Cloud computing enables computational resource-limited devices to economically outsource much computations to the cloud. Modular exponentiation is one of the most expensive operations in public key cryptographic protocols, and such operation may be a heavy burden for the resource-constraint devices. Previous works for secure outsourcing modular exponentiation which use one or two untrusted cloud server model or have a relatively large computational overhead, or do not support the 100% possibility for the checkability. In this letter, we propose a new efficient and verifiable algorithm for securely outsourcing modular exponentiation in the two untrusted cloud server model. The algorithm improves efficiency by generating random pairs based on EBPV generators, and the algorithm has 100% probability for the checkability while preserving the data privacy.

With the popularity of cloud computing services, outsourcing computation has entered a period of rapid development. Modular exponentiation is one of the most expensive operations in public key cryptographic systems, but the current outsourcing algorithms for modular exponentiations (MExps) with single server are inefficient or have small checkability. In this paper, we propose an efficient and fully verifiable algorithm for outsourcing multiple MExps with single untrusted server where the errors can be detected by an outsourcer with a probability of 1. The theory analysis and experimental evaluations also show that the proposed algorithm is the most efficient one compared with the previous work. Finally, we present the outsourcing schemes of digital signature algorithm (DSA) and attribute based encryption (ABE) as two applications of the proposed algorithm.