The complexity of class polynomial computation via floating point approximations
Andreas Enge
TL;DR
The paper analyzes the complexity of constructing CM elliptic curves by computing class polynomials through floating-point approximations of modular-function values. It develops near-optimal complex-analytic algorithms, notably an arithmetic-geometric-mean (AGM)–based Newton iteration method achieving $O(|D|^{1+ε})$ (i.e., $O(h^{2+ε})$) and a multipoint-evaluation variant with an extra logarithmic factor, together with a rigorous height bound of $O(\sqrt{|D|}\log^2|D|)$ and verifiable Monte Carlo checks. It also provides detailed class-group enumeration methods and discusses $N$-systems, along with alternative $p$-adic approaches and comparisons to other strategies. Empirical implementation up to class number $h\approx 10^5$ demonstrates feasibility and clarifies practical trade-offs between asymptotic speed and memory usage, while confirming that the output size governs the computational effort and that numerical errors can be controlled via probabilistic verification.
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. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + ε}) = O (h^{2 + ε})$ for any $ε> 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. 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.