The Secrecy Capacity Region of the Gaussian MIMO Multi-receiver Wiretap Channel

TL;DR

This paper derives the secrecy capacity region for the Gaussian MIMO multi-receiver wiretap channel, introducing new proof techniques and extending results from degraded and aligned cases to the general scenario.

Contribution

It presents the first complete characterization of the secrecy capacity region for the general Gaussian MIMO multi-receiver wiretap channel, using novel proof methods and channel enhancement techniques.

Findings

Secrecy capacity region for degraded MIMO channels established.

Extension of results to aligned MIMO channels using channel enhancement.

Capacity-achieving scheme identified as a variant of dirty-paper coding.

Abstract

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.