# Chaos-Based Physical Unclonable Functions
## 4 Testing and Results
Consequently, the minimum number of bit-strings is two, whereas the maximum results from acceptable entropy uncertainty, and as a consequence, the acceptable critical \(m\) fluctuations. For this reason, we suggest to either generate a fixed number of bit-strings (e.g., \(N=10\)) or to stop when the first bit-string differs from the others. In each of the procedures, there is always a possibility that a rare event for lower \(m\) may cause disturbance resulting in an incorrect \(m\) value. The error can be avoided by increasing the number of bit-strings, but it may be more efficient to ignore the result (detect the error with a simple cyclic redundancy check code) and repeat the key generation procedure. The number of bit-strings required to generate the \(m\) values for all of the examples presented in Figure 9 varies from 2 to 6--on average, 3.04 bit-strings per one generation. In order to evaluate limitations of the proposed solution, we estimated critical \(m\)-values for eight Artix (Xilinx) and eight Cyclone 5 (Altera/Intel (Santa Clara, CA, USA)) devices, where each consisted of three independent PUFs. This way, we were able to estimate a histogram of critical \(m\)-values (keys),  
Figure 9: Phase detector output bit-strings (\(s_{N}[m]\)) and extracted PUF keys. shown in Figure 10. The measured inter-class randomness of PUF estimated with standard deviation was \(\sigma=52.6\). Despite the limited probe size (16 devices \(\times\) 3 PUFs), one can see in Figure 10 that the \(m\) distribution is asymmetric and the most likely \(m\) values are located in the 10-50 range (obtained also in simulation). The asymmetric \(m\) distribution results strictly from the influence of noise on the trajectory of chaotic system. In such a system, the location of subsequent points (see Figure 3) is strongly affected by the slight phase fluctuations (resulting from noise processes) during subsequent system states. In other words, the longer circuit operates, the higher the number of transitions near \(C\) and \(C^{\prime}\) is. Therefore, it is unlikely to maintain the same deterministic behavior for high \(m\)-values in subsequent realizations by the system. In the proposed PUF solution, the chaotic operation results from \(C\) and \(C^{\prime}\) ranges of operation, which ensure \(1<k<2\) inclination in the chaotic map. The tolerances present in the system affect both the inclination of \(C\) and \(C^{\prime}\) ranges (sections of chaotic map) and the length of these sections. Therefore, tolerances affect the probability of entering \(C\) and \(C^{\prime}\) ranges of operation, and, in turn, the probability of entering different (random) trajectory paths. The other parameter vital to PUFs is its reliability, which identifies the intra-class randomness. The 48 PUF instances used for randomness extraction in Figure 10 were subjected to challenge-response operation multiple times in the same operating conditions. This way, the \(m\) error distribution was obtained, which is shown in Figure 11. The mean value of obtained distribution \(\mu\approx 0\), whereas the standard deviation \(\sigma=6.94\). It is worth mentioning that \(m\) error mainly results from the non-monotonic character of \(H(m)\) dependence (see Figure 8), which affects the estimate of critical \(m\)-value based on the threshold level. Nevertheless, over 80% of intra-class \(m\) error corresponds to 2-bit Hamming distance, whereas the maximum Hamming distance corresponds to 3-bits. Figure 11: Intra-class PUF (\(m\)) distribution. Reliability of proposed PUF measured on 16 devices (eight Xilinx Artix and eight Intel Cyclone V) with three PUFs each (48 instances). Figure 10: Inter-class PUF (\(m\)) distribution. Randomness of proposed PUF was measured on 16 devices (eight Xilinx Artix and eight Intel Cyclone V) with three PUFs each (48 instances). We have also investigated the influence of temperature on the critical \(m\)-value.