# Challenge-Response Pair Mechanisms and Multi-Factor Authentication Schemes to Protect Private Keys
## 4 Experimental Section and Analysis
### Combination of Multiple Factors
During our experiment, we found that, sometimes, the challenges could not  
Figure 10: Seed error distribution by fragmentation level, showing little to no impact of the fragmentation process. In this analysis, we want to verify that the error rates are well distributed within the cryptographic keys, and that the key fragmentation process is not negatively impacted by a poor distribution of the erratic bits. produce a stable key from the crypto table due to high number of flaky \"X\" cells. As the process is completely random, we could not control the process other than by increasing the number of challenges. Finally, we gathered 1970 verification test results for the False Reject Test (FRR), which verifies the right person, and 1968 verification test results for the False Acceptance Rate Test (FAR), which tries to verify with the wrong person. Table 1 represents the results of the signing verification for both FRR and FAR tests. In Table 2, we analyze the verification percentages based on gender, age range, and skin tone enrollment. As shown in Figure 11, different skin tones can represent different ethnicities. We aimed to determine whether there was any bias in our protocol toward various ethnicities. The results indicate that our protocol is highly robust and does not exhibit any bias toward gender or ethnicity. As observed, increasing the fragmentation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Type** & **Fragmentation** & **True (\%)** & **False (\%)** \\ \hline \multirow{4}{*}{FRR (False Reject Rate)} & 1 & 77.67 & 22.33 \\ \cline{2-4}  & 2 & 96.70 & 3.30 \\ \cline{2-4}  & 4 & 99.24 & 0.76 \\ \cline{2-4}  & 8 & 100 & 0 \\ \cline{2-4}  & 16 & 100 & 0 \\ \hline \multirow{4}{*}{FAR (False Acceptance Rate)} & 1 & 0 & 100 \\ \cline{2-4}  & 2 & 0.25 & 99.75 \\ \cline{1-1} \cline{2-4}  & 4 & 1.01 & 98.98 \\ \cline{1-1} \cline{2-4}  & 8 & 7.89 & 92.13 \\ \cline{1-1} \cline{2-4}  & 16 & 21.88 & 78.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: FRR and FAR verification percentages for different fragmentation levels. With higher fragmentation levels, the protocol can handle higher rates of errors. Therefore, the false rejection rate decreases with higher fragmentation, while the false acceptance rate moves in the opposite direction. Figure 11: Diversity based on gender, age, and ethnicity. level leads to achieving 100% fragmentation for FRR. This occurs because the RBC error correction method relies on an exhaustive key search, requiring all possible combinations to be tested. It is easier to detect bit flips in a 16-bit seed than in a 256-bit seed. Due to the 5 s limit imposed on the RBC for key recovery, it becomes more challenging to find the correct key at higher fragmentation levels with increased error rates. The RBC time cap not only prevents brute-force attacks but also reduces the false acceptance rate. In addition to this, we see that fragmentation level 4 would be perfect for this protocol. If we look at the FRR test, it fails to recognize the right person only 0.76% of the time, and 99.24% of the time it is able to determine the correct person. Looking at the FRR, with the