Despite it not being an armed conflict between nations, there is a war that has been waged for over 5000 years and is still being fought today. Battles have been won and lost by both sides. The battlefield is the world of cryptography. Our combatants are cryptographers, personnel who make secret codes; and cryptanalysts, personnel who try to break the secret codes; In this thesis, we examine public-key or asymmetric cryptography, the are of writing or deciphering secret codes or ciphers. We begin by taking a brief look at the overall history of cryptography. Our primary focus involves studying the mathematics behind today\u27s public-key cryptographic methods such as the theory of congruences by Carl Friedrich Gauss, Fermat\u27s little theorem, Euler\u27s phi-function, primitive roots and indices, and elliptic curves over finite fields. Once we have explore the preliminaries we will consider some of the more popular methods of encryption and decryption, for example RSA. We not only discuss how to encrypt and decrypt plain text using these methods but explain why it is hard to break the encrypted text. We conclude our study by inspecting the shortfall of these ciphers, techniques used to break the encryption, and what the future possibly holds
