# FeRHA: Fuzzy-Extractor-Based RF and Hardware Fingerprinting Two-Factor Authentication
## 6 Results and Performance Analysis
### Fuzzy Extractor Performance  
To evaluate the reliability of our implementation, we conducted tests using the fuzzy extractor in combination with the generated PUF responses and RFF. The implementation diagram of the fuzzy extractor utilizing the BCH code and syndrome concept (\(N=63\)) is illustrated in Figure 2. Our assessment of the proposed fuzzy extractor's performance is based on Table 2. This table showcases all conceivable combinations of message length \(L\) for BCH code, where the code-word length is fixed to 63 bits. Notably, the variable \(t\) signifies the maximum number of bits that the fuzzy extractor can correct for the corresponding key size. Given that FeRHA aims to mitigate the impact of noise, we have evaluated the performance when the fuzzy extractor is employed in conjunction with the PUF by considering three temperature settings: 70 \({}^{\circ}\)C, 30 \({}^{\circ}\)C, and 15 \({}^{\circ}\)C. To clarify, the PUF is used to map 64-bit challenge bit-strings to 64-bit responses at a baseline temperature of 30 \({}^{\circ}\)C. To factor in the effect of noise, the same challenge bit-strings were applied to generate 64-bit responses at a lower temperature of 15 \({}^{\circ}\)C and at a higher temperature of 70 \({}^{\circ}\)C.  
Figure 8: The PUF modeling accuracy when FeRHA is is subjected to a multi-link attack. Figure 9: The result of the analysis of FeRHA using OFMC. The notion of PUF reliability refers to how consistent its response is to a given challenge under varying operating conditions, e.g., fluctuations in temperature or voltage supply. The reliability of a PUF can be calculated using (13) and (14) below. By applying these equations, the fuzzy extractor effectively could mitigate the temperature noise in the PUF, leading to a reliability of 99.88% when the temperature is 70 \({}^{\circ}\)C and 99.8% for 15 \({}^{\circ}\)C. The key size considered in this experiment is 45. \[HD_{INTRA}=\frac{1}{m}\sum_{j=1}^{m}\frac{HD(R_{j}(n),R_{j}(n))}{n}\times 100\% \tag{13}\]  
From (13), the PUF reliability can be written as:  
\[Reliability=100\%-HD_{INTRA} \tag{14}\]  
where \(HD\) is the hamming distance, \(m\) denotes the number of calculated samples/chips, and \(n\) is the number of bits. We have repeated the same experiment using \(RFF\) data. According to ORACLE's published dataset [62], the noise is modeled as a Gaussian variable. We note that noise can cause alterations in the demodulated IQ sample pattern, resulting in a slightly different pattern from the original noise-free IQ sample pattern. To address this issue, we leveraged a fuzzy extractor to minimize the noise to below \(-13\) dB. This action ensures that the Earth Mover's Distance (EMD) between the original and altered patterns remains below the defined ORACLE threshold of 0.1. EMD was computed using the formula expressed in Equation (15). EMD is a widely used metric for measuring similarities between two multi-dimensional distributions. To clarify, consider two sets of points in a two-dimensional metric space, denoted by \(\mathbb{M}^{2}\). Let \(I\) and \(J\) be two subsets of \(\mathbb{M}^{2}\), each of equal size, denoted by \(|I|=|J|\). Let \(F\) be the set of all possible bijections, which are one-to-one and onto mappings from \(I\) to \(J\). The EMD between \(I\) and \(J\) can be calculated using this set \(F\) as follows:  
\[EMD(I,J)=\min_{f\in F}\sum_{i\in I}||i-f(i)||. \tag{15}\]  
In other words, EMD is calculated by identifying the smallest possible sum of Euclidean distances between points in two given sets (\(I\) and \(J\)), while considering all valid bijections between them (\(f:A\to J\)). EMD provides a reliable metric for measuring similarity accurately. A smaller EMD value indicates a greater level of similarity between the two patterns, and vice versa. In our experiments, EMD has consistently stayed below 0.1, reflecting a highly reliable RFF, which demonstrates the effectiveness of the employed fuzzy extractor in mitigating the noise. Overall, the reliability values discussed above