implement complex multiplication method #21
Description
Summary:
Implement the CM method to construct elliptic curves over 𝔽ₚ with a given number of points, using class invariants and Hilbert class polynomials.
Details:
Input: Desired prime p and target trace t, i.e., target #E(𝔽ₚ) = p + 1 - t.
Output: An elliptic curve over 𝔽ₚ with that order.
Tasks:
Compute or retrieve Hilbert class polynomial H_D(x) for discriminant D = t² - 4p.
Find a root of H_D(x) in 𝔽ₚ.
Reconstruct the curve from the j-invariant.
References:
Atkin–Morain CM method
Efficient j-invariant to curve generation.