The purpose of this thesis is to make a general survey of the development of cryptography and to discuss the mathematical background of cryptography. We first introduce the RSA cryptosystem which is based on the difficulty of factoring a large integer, and other cryptosystems based on the problem in number theory so called discrete logarithm problem. Then we describe some methods which can determine whether or not a given integer is a prime number and methods of efficient factoring large composite integers, and some methods of solving discrete logarithm problem on finite fields. In the second half of this thesis, we present some basic properties of elliptic curves and introduce a factorization method which is based on properties of elliptic curves over finite fields. Some elliptic curve cryptosystems will be introduced. In the final section of this thesis, we discuss some methods of counting the number of rational points on the elliptic curve over finite fields.

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