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

翻译：本文通过Rényi熵的交叉熵解释，提出了一种面向$\alpha \in [0,\infty)$的$\alpha$-泄露度量。Rényi熵最初定义为$f$均值（其中$f(t)=\exp((1-\alpha)t)$），但本文揭示它同时也是$\tilde{f}$均值交叉熵度量（其中$\tilde{f}(t)=\exp(\frac{1-\alpha}{\alpha}t)$）。最小化该Rényi交叉熵得到Rényi熵，进而可定义先验与后验不确定性度量，分别对应数据发布前后对手对敏感属性的知识增益。$\alpha$-泄露被提出为$\tilde{f}$均值先验与后验不确定性度量之差，该差恰好等于Arimoto互信息。这一框架不仅将现有$\alpha$-泄露从$\alpha\in[1,\infty)$严谨延拓至完整Rényi阶范围$\alpha\in[0,\infty)$（其中$\alpha=0$对应非随机泄露），更揭示了现有最大泄露本质上是所有$\alpha\in[0,\infty)$下初等$\alpha$-泄露的$\tilde{f}$均值，从而推广了现有的逐点最大泄露概念。