Studying Maximum Information Leakage Using Karush-Kuhn-Tucker Conditions

When studying the information leakage in programs or protocols, a natural question arises: “what is the worst case scenario?”. This problem of identifying the maximal leakage can be seen as a channel capacity problem in the information theoretical sense. In this paper, by combining two powerful theories: Information Theory and Karush-Kuhn-Tucker conditions, we demonstrate a very general solution to the channel capacity problem. Examples are given to show how our solution can be applied to practical contexts of programs and anonymity protocols, and how this solution generalizes previous approaches to this problem.

💡 Research Summary

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The paper addresses the problem of quantifying the worst‑case information leakage of programs and communication protocols by casting it as a channel capacity problem in the sense of information theory. A secret input S and an observable output O are modeled as a probabilistic channel C(o|s). The amount of leakage is the mutual information I(S;O), and the goal is to find the input distribution p(s) that maximizes this quantity under a set of constraints that reflect realistic security policies (e.g., some inputs may be forbidden, certain outputs may be disallowed, probability normalization, non‑negativity, etc.).

Traditional approaches use Lagrange multipliers to handle the normalization constraint, but they struggle with inequality constraints and with the non‑convex nature of the mutual‑information objective. The authors therefore bring in the Karush‑Kuhn‑Tucker (KKT) conditions, which provide necessary (and under mild regularity conditions, sufficient) optimality conditions for problems with both equality and inequality constraints. By formulating the leakage maximization as

 maximize L(p) = I(S;O) = Σ_{s,o} p(s) C(o|s) log