  In this paper, we investigate the energy efficiency (EE) problem under
reconfigurable intelligent surface (RIS)-aided secure cell-free networks, where
multiple legitimate users and eavesdroppers (Eves) exist. We formulate a
max-min secure EE optimization problem by jointly designing the distributed
active beamforming and artificial noise at base stations as well as the passive
beamforming at RISs under practical constraints. To deal with it, we first
divide the original optimization problem into two sub-ones, and then propose an
iterative optimization algorithm to solve each sub-problem based on the
fractional programming, constrained convex-convex procedure (CCCP) and
semi-definite programming (SDP) techniques. After that, these two sub-problems
are alternatively solved until convergence, and the final solutions are
obtained. Next, we extend to the imperfect channel state information of the
Eves' links, and investigate the robust SEE beamforming optimization problem by
bringing the outage probability constraints. Based on this, we first transform
the uncertain outage probability constraints into the certain ones by the
bernstein-type inequality and sphere boundary techniques, and then propose an
alternatively iterative algorithm to obtain the solutions of the original
problem based on the S-procedure, successive convex approximation, CCCP and SDP
techniques. Finally, the simulation results are conducted to show the
effectiveness of the proposed schemes.
