Legitimate against Illegitimate IRSs on MISO Wiretap Channels

The low-cost legitimate intelligent reflecting surfaces (IRSs) have been applied to the wiretap channel in physical layer security to enhance the secrecy rate. In practice, the eavesdropper can also deploy an IRS, namely illegitimate IRS, to deteriorate the secrecy rate. This paper studies the interplay between a trans-mitter, a legitimate IRS, and an illegitimate IRS in a multiple-input single-output (MISO) wiretap channel. We formulate a max-min secrecy rate problem, where all the information is available at the transmitter and the receivers. We aim to design an efficient transmit beamforming and phase shifting strategy of the legitimate IRS, under the worst-case secrecy rate achieved based on optimizing the phase shifting strategy of the illegitimate IRS. We propose three solution methods based on the gradient descent ascent (GDA), the alternate optimization (AO), and the mixed Nash equilibrium (NE) in zero-sum games in strategic form. Numerical results are provided to demonstrate the performance and convergence behavior of AO, GDA, and the mixed NE for continuous and discrete domains of IRSs' phase shifts.