$α$-leakage Interpretation of Rényi Capacity

Abstract: For $\tilde{f}(t) = \exp(\frac{\alpha-1}{\alpha}t)$, this paper shows that the Sibson mutual information is an $\alpha$-leakage averaged over the adversary's $\tilde{f}$-mean relative information gain (on the secret) at elementary event of channel output $Y$ as well as the joint occurrence of elementary channel input $X$ and output $Y$. This interpretation is used to derive a sufficient condition that achieves a $\delta$-approximation of $\epsilon$-upper bounded $\alpha$-leakage. A $Y$-elementary $\alpha$-leakage is proposed, extending the existing pointwise maximal leakage to the overall R\'{e}nyi order range $\alpha \in [0,\infty)$. Maximizing this $Y$-elementary leakage over all attributes $U$ of channel input $X$ gives the R\'{e}nyi divergence. Further, the R\'{e}nyi capacity is interpreted as the maximal $\tilde{f}$-mean information leakage over both the adversary's malicious inference decision and the channel input $X$ (represents the adversary's prior belief). This suggests an alternating max-max implementation of the existing generalized Blahut-Arimoto method.