Abstract
Class fields of imaginary quadratic number fields can be constructed from singular values of modular functions, called class invariants. From a computational point of view, it is desirable that the associated minimal polynomials be small. We examine different approaches to measure the size of the polynomials. Based on experimental evidence, we compare two families of class invariants suggested in the literature with respect to these criteria. Our results lead to more efficient constructions of elliptic curves for cryptography or in the context of elliptic curve primality proving (ECPP).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Draft, 1988.
A. O. L. Atkin and F. Morain. Elliptic curves and primality proving. Math. Comp., 61(203):29–68, July 1993.
B. J. Birch. Weber’s class invariants. Mathematika, 16:283–294, 1969.
J. M. Borwein and P. B. Borwein. Pi and the AGM. John Wiley, 1987.
N. Brisebarre and G. Philibert. Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. In preparation, January 2002.
H. Cohen. Advanced topics in computational number theory, volume 193 of Graduate Texts in Mathematics. Springer-Verlag, 2000.
R. Dedekind. Erläuterungen zu den vorstehenden Fragmenten. In R. Dedekind and H. Weber, editors, Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlaβ, pages 438–447. Teubner, Leipzig, 1876.
A. Enge and F. Morain. Further investigations of the generalised Weber functions. In preparation, 2001.
A. Enge and R. Schertz. Constructing elliptic curves from modular curves of positive genus. In preparation, 2001.
A. Enge and R. Schertz. Modular curves of composite level. In preparation, 2001.
Robert Fricke. Lehrbuch der Algebra, volume III-Algebraische Zahlen. Vieweg, Braunschweig, 1928.
A. Gee. Class invariants by Shimura’s reciprocity law. J. Th’eor. Nombres Bordeaux, 11:45–72, 1999.
A. Gee and P. Stevenhagen. Generating class fields using Shimura reciprocity. In J. P. Buhler, editor, Algorithmic Number Theory, volume 1423 of Lecture Notes in Comput. Sci., pages 441–453. Springer-Verlag, 1998. Third International Symposium, ANTS-III, Portland, Oregon, june 1998, Proceedings.
G. Hanrot and F. Morain. Solvability by radicals from a practical algorithmic point of view. Submitted. Available from http://www.lix.polytechnique.fr/-Labo/Francois.Morain/, November 2001.
G. Hanrot and F. Morain. Solvability by radicals from an algorithmic point of view. In B. Mourrain, editor, Symbolic and algebraic computation, pages 175–182. ACM, 2001. Proceedings ISSAC’2001, London, Ontario.
Marc Hindry and Joseph H. Silverman. Diophantine Geometry — An Introduction. Springer-Verlag, New York, 2000.
Carl Gustav Jacob Jacobi. Fundamenta nova theoriae functionum ellipticarum. In Gesammelte Werke, pages 49–239. Chelsea, New York, 2 (1969) edition, 1829.
Felix Klein. Uber die Transformationsgleichung der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades. Math. Annalen, 14:111–172, 1878. Gesammelte Mathematische Abhandlungen III:13–75.
D. Kohel. CM divisors on modular curves. In preparation, January 2002.
C. Meyer. Bemerkungen zum Satz von Heegner-Stark über die imaginärquadratischen Zahlkörper mit der Klassenzahl Eins. J. Reine Angew. Math., 242:179–214, 1970.
F. Morain. Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques. J. Théor. Nombres Bordeaux, 7:255–282, 1995.
F. Morain. Primality proving using elliptic curves: an update. In J. P. Buhler, editor, Algorithmic Number Theory, volume 1423 of Lecture Notes in Comput. Sci., pages 111–127. Springer-Verlag, 1998. Third International Symposium, ANTS-III, Portland, Oregon, june 1998, Proceedings.
F. Morain. Modular curves and class invariants. Preprint, June 2000.
R. Schertz. Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3. J. Reine Angew. Math., 286/287:46–74, 1976.
R. Schertz. Weber’s class invariants revisited. To appear in J. Théor. Nombres Bordeaux, 2001.
Reinhard Schertz. Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkörpern über einem imaginär-qudratischen Zahlkörper. Journal of Number Theory, 34(1):41–53, January 1990.
Goro Shimura. Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press, 1971.
J. H. Silverman. The Arithmetic of Elliptic Curves, volume 106 of Grad. Texts in Math. Springer, 1986.
J. H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves, volume 151 of Grad. Texts in Math. Springer-Verlag, 1994.
H. M. Stark. Counting points on CM elliptic curves. Rocky Mountain J. Math., 26(3):1115–1138, 1996.
H. Weber. Lehrbuch der Algebra, volume III. Chelsea Publishing Company, New York, 1908.
N. Yui and D. Zagier. On the singular values of Weber modular functions. Math. Comp., 66(220):1645–1662, October 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Enge, A., Morain, F. (2002). Comparing Invariants for Class Fields of Imaginary Quadratic Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_21
Download citation
DOI: https://doi.org/10.1007/3-540-45455-1_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43863-2
Online ISBN: 978-3-540-45455-7
eBook Packages: Springer Book Archive