Discrete exponential operation, such as modular exponentiation and scalar multiplication on elliptic curves, is a basic operation of many public-key cryptosystems. However, the exponential operations are considered prohibitively expensive for resource-constrained mobile devices. In this paper, we address the problem of secure outsourcing of exponentiation operations to one single untrusted server. Our proposed scheme (ExpSOS) only requires very limited number of modular multiplications at local mobile environment thus it can achieve impressive computational gain. ExpSOS also provides a secure verification scheme with probability approximately 1 to ensure that the mobile end-users can always receive valid results. The comprehensive analysis as well as the simulation results in real mobile device demonstrates that our proposed ExpSOS can significantly improve the existing schemes in efficiency, security and result verifiability. We apply ExpSOS to securely outsource several cryptographic protocols to show that ExpSOS is widely applicable to many cryptographic computations.
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