Secure Modular Exponentiation Outsource With Two Untrusted Programs and Improved Checkability

With the development of cloud computing, more and more mobile devices have the requirement to outsource expensive computations in an untrusted environment. Among all such computations, exponentiations modulo a large prime are basic and frequent operations in many discrete-logarithm-based cryptographic protocols. Currently the most efficient outsourcing algorithm of modular exponentiations is under a one-malicious version of a two-untrusted-program assumption. And the result of the algorithm is checkable with a probability at most 2/3. We here propose an efficient algorithm under a two-untrusted-program assumption and improve the probability of checkability to be about 1-2/3s for a suitable s configured by a client. Further, we provide practical results to show the energy saving about outsourcing modular exponentiations.