Distributed Functional Signature with Function Privacy and Its Application

We introduce a novel notion of distributed functional signature. In such a signature scheme, the signing key for function f will be split into n shares sk f i and distributed to different parties. Given a message m and a share sk f i , one can compute locally and obtain a pair signature f i m , σ i . When given all of the signature pairs, everyone can recover the actual value f m and corresponding signature σ . When the number signature pairs are not enough, nobody can recover the signature f m , σ . We formalize the notion of function privacy in this new model which is not possible for the standard functional signature and give a construction from standard functional signature and function secret sharing based on one-way function and learning with error assumption. We then consider the problem of hosting services in multiple untrusted clouds, in which the verifiability and program privacy are considered. The verifiability requires that the returned results from the cloud can be checked. The program privacy requires that the evaluation procedure does not reveal the program for the untrusted cloud. We give a verifiable distributed secure cloud service scheme from distributed functional signature and prove the securities which include untrusted cloud security (program privacy and verifiability) and untrusted client security.