Secure outsourcing algorithms of modular exponentiations with optimal checkability based on a single untrusted cloud server

Modular exponentiation is an expensive discrete-logarithm operation, difficult for resource-constrained users to perform locally. Fortunately, thanks to burgeoning cloud computing, users are willing to securely outsourcing modular exponentiations to cloud servers to reduce computation overhead. In this paper, we contrive a fully verifiable secure outsourcing scheme for modular exponentiation with only a single server, named MExp. MExp not only prevents users’ private information leakage during outsourcing by our new logical division method, but also eliminates collusion attacks occurring in algorithms with two untrusted servers. Moreover, our MExp allows outsourcers to detect any misbehavior with a probability of 1, which shows significant improvement in checkability when compare to other single-server-based schemes. With a view to reducing computation overhead, MExp is extended to multiple modular exponentiations, named M2Exp. The algorithm significantly diminishes the local costs of multiple modular exponentiation calculations and the checkability is still 1. Compared with existing state-of-the-art schemes, MExp and M2Exp have outstanding performance in both efficiency and checkability. Finally, MExp and M2Exp are applied to Cramer–Shoup encryptions and Schnorr signatures.