Elliptic curve method using complex multiplication method

In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p = 1 + Dv ² for some integer v, where −D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to −D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p = t ² + Dv ² and p + 1 − t is a smooth integer.