The Improvement of Elliptic Curve Factorization Method to Recover RSA’s Prime Factors

Abstract

Elliptic Curve Factorization Method (ECM) is the general-purpose factoring method used in the digital computer era. It is based on the medium length of the modulus; ECM is an efficient algorithm when the length of modulus is between 40 and 50 digits. In fact, the main costs for each iteration are modular inverse, modular multiplication, modular square and greatest common divisor. However, when compared to modular multiplication and modular square, the costs of modular inverse and greatest common divisor are very high. The aim of this paper is to improve ECM in order to reduce the costs to compute both of modular inverse and greatest common divisor. The proposed method is called Fast Elliptic Curve Factorization Method (F-ECM). For every two adjacent points on the curve, only one modular inverse and one greatest common divisor will be computed. That means it implies that the costs in both of them can be split in half. Furthermore, the length of modulus in the experiment spans from 30 to 65 bits. The experimental results show that F-ECM can finish the task faster than ECM for all cases of the modulus. Furthermore, the computation time is reduced by 30 to 38 percent.