## 5 Performance Evaluation  
We evaluated the performance of proposed architecture by implementing its structure in C++ using the original CRPs obtained from the pm-PUF [29]. The hash function used for our proposed structure is a 64-bit cycle redundancy check (64-CRC). We assumed that the tables of authentication server held \(2^{9}\), \(2^{10}\), \(2^{11}\), and \(2^{12}\) CRPs. Table 1 shows the PDBF, CRPBF, and total Bloom filter memory requirements when the sizing factor \(\alpha\) of the Bloom filter is increased to two, four, and eight in the proposed structure. The total memory requirement for the Bloom filter is \(M_{PDBF}+M_{CRPBF}\). The memory requirement for the PDBF is equal to \(2\cdot\alpha\cdot 2^{\lceil\log_{2}n_{c}\rceil}\) where \(n_{c}\) is the number of challenges in the authentication server. The memory requirement for the CRPBF is \(\alpha\cdot 2^{\lceil\log_{2}n_{cP}\rceil}\) where \(n_{cP}\) is the number of CRPs in the authentication server. Table 2 shows the simulation result for the input set which is the sum of the authentication challenges and attacker challenges. To evaluate the performance of the proposed structure, we created input challenge sets with three times the challenges of authentication servers. It is assumed that one-third of the input challenges are authentic challenges, and the remaining two-thirds are attacker challenges. If authentication succeeded, the challenge is deleted from the PDBF. In Table 2, the CRBPF false positives are equal to the number of data transmission errors in the proposed structure. When the sizing factor of the Bloom filter is increased to four, the CRPBF false positives from all sets are reduced to zero. In other words, the proposed structure only transmitted data that existed on the authentication server. Table 3 shows the simulation results for an input set consisting only of attacker challenges. We created the input challenge sets with two times the challenges of authentication servers. In this simulation, since the challenge of the authentication server is not included in the input set, deletion of the PDBF did not occur. In the proposed structure, when the sizing factor of the Bloom filter is increased to eight, the CRPBF false positives of all sets are reduced to a range of zero to three. It means that the data transmission error rate is 0-0.098%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**CRPs** & \(\mathbf{\alpha}\) & \(\mathbf{M_{PDBF}}\) & \(\mathbf{M_{CRPBF}}\) & **Total Memory** \\ \hline \multirow{2}{*}{_Set 1_ (\(2^{9}\))} & 2 & 0.25 & 0.13 & 0.38 \\  & 4 & 0.50 & 0.25 & 0.75 \\  & 8 & 1.00 & 0.50 & 1.50 \\ \hline \multirow{2}{*}{_Set 2_ (\(2^{10}\))} & 2 & 0.50 & 0.25 & 0.75 \\  & 4 & 1.00 & 0.50 & 1.50 \\  & 8 & 2.00 & 1.00 & 3.00 \\ \hline \multirow{2}{*}{_Set 3_ (\(2^{11}\))} & 2 & 1.00 & 0.50 & 1.50 \\  & 4 & 2.00 & 1.00 & 3.00 \\  & 8 & 4.00 & 2.00 & 6.00 \\ \hline \multirow{2}{*}{_Set 4_ (\(2^{12}\))} & 2 & 2.00 & 1.00 & 3.00 \\  & 4 & 4.00 & 2.00 & 6.00 \\ \hline \multirow{2}{*}{_Set 4_ (\(2^{10}\))} & 2 & 2.00 & 1.00 & 3.00 \\  & 4 & 4.00 & 2.00 & 6.00 \\ \hline \multirow{2}{*}{_Set 2_ (\(2^{10}\))} & 8 & 8.00 & 4.00 & 12.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Memory requirement of total Bloom filter (KB). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & _Set 1_ & _Set 2_ & _Set 3_ & _Set 4_ \\ \hline \multirow{2}{*}{\(\alpha\)} & 2 & 4 & 2 & 4 & 2 & 4 \\ total no. of PDBF queries & \multicolumn{2}{c}{1536} & \multicolumn{2}{c}{3072} & \multicolumn{2}{c}{6144} & \multicolumn{2}{c}{12288} \\ total no. of CRPBF queries & 524 & 512 & 1054 & 1024 & 2112 & 2048 & 4214 & 4096 \\ no. of PDBF false positives & 12 & 0 & 30 & 0 & 64 & 0 & 118 & 0 \\ no. of CRPBF false positives & 5 & 0 & 11 & 0 & 25 & 0 & 34 & 0 \\ no. of transmission data & 517 & 512 & 1035 & 1024 & 2073 & 2048 & 4130 & 4096 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bloom filter performance.