Optimization of elliptic curve operations for ECM using double & add algorithm

Nowadays the security becomes more and more important and as a need for secure data encryption grows, we have to be sure that the algorithms we are using are safe. But it is not always just about algorithm itself as about settings, for example key length. RSA, the most popular asymmetric cipher is a perfect example, because it fully depends on hardness of large numbers factorization. In this paper, we propose a novel approach for Elliptic Curve Method (ECM) which speeds-up the factorization time in affine coordinates, thanks to optimizing the calculation steps for need of a Double & Add algorithm. However, proposed equations could be used also in general Elliptic Curve Cryptography (ECC) or Elliptic Curve Digital Signature Algorithm (ECDSA), where the same principle is used and thus can make the operations faster.