  In this paper, we consider the Gaussian multiple-input multiple-output (MIMO)
multi-receiver wiretap channel in which a transmitter wants to have
confidential communication with an arbitrary number of users in the presence of
an external eavesdropper. We derive the secrecy capacity region of this channel
for the most general case. We first show that even for the single-input
single-output (SISO) case, existing converse techniques for the Gaussian scalar
broadcast channel cannot be extended to this secrecy context, to emphasize the
need for a new proof technique. Our new proof technique makes use of the
relationships between the minimum-mean-square-error and the mutual information,
and equivalently, the relationships between the Fisher information and the
differential entropy. Using the intuition gained from the converse proof of the
SISO channel, we first prove the secrecy capacity region of the degraded MIMO
channel, in which all receivers have the same number of antennas, and the noise
covariance matrices can be arranged according to a positive semi-definite
order. We then generalize this result to the aligned case, in which all
receivers have the same number of antennas, however there is no order among the
noise covariance matrices. We accomplish this task by using the channel
enhancement technique. Finally, we find the secrecy capacity region of the
general MIMO channel by using some limiting arguments on the secrecy capacity
region of the aligned MIMO channel. We show that the capacity achieving coding
scheme is a variant of dirty-paper coding with Gaussian signals.
