We investigate the reconfigurable intelligent surface (RIS) assisted downlink secure transmission where only the statistical channel of eavesdropper is available. To handle the stochastic ergodic secrecy rate (ESR) maximization problem, a deterministic lower bound of ESR (LESR) is derived. We aim to maximize the LESR by jointly designing the transmit beamforming at the access point (AP) and reflect beamforming by the phase shifts at the RIS. To solve the non-convex LESR maximization problem, we develop a novel penalty dual convex approximation (PDCA) algorithm based on the penalty dual decomposition (PDD) optimization framework, where the exacting constraints are penalized and dualized into the objective function as augmented Lagrangian components. The proposed PDCA algorithm performs double-loop iterations, i.e., the inner loop resorts to the block successive convex approximation (BSCA) to update the optimization variables; while the outer loop adjusts the Lagrange multipliers and penalty parameter of the augmented Lagrangian cost function. The convergence to a Karush-Kuhn-Tucker (KKT) solution is theoretically guaranteed with low computational complexity. Simulation results show that the proposed PDCA scheme is better than the commonly adopted alternating optimization (AO) scheme with the knowledge of statistical channel of eavesdropper.
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