Cryptography is the study of methods of sending messages in disguised form so that only the intended recipients can remove the disguise and read the message. This book presents the origin of cryptography to its recent advancement. The first chapter of this book describes the private keys cryptosystem, the second chapter describes the public key cryptosystem. In third chapter are given clear ideas of elliptic curve cryptosystem and its problem on plotting points on the elliptic curve and discussion of deterministic algorithm.This book presents the problem of handling keys in private key cryptosystem and it links between private key and public key cryptosystem. In the private key cryptosystem, I clearly present RSA cryptosystem, Rabin cryptosystem, Elgamal cryptosystem, Massey Omura Cryptosystem, Diffine Hellman key Exchange protocol and elliptic curve cryptosystem.