Secure Multicast Energy-Efficiency Maximization With Massive RISs and Uncertain CSI: First-Order Algorithms and Convergence Analysis

Reconfigurable intelligent surface (RIS) has the potential to significantly enhance the network secure transmission performance by reconfiguring the wireless propagation environment. However, due to the passive nature of eavesdroppers and the cascaded channel brought by the RIS, the eavesdroppers' channel state information is imperfect at the base station. Under channel uncertainty, the optimal phase-shift, power allocation, and transmission rate design for massive antennas and reflecting elements secure transmission are challenging to solve due to the outage probabilistic constraint with coupled variables. To fill this gap, this paper formulates a problem of energy-efficient secure transmission design with the probabilistic outage constraint. By leveraging the exponential distribution property of the received signal power, the stochastic resource allocation is equivalently transformed into a deterministic one, and the secure energy efficiency maximization problem can be iteratively solved via low complexity first-order algorithms under the alternating maximization (AM) framework. However, due to the nonsmooth problem, the convergence of the objective function value and nature of the converged solution under AM iteration are uncertain. Therefore, the convergence properties with respect to the objective function value and sequence of solutions are further established. Simulation results corroborate the convergence results of the first-order algorithms and show that the proposed algorithm achieves identical performance to the conventional method but saves at least two orders of magnitude in computation time. Moreover, the resultant RIS aided secure transmission significantly improves the energy efficiency compared to baseline schemes of random phase-shift, fixed phase-shift, and RIS ignoring CSI uncertainty.