# Pitfall of the Strongest Cells in Static Random Access Memory Physical Unclonable Functions
## 4 Proposed New Fuzzy Extractor
### ERFE Architecture
### Leakage of Strongest Cells

The samples of the responses are sent into the PUF, and the enrollment operation is performed 100 times to obtain 100 sets of helper data, which are denoted as \(H_{1}(x),H_{2}(x),H_{3}(x)\ldots H_{100}(x)\). Choosing response with maximum \(q_{y0}\) as response to be cloned, and the helper data (\(H_{1}(x)\)) generated by the chosen response and the empty cells \(e_{z}\) consist of a clone matrix \(E\). The _Diff_ is sequentially calculated according to Equation (3). The parameter \(t\) is two corresponding to \(C_{rep}(5,1,2)\). Then \(H_{i}^{\prime}(x)\) is obtained according to Equation (4). Table 5 shows the distribution of the weighted sum percentage \(P_{h}\) of each helper data calculated according to Equation (5). Due to the errors in the helper data adjustment in Equations (3) and (4), the ratio of \(P_{h}=0\) and \(P_{h}=1\) are reduced compared to \(P\) shown in Table 3. In this experiment, let \(T_{\text{th}}=0.9\), meaning a cell with \(P_{h}<0.1\) and \(P_{h}>0.9\) is a strong cell. If all of the four helper data have same power-on tendency of zero, choose the second one as strongest cell \(S(i,j)\). According to Equation (6), the value of \(e_{z}\) corresponding to the strongest cell \(S(i,j)\) or the surrounding cells around is adjusted. The elements in the clone matrix \(E\) are sequentially processed, and a final clone matrix \(K\) is obtained according to Equation (8). Table 6 shows the similarity of the response \(R\) and the clone matrix, where \(N_{cells}\) is the number of strongest cells, \(\delta_{1}\) is the similarity of the response \(R\) and initial clone matrix \(E\), \(\delta_{2}\) is the similarity of response \(R\) and the clone matrix \(K\) corrected using the distribution of strongest cells, and \(\Delta\delta\) is the repair rate obtained by leakage of strongest cells. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **SRAM1** & **SRAM2** & **SRAM3** \\ \hline \(N_{cells}\) & 23 & 14 & 22 \\ \(\delta_{1}\) & 65.8\% & 73.1\% & 59.0\% \\ \(\delta_{2}\) & 74.1\% & 79.0\% & 66.7\% \\ \(\Delta\delta\) & 8.3\% & 5.9\% & 7.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the clone matrix and response \(R\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(q_{y0}\) & **SRAM1** & **SRAM2** & **SRAM3** \\ \hline \(q_{y0}>70\%\) & 0 & 6\% & 0 \\ \(57.4\%<q_{y0}\leq 70\%\) & 7.8\% & 0 & 1.2\% \\ \(50\%<q_{y0}\leq 57.4\%\) & 56.2\% & 71\% & 79.8\% \\ \(40\%<q_{y0}\leq 50\%\) & 36\% & 23\% & 19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distribution of \(q_{y0}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(P_{h}\) & **SRAM1** & **SRAM2** & **SRAM3** \\ \hline \(0\) & 16.54\% & 19.61\% & 19.49\% \\ \(0<P_{h}<0.1\) & 30.15\% & 26.59\% & 29.53\% \\ \(0.1<P_{h}<0.3\) & 3.92\% & 3.43\% & 4.17\% \\ \(0.3<P_{h}<0.7\) &



Based on the clone matrix obtained from the weak response, the similarity increased by 8.3%, 5.9% and 7.7% respectively. Moreover, the greater the strongest-cells exist in the SRAM array, the greater the similarity increase. Notably, the above repair process can be performed only when the cloned matrix and SRAM array meet the relationship shown in Figure 4. The strongest cells distribution characteristics also fully or partially appears in matrix \(E\). ### Results of ERFE

For the data in SRAM1, Figure 11 provides the statistical distribution of the sum value according to Equation (9). The sum value is relatively concentrated in the region of (4,5,6,7), reaching 75.56%. The distribution in the region (0,2,9,11) is low, only 9.66%, meaning only 9.66% cells are valid for use in ERFE. The probability was determined to be \(P_{key}=57.66\%\) according to Equation (12). Table 7 compares ERFE and other Hard FE. \(N_{help}\) is the size of the helper data, and \(N_{\text{Mask}}\) and \(N_{\text{AddOp}}\) are the number of _Mask_ and _AddOp_ respectively. \(ERFE_{\text{ecc}}\) is the ERFE with error correction code, and \(ERFE_{\text{stable}}\) is the ERFE with stable bits.