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The current research trend shows three categories of public-key cryptographic systems for the next generation that are efficient and secure. Based on the corresponding mathematical foundation, they are Elliptic Curve Cryptosystem, Integer Factorization Systems and Discrete Logarithm Systems. This thesis emphasizes in using the Cryptosystem based on Elliptic Curves as the operations on elliptic curves have a characteristic that is extremely valuable for Cryptographic purposes: they are relatively easy to perform, and extremely difficult to reverse making it virtually impossible for an attacker to decipher the code. This dissertation discusses some of the Algorithmic approaches based on Elliptic Curves.