ECM And The Elliott-Halberstam Conjecture For Quadratic Fields

TL;DR

This paper investigates the complexity of elliptic curve methods for factorization over quadratic fields, establishing conditional results based on the Elliott-Halberstam conjecture and analyzing ECM-friendly curves with complex multiplication.

Contribution

It provides rigorous, conditional results for ECM complexity in quadratic fields using the Elliott-Halberstam conjecture, focusing on CM elliptic curves and their properties.

Findings

Conditional bounds on ECM complexity for quadratic fields

Introduction of a measure for ECM-friendliness of CM elliptic curves

Exploration of elliptic curve analogues of prime distribution conjectures

Abstract

The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more studied and implemented, especially because it allows us to use ECM-friendly curves. In the case of curves with complex multiplication (CM) we replace the heuristics by rigorous results conditional to the Elliott-Halberstam (EH) conjecture. The proven results mirror recent theorems concerning the number of primes p such thar p -- 1 is smooth. To each CM elliptic curve we associate a value which measures how ECM-friendly it is. In the general case we explore consequences of a statement which translated EH in the case of elliptic curves.