Cyber-physical systems are prone to various security threats. Systems must be equipped with robust detection strategies to ensure timely detection of attacks. One such detection strategy proposes adding a random authentication signal to the optimal control signal and uses a chi-squared failure detector on the residues of measurements to detect the attacks. In this paper, we design the authentication signal's covariance matrix to optimize the performance of the chi-squared detector. The covariance matrix is a unit rank matrix and is an outer product of a well-defined vector. This structure maximizes the probability of correctly detecting an attack while maintaining the probability of false alarm in the chi-squared detector. We corroborate our analytical results by employing simulations. The optimal chi-squared setting demonstrates significant performance gain arbitrary settings. © 2021 IEEE.