Computing modular polynomials by deformation

TL;DR

This paper introduces an efficient CRT-based algorithm for computing modular polynomials using deformation techniques, enabling quasi-linear time calculations and practical implementations for reductions modulo primes.

Contribution

The paper presents a novel CRT algorithm leveraging deformation of isogenies, achieving quasi-linear time complexity for modular polynomial computation.

Findings

Algorithm runs in quasi-linear time.

Proof-of-concept implementation demonstrates practicality.

Can compute reductions modulo p efficiently.

Abstract

We present an unconditional CRT algorithm to compute the modular polynomial $\Phi_\ell(X,Y)$ in quasi-linear time. The main ingredients of our algorithm are: the embedding of $\ell$-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 $\Phi_{\ell}$ modulo $p$ in quasi-linear time (with respect to $\ell$) $\tilde{O}(\ell^2(\log p + \log \ell)^{\mathfrak{O}})$.