This paper proposes to investigate the impact of the channel model for authentication systems based on codes that are corrupted by a physically unclonable noise such as the one emitted by a printing process. The core of such a system for the receiver, is to perform a statistical test in order to recognize and accept an original code corrupted by noise and reject any illegal copy or a counterfeit. This study highlights the fact that the probability of type I and type II errors can be better approximated, by several orders of magnitude, when using the Cramér-Chernoff theorem instead of a Gaussian approximation. The practical computation of these error probabilities is also possible using Monte-Carlo simulations combined with the importance sampling method. By deriving the optimal test within a Neyman-Pearson setup, a first theoretical analysis shows that a thresholding of the received code induces a loss of performance. A second analysis proposes to find the best parameters of the channels involved in the model in order to maximize the authentication performance. This is possible when the opponent's channel is identical to the legitimate channel but also when the opponent's channel is different, leading this time to a min-max game between the two players. Finally we evaluate the impact of an uncertainty for the receiver on the opponent channel and we show that the authentication is still possible whenever the receiver can observe forged codes and uses them to estimate the parameters of the model.