On the depth complexity of homomorphic encryption schemes

We show that secure homomorphic evaluation of any non-trivial functionality of sufficiently many inputs with respect to any CPA secure encryption scheme cannot be implemented by constant depth, polynomial size circuits, i.e. in the class AC. In contrast, we observe that certain previously studied encryption schemes (with quasipolynomial security) can be implemented in AC. We view this as evidence that encryption schemes that support homomorphic evaluation are inherently more complex than ordinary ones.