Probabilistic Opacity for Markov Decision Processes

Opacity is a generic security property, that has been defined on (non probabilistic) transition systems and later on Markov chains with labels. For a secret predicate, given as a subset of runs, and a function describing the view of an external observer, the value of interest for opacity is a measure of the set of runs disclosing the secret. We extend this definition to the richer framework of Markov decision processes, where non deterministic choice is combined with probabilistic transitions, and we study related decidability problems with partial or complete observation hypotheses for the schedulers. We prove that all questions are decidable with complete observation and $\omega$-regular secrets. With partial observation, we prove that all quantitative questions are undecidable but the question whether a system is almost surely non opaque becomes decidable for a restricted class of $\omega$-regular secrets, as well as for all $\omega$-regular secrets under finite-memory schedulers

Probabilistic Opacity in Refinement-Based Modeling

Given a probabilistic transition system (PTS) $\cal A$ partially observed by an attacker, and an $\omega$-regular predicate $\varphi$over the traces of $\cal A$, measuring the disclosure of the secret $\varphi$ in $\cal A$ means computing the probability that an attacker who observes a run of $\cal A$ can ascertain that its trace belongs to $\varphi$. In the context of refinement, we consider specifications given as Interval-valued Discrete Time Markov Chains (IDTMCs), which are underspecified Markov chains where probabilities on edges are only required to belong to intervals. Scheduling an IDTMC $\cal S$ produces a concrete implementation as a PTS and we define the worst case disclosure of secret $\varphi$ in ${\cal S}$ as the maximal disclosure of $\varphi$ over all PTSs thus produced. We compute this value for a subclass of IDTMCs and we prove that refinement can only improve the opacity of implementations

Probabilistic Disclosure: Maximisation vs. Minimisation

We consider opacity questions where an observation function provides to an external attacker a view of the states along executions and secret executions are those visiting some state from a fixed subset. Disclosure occurs when the observer can deduce from a finite observation that the execution is secret, the epsilon-disclosure variant corresponding to the execution being secret with probability greater than 1 - epsilon. In a probabilistic and non deterministic setting, where an internal agent can choose between actions, there are two points of view, depending on the status of this agent: the successive choices can either help the attacker trying to disclose the secret, if the system has been corrupted, or they can prevent disclosure as much as possible if these choices are part of the system design. In the former situation, corresponding to a worst case, the disclosure value is the supremum over the strategies of the probability to disclose the secret (maximisation), whereas in the latter case, the disclosure is the infimum (minimisation). We address quantitative problems (comparing the optimal value with a threshold) and qualitative ones (when the threshold is zero or one) related to both forms of disclosure for a fixed or finite horizon. For all problems, we characterise their decidability status and their complexity. We discover a surprising asymmetry: on the one hand optimal strategies may be chosen among deterministic ones in maximisation problems, while it is not the case for minimisation. On the other hand, for the questions addressed here, more minimisation problems than maximisation ones are decidable

Opacity for Linear Constraint Markov Chains

On a partially observed system, a secret ϕ is opaque if an observer cannot ascertain that its trace belongs to ϕ. We consider specifications given as Constraint Markov Chains (CMC), which are underspec-ified Markov chains where probabilities on edges are required to belong to some set. The nondeterminism is resolved by a scheduler, and opacity on this model is defined as a worst case measure over all implementations obtained by scheduling. This measures the information obtained by a passive observer when the system is controlled by the smartest sched-uler in coalition with the observer. When restricting to the subclass of Linear CMC, we compute (or approximate) this measure and prove that refinement of a specification can only improve opacity