Recent hardware advances, called gate camouflaging, have opened the possibility of protecting integrated circuits against reverseengineering attacks. In this paper, we investigate the possibility of provably boosting the capability of physical camouflaging of a single Boolean gate into physical camouflaging of a larger Boolean circuit.We first propose rigorous definitions, borrowing approaches from modern cryptography and program obfuscation areas, for circuit camouflage. Informally speaking, gate camouflaging is defined as a transformation of a physical gate that appears to mask the gate to an attacker evaluating the circuit containing this gate. Under this assumption, we formally prove two results: a limitation and a construction. Our limitation result says that there are circuits for which, no matter how many gates we camouflaged, an adversary capable of evaluating the circuit will correctly guess all the camouflaged gates. Our construction result says that if pseudo-random functions exist (a common assumptions in cryptography), a small number of camouflaged gates suffices to: (a) leak no additional information about the camouflaged gates to an adversary evaluating the pseudo-random function circuit; and (b) turn these functions into random oracles. These latter results are the first results on circuit camouflaging provable in a cryptographic model (previously, construction were given under no formal model, and were eventually reverse-engineered, or were argued secure under specific classes of attacks). Our results imply a concrete and provable realization of random oracles, which, even if under a hardware-based assumption, is applicable in many scenarios, including public-key infrastructures. Finding special conditions under which provable realizations of random oracles has been an open problem for many years, since a software-only provable implementation of random oracles was proved to be (almost certainly) impossible.