## 4 Simulation Results and Discussions
### Uniqueness  
Another important metric for PUF is uniqueness, which measures the capability of a PUF instance to distinguish itself from other PUF instances. The uniqueness can be measured by calculating the inter-die Harming Distance (HD). Ideally, the responses generated from two PUF instances by applying the same challenge should have half of the bits being different. Thus, the uniqueness should be 50%. Let \(R_{u}\) and \(R_{v}\) be the \(n\)-bit responses of two different chips, \(u\) and \(v\), to the same challenge \(C\), and the uniqueness \(U\) for \(m\) chips is expressed as [25]  
\[U=\frac{2}{m(m-1)}\sum_{u=1}^{m-1}\sum_{v=u+1}^{m}\frac{\mathrm{HD}(R_{u},R_{v} )}{n}\times 100\%. \tag{18}\]  
To measure the uniqueness of the designed SPUF, two hundred 100-bit-long responses are generated and their fractional HDs are calculated. The distribution is shown in Figure 15. The uniqueness is calculated to be 0.4999, which closely approaches the ideal value of 0.5. The fitted Gaussian curve of the histogram for fractional HDs has a mean value of \(\mu=0.4999\) and a standard deviation of \(\sigma=0.0503\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Description** & **Design** & **X** & **M** & **Pr(RF)** & **Pr(WF)** & **Pr(REWF)** & **BER** & **Area** & **Leakage** & **Power** \\ \hline Single-mode PUF & 1 & (180, 90, 135) & \(N.A.\) & 0.131 & 0.136 & 0.249 & \(1.37\times 10^{-7}\) & 0.45 & 25.2 & 140 \\ Single-mode memory & 2 & (500, 250, 375) & \(N.A.\) & 0.022 & 0.007 & 0.068 & \(3.29\times 10^{-2}\) & 1.28 & 73.2 & 405 \\ \hline \multirow{3}{*}{Dual-mode SPUF} & 3 & (202, 101, 152) & \(N.A.\) & 0.118 & 0.125 & 0.229 & \(9.73\times 10^{-7}\) & 0.49 & 28.5 & 158 \\  & 4 & (202, 101, 152) & \(W.M\) & 0.021 & 0.125 & 0.143 & \(9.73\times 10^{-7}\) & 0.49 & 28.5 & 154 \\  & 5 & (202, 101, 152) & \(W.S\) & 0.027 & 0.036 & **0.062** & \(9.73\times 10^{-7}\) & 0.49 & 19.8 & 143 \\  & 6 & (202, 101, 152) & \(N.S\) & 0.118 & 0.004 & 0.122 & \(9.73\times 10^{-7}\) & 0.49 & 28.5 & 155 \\ \hline Dual-mode RO PUF & 7 & – & – & – & 0.51 & \(1.46\times 10^{-10}\) & – & – & – \\ \hline \hline \end{tabular}  
* The transistor width is searched in the range of \([45,500]\) nm, and the ratio \(W_{\text{MI}}:W_{\text{MI}}:W_{\text{MI}}:W_{\text{MI}}\) is the same for the three designs so that no memory failure occurs with nominal parameter values; The power refers to the sum of dynamic power during read and write operations; The data of RO PUF was from [24]. The failure probability was calculated with 0.5% frequency tolerance and the BER was obtained with the same ECC as SPUF; The units for area, leakage and power are \(\mu\)m\({}^{2}\), nA and \(\mu\)W, respectively; Reliability can be computed from BER by 1 – BER; WLM, DVS, NBL and ABB refer to Word-Line Voltage Modulation, Dynamic Voltage Scaling, Negative Bit-Line and Adaptive Body Bias, respectively. RF and WF refer to read failure and write failure, respectively. PUF refers to physical unclonable function. SPUF refers SRAM-based PUF. RO refers to ring oscillator.  
\end{table}
Table 2: Simulation results \(\emptyset V_{DD}=0.9\) V, temperature \(=25\)\({}^{\circ}\)C.  
Figure 15: Frequency distribution of the simulated fractional inter-die Harming Distance (HD).