## 6 Performance Analysis and Implementation Results
### Operation Counts  
Tables 1 and 2 describe the operation counts of field arithmetic over \(\mathbb{F}_{p^{2}}\) and their conversion into field arithmetic over \(\mathbb{F}_{p}\) for curve Ted127-glv4 and Four\(\mathbb{Q}\) using Algorithm 1. Because both curves support the four-dimensional decomposition, the operation counts for Algorithm 1 can be compared step by step. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Operation**} & \multicolumn{8}{c}{**Ted127-glv4**} \\ \cline{2-10}  & \(\mathbf{I_{2}}\) & \(\mathbf{M_{2}}\) & \(\mathbf{S_{2}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{M_{1}}\) & \(\mathbf{S_{1}}\) & \(\mathbf{A_{1}}\) & \(\mathbf{A_{i}}\) \\ \hline Compute & - & 73 & 27 & 59.5 & 273 & - & 365 & 200 \\ Precompute & - & 63 & - & 56 & 189 & - & 301 & 126 \\ Scalar & - & - & - & - & - & - & - & - \\ decomposition & - & 704 & 256 & 835 & 2,624 & - & 4038 & 1,920 \\ Scalar & 1 & 2 & - & - & 18 & 128 & 8 & 4 \\ \hline Total Cost & 1 & 842 & 283 & 950.5 & 3,104 & 128 & 4712 & 2,250 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The operation counts of curve Four\(\mathbb{Q}\) using field arithmetic over \(\mathbb{F}_{p^{2}}\) and operation counts for conversion into field arithmetic over \(\mathbb{F}_{p}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Operation**} & \multicolumn{8}{c}{**Ted127-glv4**} \\ \cline{2-10}  & \(\mathbf{I_{2}}\) & \(\mathbf{M_{2}}\) & \(\mathbf{S_{2}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{M_{1}}\) & \(\mathbf{S_{1}}\) & \(\mathbf{A_{1}}\) & \(\mathbf{A_{i}}\) \\ \hline Compute & - & 73 & 27 & 59.5 & 273 & - & 365 & 200 \\ Precompute & - & 63 & - & 56 & 189 & - & 301 & 126 \\ Scalar & - & - & - & - & - & - & - & - \\ recoding & - & - & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The operation counts of curve Ted127-glv4 using field arithmetic over \(\mathbb{F}_{p^{2}}\) and operation counts for conversion into field arithmetic over \(\mathbb{F}_{p}\). Step 1 of Algorithm 1 computes three endomorphisms \(\phi(P),\psi(P)\), and \(\phi(\psi(P))\), and requires \(73M_{2}+27S_{2}+59.5A\) operations for Four\(\mathbb{Q}\) and \(13M_{2}+25_{2}+11.5A_{2}\) operations for curve Ted127-glv4. Step 2 requires seven ECADD operations, which require \(49M_{2}+28A_{2}\) operations for Four\(\mathbb{Q}\) and \(56M_{2}+42A_{2}\) operations for curve Ted127-glv4. However, these outputs are all converted for faster ECADD computations, which require \(14M_{2}+28A_{2}\) operations for Four\(\mathbb{Q}\) and \(7M_{2}+28A_{2}\) operations for curve Ted127-glv4. Steps 3 and 4 require only bit and integer operations for all positive scalar decomposition and fixed-length recoding operations. Step 5 requires \(1A_{2}\) operations for one point negation and one table lookup, and a conversion to extensible coordinates \((X,Y,Z,T_{a},T_{b})\) for the initial point \(Q\), which require \(2A_{2}\) operations. Steps 6 to 9 require 64 ECDBL operations, 64 ECADD operations, 64 point negations, and 64 table lookups for Four\(\mathbb{Q}\), and 65 ECDBL operations, 65 ECADD operations, 65 point negations, and 65 table lookups for curve Ted127-glv4. The operation counts of these steps are \(704M_{2}+256S_{2}+835A_{2}\) for Four\(\mathbb{Q}\) and \(715M_{2}+260S_{2}+845A_{2}\) for curve Ted127-glv4. Step 10 requires \(1I_{2}+2M_{2}\) operations for the normalization of the result point \(Q\). Variable-base scalar multiplication using the four-dimensional decomposition requires \(1I_{2}+842M_{2}+283S_{2}+950.5A_{2}\) operations for Four\(\mathbb{Q}\) and \(1I_{2}+793M_{2}+262S_{2}+929.5A_{2}\) operations for curve Ted127-glv4. The curve Ted127-glv4 requires \(49M_{2}+21S_{2}+21A_{2}\) fewer operations than Four\(\mathbb{Q}\) because the endomorphisms in curve Ted127-glv4 are efficiently computable. However, the operation counts of field inversion over \(\mathbb{F}_{p}\) for Four\(\mathbb{Q}\) and curve Ted127-glv4 are \(I_{1}=10M_{1}+126S_{1}\) and \(I_{1}=13M_{1}+126S_{1}\), respectively.