The Graph and its Role for Speeding up the Elliptic Scalar Multiplication Algorithms

Elliptic curves play a major role in cryptography. It used increasingly to form the public key cryptosystems and digital signature schemes. The smaller key sizes and the efficient implementations on the elliptic curve cryptography (ECC) gave the same level of security in comparison with the RSA (Rivest-Shamir- Adleman) and other cryptosystems. The security on the ECC depending on the hardness of the elliptic curve discrete logarithm problem (ECDLP) to solve. The efficiency of the ECC is determined based on the efficient computation of the scalar multiplication on elliptic curves defined over the finite fields. A scalar multiplication is considered as a core operation in the ECC. It is not only the main computational operation but it also forms a central time-consuming process. The efficient performances of scalar multiplication directly determine the ECC performance.