## 5 Electrical Characterization
### PUF Metrics
#### 5.3.2 Bit Error Rate  
The BER is a measurement used to quantify the reliability of a PUF. It is calculated by generating a response sequence from a set of challenges on the client device and comparing it to the response sequence generated with the same set of challenges on the server side. The server uses enrollment data to generate responses, while the client uses the PUF to generate real-time responses. The number of discrepancies between the two sequences divided by the size of the response sequence is the BER.  
The BER is similar to the intra-chip Hamming distance (\(HD_{intra}\)), which also quantifies the reliability of a PUF. The \(HD_{intra}\) of a PUF is a random variable describing the distance between two PUF responses from the same PUF using the same challenge. The BER compares the response of a PUF to that generated from enrollments. A formula for the BER is found in Equation (3). In this equation, the \(HD_{diff}\) is the Hamming distance between a cryptographic key generated on the server using a set of challenges and a cryptographic key generated on the client using the same set of challenges.  
\[\text{BER}=HD_{diff}/\text{key size} \tag{3}\]  
Since our responses are in the form of voltages, an important factor in the BER is the settling time allowed for the output signal of the comparator to settle to a state. For most comparators, the slew rate of the output is dependent on the difference between the two inputs; making comparisons between voltages with a small difference requires more time to settle. Theoretically, a higher settling time will lead to more accurate responses as the comparator will have more time to settle into the output, especially with CRPs with small voltage differences.  
To quantify the BER, we generated one million responses with different voltage differences, settling times, and resistive states. The voltage differences ranged from 1 to 12 mV in 1 mV steps, and the settling times ranged from 10 to 50 \(\upmu\)s in 10 \(\upmu\)s steps. Moreover, each test was performed in a high- and low-resistance state. The plot and values for the BER with the voltage difference, settling time, and resistive state are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(HD_{inter}\) & **Low-Resistance State** & **High-Resistance State** \\ \hline Filtered & 0.4997 & 0.4999 \\ Unfiltered & 0.4797 & 0.4849 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The \(HD_{inter}\) of the filtered and unfiltered responses in high- and low-resistance states at 23 \({}^{\circ}\)C. The filtered \(HD_{inter}\) had responses filtered for X states, while the unfiltered \(HD_{inter}\) did not.  
Based on the conducted tests, we observed that, as the voltage difference and settling time increased, the BER decreased, which aligned with our expectations. Surprisingly, the responses generated from a low-resistance state exhibited a noticeably higher error rate compared to those generated from a high-resistance state. This unexpected result suggests that the higher intra-cell variation, measured in millivolts (mV), observed in the low-resistance state contributes to greater instability in the CRPs, resulting in an impact on the BER.