## 7 Comparative Analysis with Existing Schemes
### Comparative Study of Computation Costs  
The notations \(T_{ecm}\), \(T_{fe}\), and \(T_{h}\) are used to represent the time required for an \"ECC point multiplication\", a \"fuzzy extractor function\" for biometric verification in the case of the scheme designed by [PERSON] et al. [3], and a \"one way hash function\", respectively. Based on the experimental results available in [57], we took \(T_{h}\approx 0.00032\) s, \(T_{ecm}\approx 0.0171\) s, and \(T_{fe}\approx T_{ecm}\), that is \(T_{fe}\approx 0.0171\) s. The computation costs' comparison among LDAKM-EIoT and other schemes is depicted in Table 3. In LDAKM-EIoT, during the authentication and key agreement procedure, \(D_{i}\), \(EN\), and \(CS_{j}\) incurred \(9T_{h}\approx 2.88\) ms, \(15T_{h}\approx 4.8\) ms, and \(8T_{h}\approx 2.56\) ms, respectively. The comparative results demonstrated that LDAKM-EIoT performed better than [PERSON] et al.'s scheme [3] and [PERSON] et al.'s scheme [41]. Moreover, LDAKM-EIoT needed the same computation cost as compared to that for [PERSON] et al.'s scheme [21]. However, LDAKM-EIoT required more computation cost as compared to that for [PERSON] and [PERSON]'s scheme [39] and [PERSON] et al.'s scheme [18]. However, this was accepted as LDAKM-EIoT provided extra security and functionality features as compared to those for the other compared schemes of [PERSON] et al. [3], [PERSON] et al. [21], [PERSON] and [PERSON] [39], [PERSON] et al. [41], and [PERSON] et al. [18] (see Table 4).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Protocol** & **No. of Messages** & **No. of Bits** \\ \hline LDAKM-EIoT & 4 & 2400 \\ [PERSON] et al. [3] & 3 & 2528 \\ [PERSON] et al. [21] & 4 & 2752 \\ [PERSON] and [PERSON] [39] & 4 & 2912 \\ [PERSON] et al. [41] & 4 & 3840 \\ [PERSON] et al. [18] & 4 & 2720 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Communication costs’ comparison. LDAKM-EIoT, the lightweight authentication and key management scheme for the edge based IoT environment.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & **User/Smart** & **Gateway Node/** & **Sensing Device** & **Total Cost** \\  & **IoT Device** & **Edge Node** & **/Cloud Server** & **\\ \hline \multirow{2}{*}{LDAKM-EIoT} & \(9T_{h}\) & \(15T_{h}\) & \(8T_{h}\) & \(32T_{h}\) \\  & \(\approx 2.88\) ms & \(\approx 4.8\) ms & \(\approx 2.56\) ms & \(\approx 10.24\) ms \\ \hline \multirow{2}{*}{[PERSON] et al. [3]} & \(1T_{fe}+5T_{ecm}+5T_{h}\) & \(5T_{ecm}+4T_{h}\) & \(4T_{ecm}+3T_{h}\) & \(1T_{fe}+14T_{ecm}+12T_{h}\) \\  & \(\approx 104.20\) ms & \(\approx 86.78 ms\) & \(\approx 69.36\) ms & \(\approx 260.34\) ms \\ \hline \multirow{2}{*}{[PERSON] et al. [21]} & \(11T_{h}\) & \(14T_{h}\) & \(7T_{h}\) & \(32T_{h}\) \\  & \(\approx 3.52\) ms & \(\approx 4.48\) ms & \(\approx 2.24\) ms & \(\approx 10.24\) ms \\ \hline \multirow{2}{*}{[PERSON] and [PERSON] [39]} & \(11T_{h}\) & \(7T_{h}\) & \(5T_{h}\) & \(23T_{h}\) \\  & \(\approx 3.52\) ms & \(\approx 2.24\) ms & \(\approx 1.6\) ms & \(\approx 7.36\) ms \\ \hline \multirow{2}{*}{[PERSON] et al. [41]} & \(10T_{h}\) & \(7T_{h}\) & \(19T_{h}\) & \(36T_{h}\) \\  & \(\approx 3.2\) ms & \(\approx 2.24\) ms & \(\approx 6.08\) ms & \(\approx 11.52\) ms \\ \hline \multirow{2}{*}{[PERSON] et al. [18]} & \(7T_{h}\) & \(5T_{h}\) & \(7T_{h}\) & \(19T_{h}\) \\  & \(\approx 2.24\) ms & \(\approx 1.6\) ms & \(\approx 2.24\) ms & \(\approx 6.08\) ms \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of computation costs.