  This paper studies remote state estimation in the presence of an
eavesdropper. A sensor transmits local state estimates over a packet dropping
link to a remote estimator, while an eavesdropper can successfully overhear
each sensor transmission with a certain probability. The objective is to
determine when the sensor should transmit, in order to minimize the estimation
error covariance at the remote estimator, while trying to keep the eavesdropper
error covariance above a certain level. This is done by solving an optimization
problem that minimizes a linear combination of the expected estimation error
covariance and the negative of the expected eavesdropper error covariance.
Structural results on the optimal transmission policy are derived, and shown to
exhibit thresholding behaviour in the estimation error covariances. In the
infinite horizon situation, it is shown that with unstable systems one can keep
the expected estimation error covariance bounded while the expected
eavesdropper error covariance becomes unbounded. An alternative measure of
security, constraining the amount of information revealed to the eavesdropper,
is also considered, and similar structural results on the optimal transmission
policy are derived. In the infinite horizon situation with unstable systems, it
is now shown that for any transmission policy which keeps the expected
estimation error covariance bounded, the expected amount of information
revealed to the eavesdropper is always lower bounded away from zero. An
extension of our results to the transmission of measurements is also presented.
