Elliptic Curve Cryptosystems and Scalar Multiplication

One of the most used cryptosystems in the world is the RSA system. Its popularity is due to its high security level. In the last decades, the studies have shown that the cryptosystems based on elliptic curves have the same security level as the RSA system. Besides that, the elliptic curve cryptosystems have a higher efficiency and they use shorter keys. In this paper we describe basics of the elliptic curve cryptosystems. We, also, present methods (based on the Lee et al. methods) of the scalar multiplications of the elliptic curves. In the end, we propose an efficient method for simultaneous multiple point multiplication. 2010 Mathematics Subject Classification. Primary 94A55; Secondary 11T71, 68P25.