RSA算法中Z*(φ)(n)的代数结构研究

摘要

[Abstract] Based on the theory of quadratic residues, this paper considers the algebraic structure of Z*φ(n) in the two order strong RSA algorithm. It is proved that the element α of Z*φ(n) gets maximal order if and only if gcd(α±l,n1) = 1, and the numbers of quadratic residues in the group Z*φ(n) is φ(φ(n))/8 .Z*φ(n) is divided up by the group which is composed of all quadratic residues, and all cosets form a Klein eight-group. It proves that the group Z*φ(n) can be generated by seven elements of quadratic non-residues.%应用二次剩余理论,对二阶强RSA算法中Z*(φ)(n)的代数结构进行研究,证明Z*(φ)(n)中元素a取最大阶的充要条件为gcd(a±1,n1)=1,以及任意元素的阶Z*(φ)(n)中模(φ)(n)的二次剩余个数为(φ)(φ)(n))/8,以所有二次剩余构成的群对Z*(φ)(n)进行分割,利用所有陪集构成一个Klein八元群,在此基础上证明Z*(φ)(n)可由7个二次非剩余元素生成.