中文摘要：

模指数运算是公钥密码体制中的基本运算，但造价昂贵。可验证外包计算将用户的计算代价转移给服务器，能大大提高用户的计算效率，且能有效验证服务器的计算结果。针对多个模指数运算的外包方案进行了研究，利用指数之间的相互隐藏，提出了新的两个模指数运算外包方案，并推广到多个模指数运算。方案基于两个服务器，实现了模指数运算底数和指数的隐私性，并且用户的可验证概率接近于1。与已有方案相比，所提方案同时提高了用户的计算效率和外包结果的可验证概率。理论分析和仿真实验均表明，所提方案极大地降低了用户的计算代价。

英文摘要：

Modular exponentiation is the basic operation of public key cryptosystem, but the cost is expensive. Verifiable outsourcing delegates the computing cost to servers, which can greatly improve the computing efficiency of user, and verify the result returned from servers. In study of outsourcing multiple modular exponentiations, this paper proposed a new algorithm of double modular exponentiations by hiding the exponent, and then proposed an algorithm of multiple modular exponentiations similarly. These schemes used two un-trusted servers to achieve the privacy of base and index, and the user could detect the error with probability close to 1. Compared with the previous algorithms, proposed algorithms increase the computational efficiency and verifiable probability. Finally, theoretical analysis and simulation experiments show that they greatly reduce the computational cost of user.