## 5 The Proposed Fortified-Edge 2.0 Framework
15:Step 3: Verifying a Signature
16: Given message \(M\), signature \(\sigma_{b}\), and public key \(pk\):
17: Decode: \(\sigma=\text{Base64 Decode}(\sigma_{b})\). 18: Verify using ECDSA: Verify\((\sigma,M,pk)\). 19: Return verification result (valid or invalid). ```

**Algorithm 5** ECDSA Key Management and Signing

### Analysis of Results

The binary CRP data are converted to their vector form using ML algorithms. The vectorized data are analyzed and visualized first to study the distribution of the feature values across the challenge features and response features. Figure 7 shows the distribution of feature values, showing how challenge features correlate with response features and vectorization. The correlation is estimated using the Pearson Correlation Coefficient (PCC), which measures the linear correlation between each challenge feature and the corresponding response feature. PCC is the ratio between the covariance of two variables and the product of their standard deviations. The result is always between \(-1\) and \(1\). Mathematically, it is represented by Equation (1). \[r=\frac{\text{cov}(X,Y)}{\sigma_{X}\sigma_{Y}} \tag{1}\]

where:

\(\bullet\quad\text{cov}(X,Y)\) is the covariance between variables \(X\) and \(Y\),

\(\bullet\quad\sigma_{X}\) and \(\sigma_{Y}\) are the standard deviations of \(X\) and \(Y\), respectively. Figure 7: Feature value distribution of challenges and responses. From the evaluations, the correlation values obtained are: Average Correlation: 0.2721, Maximum Correlation: 1.0000, and Minimum Correlation: \(-\)0.0964. A correlation average of 0.27 is a low to moderate value, this means the challenge feature and response features are somewhat related but largely independent. This is good for PUF-based security, as a high average correlation would mean that attackers could predict the response from the challenge easily. When selecting the challenges for authentication, those with high correlation can be omitted. A scatter plot of the discrete feature values is shown in Figure 8, which shows that the challenge and response vectors are mostly smaller integer values because of count vectorization frequencies. Many challenge and response pairs seem to occupy a small set of possible values, and a low variation across features aligns with the Pearson correlation average of 0.27 found earlier. For some pairs, as the challenge feature values increase, response feature values also seem to increase, although not strongly linearly. The evaluation of the entropy distribution of the response features is shown in Figure 9. It is seen that most of the responses generated appear in the high entropy region between 1.4 and 1.6, which implies that the feature values are well spread and less predictable. However, a small number of features have lower entropy in the range of 0.6-1.2, and these responses can be omitted from use in the authentication process. The entropy plot also shows that it varies across features, and not every feature is equally random, which is quite expected in real-world PUF response data. Figure 8: Scatter plot of extracted response features. Figure 9: Distribution of entropy for response features. The challenge vectors generated are visualized to study the distribution of features by analyzing selected challenge vectors as shown in Figure 10, which displays five randomly selected challenge vectors. Each line represents a vectorized challenge, showing feature values across indices. High variability is seen in the vectors, suggesting the diversity of challenge representation. Figure 11 shows five randomly selected response vectors; the difference in representation from challenge vectors is because of the PUF mechanism. It is seen that some response vectors have higher peak values, which reflect a stronger feature significance. This study will help select a pair of challenge responses with stronger feature representations for authentication purposes. The results of the code execution are illustrated in Figures 12 and 13.