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Accelerating the CM method

Part of: Curves Arithmetic algebraic geometry Computational number theory

Published online by Cambridge University Press:  01 August 2012

Andrew V. Sutherland
Andrew V. Sutherland*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: drew@math.mit.edu)

Abstract

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Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/Fq by obtaining a root of the Hilbert class polynomial HD(X) modulo q. We consider an approach based on a decomposition of the ring class field defined by HD, which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of HD mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to ∣D∣, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D) , which may be as small as O(∣D∣1/4 log q) . The practical efficiency of the algorithms is demonstrated using ∣D∣>1016 and q≈2256, and also ∣D∣>1015 and q≈233220. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.

MSC classification

Secondary: 11Y16: Algorithms; complexity 11G15: Complex multiplication and moduli of abelian varieties 11G20: Curves over finite and local fields 14H52: Elliptic curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 172 - 204
Copyright
Copyright © London Mathematical Society 2012

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