On the evaluation of modular polynomials
This work addresses computational bottlenecks in elliptic curve cryptography and number theory, offering incremental improvements for applications like point counting and endomorphism ring computation.
The authors tackled the problem of computing modular polynomials for elliptic curves without assuming the polynomial is given, resulting in new algorithms that set a point-counting record for primes over 5,000 digits and evaluated a modular polynomial at a high level of 100,019.
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.