# A Privacy-Preserving Record Linkage Method Based on Secret Sharing and Blockchain
## 4 Methods
### Output and Audit Module
#### 4.4.4 Record Keeping and Provenance
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Participants = 3** & **Participants = 5** & **Participants = 7** & **Participants = 9** \\ \hline \multirow{3}{*}{SSB-PPRL} & 0.77 (Mod-1) & 0.73 (Mod-1) & 0.67 (Mod-1) & 0.61 (Mod-1) \\  & 0.72 (Mod-2) & 0.67 (Mod-2) & 0.61 (Mod-2) & 0.54 (Mod-2) \\  & 0.66 (Mod-3) & 0.61 (Mod-3) & 0.56 (Mod-3) & 0.48 (Mod-3) \\ \hline \multirow{3}{*}{Han} & 0.71 (Mod-1) & 0.66 (Mod-1) & 0.61 (Mod-1) & 0.55 (Mod-1) \\  & 0.64 (Mod-2) & 0.61 (Mod-2) & 0.56 (Mod-2) & 0.51 (Mod-2) \\  & 0.57 (Mod-3) & 0.55 (Mod-3) & 0.48 (Mod-3) & 0.44 (Mod-3) \\ \hline \multirow{3}{*}{Yao} & 0.87 (Mod-1) & 0.83 (Mod-1) & 0.78 (Mod-1) & 0.73 (Mod-1) \\  & 0.81 (Mod-2) & 0.77 (Mod-2) & 0.72 (Mod-2) & 0.68 (Mod-2) \\  & 0.74 (Mod-3) & 0.71 (Mod-3) & 0.66 (Mod-3) & 0.63 (Mod-3) \\ \hline \multirow{3}{*}{Vatsalan} & 0.75 (Mod-1) & 0.73 (Mod-1) & 0.66 (Mod-1) & 0.61 (Mod-1) \\  & 0.71 (Mod-2) & 0.67 (Mod-2) & 0.61 (Mod-2) & 0.55 (Mod-2) \\ \cline{1-1}  & 0.65 (Mod-3) & 0.61 (Mod-3) & 0.54 (Mod-3) & 0.49 (Mod-3) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: F-score values for three perturbed datasets changing with participant numbers. Figure 9: The F-score variation with the number of participants in the three perturbed datasets. #### 6.2.3 Security Evaluation

In this section, we set the number of secret shares t = 10, meaning each Bloom filter is decomposed into 10 Boolean shares, distributed across 10 computing nodes. We evaluated the ability of colluding nodes to reconstruct the Bloom filter by varying the total number of nodes N (50, 100, 200) and the number of colluding nodes c (starting from 10 and increasing to 0.8 \(\times\) N). The experimental results are shown in Table 6 below. The experimental results demonstrate how the total number of nodes \(N\) and the number of colluding nodes \(c\) affect the reconstruction rate under a fixed number of secret shares \(t=10\). The reconstruction rate, defined as the probability of colluding nodes recovering a Bloom filter, depends on the likelihood of collecting all \(t\) shares and guessing the correct permutation. The theoretical reconstruction rate is given by

\[\text{Reconstruction rate}=\frac{\binom{c}{t}}{\binom{N}{t}}\times\frac{1}{t!} \tag{19}\]

When c < t, the reconstruction rate is 0, as insufficient shares are collected. In the experiments, t = 10, with permutation possibilities t! = 3,628,800, significantly reducing the attack success rate. When c = t = 10 (i.e., the number of colluding nodes equals the number of secret shares), the theoretical rate is extremely low (e.g., \(2.683\times 10^{-17}\) for N = 50). Experimental values are 0, which is consistent with such a low theoretical probability of successful reconstruction. When c \(\geq\) t, the theoretical reconstruction rate increases with c. However, it remains very small for lower c values and becomes more significant (though still low) for higher c values (e.g., 80% node collusion). For example, for N = 50, c = 40 (80% collusion), the corrected theoretical rate is \(2.274\times 10^{-8}\), with an experimental value of \(2\times 10^{-7}\). For N = 200, c = 160 (80% collusion), the corrected theoretical rate is \(2.601\times 10^{-8}\), with an experimental value of \(1\times 10^{-7}\). Permutation difficulty ensures security. Increasing N (total nodes) reduces the theoretical reconstruction rate, e.g., for c = 40, the rate drops from \(2.274\times 10^{-8}\) at N = 50 to \(1.04\times 10^{-14}\) at N = 200, due to lower collection probability. Increasing c (colluding nodes) raises the theoretical rate. The permutation factor t! ensures these theoretical rates remain very low (e.g., in the order of \(10^{-8}\) or lower for the parameters tested). The experimental rates observed are in the order of \(10^{-7}\) for high collusion scenarios.