## 4 Data Set Analysis for Simulation  
In this paper, we utilized a data set obtained from a pm-PUF circuit for simulation [29]. As shown in Figure 5(a), the pm-PUF circuit consists of a 64 \(\times\) 64 memristor crossbar array with row and column control blocks made up of multiplexers, random pulse generators, row and column decoder blocks, and voltage sense amplifiers. The line resistance is 1.2 \(\Omega\) and each memristor in the array follows the electrical characteristics of a Ta\({}_{2}\)O\({}_{5}\) memristor device. The characteristics were modeled in [58] and coded in Verilog-A for HSPICE circuit simulation. The pm-PUF applies a bank design in which several unit blocks are integrated to form a large structure and therefore can generate a multi-bit response in a single cycle. The pm-PUF bank design is described in Figure 5(b). The entire PUF is composed of \(N\) small PUF blocks. Each small block receives a corresponding challenge bit string and generates a response bit string. In the simulated circuit, a 64 \(\times\) 64 array consists of 8 blocks of 8 \(\times\) 64 cells. We maximized resistance variation by applying various pulses with varied pulse widths and used this random variation as an entropy source. The pulse amplitude was set to 1.5 V, which is higher than the positive threshold voltage (1 V). The read voltage was set to 0.75 V, which is lower than the threshold voltage. The entire challenge consisted of 30 bits with 6 bits used as row select bits and 24 bits used as column select bits, allowing one memristor cell to be selected in each block. Two bits of the state of selected cell are output as its response. A total of eight blocks generates a 16-bit response. Therefore, the CRP data set consisted of 30-bit challenge and 16-bit response pairs. We evaluated the CRP data set in terms of randomness, diffuseness, uniqueness, and steadiness. Mathematical expressions of these performance metrics are described in detail in [29,41]. The notation used in this paper is used in [29,41]. * **Randomness**: Let \(p_{n}\) be the relative frequency of 1 appearing in all the response bits generated in device \(n\). Then, \(p_{n}\) is given by \[p_{n}=\frac{1}{K\cdot T\cdot L}\cdot\sum_{k=1}^{K}\sum_{t=1}^{T}\sum_{l=1}^{L}b _{n,k,t,l}\] (3) Then, the device randomness \(H_{n}\) is defined as follows [29,41]: \[H_{n}=-\text{log}_{2}max(p_{n},1-p_{n})\] (4) When \(p_{n}\) = 0.5, \(H_{n}\) takes the highest value 1 and is considered as ideal value. We randomly applied 5500 different challenges and investigated 5500 sets of 16-bit length responses. The randomness of the PUF is 0.9828 (98.28%), which is slightly lower than the ideal value of 1. The probability for a 0 response is 50.6% and for a 1 is 49.4% as shown in Figure 7a. * **Diffuseness**: Let \(d_{n,l}\) be the sum of Hamming distance of the possible bit-combinations in response [29,41]: \[d_{n,l}=\sum_{i=1}^{K-1}\sum_{j=i+1}^{K}(b_{n,i,l}\oplus b_{n,j,l})\] (5) Then, since each bit of \(L\)-bit responses is independent of other bits, the upper bound of the sum of Hamming distance of the possible response combination is \(L\cdot(K/2)^{2}\). Therefore, the device diffuseness \(D_{n}\) is defined as follows [29,41]. \[D_{n}=\frac{1}{L}\sum_{l=1}^{L}\frac{d_{n,l}}{(K/2)^{2}}\] (6) When the sum of Hamming distance reaches the upper bound, \(D_{n}\) takes the highest value 1 and is considered as the ideal value. We randomly applied 5500 different challenges and investigated 5500 sets of 16-bit length responses. The diffuseness of the PUF is 0.9871, which is close to the  
Figure 6: Simplified pulsewidth-based memristive PUF (pm-PUF) architecture [29]. (**a**) Simplified pulsewidth-based memristive PUF (pm-PUF) architecture. (**b**) A concept of bank design for pm-PUF. ideal value of 1. The distribution of the intra-HD among the obtained responses is shown in Figure 7b; it is mean is 49.29%. * **Uniqueness**: The upper bound of the sum of Hamming distance of the possible responses among all devices is \(L\cdot(N/2)^{2}\).