Abstract: This paper gives a generalization of Pollard’s [(p-1)]-factorization method so that the generalized algorithm not only works more efficiently，but also works on some integers on which the original Pollard’s [(p-1)]-factorization algorithm does not work;based on the [(p-1)]-factorization method，it proposes a higher order upgrade factorization algorithm.This paper further proposes a measure on the robustness of prime numbers in terms of resisting factorization.Under this new measure，this paper proposes the concept of stability order of prime numbers，which means that those numbers satisfying Rivest’s condition are robust only against the second order upgrade factorization.

Key words: integer factorization, the (p-1)-method, RSA cryptography

摘要： 对Pollard的(p-1)-整数分解算法进行了修改，使其在提高了运行速度的同时，也适用于一些不满足原始(p-1)-整数分解算法的局限条件的数;在(p-1)-分解算法基础上，进一步提出了一种高阶升级分解算法;并给出了在对抗整数分解方面，素数好坏的一种度量方法，在这种新度量方法下，提出了素数稳定阶数的概念，从而说明满足Rivest 条件的数仅仅在对抗二级升级算法时是安全的。

关键词: 整数分解, (p-1)-算法, RSA 密码体制