The complexity of class polynomial computation via floating point approximations
Andreas Enge · Mathematics of Computation · 2008
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 \[ O ( | D | log 3 | D | M ( | D | log 2 | D | ) ) ⊆ O ( | D | log 6 + ε | D | ) ⊆ O ( h 2 + ε ) O \left ( \sqrt {|D|} \log ^3 |D| \, M \left ( \sqrt {|D|} \log ^2 |D| \right ) \right ) \subseteq O \left (|D| \log ^{6 + \varepsilon } |D| \right ) \subseteq O \left ( h^{2 + \varepsilon } \right ) \] for any ε > 0 \varepsilon > 0 , where D D is the CM discriminant, h h is the degree of the class polynomial and M ( n )</