Computing modular polynomials in quasi-linear time

TL;DR

This paper presents a quasi-linear time algorithm for computing modular polynomials using floating point evaluation and interpolation, significantly improving efficiency for large levels and various modular functions.

Contribution

The authors introduce a novel algorithm with nearly linear complexity for computing modular polynomials, applicable to prime, composite levels, and various modular functions.

Findings

Algorithm achieves near-linear complexity in polynomial size.

Successfully computes high-level modular equations in record time.

Distributed implementation confirms theoretical efficiency.

Abstract

We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomials $\Phi_\ell$ of prime level $\ell$ in time O (\ell^3 \log^4 \ell \log \log \ell). Besides treating modular polynomials for $\Gamma^0 (\ell)$, which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schl\"afli polynomials and their generalisations. Our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors.