Abstract: Modular exponentiation is one of the most fundamental operations in many encryption and signature systems.Due to the heavy computation cost of modular exponentiation,many schemes have been put forward to securely outsource modular exponentiation to cloud.However,most of the existing approaches need two non-colluded cloud servers to implement the secure modular exponentiation,resulting in private data leakage once the cloud servers collude.Besides,most existing schemes assume both base and exponent in modular exponentiation are private,which does not conform to many real-world applications.Usually,in order to reduce the overhead of computation,only the sensitive messages should be privately protected.To solve the above problems,this paper proposed two secure outsourcing schemes based on fixedbase (public base and private exponent) or fixed exponent(private base and public exponent),respectively.In the proposed schemes,the client only needs one cloud server,thus avoiding collusion attack of two servers.Theoretical analysis and experimental results reveal the security and efficiency of the proposed schemes.

Key words: Cloud computing, Modular exponentiations, Outsourcing-secure algorithm, Single server