Efficient algorithms for implementing the elliptic curve cryptosystems

Elliptic Curve Cryptography (ECC) represents a different way to do public-key cryptography and it offers certain advantages over the traditional cryptosystems. This thesis deals with the efficient algorithms for implementing the ECC and its applications, and how to find the cryptographically secure elliptic curves that can be used in the commercial applications. Efficient algorithms for the elliptic curves can be classified into the arithmetic under the given finite fields. In this thesis, we explored the possibilities using the elliptic curve cryptography in the commercial applications and implemented the elliptic curve systems over the finite field F(2"). Based on different algorithms over the finite field F(Z'), we have implemented some security protocols such as the ECC versions of the Diffie-Hellman key exchange protocol and the ElGamal cryptographic scheme. In the end, we described the structured approach to find such curves using the complex multiplication ("CM") algorithm. A list of the secure elliptic curves with different bits has been provided over the field F(p). Schoof algorithm has also been discussed and some examples have been provided to count the number of points over the randomly generated elliptic curves