机构地区: 西安电子科技大学通信工程学院综合业务网理论与关键技术国家重点实验室
出 处: 《电子学报》 2011年第1期242-246,共5页
摘 要: 本文首次应用二次剩余理论对RSA中的代数结构进行了研究.计算出了Zn*中模n的二次剩余和二次非剩余的个数,对它们之间的关系进行了分析,并用所有二次剩余构成的群对Zn*进行了分割,证明了所有陪集构成的商群是一个Klein四元群.对强RSA的结构进行了研究,证明了强RSA中存在阶为φ(n)/2的元素,并且强RSA中Zn*可由三个二次非剩余的元素生成.确定了Zn*中任意元素的阶,证明了Zn*中所有元素阶的最大值是lcm(p-1,q-1),并且给出了如何寻找Zn*中最大阶元素方法.从而解决了RSA中的代数结构. Based on the theory of quadratic residues,the algebra structure of RSA arithmetic is researched in this paper.This work calculates numbers of quadratic residues and non-residues in the group Zn^* and investigates their relationship.Zn^* is divided up by the group made up with all quadratic residues in Zn^* and all cosets form a quotient group of order 4 which is a Klein group.Studyed the structure of strong RSA further,it shows that the element of order (n)/2 exists and the group Zn^* can be generated by three elements of quadratic non-residues.Let the facterization n=p·q,the order of each element can be calculated,and the biggest order of all element is lcm(p-1,q-1) in Zn^*.It also shows how to find the element of the biggest order.So the algebra structure of RSA arithmetic is solved.