Method for constructing elliptic curves using complex multiplication and its optimizations

TL;DR

This paper discusses a method for constructing elliptic curves over finite fields using complex multiplication, focusing on optimizing the computational process involved in polynomial reconstruction.

Contribution

It provides a detailed theoretical analysis and optimization strategies for the complex multiplication method in elliptic curve construction.

Findings

Proves new theoretical results on polynomial reconstruction

Offers optimized algorithms for elliptic curve construction

Enhances efficiency of complex multiplication-based methods

Abstract

Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer coefficients. We will prove theoretical results and give a detailed account of the method itself and how one can use a divisor of the mentioned polynomial with coefficients in some extension of the field of rational numbers.