Security and privacy with K -step opacity for finite automata via a novel algebraic approach

Abstract

As an important secretive attribute, opacity of cyber-physical systems (CPSs) has attracted considerable attention. Existing works on opacity mainly concentrate on the formal language method by assuming that the intruder tracks partial knowledge of observable transitions. In this paper, under the framework of Boolean semi-tensor product (BSTP) of matrix, we extend the verification of opaque property to algebraic mechanisms that have great advantages in problem description and solution. First, we show that how $K$ -step opacity problem of nondeterministic finite automata (NFAs) can be transformed to the construction problem of a polynomial matrix that characterizes the state estimation eavesdropped by malicious intruders within the last $K$ observations. Second, the necessary and sufficient condition of verifying $K$ -step opacity is obtained, which is equivalent to validate the composition of elements within the polynomial matrix. Finally, the effectiveness of this result is demonstrated by an illustrative example. Taking together, the matrix-based method will be useful to deliver a novel theoretical tool for investigating the privacy-preserving problem in the related area of CPSs.