Non-uniform distributions in quantitative information-flow

Quantitative information-flow analysis (QIF) determines the amount of information that a program leaks about its secret inputs. For this, QIF requires an assumption about the distribution of the secret inputs. Existing techniques either consider the worst-case over a (sub-)set of all input distributions and thereby over-approximate the amount of leaked information; or they are tailored to reasoning about uniformly distributed inputs and are hence not directly applicable to non-uniform use-cases; or they deal with explicitly represented distributions, for which suitable abstraction techniques are only now emerging. In this paper we propose a novel approach for a precise QIF with respect to non-uniform input distributions: We present a reduction technique that transforms the problem of QIF w.r.t. non-uniform distributions into the problem of QIF for the uniform case. This reduction enables us to directly apply existing techniques for uniform QIF to the non-uniform case. We furthermore show that quantitative information flow is robust with respect to variations of the input distribution. This result allows us to perform QIF based on approximate input distributions, which can significantly simplify the analysis. Finally, we perform a case study where we illustrate our techniques by using them to analyze an integrity check on non-uniformly distributed PINs, as they are used for banking.