## 6 Computation Overhead Analysis
### Proposed RCA Method  
The computation time at senders including the aggregation time of \(sk_{aggr}\), and \(pk_{aggr}\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Security Methods**} & **Signature Generation Time** & **Signature Verification Time Per Message for Single Zone Case (ms)** & **Signature Generation Time Per Message for In Receivers in n Multi-Zones (ms)** & **Signature Verification Time Per Message for Each Receiver in n Multi-Zones (ms)** \\ \hline (l) [PERSON] et al. [19] & 2 IMG1 + 2 TAG1 = 0.9594 & 3 TP + TMG1 + TM2P = 7.8305 & 0.9594n & 3 nT + nT3G1 + nT3P2 = 952/n \\ \hline (2) [PERSON] et al. [20] & TAG1 + 2 TMG1 + TM2P = 1.1 & 3 TP + TMG1 + 2 TM2P + TAG1 & 1.1n & 3 nT + nT3G1 + 2 nT3P2 + \\  & & & & = nTAG1 + 2 nT3P2 + \\ \hline (3) [PERSON] et al. [21] & 4 TMG1 + 2 TAG1 = 1.93 & 3 TP + 3 TMG1 + TAG1 = 8.782 & 1.93n & 3 nT + nT3G1 = 8.78n \\ \hline (4) [PERSON] et al. [22] & 3 TP + TM2P = 7.473 & 5 TP + 8 TMG1 = 16.132 & 7.473n & 5 nTP + 8 nT3G1 = 16.132n \\ \hline (5) [PERSON] et al. [23] & 5 TMG1 + TAG1 + TH & 3 TP + TMG1 + TH T4P2 = 2.549 & 3n TP + nT3G1 + TM2P = 7.952n \\  & + TM2P = 2.549 & 7.8365 & 2.549n & 4 nTP + 2 nTM2P = 10.054n \\ \hline (6) [PERSON] et al. [24] & 5 TP + STMG1 = 9.7275 & 4 TP + 2 TM2P = 10.054 & 9.7275n & 4 nTP + 2 nTM2P = 10.054n \\ \hline The proposed TCA & TM2P + TMG1 = 0.614 & 2 TP + TM2P = 4.9 + 0.135 = 5.035 & 0.614n & 2 TP = 5.035 \\ \hline The proposed SCA & TM2P + TMG1 = 0.614 & 2 TP + TM2P = 4.9 + 0.135 = 5.035 & 0.614n & 2 TP = 5.035 \\ \hline \multirow{3}{*}{The proposed RCA} & \( \begin{array}{c}(n-1)TAFp+\\ (n-1)\text{TAG2}+\text{TM2P}+\\ \text{TMG1}=\\ 0.600\pm 0.014n\end{array} \) & \( \begin{array}{c}(n-1)\text{TAG2}+\text{TM2P}+2\text{TP}\\ 0.013\left(n-1\right)+0.135+4.9=\\ 5.014+0.013n\end{array} \) & \(\begin{array}{c}(n-1)\text{TAG2}+\text{TM2P}+2\text{TP}\\ 0.013\left(n-1\right)+0.135+4.9=\\ 5.0  
\(sk_{aggr}\) consumes \(n-1\) addition operations over \(F_{p}:(n-1)\)TAFP = 0.001 \((n-1)\)ms. While \(pk_{aggr}\) needs \(n-1\) addition operations in a group \(G_{2}\) = \((n-1)\)TAG2 = 0.013 \((n-1)\) ms. Therefore, the generation time of single signature \(\sigma_{aggr}=sk_{aggr}\). \(H(m)\) requires mapping to a point in the group \(G_{1}\), and one scalar multiplication over \(G_{1}\) and the aggregation time of \(sk_{aggr}\), and \(pk_{aggr}\). Therefore, the signature generation time is TM2P + TMG1 +\((n-1)\)TAFP + \((n-1)\)TAG2 = 0.001 \((n-1)+0.013\)\((n-1)\) = 0.600 ms + n*0.014 ms. While the signature verification per message requires the aggregation of \(g_{2}\)\({}_{aggr}\) that requires \(n-1\) addition in \(G_{2}\), mapping to a point in the group \(G_{1}\), and one scalar multiplication over \(G_{1}\). Therefore, the signature verification time is \((n-1)\)TAG2 + TM2P + 2 TP = 0.013 \((n-1)+0.135\) + 4.9 = 5.014 + n*0.013 ms. Thus, the total computation time for the RCA method including the signature generation time, the verification time, and the aggregation time of \(sk_{aggr}\), \(pk_{aggr}\) at the transmitter, and \(g_{2}\)\({}_{aggr}\) time at receiver is 0.600 ms + n*0.014 ms + 5.014 + n*0.013 ms. Therefore, the total computation time is 5.614 ms + (0.027n) ms. The computation time of SCA depends on the number of destination zones \(n\). From the described analysis, we found that the TCA method computation time is the lowest among the other proposed methods (SCA and RCA). While the RCA method introduces a little overhead, as compared to the high overhead of the SCA method that requires the generation of n signatures at transmitters. Table 5 compares the computation overhead of the three proposed authentication methods and the six previous methods [19; 20; 21; 22; 23; 24], for single-zone and multi-zone scenarios. In a multi-zone scenario, the six previous methods incur a high computation overhead to sign \(n\) messages for \(N\) destination zones. In contrast, the three proposed methods in a multi-zone scenario send a single message with a single short signature.