[Paper Review] K-Step Opacity in Discrete Event Systems: Verification, Complexity, and Relations

Jiří Balun, Tomáš Masopust|arXiv (Cornell University)|Sep 5, 2021

Petri Nets in System Modeling23 references4 citations

TL;DR

This paper presents a novel algorithm for verifying K-step opacity in discrete event systems with time complexity O((n + m)2^n), independent of K, significantly improving upon prior methods whose complexity scales with K. It further introduces polynomial-time, neutral-state-free transformations between K-step, current-state, and infinite-step opacity, enabling efficient interconversion without increasing system size or compromising determinism.

ABSTRACT

Opacity is a property expressing whether a system may reveal its secret to a passive observer (an intruder) who knows the structure of the system but has a limited observation of its behavior. Several notions of opacity have been studied, including current-state opacity, K-step opacity, and infinite-step opacity. We study K-step opacity that generalizes both current-state opacity and infinite-step opacity, and asks whether the intruder cannot decide, at any time, whether or when the system was in a secret state during the last K observable steps. We design a new algorithm deciding K-step opacity the complexity of which is lower than that of existing algorithms and that does not depend on K. We then compare K-step opacity with other opacity notions and provide new transformations among the notions that do not use states that are neither secret nor non-secret (neutral states) and that are polynomial with respect to both the size of the system and the binary encoding of K.

Motivation & Objective

• To develop a more efficient algorithm for verifying K-step opacity in discrete event systems, with complexity independent of K.
• To eliminate the dependency of existing verification algorithms on the value of K, which previously led to exponential growth in computational cost.
• To provide polynomial-time transformations between K-step opacity, current-state opacity, and infinite-step opacity that do not rely on neutral states (states neither secret nor non-secret).
• To preserve determinism during transformations, ensuring practical applicability to real-world systems.
• To establish a unified framework for comparing and converting among different opacity notions using only the system size and the binary encoding of K.

Proposed method

• Design a new verification algorithm based on observer construction with time complexity O((n + m)2^n), where n is the number of states and m is the number of transitions, independent of K.
• Construct a family of automata A_K of size polynomial in log K, where the observer has a unique path of non-marked states of length exactly K, and all other states are marked.
• Use a base-4 representation of K to decompose the problem into smaller, manageable sub-automata A_{i,i} of size polynomial in i.
• Connect multiple copies of these sub-automata using a new event c, ensuring a unique path of length K through non-marked states.
• Introduce a deterministic construction via unobservable events to transform any NFA into a DFA while preserving K-step opacity and secret state structure.
• Prove that the determinization process preserves K-step opacity by maintaining the reachability and secret status correspondence between original and transformed states.

Experimental results

Research questions

• RQ1 Can K-step opacity be verified with complexity independent of K, and if so, how can this be achieved?
• RQ2 What is the minimal system size required to realize a unique observer path of length K consisting only of non-marked states?
• RQ3 Can transformations between K-step, current-state, and infinite-step opacity be performed in polynomial time without introducing neutral states?
• RQ4 Is it possible to preserve determinism during transformations between different opacity notions while maintaining correctness?
• RQ5 How does the binary encoding of K influence the complexity of transformations between opacity types?

Key findings

• The proposed algorithm for verifying K-step opacity runs in O((n + m)2^n) time, which is independent of K and strictly better than previous methods that scale with K.
• A family of automata A_K was constructed such that its observer has a unique path of non-marked states of length exactly K, with size polynomial in log K.
• The construction uses base-4 decomposition of K to generate sub-automata A_{i,i}, each of which is of size polynomial in i, and composes them via c-transitions.
• Transformations between K-step, current-state, and infinite-step opacity are polynomial in both the system size and the binary encoding of K, without using neutral states.
• The determinization procedure preserves K-step opacity by introducing unobservable events and new states that maintain secret state reachability and observer behavior.
• The transformations are proven correct and preserve determinism, ensuring that the resulting systems remain suitable for practical verification and deployment.