An eavesdropping game with SINR as an objective function

We examine eavesdropping over wireless channels, where secret communication in the presence of an eavesdropper is formulated as a zero-sum game. In our problem, the legitimate receiver does not have complete knowledge about the environment, i.e. does not know the exact values of the channels gains, but instead knows just their distribution. To communicate secretly, the user must decide how to transmit its information across subchannels under a worst-case condition and thus, the legal user faces a max-min optimization problem. To formulate the optimization problem, we pose the environment as a secondary player in a zero-sum game whose objective is to hamper communication by the user. Thus, nature faces a min-max optimization problem. In our formulation, we consider signal-to-interference ratio (SINR) as a payoff function. We then study two specific scenarios: (i) the user does not know the channels gains; and (ii) the user does not know how the noise is distributed among the main channels. We show that in model (i) in his optimal behavior the user transmits signal energy uniformly across a subset of selected channels. In model (ii), if the user does not know the eavesdropper's channel gains he/she also employs a strategy involving uniformly distributing energy across a subset of channels. However, if the user acquires extra knowledge about environment, e.g. the eavesdropper's channel gains, the user may better tune his/her power allocation among the channels. We provide criteria for selecting which channels the user should transmit on by deriving closed-form expressions for optimal strategies for both players.