## 3 Results and Discussion
### Communication Cost  
The communication cost comparison of the proposed scheme and other existing schemes regarding the total number of bits required for that transmitted message is illustrated in Table 4. For computing the communication cost, we assume that the identities \(HGW_{ID},FGW_{ID},PID_{i}\), and \(U_{ID}\) are all 160 bits in length; the hash output is 160 bits long; and the identities SN ID, random number/nonce, and timestamp are all 32 bits. For a 128-bit plaintext block, symmetric encryption/decryption (using AES-128) requires 128 bits. In the [PERSON] et al. [45] scheme, the messages \(Msg_{1}=<TID_{U},M_{1},Y_{1},Sign_{y1},T_{1}>\) with \(|TID_{U}|=160\) bits, \(|M_{1}|=320\) bits, \(|Y_{1}|=320\) bits, \(|Sign_{y1}|=256\) bits and \(|T_{1}|=32\) bits, requires 1088 bits; \(Msg_{2}=<TID_{C},M_{2},Y_{1},Y_{2},Sign_{y2},T_{2}>\) with \(|TIDC|=160\) bits, \(|M_{2}|=256\) bits, \(|Y_{1}|=320\) bits, \(|Y_{2}|=320\) bits, \(|Sign_{y2}|=256\) bits and \(|T_{2}|=32\) bits, needs 1344 bits; \(Msg_{3}=<TID_{C}^{*},M_{3},M_{4},Y_{3},Sign_{y3},T_{3}>\) with \(|TID_{5}^{*}|=256\) bits, \(|M_{3}|=256\) bits, \(|M_{4}|=256\) bits, \(|Y_{3}|=320\) bits, \(|Sign_{y3}|=256\) bits and \(|T_{3}|=32\) bits, demands 1376 bits; and \(Msg4=<TID_{U}^{*},M_{3},M_{4},M_{5},Y_{3},Y_{4},Sign_{y4},T_{3},T_{4}>\) with \(|TID_{U}^{*}|=256\) bits, \(|M_{3}|=256\) bits, \(|M_{4}|=256\) bits, \(|S_{2}|=526\) bits, \(|Y_{3}|=320\) bits, \(|Y_{4}|=320\) bits, \(|Sign_{y4}|=256\) bits, \(|T_{3}|=32\) bits and \(|T_{4}|=32\) bits, requires 1984 bits. The cumulative communication cost in this scheme amounts to 5792 bits. Furthermore, the Dhillon and Kalra [38] scheme has four messages, which are transmitted among the participant's entities, costing 4016 bits. In the [PERSON]. [PERSON] et al. [41] scheme, the user transmitting \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{3}{c}{**Computation Cost (ms)**} & \multirow{2}{*}{**Communication Cost**} \\ \cline{2-2} \cline{4-5}  & **User** & **Sensor** & **Gateway** & **Total** \\ \hline [PERSON] et al. [45] & \(13T_{h}+4T_{ccm}+T_{ccm}+T_{fe}\) & \(9T_{h}+4T_{ccm}+T_{cca}\) & \(12T_{h}+6T_{ccm}+2T_{cca}\) & 32.13 ms & 5792 bits \\ [PERSON] and [PERSON] [38] & \(10T_{h}+1T_{E/D}\) & \(6T_{h}+2T_{E/D}\) & \(7T_{h}+2T_{E/D}\) & 20.382 ms & 4016 bits \\ [PERSON]. [PERSON] et al. [41] & \(4T_{H}+2T_{ccm}\) & \(4T_{H}+1T_{ccm}+1T_{E/D}\) & \(7T_{H}+1T_{ccm}+2T_{E/D}\) & 20.4885 ms & 2752 bits \\ [PERSON] et al. [28] & \(14T_{H}+1T_{fe}\) & \(9T_{H}\) & \(5T_{H}\) & 27.2743 ms & 2400 bits \\ Proposed scheme & \(9T_{H}+1T_{fe}+1T_{E/D}\) & \(3T_{H}+1T_{E/D}\) & \(5T_{H}+2T_{E/D}\) & 17.6651 ms & 1696 bits \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the computation and communication costs of schemes. the message \(M_{1}=h(storedDG||R_{U11}||T11|K)\) that requires 256 bits, and receives \(M_{6}=h(receivedID_{N}||ID_{G}||storedDG||T_{4}),M_{7}=h(M_{6}||RG_{1}||received RM||K)\) with 352 bits size length. Likewise, the gateway transmits the message \(M_{1}^{\prime}=h(storedDG||receivedRU_{11}||\)\(receivedT_{1}||K)\) and receives \(M_{3}=h(M_{2}||RG_{1}||receivedRU_{1}||L)\) with a size of 1376 bits. The sensor node transmits and receives the messages \(M_{4}=h(ID_{N}||receivedRG_{1}||storedAN||T_{3})\) and \(M_{5}=h(M_{4}||RN||L)\) that cost approximately 832 bits. Therefore, the total communication cost of [PERSON]. [PERSON] et al. [41] is 2752 bits. In [PERSON] et al. [28], four messages are exchanged between entities, the user transmits the login request message \(\{HID_{i},C_{i},VUL_{i}\}\) and receives an authentication message \(\{H_{i},M_{Cu}\}\) from GWNs. While the gateway sends the message \(\{HID_{i},C_{i},PID_{j},D_{i},VS_{j}\}\) to control server and receives \(\{M_{cg},M_{Cu},E_{i},F_{i}\}\) from the GWN. Therefore, the total communication cost of [PERSON] et al. [28] is 2400 bits.