# Pitfall of the Strongest Cells in Static Random Access Memory Physical Unclonable Functions
## 4 Proposed New Fuzzy Extractor
### ERFE Architecture
\(p_{b}\) in \(ERFE_{\text{stable}}\) is 0.01, and the others are 0.15. Min-entropy of all FEs are 95%, a total of 128-bit keys are extracted, entropy loss in randomness extraction is 128-bit, \(\gamma=256\), and \(BCH\) is BCH code. We provide the results of \(ERFE_{\text{stable}}\) obtained using the method proposed in [PERSON] et al. [21]: 59.26% of the cells are stable bit and can be used as the input for ERFE. \(N_{pqf}=256\times 11\div 0.0966\div 0.5926=49,126\), \(N_{help}=N_{\text{Mask}}+N_{\text{AddOp}}=49,126\div 11+256=4722\), and \(P_{fail}\) = 3.5 \(\times\) 10\({}^{-7}\). We also obtained a result of \(P_{fail}\) as low as 1.8 \(\times\) 10\({}^{-8}\) with an error correcting code \(C_{rep}(9,1,4)\), and \(p_{b}=0.15\). In this case, \(N_{pqf}=256\times 11\div 0.0966=29,161\), \(N_{help}=N_{\text{Mask}}+N_{\text{AddOp}}+N_{\text{rep}}=29,161\div 11+256+256\times 11 \div 9\times 8=5410\), and \(N_{\text{rep}}\) is the size of helper data generated by \(C_{rep}(9,1,4)\). Because only 9.66% of cells are valid, ERFE needs a larger \(N_{pqf}\) than other Hard FEs. Figure 11: Distribution of the sum value. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Scheme** & \(\mathbf{C_{1}(n_{1}\)**,\(k_{1}\)**,\(t_{1}\))} & \(\mathbf{C_{2}(n_{2}\)**,\(k_{2}\)**,\(t_{2}\))} & \(\mathbf{l_{2}}\) & \(N_{pqf}\) & \(N_{help}\) & \(P_{fail}\) & **Slice** \\ \hline Hard FE [15] & \(C_{rep}(3,1,1)\) & \(BCH(977,232,102)\) & 3 & 8793 & 8097 & \(1.9\times 10^{-7}\) & 41 [14] \\ Hard FE [15] & \(C_{rep}(5,1,2)\) & \(BCH(982,502,53)\) & 1 & 4910 & 4408 & \(8.2\times 10^{-7}\) & 41 [14] \\ Hard FE [15] & \(C_{rep}(7,1,3)\) & \(BCH(817,542,28)\) & 1 & 5719 & 5177 & \(4.9\times 10^{-7}\) & 41 [14] \\ \(ERFE_{\text{ecc}}\) & \(C_{rep}(9,1,4)\) & - & - & 29,161 & 5410 & \(1.8\times 10^{-8}\) & 80 \\ \(ERFE_{\text{stable}}\) & - & - & - & 49,126 & 4722 & \(3.5\times 10^{-7}\) & 39 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Result of different fuzzy extractors (FEs). We synthesized and mapped ERFE in the Xilinx Spartan-3E-500 FPGA platform. [PERSON] et al. [14] provided the FPGA resources of \(C_{rep}(3,1,1)\), \(C_{rep}(5,1,2)\), \(C_{rep}(7,1,3)\), and \(C_{rep}(9,1,4)\) using 41 slices. ERFE almost requires the same slice resources as repetition code as shown in Table 7. However, the BCH algorithms themselves are much more complex: thus, that their hardware complexity is expected to be similarly higher. We estimate that ERFE is much more efficient than Hard FE in terms of hardware resource requirements. Our proposed ERFE is space efficient and suitable for use in resources-limited device such as Radio Frequency Identification (RFID) and IoT and so on. ## 6 Conclusions

In this paper, we analyzed the physical challenge space of SRAM PUFs, and found a weak response in the SRAM PUF based on a repetition code. The presence of strongest cells in SRAM can cause public helper data generated by a repetition code to leak more information about the responses. Using the helper data generated by the weak responses, the leftover entropy of PUF was zero. Our research experimentally confirmed that the pitfall of using repetition codes [15] exists, and the distribution of strongest cells causes helper data to reveal more information about PUFs' responses, which further decreases the leftover entropy of the PUF based on repetition codes. We proposed ERFE which does not leak information of the strongest cells. ERFE uses cells' sum value as the key with 4-bit error tolerance ability. ERFE is as lightweight as a repetition code, and is also very suitable for implementation in FPGA or software. Future work will include reducing the size of \(N_{buf}\), and testing ERFE with additional types of SRAM chips and other types of PUFs. The work presented here was completed in collaboration between all authors. [PERSON] and [PERSON] co-prepared the manuscript, co-designed the clone method and the ERFE. [PERSON] and [PERSON] performed all the experiments. [PERSON] proposed the ideas and revised the manuscript.