公钥RSA加密算法的改进与实现

发布时间：2018-04-05 10:21

本文选题：RSA加密算法　切入点：三个素数因子　出处：《安徽大学》2014年硕士论文

【摘要】：随着网络信息技术的高速发展,其安全问题越来越凸显出来。而网络信息安全重要理论基础是密码学。其中,RSA加密算法是被应用最广泛的公钥加密算法,它既可以实现数据加密中,又可以实现数字签名。但是近些年来,随着因子分解技术的不断提高(即对大数分解能力的提高),它的安全性受到了威胁。为了保证RSA加密算法的安全性,人们将算法中密钥的位数不断地增加。例如,在2009年12月,研究人员完成了768位RSA算法(即232数位数字的RSA-768密钥)的因式分解,同时研究人员也表示现在普遍使用的1024位密钥预计也将会在10年内攻破,在未来几年中需要将RSA算法中的1024位密钥过渡到2048位密钥。虽然密钥位数的增加确保了其安全性,但是也导致了算法效率大大的下降,主要表现在选择密钥的困难性上和加解密计算的复杂性上,这大大限制了它的应用范围。为了提高RSA加密算法的效率,本文首先介绍和分析了RSA加密算法,并对RSA加密算法做出了改进,内容大致如下： 1)介绍了传统的RSA加密算法,对于密钥位数的增加所带来的问题,本文采用了三个素数因子的RSA加密算法,这样虽然增加了素数因子的个数但是减少了素数因子的位数,从而降低了选择大素数因子的困难性,同时也提高了其选择的效率。 2)在加密过程中介绍了常用的加密算法：传统的平方乘算法、基于乘同余对称性的SMM算法、2K进制算法以及扩展2K进制算法,然后分析了这些算法的优缺点,最后在加密算法过程中将SMM算法和扩展的2K进制算法进行组合,这样可以将它们的优点进行结合形成一种新的组合算法。 3)在解密过程中介绍了基于中国剩余定理的解密算法,与以前算法不同点是在本文中采用了三个素数因子,分析了其解密过程,将解密算法中的取模运算转化为对三个较小素数因子的取模运算,减少了解密算法的计算量。 4)设计并实现了RSA加解密软件,软件中可以选择不同的算法进行加密和解密并测试出其所需的时间,将改进后的算法与传统的算法从效率上进行比较, 最后得出改进后的算法在效率上有了一定程度的提高。
[Abstract]:With the rapid development of network information technology, its security problems become more and more prominent.The important theoretical foundation of network information security is cryptography.Among them, RSA encryption algorithm is the most widely used public key encryption algorithm, it can not only realize data encryption, but also realize digital signature.However, in recent years, with the constant improvement of factorization technology (that is, the ability to decompose large numbers), its security is threatened.In order to ensure the security of RSA encryption algorithm, people increase the number of keys in the algorithm.For example, in December 2009, researchers completed factorization of the 768 bit RSA algorithm, or the 232-digit RSA-768 key, while researchers said that the now widely used 1024 bit key is also expected to break through within 10 years.In the next few years, the 1024 bit key in RSA algorithm needs to be transferred to 2048 bit key.Although the increase in the number of key bits ensures its security, it also leads to a great decrease in the efficiency of the algorithm, which is mainly reflected in the difficulty of selecting the key and the complexity of encryption and decryption computation, which greatly limits its application scope.In order to improve the efficiency of RSA encryption algorithm, this paper first introduces and analyzes the RSA encryption algorithm, and RSA encryption algorithm has been improved, the content is as follows:1) the traditional RSA encryption algorithm is introduced. For the problem of increasing the number of key bits, the RSA encryption algorithm with three prime factors is adopted in this paper, which increases the number of prime factors but reduces the number of prime factors.Therefore, the difficulty of selecting large prime factor is reduced, and the efficiency of selection is also improved.2) in the process of encryption, the common encryption algorithms are introduced: the traditional square multiplication algorithm, the SMM algorithm based on multiplicative congruence symmetry and the extended 2K-ary algorithm. Then, the advantages and disadvantages of these algorithms are analyzed.Finally, the SMM algorithm and the extended 2K binary algorithm are combined in the process of encryption, which can combine their advantages to form a new combination algorithm.3) in the process of decryption, the decryption algorithm based on the Chinese residue theorem is introduced. Different from the previous algorithm, three prime factors are used in this paper, and the decryption process is analyzed.In order to reduce the computation of decryption algorithm, the modulo operation in decryption algorithm is transformed into that of three smaller prime number factors.4) the RSA encryption and decryption software is designed and implemented. The software can choose different algorithms to encrypt and decrypt and test the time required. The improved algorithm is compared with the traditional algorithm in terms of efficiency.Finally, it is concluded that the efficiency of the improved algorithm has been improved to a certain extent.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TN918.4

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