On the Secrecy Capacity of Rank-Deficient Compound Wiretap Channels

The secrecy capacity of compound wiretap channels is studied. Using the known lower bounds for non-degraded discrete memoryless channels and establishing the converse, the secrecy capacity is established under the less capable and noisier conditions. The compound capacity is shown to be equal to the worst-case channel capacity when there exists a saddle- point in the mutual information difference. Earlier results on the secrecy capacity of compound Gaussian MIMO wiretap channels under the spectral norm constraint are extended to the case of rank-deficient eavesdroppers (which may model a massive MIMO base station) without the degradedness assumption. Its compound secrecy capacity is established in a closed form and the optimal signaling is identified: the compound capacity equals the worst-case channel capacity thus establishing the saddle-point property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is omnidirectional (i.e. isotropic over the sub-space spanned by the active eigenmodes of the legitimate channel). The case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is also studied and its compound secrecy capacity is established.