  In this paper, we consider a remote state estimation problem in the presence
of an eavesdropper. A smart sensor takes measurement of a discrete linear
time-invariant (LTI) process and sends its local state estimate through a
wireless network to a remote estimator. An eavesdropper can overhear the sensor
transmissions with a certain probability. To enhance the system privacy level,
we propose a novel encryption strategy to minimize a linear combination of the
expected error covariance at the remote estimator and the negative of the
expected error covariance at the eavesdropper, taking into account the cost of
the encryption process. We prove the existence of an optimal deterministic and
Markovian policy for such an encryption strategy over a finite time horizon.
Two situations, namely, with or without knowledge of the eavesdropper
estimation error covariance are studied and the optimal schedule is shown to
satisfy the threshold-like structure in both cases. Numerical examples are
given to illustrate the results.
