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.
Highlights
RSA [1] is one of the most well-known cryptography algorithms in the digital computer era. It is classified as asymmetric key cryptography or public key cryptography [2]. This algorithm which was proposed in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman, can be used to secure secret information transferred via an unsecure channel as well as sign digital signatures
The other group is known as general-purpose factoring, and the algorithms are based on the length of the modulus
Elliptic Curve Factorization Method (ECM) requires a large number of modular inverses and greatest common divisors, which are known as the expensive costs
Summary
RSA [1] is one of the most well-known cryptography algorithms in the digital computer era It is classified as asymmetric key cryptography or public key cryptography [2]. Assuming that two strong prime factors and 1024 bits of the modulus are selected, none of proposed integer factorization algorithms can break RSA in polynomial time by using the digital computer. ECM requires a large number of modular inverses and greatest common divisors, which are known as the expensive costs. That is, it takes a significant amount of time to finish the task. The proposed method is known as the Fast Elliptic Curve Factorization Method (F-ECM).
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