# Protecting Private Communications in Cyber-Physical Systems through Physical Unclonable Functions
## 4 Performance Evaluation and Results
### Results  
In this section, results of the described experiments in the previous section are presented and discussed. In order to remove from the simulation results (as much as possible) fluctuations in the simulation execution process caused by exogenous variables (e.g., delays operations performed by the operating systems), the average of all obtained results from the 12 performed simulation repetitions are employed to calculate the final results. Table 1 shows the result from the first experiment. As can be seen, all tests were successful guaranteeing the randomness of the proposal. Besides, obtained \(p\)-values are very high (in several tests around the unit), so the randomness level of the evaluated key generator output is also very high. Figure 10 shows the results from the second experiment. The Bit Error Rate (BER) is evaluated according to the standard expression (25), being \(N_{error}\) the number of bit errors in each time period. \[\text{BER }(\%)=\frac{N_{error}}{N_{bits}} \tag{25}\]  
As can be seen, the bit error rate (BER) is always below 0.01%. BER evolution follows a sigmoid curve, where there is an initial period (around 150 h) when BER is stable. Later, BER starts going down until it reaches the value \(5\times 10^{-6}\%\) (approximately) and gets stable another time. This behavior is coherent with the PUF behavior. In fact, as the proposed PUF is not generating directly the key stream, but a seed for a PNRG (which only changes when long-term changes in the PUF response are detected), and as PUF evolves and changes quite slowly, BER may present very low values. At first, PUF starts generating signals which may still present some very small fluctuations caused by the transitory period in the magnetic field and induction and PUF response. As time passes, and EB effect starts making the resonance frequency evolve, the small fluctuations tend to disappear and BER goes down. However, there is a minimum value for BER due to numerical processing, etc. This also explains why obtained values are very low, as only errors caused by the encryption mechanism are considered. Besides, only a few bits in packets are corrupted, when fluctuations in the magnetic fields caused very fast and transitory drifts in the PUF behavior. No structural errors are detected. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test** & \(p\)**-Value** & **Result** \\ \hline Runs & 0.973 & Successful \\ Frequency Monobit & 0.974 & Successful \\ Overlapping Template Matching & 0.319 & Successful \\ Frequency Test within a Block & 0.654 & Successful \\ Longest Run of Ones in a Block & 0.807 & Successful \\ Universal & 0.388 & Successful \\ Linear Complexity & 0.309 & Successful \\ Binary Matrix Rank & 0.419 & Successful \\ Serial & 0.999 & Successful \\ Discrete Fourier Transform & 0.215 & Successful \\ Random Excursions Test * & 0.461 & Successful \\ Random Excursions Variant Test * & 0.399 & Successful \\ Approximate Entropy & 0.921 & Successful \\ Non-Overlapping Template Matching & 0.979 & Successful \\ Cumulative Sums & 0.955 & Successful \\ \hline \hline \end{tabular}  
* Several different values for \(p\)-value where obtained. Result is the mean value.  
\end{table} 
Table 1: NIST test results. First experiment. On the other hand, it can be also seen, as the number of EB devices in the PUF is increased, the initial BER is also higher. In fact, as more different magnetic materials are considered, the power and energy associated to fluctuations in magnetic materials are also higher. Consequently, more errors are induced. In any case, the obtained BER values are similar to those associated to traditional encryption and communication systems [47]. It is important to note that, contrary to other works where real implementations are employed to evaluate BER [34], in this case we are using a simulation scenario where only some effects are considered (in particular, only effects related to EB phenomenon are analyzed). For example, fluctuations in the electrical challenges are not considered.