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Action of Modular Correspondences around CM Points

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

We study the action of modular correspondences in the p-adic neighborhood of CM points. We deduce and prove two stable and efficient p-adic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over \( \mathbb{F}_q \) can be computed in probabilistic time ≪ exp((log q)1/2+ε) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.

The GRIMM is supported by the French Ministry of Research through Action Concertée Incitative CRYPTOLOGIE, by the Direction Centrale de la Sécurité des Systèmes d’Information and by the Centre Électronique de L’ARmement.

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Author information

Authors and Affiliations

  1. Groupe de Recherche en Informatique et Mathématiques du Mirail, Université de Toulouse II, 5 allées Antonio Machado, 31058, Toulouse, France

    Jean-Marc Couveignes & Thierry Henocq

Authors
  1. Jean-Marc Couveignes
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  2. Thierry Henocq
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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Couveignes, JM., Henocq, T. (2002). Action of Modular Correspondences around CM Points. 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_19

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  • DOI: https://doi.org/10.1007/3-540-45455-1_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

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