Binary Maximal Leakage

Robinson D. H. Cung, Yanina Y. Shkel, Ibrahim Issa

Abstract: Given two random variables $X$ and $Y$, the maximal leakage $\mathcal{L}(X\rightarrow Y)$ from $X$ to $Y$ was recently proposed as an operational privacy measure. Maximal leakage quantifies the multiplicative increase of the probability of correctly guessing any randomized function of $X$- after observing $\mathrm{Y}$-. This work investigates the properties of maximal leakage in the situation where only certain functions of $X$- are assumed to be of interest to the adversary; specifically, the focus is on measuring maximal leakage with respect to all binary functions of $X$. A definition for binary leakage $\mathcal{L}_{2}^{\ast }(X\rightarrow Y)$ is proposed and a characterization theorem for this new measure is derived. The new privacy measure is shown to satisfy standard properties, such as composition theorems and the data processing inequalities. Many of the stated results naturally extend to $C_{k}^{\ast }(X\rightarrow Y)$ which assumes the function of interest is $k$-valued. Finally, a relation between the binary leakage and the Dobrushin coefficient is established, and possible applications of this relation are explored.