Exchanging information electronically in a secure way become very important in the age of computers. Over fifty years of research and implementation were devoted to developing methods and systems for such a secure communicat ion. These systems are common ly known as cryptosystems. Among the cryptosystems currently in use, the public key cryptosystems have a special place. These are cryptosystems which allow any user, completely unaware of the methods of encryption and decryption, i.e. of the

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... ricate details of the system, to use them for variety of purposes in their daily life. The most common examp les of this are digital signatures and electronic banking transactions. Here we provide an overview of different cryptosystems, including but not limited to an examination and comparison of five different influential public key cryptosystems. This examinat ion includes an in-depth look at: the RSA algorithm, the Diffie-Hellman key exchange protocol, the elliptic curve cryptosystems, the ElGamal encryption method and the knapsack approach. We provide the necessary mathemat ical and number theoretic preliminaries. Fu rther, we introduce each cryptosystem starting with the relevant definitions, theorems and properties. We discuss the encryption and the decryption process, the creation and exchange of the keys, present plenty of examples, as well as discussion of the implementation issues and the known strengths and weaknesses of each cryptosystem against cryptanalytic attacks.