  Opacity is a security property formalizing the information leakage of a
system to an external observer, namely intruder. The conventional opacity that
has been studied in the Discrete Event System (DES) literature usually assumes
passive intruders, who only observe the behavior of the system. However, in
many cybersecurity concerns, such as web service, active intruders, who are
capable of influencing the system's behavior beyond passive observations, need
to be considered and defended against. We are therefore motivated to extend the
opacity notions to handle active intruders. For this, we model the system as a
non-deterministic finite-state transducer. It is assumed that the intruder has
a full knowledge of the system structure and is capable of interacting with the
system by injecting different inputs and observing its responses. In this
setup, we first introduce reactive current-state opacity (RCSO) notion
characterizing a property that the system does not leak its secret state
regardless of how the intruder manipulates the system behavior. We furthermore
extend this notion to language-based and initial-state reactive opacity
notions, and study the relationship among them. It turns out that all the
proposed reactive opacity notions are equivalent to RCSO. We therefore focus on
RCSO and study its verification problem. It is shown that the RCSO can be
verified by constructing an observer automaton.
