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We present an unconditional CRT algorithm to compute the modular polynomial _ (X, Y) in quasi-linear time. The main ingredients of our algorithm are: the embedding of -isogenies in smooth-degree isogenies in higher dimension, and the computation of m-th order deformations of isogenies. We provide a proof-of-concept implementation of a heuristic version of the algorithm demonstrating the practicality of our approach. Our algorithm can also be used to compute the reduction of _ modulo p in quasi-linear time (with respect to) O (² (p +) ^O).
Kunzweiler et al. (Tue,) studied this question.
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