For a physical layer message authentication procedure based on the comparison of channel estimates obtained from the received messages, we focus on an outer bound on the type I/II error probability region. Channel estimates are modelled as multivariate Gaussian vectors, and we assume that the attacker has only some side information on the channel estimate, which he does not know directly. We derive the attacking strategy that provides the tightest bound on the error region, given the statistics of the side information. This turns out to be a zero mean, circularly symmetric Gaussian density whose correlation matrices may be obtained by solving a constrained optimization problem. We propose an iterative algorithm for its solution: Starting from the closed form solution of a relaxed problem, we obtain, by projection, an initial feasible solution; then, by an iterative procedure, we look for the fixed point solution of the problem. Numerical results show that for cases of interest the iterative approach converges, and perturbation analysis shows that the found solution is a local minimum.