  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.
