Reducing the computational cost of cryptographic computations for resource-constrained devices is an active research area. Outsourcing the computation to an external server securely provides a practical solution. In particular, modular exponentiations are the most expensive computation of many cryptographic protocols. Outsourcing the modular exponentiations to a single, external and potentially untrusted cloud server securely while ensuring the privacy is the most realistic scenario. In this paper, we propose new efficient outsourcing algorithms for modular exponentiations using only one untrusted cloud server. These algorithms cover public-base & private-exponent, private-base & public-exponent, private-base & private-exponent and more generally private-base & private-exponents simultaneous modular exponentiations. Our algorithms are the most efficient outsourced computation algorithms using a single untrusted server. Furthermore, our algorithms provide the best
checkability property with predetermined parameters. Finally, we give two different applications for outsourcing within the realms of Oblivious Transfer Protocols and Blind Signatures.
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