  This paper studies the capacity of the multi-antenna or multiple-input
multiple-output (MIMO) secrecy channels with multiple eavesdroppers having
single/multiple antennas. It is known that the MIMO secrecy capacity is
achievable with the optimal transmit covariance matrix that maximizes the
minimum difference between the channel mutual information of the secrecy user
and those of the eavesdroppers. The MIMO secrecy capacity computation can thus
be formulated as a non-convex max-min problem, which cannot be solved
efficiently by standard convex optimization techniques. To handle this
difficulty, we explore a relationship between the MIMO secrecy channel and the
recently developed MIMO cognitive radio (CR) channel, in which the
multi-antenna secondary user transmits over the same spectrum simultaneously
with multiple primary users, subject to the received interference power
constraints at the primary users, or the so-called ``interference temperature
(IT)'' constraints. By constructing an auxiliary CR MIMO channel that has the
same channel responses as the MIMO secrecy channel, we prove that the optimal
transmit covariance matrix to achieve the secrecy capacity is the same as that
to achieve the CR spectrum sharing capacity with properly selected IT
constraints. Based on this relationship, several algorithms are proposed to
solve the non-convex secrecy capacity computation problem by transforming it
into a sequence of CR spectrum sharing capacity computation problems that are
convex. For the case with single-antenna eavesdroppers, the proposed algorithms
obtain the exact capacity of the MIMO secrecy channel, while for the case with
multi-antenna eavesdroppers, the proposed algorithms obtain both upper and
lower bounds on the MIMO secrecy capacity.
