Efficiently Computable Complex Multiplication of Elliptic Curves.

After the GLV and GLS methods were proposed, a great deal of attention was devoted to the efficiently computable complex multiplication of elliptic curves. In this paper, we mainly compute the rational expressions of complex multiplications of elliptic curves. Our contribution involves two aspects. We propose a new algorithm to effectively compute the rational expression of a complex multiplication over finite field. We also present a method to construct an elliptic curve with efficiently computable complex multiplication by a given quadratic imaginary field. Specifically, we show all 13 classes of elliptic curves with complex multiplication over quadratic imaginary fields. This is the first systematic study of the expression of complex multiplication.