An Algorithmic Construction of Finite Elliptic Curves of Order Divisible by a Large Prime

Abstract

Given a square-free integer Δ < 0, we present an algorithm constructing a pair of primes p and q such that q|p + 1-t and 4p-t2 = Δf2, where |t| ≤ 2√p for some integers f, t. Together with a CM method presented in the paper, such primes p and q are used for a construction of an elliptic curve E over a finite field $\mathbb{F}_p$ such that the order of E is divisible by a large prime. It is shown that our algorithm works in polynomial time.