Secure Degrees of Freedom for the MIMO Wire-Tap Channel with a Multi-Antenna Cooperative Jammer

In this paper, a multiple antenna wire-tap channel in the presence of a multi-antenna cooperative jammer is studied. In particular, the secure degrees of freedom (s.d.o.f.) of this channel is established, with $N-{t}$ antennas at the transmitter, $N-{r}$ antennas at the legitimate receiver, and $N-{e}$ antennas at the eavesdropper, for all possible values of the number of antennas, $N-{c}$ , at the cooperative jammer. In establishing the result, several different ranges of $N-{c}$ need to be considered separately. The lower and upper bounds for these ranges of $N-{c}$ are derived, and are shown to be tight. The achievability techniques developed rely on a variety of signaling, beamforming, and alignment techniques, which vary according to the (relative) number of antennas at each terminal and whether the s.d.o.f. is integer valued. Specifically, it is shown that, whenever the s.d.o.f. is integer valued, Gaussian signaling for both transmission and cooperative jamming, linear precoding at the transmitter and the cooperative jammer, and linear processing at the legitimate receiver, are sufficient for achieving the s.d.o.f. of the channel. By contrast, when the s.d.o.f. is not an integer, the achievable schemes need to rely on structured signaling at the transmitter and the cooperative jammer, and joint signal space and signal scale alignment. The converse is established by combining an upper bound, which allows for full cooperation between the transmitter and the cooperative jammer, with another upper bound which exploits the secrecy and reliability constraints.