Honeyword is a representative ``honey" technique to detect intruders by luring them with decoy data. This kind of honey technique blends a primary object (from distribution $P$) with decoy samples (from distribution $Q$). In this research, we focus on two key Honeyword security metrics: the flatness function and the success-number function. Previous researchers are engaged in designing experimental methods to estimate their values. We've derived theoretical formulas on both metrics of the strongest $\mathcal{A}$ using the optimal guessing strategy, marking a first in the field. The mathematical structures of these metrics are intriguing: the flatness function has an expression as $\epsilon(i)=\sum_{j=1}^{i}\int_{0}^{+\infty}\tbinom{k-1}{j-1} f(x)G^{k-j}(x)(1-G(x))^{j-1}dx$. In particular, the most important one, $\epsilon(1)$ is $\frac{1}{k}(M-\int_{0}^{M}G^k(x)dx)+b$, where $M=\max_{x: Q(x)\neq 0}\frac{P(x)}{Q(x)}$, $b=\sum_{x: Q(x)=0}P(x)$, and $G$ is a cumulative distribution function derived from $P$ and $Q$. This formula provides a criterion to compare different honey distributions: the one with smaller $M$ and $b$ is more satisfactory. The mathematical structure of the success-number function is a series of convolutions with beta distribution kernels: $\lambda_U(i)=U\sum_{j=1}^{i}\int_{\frac{1}{k}}^{1} \frac{\phi(x)}{1-\phi(x)} \tbinom{U-1}{j-1} x^{U-j}(1-x)^{j-1}dx$, where $U$ is the number of users in the system and $\phi(x)$ is a monotonically increasing function. For further elaboration, we made some representative calculations. Our findings offer insights into security assessments for Honeyword and similar honey techniques, contributing to enhanced security measures in these systems.

翻译：蜜词是一种代表性的“蜜”技术，通过诱饵数据吸引入侵者以实现检测。该蜜技术将主要对象（来自分布$P$）与诱饵样本（来自分布$Q$）混合。本研究聚焦于蜜词的两项关键安全度量：平坦度函数与成功次数函数。以往研究者致力于设计实验方法估算其数值，而我们则推导了最强攻击者$\mathcal{A}$采用最优猜测策略时这两项度量的理论公式，这在领域内尚属首次。这些度量的数学结构颇具趣味：平坦度函数可表示为$\epsilon(i)=\sum_{j=1}^{i}\int_{0}^{+\infty}\tbinom{k-1}{j-1} f(x)G^{k-j}(x)(1-G(x))^{j-1}dx$。其中最重要的$\epsilon(1)$值为$\frac{1}{k}(M-\int_{0}^{M}G^k(x)dx)+b$，此处$M=\max_{x: Q(x)\neq 0}\frac{P(x)}{Q(x)}$，$b=\sum_{x: Q(x)=0}P(x)$，$G$为由$P$和$Q$导出的累积分布函数。该公式提供了比较不同蜜分布的准则：$M$和$b$较小的分布更为理想。成功次数函数的数学结构则是一系列包含Beta分布核的卷积：$\lambda_U(i)=U\sum_{j=1}^{i}\int_{\frac{1}{k}}^{1} \frac{\phi(x)}{1-\phi(x)} \tbinom{U-1}{j-1} x^{U-j}(1-x)^{j-1}dx$，其中$U$为系统用户数，$\phi(x)$为单调递增函数。为深入阐述，我们进行了若干代表性计算。研究结果为蜜词及类似蜜技术的安全评估提供了见解，有助于增强这些系统的安全措施。