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The book explores the basics of Basic Number Theory and the basics of classical and public key cryptography. It also discusses how the methods, Shank's Baby-step Giant-step algorithm and Pollard rho algorithm are used in solving the discrete logarithm problem. Also the Lucas sequences are introduced and some of the basic properties are discussed briefly. The ideas of generalized discrete logarithm problem based on semi-group/group actions adapted and implemented with Lucas sequences are dealt with. It also emphasizes on the possibilities of extensions of Shank's Baby-step Giant-step and Pollard rho attacks. Diffie-Hellman protocol introduces a practical solution to the key distribution problem by allowing two parties to share a secret key by exchanging information over an open channel. The Pollard rho method with group actions, its possible extensions on matrix action and how this Pollard rho method can be adapted with semi-group with large subgroup is also dealt with. Two generalized discrete logarithm problems, one with Lucas group action and the other with Lucas semi-group action which give rise to the Lucas group action problem and Lucas semi-group action problem are discussed.