【作者】 张俊；

【导师】 王瑞；

【作者基本信息】 云南大学 ， 计算机应用技术， 2017， 硕士

【摘要】 在单向陷门函数思想第一次提出后,人们继而提出公钥密码体制思想,RSA密码是利用数论方法实现了单向陷门函数思想,促进了公钥密码体制的发展,诸如椭圆曲线密码体制、EIGamal密码体制、组合密码等等。破译者的破译手段层出不穷,尝试不断的攻击、对相关算法的不断突破,同时随着计算机科学的不断发展,借助计算机已将原来密钥长度认为是不可能被攻破的变为可能被攻破的密钥长度。这使得已有算法的改进与算法的突破迫在眉睫,否则,将会造成不可估量的严重后果。通常建立一种可靠安全的密码体制在于数学困难性问题的设计,离散对数问题或大整数分解等。本文基于上述问题,我们找到一种方法用双曲线算术并扩充到双曲线丛的密码算法称之为双曲线丛密码算法,在模意义下的Pell方程,用方程的单位群与一组元素做陪集运算得到双曲线丛群,利用双曲线丛群上的点进行加密和解密。曲线丛密码算法兼顾了 RSA密码算法和椭圆曲线密码算法的优点,是一个同时基于离散对数问题和大整数分解两个困难问题设计的,选择不同双曲线丛的群具有多样性,双曲线丛与双曲线相比,双曲线丛的密码空间更大、密钥空间也更大。与传统的RSA相比,RSA的密钥太长、模太大、运算速度慢,在相同安全性下双曲线丛的密钥比RSA的密钥小的多。与椭圆曲线相比,椭圆曲线的解一般情况下只有估计式并且加法群不太完善,双曲线丛用二次方程代替了椭圆曲线的三次方程使得运算更简洁,用乘法群代替加法群使得计算更简单。当然,方案初次提出还有不完备的地方有待进一步研究与探讨。更多还原

【Abstract】 After the first one of the ideas of the one-way trap function,the people then put forward the idea of public key cryptosystem.The RSA cryptography realizes the idea of unidirectional trap function by means of number theory,which promotes the development of public key cryptosystem,such as elliptic curve cryptography System,EIGamal password system,the combination of passwords and so on.The cryptographer began to study the cryptographic system from another direction,making some cryptographic algorithms overcome and some cryptographic algorithms improved.At the same time with the continuous development of computer science,with the computer has been the original key length that is impossible to be broken into the key length may be broken.This makes the improvement of the algorithm and the algorithm of the breakthrough is imminent,otherwise,will cause immeasurable serious consequences.Usually establish a reliable and secure cryptography system is the design of mathematical difficulties,discrete logarithm problems or large integer decomposition.Based on the above-mentioned problem,we find a hyperbolic algorithm and a hyperbolic cluster cipher algorithm,which is a hyperbolic cluster cipher algorithm.It is a Pell equation in the modular sense,using the unit group of hyperbolic clusters and a group of elements Do coset operations.The algorithm of curve plexer cryptography takes into account the advantages of RSA cryptography and elliptic curve cryptography,which is based on two difficult problems:discrete logarithm problem and large integer decomposition.The diversity of hyperbolic clusters is different.Compared with the hyperbolic curve,the hyperbolic cluster has a larger code space and a larger key space.Compared with the traditional RSA,the RSA key is too long,the module is too large,the operation speed is slow,and the key of the hyperbolic cluster is much smaller than the RSA key under the same security.Compared with the elliptic curve,the solution of the elliptic curve is only in the general case and the additive group is not perfect.The hyperbolic cluster replaces the cubic equation of the elliptic curve with the quadratic equation so that the operation is more concise,and the multiplication group is used to replace the additive group simpler.Of course,the program for the first time there are incomplete places to be further research and discussion.更多还原