  Opacity is a confidentiality property that characterizes the non-disclosure
of specified secret information of a system to an outside observer. In this
paper, we consider the enforcement of opacity within the discrete-event system
formalism in the presence of multiple intruders. We study two cases, one
without coordination among the intruders and the other with coordination. We
propose appropriate notions of opacity corresponding to the two cases,
respectively, and propose enforcement mechanisms for these opacity properties
based on the implementation of insertion functions, which manipulates the
output of the system by inserting fictitious observable events whenever
necessary. The insertion mechanism is adapted to the decentralized framework to
enforce opacity when no coordination exists. Furthermore, we present a
coordination and refinement procedure to synthesize appropriate insertion
functions to enforce opacity when intruders may coordinate with each other by
following an intersection-based coordination protocol. The effectiveness of the
proposed opacity-enforcement approaches is validated through illustrative
examples.
