Abstract: Based on the projective definition of elliptic curves, the structure of points on the curve is researched deeply in this paper. Projective and affine definitions of the addition operation are discussed and a error in [1] and [3] is pointed out. Furthermore, the relation between two affine definitions of the addition operation is given. In conclusion, the essential reason is analyzed when the addition operation is not defined by discussing these cases.

Key words: elliptic curves over , the addition operation

摘要： 基于环 上椭圆曲线的射影化定义，深入研究了 上椭圆曲线点群中点的结构形式，讨论了点的加法的射影化定义和仿射化定义，指出在仿射化定义条件下文献[1]和[3]在定义点的加法时存在的一个认识上的错误，给出了仿射化定义条件下两种定义方法之间的区别与联系，最后针对仿射化定义条件下某些点的加法无意义的情形进行了讨论，并分析了导致这种情形的根本原因。

关键词: 上的椭圆曲线, 点的加法