Three New Versions of Strong Opacity Based on the Security Quantification Consideration

Privacy security research is one of the most important issues for the cyber-physical systems, and opacity as an information flow property, provides a new perspective for studying privacy security in complex information interaction scenarios. In the context of the discrete event systems, vital privacy can be modeled as secret states and based on the available observations (i.e., the sequence of observable events), intruders would analyse whether something secret has occurred. In this article, three kinds of strong W-limited opacity, named strong W, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(W,M)$ </tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(W,\infty)$ </tex-math></inline-formula> opacity for the partially observed discrete event system are investigated based on the security quantification consideration. It is assumed that the vicious eavesdropper has a complete knowledge of given systems’ structure and analyses something secret by limited observable information. Meanwhile, in addition to the current and delayed information analysis, predictive information analysis is first added to the strong opacity setting. W denotes the limited observable information and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M~(\infty)$ </tex-math></inline-formula> represents the finite (infinite) predictability. Subsequently, a pivotal technique called labeled W-limited state estimator is constructed, based on which, verifications for the three kinds of strong W-limited opacity are accomplished with the help of secret-labeled states and secret-reaching states. The current work shows that the complexity is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(2^{2|X|}(W+1)|\Sigma _{o}|)$ </tex-math></inline-formula>. Finally, the upper bounds of W and M are also discussed, and it is proved to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$|X|(2^{|X|}-1)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{|X\backslash X_{S}|}-1$ </tex-math></inline-formula>. (X is the state set, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X_{S}$ </tex-math></inline-formula> is the secret state set, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Sigma _{o}$ </tex-math></inline-formula> denotes the observable events set.)