A Game-theoretic Approach to Covert Communications in the Presence of Multiple Colluding Wardens

In this paper, we address the problem of covert communication under the presence of multiple wardens with a finite blocklength. The system consists of Alice, who aims to covertly transmit to Bob with the help of a jammer. The system also consists of a Fusion Center (FC), which combines all the wardens' information and decides on the presence or absence of Alice. Both Alice and jammer vary their signal power randomly to confuse the FC. In contrast, the FC randomly changes its threshold to confuse Alice. The main focus of the paper is to study the impact of employing multiple wardens on the trade-off between the probability of error at the FC and the outage probability at Bob. Hence, we formulate the probability of error and the outage probability under the assumption that the channels from Alice and jammer to Bob are subject to Rayleigh fading, while we assume that the channels from Alice and jammer to the wardens are not subject to fading. Then, we utilize a two-player zero-sum game approach to model the interaction between joint Alice and jammer as one player and the FC as the second player. We derive the pay-off function that can be efficiently computed using linear programming to find the optimal distributions of transmitting and jamming powers as well as thresholds used by the FC. The benefit of using a cooperative jammer is shown by means of analytical results and numerical simulations to neutralize the advantage of using multiple wardens at the FC.