# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## The complexity of class polynomial computation via floating point approximationsHTML articles powered by AMS MathViewer

by Andreas Enge
Math. Comp. 78 (2009), 1089-1107

## 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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