In this paper, we compare two methods that can be used by the anti-counterfeiting industry to protect physical objects, which are either based on an object's natural randomness or on artificial randomness embedded on the object. We show that the considered verification architectures rely either on a comparison between an enrolled fingerprint and an extracted one or between a tag and a fingerprint. We compare these setups from detection-theoretic perspectives for both types of architectures. Authentication performance using false and miss error probabilities of the two systems are analysed and then compared using two practical setups. We highlight the advantages and limitations of each architecture. These theoretical results derived for binary fingerprints are useful to construct and optimise practical methods and to help select the appropriate architecture.