  The problem of maximizing secrecy rate of multiple-input multiple-output
multiple-eavesdropper (MIMOME) channels with arbitrary numbers of antennas at
each node is studied in this paper. First, the optimization problem
corresponding to the secrecy capacity of the MIMOME channel is converted to an
equivalent optimization based on Givens rotations and eigenvalue decomposition
of the covariance matrix. In this new formulation, precoder is a rotation
matrix which results in a positive semi-definite (PSD) covariance matrix by
construction. This removes the PSD matrix constraint and makes the problem
easier to tackle. Next, a Broyden-Fletcher-Goldfarb-Shanno (BFGS)-based
algorithm is developed to find the rotation and power allocation parameters.
Further, the generalized singular value decomposition (GSVD)-based precoding is
used to initialize this algorithm. The proposed rotation-BFGS method provides
an efficient approach to find a near-optimal transmit strategy for the MIMOME
channel and outperforms various state-of-the-art analytical and numerical
methods. In particular, the rotation BFGS precoding achieves higher secrecy
rates than the celebrated GSVD precoding, with a reasonably higher
computational complexity. Extensive numerical results elaborate on the
effectiveness of the rotation-BFGS precoding. The new framework developed in
this paper can be applied to a variety of similar problems in the context of
multi-antenna channels with and without secrecy.
