  Opacity and attack detectability are important properties for any system as
they allow the states to remain private and malicious attacks to be detected,
respectively. In this paper, we show that a fundamental trade-off exists
between these properties for a linear dynamical system, in the sense that if an
opaque system is subjected to attacks, all attacks cannot be detected. We first
characterize the opacity conditions for the system in terms of its weakly
unobservable subspace (WUS) and show that the number of opaque states is
proportional to the size of the WUS. Further, we establish conditions under
which increasing the opaque sets also increases the set of undetectable
attacks. This highlights a fundamental trade-off between security and privacy.
We demonstrate application of our results on a remotely controlled automotive
system.
