此篇論文的主要目標是對現今密碼學的發展做一個概略的介紹，並對其中關於數論應用的部份做一個整理。其中包括根據整數分解的困難性所建立的 RSA 密碼系統，以及根據離散對數問題所建立的各種密碼系統。此外，並介紹一些判定質數的方法，以及一些有效分解整數的演算。針對解離散對數的問題，我們也做一些介紹。另外，在論文的後半段，我們簡單的描述一些關於橢圓曲線的性質，並介紹橢圓曲線在整數分解，及密碼學上的應用。最後，我們介紹一些計算橢圓曲線在有限體上有理點個數的方法。除此之外，我們也針對以上所提及的演算法，討論關於其複雜度得問題，使我們可以對各演算法做一個時間上的比較。 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.
