基于剩余类环Z_n上圆锥曲线的公钥密码体制

Public-key Cryptosystem Based on the Conic Curves over Z_n

为了实现更高效的曲线上的密码体制,讨论了当n为两个素数的乘积时剩余类环Zn上圆锥曲线Cn(a,b)的基本性质,证明Cn(a,b)中用映射方式和以坐标方式定义的两种运算是一致的,该运算使得Cn(a,b)的有理点构成Abel群。给出了在Cn(a,b)上寻找基点的简单方法,并给出RSA和ElGamal密码体制在Cn(a,b)上的模拟。这两类密码体制的安全性基于大数分解和有限Abel群(Cn(a,b),)上离散对数问题的困难性,具有明文嵌入方便、运算速度快、易于实现等优点。

In order to get more efficient cryptosystem over curves , this paper discussed some basic properties of conic Cn （ a, b） over the residue class ring Zn, where n is the product of two primes . We proved that the rational points of Cn （ a, b） form an abelian group, whose operation may be given in two ways： one by reduction map and another by a formula with respect to the coordinates. And we also provided a simple method to find a base point. As applications, we gave analogues of RSA and EIGamal cryptosystem over Cn （ a, b ）. The two analogues are easy to implement and their seeurities are based on the difficulty of integer factorization and the discrete logarithm over Cn（ a, b ） respectively.