A new scale-invariant homomorphic encryption scheme

We propose a new fully homomorphic encryption over the integers. The proposed scheme is not only efficient in the sense that it enables scale-invariant multiplications, but it also has an interesting property, which is that all evaluations over encrypted messages can be performed with only partial information about the underlying message space. For example, it can be public that the message space M is a set of positive integers less than some integer g, but the exact value of g will be provably hidden from all public information under a reasonable assumption. We call this property, informally, message space hidability. Such message space hidability has an immediate and interesting application, namely, homomorphic commitment to large integers.