# Mathematical Foundations and Implementation of CONIKS Key Transparency
## 3 Results
### Performance Metrics and Evaluation  
The performance metrics and evaluation section focuses on analyzing the data collected from the experimental setup to assess the efficiency and effectiveness of the CONIKS system. Key metrics include the time required for hash computation, tree construction, and digital signature operations. These metrics provide insight into the system's operational efficiency and its ability to handle various cryptographic tasks. By comparing the experimental data with theoretical expectations, we can evaluate how well the CONIKS system performs relative to established cryptographic standards and identify areas where performance meets or exceeds expectations. This section also includes a detailed evaluation of the results, with visual representations such as tables and graphs to illustrate the findings. The analysis covers aspects such as the scalability of Merkle tree construction with increasing leaf nodes, the impact of key size on digital signature performance, and the overall alignment of experimental results with theoretical models (see Table 1). This comprehensive evaluation helps to understand the practical implications of the system's performance and highlights any discrepancies or unexpected results that may warrant further investigation. Table 2 provides a comparison of digital signature operations across different key sizes, including RSA key sizes of 2048 and 4096 bits, as well as ECDSA with a 256-bit key. The data reveal significant differences in key generation, signing, and verification times. Specifically, RSA keys with larger bit sizes result in longer key generation and signature times, reflecting the increased computational complexity. In contrast, ECDSA with a smaller 256-bit key shows faster operation times for both signing and verification, illustrating a favorable trade-off between security and performance. This comparison emphasizes the impact of key size on the efficiency of digital signature operations and the importance of choosing an appropriate key size based on the specific security and performance requirements of the application. Figure 3 shows the impact of key size on the time required for generating keys, signing messages, and verifying signatures. RSA key sizes of 2048 and 4096 bits are compared with ECDSA using a 256-bit key. The graph highlights the trade-offs between security \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Number of Leaves** & **Hash Computation Time (ms)** & **Tree Construction Time (ms)** & **Root Hash Calculation Time (ms)** \\ \hline
10 & 2.5 & 15 & 5 \\
50 & 8.0 & 45 & 12 \\
100 & 15.0 & 85 & 22 \\
500 & 60.0 & 400 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Merkle tree construction time. \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Key Size (bits)** & **Key Generation Time (ms)** & **Signing Time (ms)** & **Verification Time (ms)** \\ \hline
2048 & 1500 & 10 & 12 \\
4096 & 4000 & 20 & 25 \\
256 (ECDSA) & 500 & 8 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Digital signature operations. and performance, with larger keys resulting in increased time for key generation and signature operations. In evaluating the efficiency of cryptographic operations, comparing experimental results with theoretical expectations is crucial. Table 3 presents a comparative analysis of time complexities for various operations, including Merkle tree construction, hash computation, and digital signature processes. The theoretical time complexities are contrasted with the experimental data obtained from our implementation, showing deviations that provide insight into the performance and accuracy of the system.