## 5 Electrical Characterization
### PUF Metrics  
#### 5.3.1 Inter-Distance  
Correlation between MRAM devices can cause security risks, as the responses of one chip can give similar responses to those of other chips. To ensure no correlation between chips, we quantified the inter-chip Hamming distance (\(HD_{inter}\)), also known as the inter-distance metric. \(HD_{inter}\) is a random variable that describes the distance between two PUF responses to the same challenge for two different PUFs. The inter-distance value of a PUF is also a vital PUF metric as it allows one to quantify the uniqueness of a PUF [39]. Specifically, it represents the Hamming distance (\(HD_{i,j}\)) between a stream of responses from two distinct PUFs when subjected to the same challenges. The resulting distance is then divided by the total number of responses to produce the \(HD_{inter}\). Ideally, the \(HD_{inter}\) of a PUF should be \(\mathbf{0.50}\). The equation for the \(HD_{inter}\) is shown in Equation (2).  
\[HD_{inter}=HD_{i,j}/\text{stream size} \tag{2}\]  
To quantify the \(HD_{inter}\) value of our PUF design, we carried out an experiment involving the generation of 15 response sequences. Each sequence consisted of a response obtained from a unique PUF configuration, where a configuration was defined as any unique _pair_ of MRAM devices. These responses were acquired from the PUF in both its high- and low-resistance states. Each response sequence was \(2^{20}\) bits long (1 Mb response), ensuring a substantial amount of data for analysis.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**MRAM Response Statistics** & **High–Resistance State** & **Low-Resistance State** \\ \hline Maximum & \(T\times 1.203\) & \(L\times 1.3666\) \\ Average & \(T\) & \(L=T\times 0.8521\) \\ Minimum & \(T\times 0.8914\) & \(L\times 0.8980\) \\ Range & \(T\pm 0.3116\) & \(L\times 0.4677\) \\ Average bit-line inter-cell variation & \(T\times 0.005794\) & \(L\times 0.005261\) \\ Average intra-cell variation & \(T\times 0.000231\) & \(L\times 0.000203\) \\ Inter-cell to intra-cell & 25.03 & 25.88 \\ variation ratio & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relative analog response statistics for 100 MRAM devices at 23 \({}^{\circ}\)C.  
Furthermore, as part of our analysis, we generated two sets of sequences: one with unfiltered challenge-response pairs (CRPs) and another with CRPs filtered for X CRPs. We then calculated the average \(HD_{inter}\) for each set.  
For the unfiltered CRPs, the average \(HD_{inter}\) was found to be **0.4797** in a low-resistance state and **0.4849** in a high-resistance state. On the other hand, for the filtered CRPs, where CRPs classified as X were removed, the average \(HD_{inter}\) was determined to be **0.4997** in a low-resistance state and **0.4999** in a high-resistance state.  
The results of these tests can be found in Table 2. From these results, we can safely conclude that the responses obtained from both states exhibited similar \(HD_{inter}\) values, indicating that the resistive state of the MRAM devices does not significantly impact the uniqueness of the PUF responses.  
Filtering ternary states yielded a more desirable \(HD_{inter}\) value. This indicates that including ternary states in the MRAM PUF responses can introduce a bias towards 0 or 1, reducing the overall uniqueness. In other words, the presence of ternary states in the responses can potentially compromise the distinctiveness of the PUF, highlighting the importance of eliminating or minimizing such states to achieve higher uniqueness and reliability, as well as to minimize error.