## 6 Conclusions  
We proposed a novel TAPUF design using two MRAM devices. This design uses extracted resistances to produce reliable responses at 33.33 kHz and characterizes analog and digital responses. We enrolled 30 MRAM devices in high- and low-resistance states at varying temperatures and characterized the analog responses. Additionally, we used real-time responses to calculate the \(HD_{inter}\) and BER of the TAPUF design.  
The electrical characterization of the PUF responses provided valuable insights into the PUF's performance. One important finding was that the CMOS series resistance significantly impacted the PUF responses. However, this issue was successfully addressed by adjusting the PUF responses to ensure they all had the same median value. This modification enabled the MRAM TAPUF to maintain its entropy while accommodating a large number of CRPs (\(n^{2}\)).  
Furthermore, the inter-cell to intra-cell variation ratio of the PUF was excellent in both states, indicating consistent behavior across cells. Additionally, minimal difference was observed between the two states. When comparing the two states, the high-resistance state stood out with better metrics, particularly regarding the BER. At a challenge-pair voltage difference of 10 mV and a temperature of 23 \({}^{\circ}\)C, the high-resistance state demonstrated exceptional performance with an inter-distance value of 0.4998, an entropy value of 0.9999, and a BER below 10\({}^{-6}\).  
Furthermore, this PUF design exhibited reliability across a temperature range of 0 \({}^{\circ}\)C to 80 \({}^{\circ}\)C. When enrollments were conducted at the same temperature as used for key generation, the BER remained extremely low. However, as the temperature difference between the enrollment temperature and the key generation temperature increased, an increase in the BER was observed. It is important to note that, even with significant temperature variations, the BER remained within acceptable limits, with a value of 9 \(\times\) 10\({}^{-5}\) at 0 \({}^{\circ}\)C and 80 \({}^{\circ}\)C.  
Additionally, the entropy and \(HD_{inter}\) remained close to the ideal values under these temperature conditions. This indicated that the system maintained a reliable level of performance even under varying temperature conditions.  
Compared to previously proposed MRAM PUF designs, this particular design demonstrated comparable, if not better, PUF metrics, as shown in Table 6. These findings highlight the promising performance and reliability of this MRAM PUF design, suggesting its potential superiority over existing designs.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**PUF Metrics** & \begin{tabular}{c} **MRAM** \\ **TAPUF** \\ \end{tabular} & \begin{tabular}{c} **STT MRAM** \\ **PUF [30]** \\ \end{tabular} & \
\begin{tabular}{c} **Geometry-Based** \\ **MRAM PUF [33]** \\ \end{tabular} &
\begin{tabular}{c} **Write Current STT** \\ **MRAM PUF [17]** \\ \end{tabular} \\ \hline CRPs & \(n^{2}\) & \(n/2\) & \(n\) & \(n\) \\ BER or \(HD_{intra}\) & \(<\)10\({}^{-6}\) * & \(6.6\times 10^{-6}\) & 0.0225 & 0.05–0.07 \\ \(HD_{inter}\) & 0.4999 * & 0.501 & 0.47 & 0.499–0.501 \\ Entropy density \(H(X)\) & 0.99999 * & 0.985 & 0.99 & 0.994–0.999 \\ \hline \hline \end{tabular}
\end{table}
Table 6: PUF metrics of previous MRAM PUFs compared to our proposed design. \(n\) = the number of MRAM cells on a single MRAM device. Metrics denoted with * are for responses with X states filtered out.