Secure Outage Analysis of RIS-Assisted Communications with Discrete Phase Control

This correspondence investigates a reconfigurable intelligent surface (RIS)-assisted wireless communication system with security threats. The RIS is deployed to enhance the secrecy outage probability (SOP) of the data sent to a legitimate user. By deriving the distributions of the received signal-to-noise-ratios (SNRs) at the legitimate user and the eavesdropper, we formulate, in a closed-form expression, a tight bound for the SOP under the constraint of discrete phase control at the RIS. The SOP is characterized as a function of the number of antenna elements, $N$, and the number of discrete phase choices, $2^b$. It is revealed that the performance loss in terms of SOP due to the discrete phase control is ignorable for large $N$ when $b\!\geq\!3$. In addition, we explicitly quantify this SOP loss when binary phase shifts with $b\!=\!1$ is utilized. It is identified that increasing the RIS antenna elements by $1.6$ times can achieve the same SOP with binary phase shifts as that by the RIS with ideally continuous phase shifts. Numerical simulations are conducted to verify the accuracy of these theoretical observations.