基于多素数和参数替换的改进RSA算法研究

为了提高RSA公钥算法在消息加密过程中的安全性，在深入分析传统RSA算法的基础上，对其进行一些改进性研究，提出了一种比传统RSA算法更加有效的方法优化其安全性。在将传统RSA改进为四素数RSA的基础上，再运用数学变换进行参数替换，消除了在公钥中对传输两个随机素数的乘积n的需要，引入了一个新的参数x代替原参数n。针对改进后的算法在运算效率方面的不足，采用中国剩余定理( Chinese remainder theorem，CRT)优化大数模幂运算。实验结果证实了改进算法的可行性，为通过公钥加密消息发送和接收提供了更安全的路径；同时，对改进算法与传统RSA 和四素数RSA算法的解密(签名)时间进行比较分析。实验结果表明改进后的算法对消息发送方和接收方之间签名效率也有一定程度的优化。

Research on improved RSA algorithm based on multi-prime number and parameter substitution

In order to achieve greater safety in RSA encryption algorithm in the data encryption process, this paper made some improvements based on in-depth analysis on the traditional RSA algorithm. It put forward a more secure method with changing traditional double primes into four primes in the traditional RSA algorithm and eliminating the requirement for transferring the product n of two random primes in public key on the basis of parameter substitution. Instead, this paper replaced the original parameter n to a new introduced parameter x. In order to boost the efficiency in the introduced algorithm, this paper used the CRT(Chinese remainder theorem) to expedite the exponentiation of large numbers. The results demonstrates the feasibility of the improved algorithm, provides a safer path for messages transmission through public key encryption. At the same time, this paper carried out a comparative analysis of encryption and decryption time among the improved algorithm and the traditional double primes and four primes RSA algorithms. The results shows that the improved algorithm to a certain degree enhanced the encryption and decryption efficiency between the message sender and receiver.