We address the problem of characterising the security of a program against unauthorised information flows. Classical approaches are based on non-interference models which depend ultimately on the notion of process equivalence. In these models confidentiality is an absolute property stating the absence of any illegal information flow. We present a model in which the notion of non-interference is approximated in the sense that it allows for some exactly quantified leakage of information. This is characterised via a notion of process similarity which replaces the indistinguishability of processes by a quantitative measure of their behavioural difference. Such a quantity is related to the number of statistical tests needed to distinguish two behaviours. We also present two semantics-based analyses of approximate non-interference and we show that one is a correct abstraction of the other.

We address the problem of characterising the security of a program against unauthorised information flows. Classical approaches are based on non-interference models which depend ultimately on the notion of process equivalence. In these models confidentiality is an absolute property stating the absence of any illegal information flow. We present a model in which the notion of non-interference is approximated in the sense that it allows for some exactly quantified leakage of information. This is characterised via a notion of process similarity which replaces the indistinguishability of processes by a quantitative measure of their behavioural difference. Such a quantity is related to the number of statistical tests needed to distinguish two behaviours. We also present two semantics-based analyses of approximate non-interference and we show that one is a correct abstraction of the other.

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Abstract. We introduce a characterisation of probabilistic transition systems (PTS) in terms of linear operators on some suitably defined vector space representing the set of states. Various notions of process equivalences can then be re-formulated as abstract linear operators related to the concrete PTS semantics via a probabilistic abstract interpretation. These process equivalences can be turned into corresponding approximate notions by identifying processes whose abstract operators "differ" by a given quantity, which can be calculated as the norm of the difference operator. We argue that this number can be given a statistical interpretation in terms of the tests needed to distinguish two behaviours.

In this paper we lay the semantic basis for a quantitative security analysis of probabilistic systems by introducing notions of approximate confinement based on various process equivalences. We re-cast the operational semantics classically expressed via probabilistic transition systems (PTS) in terms of linear operators and we present a technique for defining approximate semantics as probabilistic abstract interpretations of the PTS semantics. An operator norm is then used to quantify this approximation. This provides a quantitative measure ε of the indistinguishability of two processes and therefore of their confinement. In this security setting a statistical interpretation is then given of the quantity ε which relates it to the number of tests needed to breach the security of the system.

We show how the framework of probabilistic abstract interpretation can be applied to statically analyse a probabilistic version of the λ-calculus. The resulting analysis allows for a more speculative use of its outcomes based on the consideration of statistically defined quantities. After introducing a linear operator based semantics for our probabilistic λ-calculus Λp, and reviewing the framework of abstract interpretation and strictness analysis, we demonstrate our technique by constructing a probabilistic (first-order) strictness analysis for Λp.

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