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Honeyword is a representative ``honey" technique to detect intruders byluring them with decoy data. This kind of honey technique blends a primaryobject (from distribution $P$) with decoy samples (from distribution $Q$). Inthis research, we focus on two key Honeyword security metrics: the flatnessfunction and the success-number function. Previous researchers are engaged indesigning experimental methods to estimate their values. We've derivedtheoretical formulas on both metrics of the strongest $\mathcal{A}$ using theoptimal guessing strategy, marking a first in the field.  The mathematical structures of these metrics are intriguing: the flatnessfunction 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 acumulative distribution function derived from $P$ and $Q$. This formulaprovides a criterion to compare different honey distributions: the one withsmaller $M$ and $b$ is more satisfactory. The mathematical structure of thesuccess-number function is a series of convolutions with beta distributionkernels: $\lambda_U(i)=U\sum_{j=1}^{i}\int_{\frac{1}{k}}^{1}\frac{\phi(x)}{1-\phi(x)} \tbinom{U-1}{j-1} x^{U-j}(1-x)^{j-1}dx$, where $U$ isthe number of users in the system and $\phi(x)$ is a monotonically increasingfunction. For further elaboration, we made some representative calculations.Our findings offer insights into security assessments for Honeyword and similarhoney techniques, contributing to enhanced security measures in these systems.

Reveal the Mathematical Structures of Honeyword Security Metrics