中文摘要：

云计算是一种新兴的计算模式，它为云用户提供了强大的计算环境，但同时也引起了用户安全性和隐私性问题的关注．模幂运算是大多数现行的密码系统的基本运算之一，也是公钥密码系统在计算资源限制型设备上的计算瓶颈所在．使用传统的平方-乘算法计算一个n比特的指数的模幂运算，平均需要1．5n个模乘，对于资源有限型用户（或设备，例如智能卡）来说，这个计算量是个很沉重的负载．外包计算是云计算模式的优点之一，它使得云用户的计算能力不再受限于各自的资源约束型设备，通过外包工作负载给云，云用户可以使用云提供的无限资源来完成高代价的计算．本文围绕“外包模幂运算”这个问题展开研究，为椭圆曲线的点的乘法的计算问题，提出了一个安全的外包计算方案，并且将本文的方案应用于加速椭圆曲线数字签名的验证．

英文摘要：

Cloud computing is a novel computing paradigm which provides powerful computing resources for cloud users. However, this paradigm also brings new challenges for security and privacy. The modular exponentiation is one of basic operations among most of current cryptosystems.It also presents the computational bottleneck in most public-key cryptography on computationally limited devices.Without outsourcing,a device using the traditional square-and-multiply algorithm need 1 .5 n modular multiplications on average to carry out a modular exponentiation for n -bit exponents.Thus,it is a very time -consuming operation for device with limited computational resources,such as smart card.Cloud computing has a great deal of computational resources and then the computation/storage intensive tasks can be performed through being delegated to resource-abundant cloud servers.A practically and securely outsourced scheme was present,which was used for computing point multiplication on elliptic curves,and was applied to speeding up signature verification of the ECDSA （Elliptic Curves Digital Signature Algorithm）.