  We study the problem of securely communicating a sequence of information bits
with a client in the presence of multiple adversaries at unknown locations in
the environment. We assume that the client and the adversaries are located in
the far-field region, and all possible directions for each adversary can be
expressed as a continuous interval of directions. In such a setting, we develop
a periodic transmission strategy, i.e., a sequence of joint beamforming gain
and artificial noise pairs, that prevents the adversaries from decreasing their
uncertainty on the information sequence by eavesdropping on the transmission.
We formulate a series of nonconvex semi-infinite optimization problems to
synthesize the transmission strategy. We show that the semi-definite program
(SDP) relaxations of these nonconvex problems are exact under an efficiently
verifiable sufficient condition. We approximate the SDP relaxations, which are
subject to infinitely many constraints, by randomly sampling a finite subset of
the constraints and establish the probability with which optimal solutions to
the obtained finite SDPs and the semi-infinite SDPs coincide. We demonstrate
with numerical simulations that the proposed periodic strategy can ensure the
security of communication in scenarios in which all stationary strategies fail
to guarantee security.
