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Computing Modular Polynomials

Published online by Cambridge University Press:  01 February 2010

Denis Charles  and
Kristin Lauter
Denis Charles
Affiliation:
Department of Computer Science, University of Wisconsin-Madison, Madison, WI-53706, USA, cdx@cs.wise.edu, http://www.cs.wisc.edu/~cdx
Kristin Lauter
Affiliation:
Microsoft Research, One Microsoft Way, Redmond. WA - 98052, USA, klauter@microsoft.com, http://research.microsoft.com/~klauter

Abstract

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This paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 195 - 204
Copyright
Copyright © London Mathematical Society 2005

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