## 5 Electrical Characterization
### Response Characteristics  
To gain a comprehensive understanding of the electrical characteristics of the MRAM devices' relative resistances, we conducted a thorough analysis involving 30 MRAM devices. In these tests, we examined various parameters, including the mean, intra-cell variation, and inter-cell variation of the cells' resistance values. Additionally, we evaluated the resistances by testing both the low-resistance and high-resistance states. For each MRAM device, we performed ten separate reads, each consisting of 10,240 words. The average of these ten reads was considered the cell's resistance. The intra-cell variation is a measurement that measures the variation in a cell's resistance over multiple queries. It was calculated by taking the standard deviation of the ten reads. Inter-cell variation measures the variations in cells within the same device. We assessed the inter-cell variation in the PUF by calculating the standard deviation between the average resistances of MRAM cells in the same device. Figure 3: MRAM hardware circuitry on Nucleo-144. Series Resistance  
As discussed in Section 2.3, it is evident that the CMOS circuitry employed for accessing MRAM cells has a significant impact on the resistance obtained from these cells. Upon enrolling multiple devices, we made an intriguing discovery: MRAM cells situated on the same bit-line (such as 0, 1, 2,... 15) exhibited similar resistances. This implies that a cell located on bit \(b\) at address \(a\) will have resistances similar to one on the same bit \(b\) even at a different address \(c\). This influence can lead to responses that are less \"random\" with a less-than-normal distribution. Addressing the influence of the bit-line circuitry on the resistance of each MRAM cell is an important consideration. One approach to tackle this issue is to treat each bit-line as a separate PUF, ensuring that the resistances are independent for each bit-line. However, this approach is limited as it significantly reduces the number of available challenge responses for the PUF and requires carefully selecting MRAM devices with similar relative resistances. Alternatively, the resistance of all bit-lines can be adjusted to ensure they possess the same median value. This adjustment is necessary to mitigate the impact of the varying bit-line series resistance and maintain consistency in the characteristics of the MRAM cells across different addresses. This method requires additional circuitry and enrollment time but it does not negatively impact the number of CRPs, nor does it require careful device pairing. Furthermore, since the resistances of specific bit-lines are indiscriminately shifted, the overall entropy of the MRAM cells remains unaffected. Since the relative resistance of cells is measured in the form of a voltage drop, a voltage value can be added to shift the values so that all cells from different bit-lines have the same median values. This adjustment involves selectively modifying the resistances from specific bit-lines to shift their median values. In this study, we achieved this manipulation by incorporating a digital-to-analog converter (DAC) connected to the voltage reference pins of our amplifiers. Initially, we conducted an enrollment process with a pair of MRAM devices to identify the bit-line with the highest median voltage value. This bit-line was chosen as the reference. Subsequently, all cells from other bit-lines were adjusted upward to achieve an equal median voltage value as the reference bit-line. These adjustments were then saved on the client's device for future utilization. Figure 4 compares the relative resistances of a single MRAM device with and without the applied shift in the bit-line median values. The inter-cell to intra-cell ratio is the inter-cell variation divided by the intra-cell variation. Ideally, we want the inter-cell to intra-cell ratio to be as high as possible to  
Figure 4: Relative adjusted bit-line medians. minimize the BER. This means we want to maximize inter-cell variation and minimize intra-cell variation.