Efficient outsourcing schemes of modular exponentiations with checkability for untrusted cloud server

Abstract

Modular exponentiation is one of basic operations among most of current cryptosystems. In previous work, in order to detect the dishonest behavior of cloud servers, modular exponentiations can only be securely outsourced by using two untrusted cloud servers. However, these results cannot resist the collusion attack by the untrusted cloud servers. In this paper, we present two secure outsourcing schemes for modular exponentiations, which enable users to securely outsource modular exponentiations to a single untrusted cloud server and detect the dishonest behavior of untrusted cloud server. The first one is a secure outsourcing scheme for variable base-variable exponent modular exponentiation, while the second is for simultaneous modular exponentiations. Compared with other proposed schemes, our outsourcing schemes are superior in both efficiency and checkability. Moreover, our schemes are secure without any cryptographic assumptions. In addition, we give two applications for our outsourcing schemes, one is to construct an outsourcing scheme for Cramer–Shoup encryptions, and the other is to design an outsourcing scheme for Schnorr signatures.