficients dans un anneau henselien. Ann. Sci. École Norm. Sup., 6(4):553-604, 1973.
11. G. Frey and H.-G. Rück. A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp., 62(206):865-874, 1994.
12. S. Galbraith. Supersingular curves in cryptography. In C. Boyd, editor, Advances in Cryptology - ASIACRYPT2001, pages 495-513. Volume 2248 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2001.
13. S. Galbraith. Weil descent of Jacobians. Discrete Appl. Math., 128(1):165-180, 2003.
14. S. Galbraith, S. Paulus, and N. Smart. Arithmetic on superelliptic curves. Math. Comp., 71(237):393-405, 2002.
15. P. Gaudry and N. Gúrel. An extension of Kedlaya's algorithm for counting points on superelliptic curves. In C. Boyd, editor, Advances in Cryptology - ASIACRYPT2001, pages 480-494. Volume 2248 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2001.
16. P. Gaudry and R. Harley. Counting points on hyperelliptic curves over finite fields. In Wieb Bosma, editor, Algorithmic Number Theory, 4th International Symposium. ANTS-IV, pages 313-332. Volume 1838 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2000.
17. P. Gaudry and É. Schost. Construction of secure random curves of genus 2 over prime fields. In C. Cachin and J. Camenisch, editors, Advances in Cryptology - EUROCRYPT 20004, pages 239-256. Volume 3027 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2004.
18. A. Karatsuba and Y. Ofman. Multiplication of multidigit numbers on automata. Soviet Phys. Dokl., 7:595-596, 1963.
19. K.S. Kedlaya. Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc., 16:323-338, 2001.
20. N. Koblitz. Elliptic curve cryptosystems. Math. Comp., 48(177):203-209, 1987.
21. N. Koblitz. Hyperelliptic cryptosystems. J. Cryptology, 1(3):139-150, 1989.
22. A.G.B. Lauder and D. Wan. Counting points on varieties over finite fields of small characteristic. In J.P. Buhler and P. Stevenhagen, editors, Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. Cambridge University Press, Cambridge, 2002.
23. A.G.B. Lauder and D. Wan. Computing zeta functions of Artin-Schreier curves over finite fields. LMS J. Comput. Math., 5:34-55 (electronic), 2002.
24. A.G.B. Lauder and D. Wan. Computing zeta functions of Artin-Schreier curves over finite fields, II. J. Complexity, 20:331-349, 2004.
25. R. Lercier. Algorithmique des courbes elliptiques dans les corps finis. Ph.D. thesis, Laboratoire d'Informatique de l'École polytechnique (LIX), 1997.
26. R. Lercier and D. Lubicz. A quasi quadratic time algorithm for hyperelliptic curve point counting. Preprint, 2003.
27. J.-F. Mestre. Lettre adressée à Gaudry et Harley, December 2000. Available at http://www.math.jussieu.fr/~mestre/.
28. J.-F. Mestre. Algorithmes pour compter des points en petite caractéristique en genre 1 and 2. Available at http://www.math.univ- rennes1.fr/crypto/2001-02/mestre.ps.
29. V. S. Miller. Use of elliptic curves in cryptography. In H. C. Williams, editor, Advances in Cryptology - CRYPTO 1985, pages 417-426. Volume 218 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1986.
30. R.T. Moenck. Fast computation of GCDs. Fifth Annual ACM Symposium on Theory of Computing, pages 142-151, 1973.
31. P. Monsky. Formal cohomology, II: The cohomology sequence of a pair. Ann. of Math., 88:218-238, 1968.
32. P. Monsky. Formal cohomology, III: Fixed point theorems. Ann. of Math., 93:315-343, 1971.
33. P. Monsky. p-Adic Analysis and Zeta Functions. Lectures in Mathematics, Department of Mathematics Kyoto University. 4. Tokyo, Japan, 1970.
34. P. Monsky and G. Washnitzer. Formal cohomology, I. Ann. of Math., 88:181-217, 1968.
35. J. Pila. Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp., 55(192):745-763, 1990.
36. T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc., 15:247-270, 2000.
37. A. Schönhage and V. Strassen. Schnelle Multiplikation grosser Zahlen. Computing (Arch. Elektwn. Rechnen), 7:281-292, 1971.
38. R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483-494, 1985.
39. M. van der Put. The cohomology of Monsky and Washnitzer. Mém. Soc. Math. France, 23:33-60, 1986.
40. F. Vercauteren. Computing zeta functions of hyperelliptic curves over finite fields of characteristic 2. In M. Yung, editor, Advances in Cryptology - CRYPTO 2002, pages 369-384. Volume 2442 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2002.
41. F. Vercauteren. Computing Zeta Functions of Curves over Finite Fields. Ph.D. thesis, Katholieke Universiteit Leuven, November 2003. Available at http://www.cs.bris.ac.uk/~frederik/.
42. D. Wan. Computing zeta functions over finite fields. In Finite Fields: Theory, Applications, and Algorithms, pages 131-141. Volume 225 of Contemporary Mathematics, American Mathematical Society, Providence, RI, 1999.