A Cross Entropy Interpretation of Renyi Entropy for $\alpha$ -leakage

This paper proposes an $\alpha$ -leakage measure for $\alpha\in\lceil 0, \infty)$ by a cross entropy interpretation of Renyi entropy. While Renyi entropy was originally defined as an f -mean for $f(t)=\exp((1-\alpha{)}t)$, we reveal that it is also a $\tilde{f}\cdot$ mean cross entropy measure for $\vec{f}(t)=\exp \left(\frac{1-\alpha}{\alpha} t\right)$. Minimizing this Renyi cross-entropy gives Renyi entropy. This is used to define the prior and posterior uncertainty measures corresponding to the adversary's knowledge gain on sensitive attribute before and after data release, respectively. The a-leakage is proposed as the difference between $\hat{f}$ -mean prior and posterior uncertainty measures, which is exactly the Arimoto mutual information. This not only extends the existing $\alpha$ -leakage from $\alpha\in\lceil 1, \infty)$ to the overall Renyi order range $\alpha\in\lceil 0, \infty)$ in a well-founded way with $\alpha=0$ referring to nonstochastic leakage, but also reveals that the existing maximal leakage is a $\tilde{f}\cdot$ mean of an elementary $\alpha$ -leakage for all $\alpha\in\lceil 0, \infty)$, generalizing the existing pointwise maximal leakage.