# CNT-PUFs: Highly Robust and Heat-Tolerant Carbon-Nanotube-Based Physical Unclonable Functions +
## 5 Evaluation of the Fabricated CNT-PUFs
### Evaluation of CNT-PUFs under Normal Conditions
#### 5.2.4 Robustness  
When prompted multiple times by an identical challenge \(C\), the robustness of a PUF \(p\) guarantees that the response is always \(R_{p}(C)\). To determine the robustness, we calculated the intra-device Hamming distance between PUF responses to the same challenge. As mentioned in the previous section, the cells of the CNT-PUFs were fabricated in such a way that conductive and non-conductive cells can be distinguished from each other exceptionally well. Figure 10 plots the relative proportion of unstable responses that occurred in 28 repeated measurements of three different 144-bit CNT-PUFs for all three channel widths as a function of the threshold value selected in each case. It can be seen that a wide range of thresholds can be set in order to ensure stable responses: Values for \(I_{th}\) between 7 pA and 100 pA led to almost no misclassified responses, while \(I_{th}\) values outside this range led to unstable responses, as is evident in Figure 10. If the threshold was set in the former range, the maximum (worst) intra-device Hamming distance was \(0.03=3\%\), which corresponds to a maximum of five erroneous bits. This result indicated that, even in the worst case, within the aforementioned range, fully stable responses of at least 139 bits can be achieved, which means that cryptographic tokens, such as keys, of a length of 128 bits can be easily derived from the examined CNT-PUFs. Overall, based on the analysis of the 28 repeated measurements, an average intra-device Hamming distance of 0.06% can be achieved with a threshold \(I_{th}=20\) pA. The number of measurement cycles represents a compromise aimed at capturing a statistically profound database to allow a suitable PUF evaluation within a reasonable measurement time. Moreover, considering the high ratio between conductive and non-conductive cells, as illustrated in Figure 4, it is unlikely that additional factors, such as noise introduced by additional measurements, have an impact on the classification result, assuming that the threshold \(I_{th}\) is positioned within the window of high stability, indicated by the dashed grey lines in Figure 10. \begin{table}
\begin{tabular}{l c c l} \hline \hline \multirow{2}{*}{**Statistical Test**} & \begin{tabular}{c} **Average** \\ _p-Value_ \\ \end{tabular} & **Passed/Total** & \begin{tabular}{c} **Input Length Recommendation** \\ **Test Parameters** \\ \end{tabular} \\ \hline Frequency (Monobit) & 0.28 & 25/30 & \(n\geq 100\) bits \\ \hline Frequency Test within a Block & 0.32 & 28/30 & \begin{tabular}{c} \(n\geq 100\) bits \\ \(M=20\) bits (block length) \\ \end{tabular} \\ \hline Runs Test & 0.43 & 30/30 & \(n\geq 100\) bits \\ \hline Test for the Longest Run of Ones in a Block & 0.41 & 27/30 & \begin{tabular}{c} \(n\geq 128\) bits \\ \(m=8\) bits (block length) \\ \end{tabular} \\ \hline Binary Matrix Rank Test & / & / & \(n\geq\)**38,912 bits** \\ \hline Discrete Fourier Transform (Spectral) Test & / & / & \(n\geq\)**1000 bits** \\ \hline Non-overlapping Template Matching Test & 0.86 & 30/30 & \(m=9\) bits (template size) \\ \hline Overlapping Template Matching Test & / & / & \(n\geq\)**10\({}^{b}\) bits** \\ \hline [PERSON]’s “Universal Statistical” Test & / & / & \(n\geq\)**387,840 bits** \\ \hline Linear Complexity Test & / & / & \(n\geq\)**10\({}^{b}\) bits** \\ \hline Serial Test & 0.36 & 30/30 & \
\begin{tabular}{c} \(m<\log_{2}n]-2\) \\ \(m=4\) bits (block length) \\ \end{tabular} \\ \hline Approximate Entropy Test & 0.33 & 27/30 &
\begin{tabular}{c} \(m<\log_{2}n]-5\) \\ \(m=2\) bits (block length) \\ \end{tabular} \\ \hline Cumulative Sums (Cusum) Test & 0.30 & 26/30 & \(n\geq\)**100 bits** \\ \hline Random Excursions Test & / & / & \(n\geq\)**10\({}^{b}\) bits** \\ \hline Random Excursions Variant Test & / & / & \(n\geq\)**10\({}^{b}\) bits** \\ \hline \hline \end{tabular}
\end{table}