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我们需要知道求幂运算的逆运算的难度。We need to know how hard it is to reverse the exponentiation.

首先我会写一个,迭代乘法的过程。So I'm going to build something that's going to do iterative exponentiation.

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模幂算法广泛应用于公开密钥加密技术。Modular exponentiation algorithm is widely used in public key encryption technology.

看我怎么用这个小技巧,来求a的b次方的。So let me show you another way I could do exponentiation. Taking an advantage of a trick.

如果我们运用求幂运算来加密和解密，对手就可以运用对数进行攻击。If we use exponentiation to encrypt or decrypt , the adversary can use logarithm to attack.

综述了rsa密码算法中模幂运算的主要攻击方法及其防御措施。First countermeasures for the exponentiation computation of RSA cryptographic algorithm were summarized.

模幂运算是RSA公钥密码算法中最基本也是最耗时的运算。Modular exponentiation is the most common fundamental and time consuming operation in RSA public-key cryptosystems.

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此算法对基于多重取幂密码协议的快速运作是很有效用的。The algorithm is useful to speed up the operation for some cryptographic protocols based on multiple exponentiation.

RSA算法的解密性能与大数模幂运算的实现效率有着直接的关系。The performance of RSA decryption has direct relationship with the efficiency of modular exponentiation implementation.

模幂算法采用从右到左扫描指数的方法，可以使得两次模乘运算同时进行。Modul ar exponentiation algorithm scans encryption from right to sot, so t wo modular multiplications can be processed parallelly.

模乘幂运算是公钥密码体制中最常用的基本运算，提高其运算速度可有效地提高公钥密码算法的加解密效率。A modular exponentiation for very large integers is one of the most common fundamental operations in many public-key cryptosystems.

在设计中还采用了窗口法减少模幂运算过程中所需进行的模乘运算次数，大大提高了处理速度。The window method is used to significantly reduce the number of modular multiplications needed for completing the modular exponentiation.

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这可以允许将输入和输出同时按许多不同的单位表达，并通过乘法、除法及求幂运算组合起来。This allows input and output to be expressed in terms of many different units simultaneously, combined via multiplication, division, and exponentiation.

算法的硬件结构由模乘控制器、模幂控制器、数据寄存器和模乘运算单元构成。The hardware architecture is made up of modular controller, modular exponentiation controller, data register, and modular multiplication operation units.

大数模幂在密码学领域有广泛的应用，它是公钥密码的基础。Modular exponentiation of larger-number has universal application in cryptography, and it is the base operation in most public-key cryptography algorithms.

另外，群体特性通过微分运算及其逆运算所得到的李代数的代数结构而得到了解释。Lie's Theory Within the framework of Lie' Theory, we associate infinitesimal transformations making up a Lie algebra with finite operations which are obtained from the previous ones by exponentiation.