Modular exponentiation, as a fundamental operation used in many public-key cryptosystems, has always be considered to be very time-consuming. It is difficult for some devices with limited computation capability, such as mobile devices and low-cost radio frequency identification (RFID) tags, to perform large-scale modular exponentiations. In cryptosystems, one typical case of modular exponentiation is that the modulus is a composite number. For instance, in RSA algorithm, the modulus is the product of two distinct prime numbers. In this paper, we investigate how to securely and efficiently outsource composite modular exponentiations and put forward two secure outsourcing algorithms for composite modular exponentiations based on single untrusted cloud. The first algorithm, named MCExp, is designed for outsourcing single composite modular exponentiation, i.e. |$u^a$| mod |$N$|⁠. The second algorithm, named SMCExp, is designed for outsourcing simultaneous composite modular exponentiation, i.e. |$\prod ^{n}_{i=1}u^{a_i}_{i}$| mod |$N$|⁠. Different from algorithms based on two untrusted servers, the proposed algorithms are very practical because they avoid the strong assumption that there must exist two servers without collusion. The proposed algorithms not only protect the privacy of the exponent and the base simultaneously, but also enable users to verify the correctness of the result returned by the cloud with high probability. Compared with using the square-and-multiply algorithm, the user can achieve higher efficiency by using the proposed algorithms. Besides, we prove the security of our algorithms and conduct several experiments to demonstrate the efficiency of the proposed algorithms. Finally, we show that the proposed algorithms can be used to construct the secure outsourcing algorithms for Shamir’s identity-based signature and identity-based multi-signature.

中文翻译：

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基于单个不可信云的组合模块化幂运算安全外包算法

作为许多公钥密码系统中使用的基本操作，模幂运算一直被认为是非常耗时的。对于某些计算能力有限的设备，例如移动设备和低成本射频识别（RFID）标签，很难执行大规模的模幂运算。在密码系统中，模幂的一种典型情况是模数是一个复合数。例如，在RSA算法中，模数是两个不同质数的乘积。在本文中，我们研究了如何安全有效地外包组合模块化指数，并针对基于单个不可信云的复合模块化指数提出了两种安全外包算法。第一个算法称为MCExp，| $ u ^ a $ | mod | $ N $ |⁠。第二种算法称为SMCExp，用于外包同时进行的组合模块化幂运算，即| $ \ prod ^ {n} _ {i = 1} u ^ {a_i} _ {i} $ | mod | $ N $ |⁠。与基于两个不受信任的服务器的算法不同，该算法非常实用，因为它们避免了必须存在两个没有串通的服务器的强烈假设。所提出的算法不仅可以同时保护指数和基数的隐私，还可以使用户以较高的概率验证云返回结果的正确性。与使用平方乘算法相比，使用本算法可以使用户获得更高的效率。此外，我们证明了算法的安全性，并进行了几次实验以证明所提出算法的有效性。最后，我们证明了所提出的算法可用于构建Shamir基于身份的签名和基于身份的多重签名的安全外包算法。