# Chaos-Based Physical Unclonable Functions
## 4 Testing and Results
For this purpose, 100 realizations (runs), of \(m=250\) steps long each (within each PUF instance) were used to estimate \(H(m)\) dependence for two operating temperatures (i.e., 260 K and 300 K). Each PUF response requires multiple runs (realizations \(n\)) of chaotic system iterations (\(m\)). Therefore, each PUF response requires \(m\times r\) iterations, where each \(m\)'th iteration length results from the corresponding SCRO frequency (1). It is obvious that each PUF response requires a \(\frac{2 mr}{f_{1}+f_{2}}\) interval; therefore, when 400 MHz average frequency of FPGAs is assumed, a single PUF response requires at least 25 \(\upmu\)s without post-processing (e.g., entropy calculation). Each 25 \(\upmu\)s experiment (single PUF response extraction) was repeated multiple times in hardware (FPGA), in order to verify the PUF stability and reliability. The example \(H(m)\) results for two FPGA instances at 260 K and 300 K are shown in Figure 12. One can see that, despite the temperature change, the critical \(m\)-value (PUF response) is rather invariant over temperature (Figure 12a-d). It turns out that the temperature mainly affects the amplitude of local \(H\) fluctuations; however, the global moment of rapid entropy increase remains constant in a particular device. The results in Figure 12 show the temperature influence on the \(H(m)\) dependence, whereas, to evaluate the temperature impact on multiple instances, we needed to extract critical \(m\)-values (PUF responses) from multiple PUFs implemented in eight Artix and eight Cyclone V devices in 300 K and 260 K, respectively. The PUF responses (\(\{m\mid H(m)=0.5\}\)) corresponding to particular instances were used to build a regression plot in Figure 13. One can see that PUF responses can be easily approximated with a linear function. The inclination coefficient (0.87) in Figure 13 proves that PUF responses (\(m\)) in higher temperature (300 K) are slightly shorter (smaller) than in the case of lower temperature (260 K). These results are in accordance with Monte Carlo analysis discussed in Section 3. Figure 12: Example entropy vs. \(m\) relationships (100 circuit runs) of two circuits K1 (**a,b**) and K2 (**c,d**) at 300 K (**a,c**) and at 260 K (**b,d**). In order to evaluate the PUF performance using common metrics (see, for example, [7; 49]), we have combined three \(m\)-values (each represented by 8 bits) into one 24-bit PUF key. The differences between such keys (bit sequences) were measured with the use of Hamming distance--the number of different bits between two keys. Figure 14 shows normalized probability of occurrence of particular Hamming distances between keys in percentages for both inter-class (keys compared between different chips) and intra-class (keys repeatedly sampled within the same chip) distributions. The influence of the temperature change on such keys can be observed in Figure 15. Based on these results (and following common PUF metrics [7; 49]), we were able to estimate the basic PUF keys parameters:  
* Uniqueness: 41.16%,
* Reliability: 91.33%,
* Uniformity: 36.50%. It is worthwhile to mention that such a chaotic circuit amplifies microscopic differences between instances in the way that even the device manufacturer cannot predict the results. Moreover, since every change in the implementation results in different keys, the number of PUF challenges within even a simple device strives for infinity. An invasive attempt to measure transistors and paths geometry, material heterogeneity as well as many other device parameters most likely would change the unique trajectory and destroy the keys. On the other hand, even if someone had succeeded, it would have been impossible to reconstruct an instance or to create simulation model that would have resulted in exactly the same chaotic trajectory and the same cryptographic keys. Figure 13: Regression of generated \(m\) values (PUF) keys in the same instances under different operating temperature.