Capacity-Equivocation Region of the Gaussian MIMO Wiretap ...

5702 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 9, SEPTEMBER 2012
since 2 is convex by definition, it can be characterized by
solving the following optimization problem: 3
(29)
for all , and all possible common
message rates , which is bounded as follows:
(30)
where are the single-user capacities for the legitimate
user and the eavesdropper channels, respectively, i.e.,
(31)
(32)
We note that the optimization problem in (29) can be expressed
in the following more explicit form:
(33)
(34)
We also consider the Gaussian rate region which
is defined by (35) at the bottom of the page, where
are given as follows:
(36)
(37)
(38)
(39)
2Although is originally defined for the general, not necessarily
aligned, Gaussian wiretap channel with public messages, here we use
to denote the capacity region of the aligned Gaussian MIMO wiretap channel
with public messages as well.
3Although characterizing by solving the following optimization
problem:
(28)
for all seems to be more natural, we find working with (29) more
convenient. Here, we characterize by solving (29) for all for all
fixed feasible .
To provide the converse proof, i.e., to prove the optimality of
jointly Gaussian for the optimization problem in
(33)–(34), we will show that
where is defined as
We show (40) in two parts:
1)
2) .
A.
In this case, can be written as
(40)
(41)
(42)
(43)
where we use the fact that , and the secret message
rate can be given up in favor of the private message rate
. In other words, we use the fact that when ,the
maximum of is given by ,where
is an achievable public message rate since the secret
message can be converted into a public message. This optimization
problem gives us the capacity region of the two-user
Gaussian MIMO broadcast channel with degraded message sets,
where a common message is sent to both users, and a private
message, on which there is no secrecy constraint, is sent to one
of the two users [13]. The optimization problem for this case
given in (42)–(43) is solved in [10] by showing the optimality
of jointly Gaussian ,i.e., . This completes
the converse proof for the case .
B.
In this case, we first study the optimization problem in (41).
We rewrite as follows:
(44)
(45)
where we use the fact that since , the secret message
rate should be set as high as possible to maximize
,i.e.,weshouldset .Let
be the maximizer for this optimization problem. The necessary
(35)