  We investigate the optimality and power allocation algorithm of beam domain
transmission for single-cell massive multiple-input multiple-output (MIMO)
systems with a multi-antenna passive eavesdropper. Focusing on the secure
massive MIMO downlink transmission with only statistical channel state
information of legitimate users and the eavesdropper at base station, we
introduce a lower bound on the achievable ergodic secrecy sum-rate, from which
we derive the condition for eigenvectors of the optimal input covariance
matrices. The result shows that beam domain transmission can achieve optimal
performance in terms of secrecy sum-rate lower bound maximization. For the case
of single-antenna legitimate users, we prove that it is optimal to allocate no
power to the beams where the beam gains of the eavesdropper are stronger than
those of legitimate users in order to maximize the secrecy sum-rate lower
bound. Then, motivated by the concave-convex procedure and the large dimension
random matrix theory, we develop an efficient iterative and convergent
algorithm to optimize power allocation in the beam domain. Numerical
simulations demonstrate the tightness of the secrecy sum-rate lower bound and
the near-optimal performance of the proposed iterative algorithm.
