Homomorphic Evaluation Requires Depth

We show that homomorphic evaluation of any non-trivial functionality of sufficiently many inputs with respect to any CPA secure homomorphic encryption scheme cannot be implemented by circuits of polynomial size and constant depth, i.e., in the class AC. In contrast, we observe that there exist ordinary public-key encryption schemes of quasipolynomial security in AC assuming noisy parities are exponentially hard to learn. We view this as evidence that homomorphic evaluation is inherently more complex than basic operations in encryption schemes.