## 4 Experimental Results
### Results  
To build the neural fuzzy extractor, a key length of 1024 bits was chosen, providing a sufficient level of cryptographic security and preventing the likelihood of successful cryptanalysis using modern computing power.  
The only configurable parameter of the neural fuzzy extractor is the number of synapses of one neuron \(\eta\). This parameter directly affects the level of the first type of error, the False Rejection Rate (FRR). Increasing the number of synapses allows the neuron to more accurately represent and distinguish the features of training images, which helps to reduce the FRR. Thus, the more the synapses, the better the system copes with identifying legitimate users. Which leads to a decrease in the number of false rejections. The number of neurons similarly affects the FAR. Although FAR is also important, in this context, its reduction is not a priority, since the focus is on minimizing FRR.  
As part of the experimental evaluation, different values of \(\eta\) were tested in order to find the optimal balance between FAR and FRR and the best EER value (Table 2).  
Table 1 shows that the minimum error for the cosine-based neuron network on both datasets is achieved at \(\eta=4\) and is 0.055 for AIC-spkr-130 and 0.15 for RedDots, respectively, which indicates limited effectiveness of measure (3) with increasing model complexity. Using cotangent as the basis of neurons leads to some decrease in authentication errors, especially with a small number of synapses. The lowest error values are recorded at \(\eta=4\): 0.044 for AIC-spkr-130 and 0.061 for RedDots. The combination of neurons based on two measures (cosine and cotangent) showed the best results, demonstrating the minimum error EER \(\approx\) 0.021 for AIC-spkr-130 and EER \(\approx\) 0.032 for RedDots. Obvi  
ously, the combination of these two functionals is optimal in terms of error minimization. This is because of the correct selection of metrics underlying neurons that depend on the correlation of feature pairs. Thus, the hypothesis that the trigonometric functions of cosine and cotangent are better suited for low and highly correlated feature pairs is confirmed.  
It is worth mentioning that increasing the number of synapses in the cases of both datasets does not lead to a significant decrease in EER, but causes its growth. This may indicate that increasing the complexity of the model does not always contribute to improving its characteristics. Thus, increasing the number of neurons and, accordingly, the key length is more advantageous than increasing the number of inputs to the trigonometric neuron.  
The best experimental results presented in the table are shown in Figure 9, which characterize the dependence of FRR and FAR on the Hamming distance (the number of positions in which the \(i\)-th generated code differs from the code associated with the biometric image). The maximum value of the Hamming distance is equal to the key length. The intersection point of the obtained graphs will describe the EER value for a given ratio of neural fuzzy extractor configurations.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Measure** & \multicolumn{2}{l}{**Number of Synapses, \(\eta\)**} & **Dataset** \\ \hline \multicolumn{3}{c}{**AIC-spkr-130**} & **RedDots** \\ \hline \multirow{3}{*}{Cosine} & 4 & 0.055 & 0.15 \\ \cline{2-4}  & 6 & 0.061 & 0.147 \\ \cline{2-4}  & 8 & 0.081 & 0.155 \\ \hline \multirow{3}{*}{Cotangent} & 4 & 0.044 & 0.061 \\ \cline{2-4}  & 6 & 0.049 & 0.117 \\ \cline{2-4}  & 8 & 0.047 & 0.132 \\ \hline \multirow{3}{*}{Cosine + cotangent} & 4 & 0.021 & 0.032 \\ \cline{2-4}  & 6 & 0.061 & 0.069 \\ \cline{2-4}  & 8 & 0.065 & 0.071 \\ \hline \hline \end{tabular}
\end{table}
Table 2: EER values at different numbers of synapses of the fuzzy extractor neuron.  
Figure 9: _Cont._