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"It is insufficient to protect ourselves with laws; we need to protect ourselves with mathematics. There are two kinds of cryptography in this world: cryptography that will stop your kid sister from reading your files, and cryptography that will stop major governments for reading your files".(Bruce Schneir). Discrete Logarithm Problem (DLP) forms the basis for many cryptographic systems today. Many algorithms have been defined in literature for solving the DLP but only Index-Calculus methods offer sub-exponential time complexities for DLP on finite fields with properly chosen parameters. In this thesis, the focus is on the DLP on elliptic curves, i.e., the Elliptic Curve Discrete Logarithm Problem (ECDLP). The work studies the use of ECDLP in developing cryptosystems in general and on signature schemes in particular. It also generalises and extends an algorithm that uses partial knowledge of the secret key. Such knowledge may be available through implementation details or side channel analysis or both.