  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.
