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This paper studies the capacity of the multi-antenna or multiple-inputmultiple-output (MIMO) secrecy channels with multiple eavesdroppers havingsingle/multiple antennas. It is known that the MIMO secrecy capacity isachievable with the optimal transmit covariance matrix that maximizes theminimum difference between the channel mutual information of the secrecy userand those of the eavesdroppers. The MIMO secrecy capacity computation can thusbe formulated as a non-convex max-min problem, which cannot be solvedefficiently by standard convex optimization techniques. To handle thisdifficulty, we explore a relationship between the MIMO secrecy channel and therecently developed MIMO cognitive radio (CR) channel, in which themulti-antenna secondary user transmits over the same spectrum simultaneouslywith multiple primary users, subject to the received interference powerconstraints at the primary users, or the so-called ``interference temperature(IT)'' constraints. By constructing an auxiliary CR MIMO channel that has thesame channel responses as the MIMO secrecy channel, we prove that the optimaltransmit covariance matrix to achieve the secrecy capacity is the same as thatto achieve the CR spectrum sharing capacity with properly selected ITconstraints. Based on this relationship, several algorithms are proposed tosolve the non-convex secrecy capacity computation problem by transforming itinto a sequence of CR spectrum sharing capacity computation problems that areconvex. For the case with single-antenna eavesdroppers, the proposed algorithmsobtain the exact capacity of the MIMO secrecy channel, while for the case withmulti-antenna eavesdroppers, the proposed algorithms obtain both upper andlower bounds on the MIMO secrecy capacity.

On the Relationship Between the Multi-antenna Secrecy Communications and Cognitive Radio Communications