Euler functions have a wide range of applications in number theory. The RSA public key cryptosystem is also designed based on the properties of the Euler function. In this paper, the Euler function of two arbitrary number's product is studied, and further any amount of number's product is analyzed. The extended form of Euler function based on product property are given by using the set inclusion-exclusion principle. This form of expansion has high practical value in real application. With the help of the extended form of Euler function based on product property, a RSA-like public key encryption and digital signature algorithm based on the problem of large integer decomposition difficulty is constructed, and its security is analyzed. It is found that such public key algorithm and digital signature have high security and practical value.%欧拉函数在数论中有着广泛的应用,在密码学中,RSA公钥密码体制也是基于欧拉函数的性质设计出来的.本文探索了任意两个数乘积的欧拉函数分解表示.在此基础上,对任意若干个数乘积的欧拉函数进行了分析,利用集合容斥原理给出了基于乘积性质的欧拉函数的扩展形式表示.这种扩展形式在现实应用中具有很高的实用价值.并且借助基于乘积性质的欧拉函数的扩展形式,构造出一种类似RSA的基于大整数分解困难问题的公钥加密及数字签名算法,对其安全性进行了分析,验证了算法的安全性和实用价值.
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