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Mathematics of Computation

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The complexity of class polynomial computation via floating point approximations

Author: Andreas Enge
Journal: Math. Comp. 78 (2009), 1089-1107
MSC (2000): Primary 11Y16; Secondary 11G15
Published electronically: November 17, 2008
MathSciNet review: 2476572
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Abstract: We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time \[ O \left ( \sqrt {|D|} \log ^3 |D| M \left ( \sqrt {|D|} \log ^2 |D| \right ) \right ) \subseteq O \left (|D| \log ^{6 + \varepsilon } |D| \right ) \subseteq O \left ( h^{2 + \varepsilon } \right ) \] for any $\varepsilon > 0$, where $D$ is the CM discriminant, $h$ is the degree of the class polynomial and $M (n)$ is the time needed to multiply two $n$-bit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials.

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Additional Information

Andreas Enge
Affiliation: INRIA Saclay–Île-de-France & Laboratoire d’Informatique (CNRS/UMR 7161), École polytechnique, 91128 Palaiseau Cedex, France

Received by editor(s): April 24, 2007
Received by editor(s) in revised form: May 6, 2008
Published electronically: November 17, 2008
Article copyright: © Copyright 2008 by Andreas Enge