Copyright © The Author 2007. Published by Oxford University Press.
Kedlaya's Algorithm in Larger Characteristic
Department of Mathematics, Harvard University, 1 Oxford St, Cambridge MA 02138, USA
Correspondence: Correspondence to be sent to: dmharvey{at}math.harvard.edu
We show that the linear dependence on p of the running time of Kedlaya's point-counting algorithm in characteristic p may be reduced to p1/2.
References
- Bostan A., Pierrick G., Eric S. Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier-Manin Operator. SIAM Journal on Computing (2007) 36(no. 6):1777–1806.[CrossRef][ISI]
- Chudnovsky D. V., Chudnovsky G. V. Approximations and Complex Multiplication According to Ramanujan. (1988) Boston, MA: Academic Press. 375–472. in Ramanujan Revisited (Urbana-Champaign, Ill. 1987).
- Cohen H., ed. Handbook of Elliptic and Hyperelliptic Curve Cryptography (2006) Boca Raton, FL: Chapman & Hall/CRC. Discrete Mathematics and its Applications (Boca Raton).
- Edixhoven B. (2003) http://www.math.leidenuniv.nl/
edix/oww/mathofcrypt/carls_edixhoven/kedlaya.pdf (retrieved Oct 25th 2006). - Gaudry P., Gürel N. Counting Points in Medium Characteristic Using Kedlaya's Algorithm. Experimental Mathematics (2003) 12(no. 4):395–402.[ISI]
- Hanrot G., Quercia M., Zimmermann P. The Middle Product Algorithm. I. Applicable Algebra in Engineering Communication and Computing (2004) 14(no. 6):415–438.[CrossRef][ISI]
- Hubrechts H. Point counting in families of hyperelliptic curves. (2007) arXiv math.NT/0601438v3 (retrieved Jan 29th 2007). to appear in Foundations of Computational Mathematics.
- Kedlaya K. S. Counting Points on Hyperelliptic Curves Using Monsky-Washnitzer Cohomology. Journal of the Ramanujan Mathematical Society (2001) 16(no. 4):323–338.
- Kedlaya K. S. Computing Zeta Functions Via p-Adic Cohomology. In: Algorithmic Number Theory (2004) Berlin: Springer. 1–17. Vol. 3076 of Lecture Notes in Computer Science.
- Mazur B., Stein W., Tate J. Computation of p-Adic Heights and Log Convergence. (2006) 577–614. Documenta Mathematica (Extra Volume: John H. Coates' Sixtieth Birthday).
- Shoup V. NTL: A Library for Doing Number Theory. (2007) http://www.shoup.net/ntl/.
- Stein W., Joyner D. Sage: System for Algebra and Geometry Experimentation. Communications in Computer Algebra (ACM SIGSAM Bulletin) (2005) 39(no. 2):61–64.
- Strassen V. Gaussian Elimination is Not Optimal. Numerische Mathematik (1969) 13:354–356.[CrossRef][ISI]
| ||||||||||||||||||||||||||||||||||||||||||||||||||