Undetectable Attacks on Boolean Networks

In this paper, driven by the ever-increasing cyber-security threats, we study the undetectable attack problems for Boolean networks (BNs), which model distributed systems with a limited capacity of storage and bandwidth of communication. Given a consistent monitor, undetectable attacks are formalized for BNs as those do not yield an output sequence out of the nominal output sequence set. By the graph-theoretic approach, undetectable attacks are characterized by a reachability problem of a directed cycle in the augmented transition graph. On the other hand, the algebraic approach also derives a necessary and sufficient criterion for undetectable attacks by testing the existence of the nonzero elements in the constructed matrix. While all these derived results are only computationally efficient for relatively small-size BNs. The detection of attack signals is indeed NP-hard. In other words, there is no polynomial-time algorithm to check the detectability of an attack signal or an attack node set unless NP=P.