## 3 Results and Discussion
### Computation Cost  
In Table 4, this subsection compares the proposed scheme in terms of computation cost in the login and authentication phases with other related schemes, e.g., [PERSON] et al. [41], [PERSON] and [PERSON] [38], [PERSON] et al. [28], and [PERSON] et al. [45]. In WSN, the sensor node has the most limited resources such as the user's smart card, sensor node, and GWN (base station). We used the hardware platform primarily on the previous studies to calculate execution time, including an Intel dual-core processor with a clock speed of 2.20 GHz, Ubuntu 12.04.1 LTS 32-bit Operating System, and 2 GB memory. The approximate execution time for various cryptographic operations by using the cryptography PBC library (version 0.5.12) is based on the GMP Library (version 5.0.5) reported in [41]. The execution time in ms is required for each primitive operation as noted in Table 4. During the login and authentication phase of the [PERSON] et al. [45] scheme, a user requires \(13T_{h}+4T_{ccm}+T_{cot}+T_{fc}=15.473\) ms, an IoT smart device (sensor node) requires \(9T_{h}+4T_{ccm}+T_{cca}=11.949\) ms, and a gateway node requires \(12T_{h}+6T_{ccm}+2T_{cca}=4.708\) ms. In the Dhillon and Kalra [38] scheme, the user needs to perform \(10T_{h}+1T_{E/D}\), and the sensor applies \(6T_{h}+2T_{(E/D)}\). In the gateway side, \(7T_{h}\), and \(2T_{E/D}\) are needed, so, the total computation cost in Dhillon and Kalra [38] is 20.382 ms.  
In the [PERSON] et al. [41] scheme, the user requires \(4T_{H}+2T_{ccm}\), and the gateway nodes require \(7T_{H}+1T_{ccm}\). Likewise, in the sensor side, the computation cost is \(4T_{H}+1T_{ccm}\), thus, the total computation cost in the [PERSON]. [PERSON] et al. [41] scheme can be represented as \(15T_{H}+4T_{ccm}\) and the estimation time is \(8.9730\) ms. In the [PERSON] et al. [28] scheme, the user needs to perform hash function operation \(14T_{H}\) and one-time fuzzy extractor operation \(1T_{fe}\). At the sensor side, nine times hash operations \(9T_{H}\) are required. Likewise, five times hash operations \(5T_{H}\) are needed at the gateway side. Therefore, the total computation cost of the [PERSON] et al. [28] scheme is \(28T_{H}+1T_{fe}\equiv 27.2743 ms\). The proposed scheme, on the other hand, has a total computational cost of \(17T_{H}+1T_{fe}+4T_{E/D}\) for Case 1, and \(19T_{H}+1T_{fe}+5T_{E/D}\) for Case 2. The user is not required to employ symmetric encryption/decryption (e.g., AES) because of the computational efficiency of both the fuzzy extractor operation and symmetric encryption/decryption (e.g., AES). The computational cost of a resource-constrained sensor node is \(3T_{H}+1T_{E/D}\) in Case 1, whereas it is \(4T_{H}+1T_{E/D}\) in Case 2. Both the hash function and symmetric encryption/decryption are very efficient, which makes the proposed scheme very efficient for resource-constrained sensor nodes in WSNs. Table 4 compares the computation costs required in different schemes during the \"login and authentication phase. For instance, it shows that the [PERSON] et al. [45], [PERSON] [38], and [PERSON] et al. [41] require the computation costs \(32.13\)ms, \(32.13\)ms, and \(20.4885\)ms, respectively. According to the comparison, the proposed scheme has less computation cost when compared to the existing schemes. Furthermore, due to the utilization of the fuzzy extractor technique, it provides \"superior security and more functionality features when compared with all authentication schemes\".