Secure Transmission with Multiple Antennas: The MIMOME Channel

The Gaussian wiretap channel model is studied when there are multiple antennas at the sender, the receiver and the eavesdropper and the channel matrices are fixed and known to all the terminals. The secrecy capacity is characterized using a Sato type upper bound. A computable characterization of the capacity is provided. The secrecy capacity is studied in several interesting regimes. The high SNR secrecy capacity is attained by simultaneously diagonalizing the channel matrices (using the generalized singular value decomposition) and independently coding across the resulting parallel channels. An explicit closed form solution in terms of the generalized singular values of the channel matrices is provided in this regime. In addition to the capacity achieving scheme we also study achievable rates from a synthetic noise transmission strategy. Interestingly this rate can also be expressed in terms of the generalized singular values in the high SNR regime. Necessary and sufficient conditions for the secrecy capacity to be zero are provided and further studied when the entries in the channel matrices are sampled i.i.d. CN(0, 1), and the dimensions of the matrices go to infinity. In this regime, we show rather interestingly that an asymmetric allocation of dividing the antennas between the sender and receiver in the ratio 2 : 1 maximizes the number of antennas required by the eavesdropper for the secrecy capacity to be zero.