Measuring Information Leakage in Non-stochastic Brute-Force Guessing

This paper proposes an operational measure of non-stochastic information leakage to formalize privacy against a brute-force guessing adversary. The information is measured by non-probabilistic uncertainty of uncertain variables, the non-stochastic counterparts of random variables. For $X$ that is related to released data $Y$, the non-stochastic brute-force leakage is measured by the complexity of exhaustively checking all the possibilities of the private attribute $U$ of $X$ by an adversary. The complexity refers to the number of trials to successfully guess $U$. Maximizing this leakage over all possible private attributes $U$ gives rise to the maximal (i.e., worst-case) non-stochastic brute-force guessing leakage. This is proved to be fully determined by the minimal non-stochastic uncertainty of $X$ given $Y$, which also determines the worst-case attribute $U$ indicating the highest privacy risk if $Y$ is disclosed. The maximal non-stochastic brute-force guessing leakage is shown to be proportional to the non-stochastic identifiability of $X$ given $Y$ and upper bounds the existing maximin information. The latter quantifies the information leakage when an adversary must perfectly guess $U$ in one-shot via $Y$. Experiments are used to demonstrate the tradeoff between the maximal non-stochastic brute-force guessing leakage and the data utility (measured by the maximum quantization error) and to illustrate the relationship between maximin information and stochastic one-shot maximal leakage.