On the evaluation of modular polynomials

TL;DR

This paper introduces two algorithms for directly computing modular polynomials related to elliptic curves without needing the polynomial beforehand, demonstrating practical efficiency in large-scale computations.

Contribution

The paper presents novel algorithms for computing modular polynomials directly from elliptic curves, improving efficiency and applicability in cryptographic and number theory computations.

Findings

Achieved a new record in point counting for large primes.

Successfully computed a modular polynomial of level 100,019.

Algorithms are adaptable to other modular polynomial computations.

Abstract

We present two algorithms that, given a prime ell and an elliptic curve E/Fq, directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the j-invariants of the elliptic curves that are ell-isogenous to E. We do not assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point counting and the computation of endomorphism rings. We demonstrate the practical efficiency of the algorithms by setting a new point-counting record, modulo a prime q with more than 5,000 decimal digits, and by evaluating a modular polynomial of level ell = 100,019.