Comparatively Study of ECC and Jacobian Elliptic Curve Cryptography

- Elliptic curves were first proposed as a basis for public key cryptography in the mid-1980s independently by Koblitz and Miller. Elliptic curve cryptography (ECC) algorithm is practical than existing security algorithms. Because of this fact, it showed real attraction to portable devices (handheld devices) manufacturers and the security of their systems. In fact, through these devices, anyone can access either email, or do bank transaction or buy anything on internet using credit cards with high security standards. Elliptic curve algorithm is promising to be the best choice of these handhelds or similar devices because of low computing power and fast execution. ECC further gives very high security as compared to similar crypto systems with less size of key. For example, 160 bit ECC system is believed to provide same level of security as 1024 bit RSA [5, 6]. The other two ECC scalar multiplication modular operations are inversion and multiplication. Inversion is known to be the complex and very expensive operation [7], its cost is reduced by converting the normal (x, y) affine coordinate system to projective coordinate system (X, Y, Z) which will add-up more modular multiplications to the process, i. e. it will increase the number of multiplications in both ECC point doubling and adding processes operations to reduce the inversion complexity