  Consider a two-way communication scenario where two single-antenna nodes,
operating under full-duplex mode, exchange information to one another through
the aid of a (full-duplex) multi-antenna relay, and there is another
single-antenna node who intends to eavesdrop. The relay employs artificial
noise (AN) to interfere the eavesdropper's channel, and amplify-forward (AF)
Alamouti-based rank-two beamforming to establish the two-way communication
links of the legitimate nodes. Our problem is to optimize the rank-two
beamformer and AN covariance for sum secrecy rate maximization (SSRM). This
SSRM problem is nonconvex, and we develop an efficient solution approach using
semidefinite relaxation (SDR) and minorization-maximization (MM). We prove that
SDR is tight for the SSRM problem and thus introduces no loss. Also, we
consider an inexact MM method where an approximately but computationally cheap
MM solution update is used in place of the exact update in conventional MM. We
show that this inexact MM method guarantees convergence to a stationary
solution to the SSRM problem. The effectiveness of our proposed approach is
further demonstrated by an energy-harvesting scenario extension, and by
extensive simulation results.
