本文简要论述了椭圆曲线的基础理论，介绍了椭圆曲线上的运算并对实数域上的椭圆曲线和以及其运算构成阿贝尔群进行了检验；说明了椭圆曲线上的数乘运算并对算点的倍乘的算法给出了描述；进一步将实数域上的椭圆曲线推广到定义在有限域上或模p的椭圆曲线上，并对特征为2的椭圆曲线进行运算举例；运用二次剩余的知识对椭圆曲线上的整点数量进行估计并描述了椭圆曲线上的群结构性质；介绍了椭圆曲线密码体制中的Diffie-Hellman型密钥交换体制和ElGamal密码系统并介绍了对椭圆曲线离散对数问题的攻击方法。

In this paper, the basic theory of elliptic curves is briefly summarized. The operations of point addition on elliptic curves over real number field are introduced and an elliptic curve and the operations on it constitute an abelian group is examined. The multiplicative operation on elliptic curve is explained and the algorithm of computation of point multiples is described. Further, the elliptic curve over real number field is extended to the elliptic curve over the finite field or modulo p, and the elliptic curve with characteristic 2 is given as an example. The order of an elliptic curve is estimated using knowledge of quadratic residual and the group structure property of elliptic curve is described. Diffie–Hellman cryptosystem and Elgamal cryptosystem are introduced and also the attack method of elliptic curve discrete logarithm problem is introduced.