# Light-Weighted Password-Based Multi-Group Authenticated Key Agreement for Wireless Sensor Networks
## 5 Performance Analyses and Comparisons  
The performance of the proposed protocol in communication was compared with that of related approaches. Table 2 presents a performance comparison of the group authenticated key agreement (GAKA) protocols of [PERSON] et al. [3], [PERSON] et al. [7], [PERSON] [8], and [PERSON] [13] and the protocol that was proposed herein, where \(T_{chao}\) denotes the time required to execute a Chebyshev chaotic map operation; \(T_{sym}\) denotes the time required to execute a symmetric encryption/decryption operation; \(T_{exp}\) denotes the time required to execute a modular exponential operation, \(T_{sign/veri}\) denotes the time required to execute a signing/verifying operation in the public key system, and \(T_{chao}<T_{sym}<T_{exp}\) (\(\approx T_{sign/veri}\)) [35; 36]. The first comparison concerned computations. These GAKA protocols [3; 7; 8; 13; 14] require many time-consuming modular exponential computations or scalar multiplications on elliptic curves to realize authentication and negotiate group keys. Only the proposed GAKA protocol was developed using extended chaotic map operations and did not have a heavy computational burden. Thus, the proposed GAKA protocol was more efficient than the other GAKA protocols. The second comparison concerned the realization of user authentication in each protocol. The protocols of [PERSON] et al. [7] and [PERSON] and [PERSON] [8] realize authentication using users' public keys. The GAKA protocols of [PERSON] et al. [3], [PERSON] and [PERSON] [13], and [PERSON] et al. [14] as well as the proposed GAKA protocol realize authentication using users' passwords. However, in the GAKA protocols of [PERSON] et al. [3] and [PERSON] and [PERSON] [13], all users share the same password so their protocols do not ensure users' privacy. The third comparison concerned whether the protocol required the maintenance of users' public keys. The protocols of [PERSON] et al. [7] and [PERSON] and [PERSON] [8] employ users' public keys, and thus require extra equipment to store long-term secret keys and the results of time-consuming exponential computations in clients. The GAKA protocols of [PERSON] et al. [3], [PERSON] and [PERSON] [13], and [PERSON] et al. [14] as well as the proposed GAKA protocol are password-based authentication protocols. Each user remembers only his weak password without the need for any extra equipment to store long-term secret keys. The fourth comparison involved whether the protocol was suitable for hypergraphs. The GAKA protocols [3; 7; 8; 13] consider only a single group, and are difficult to extend to multiple groups. The protocol of [PERSON] et al. [7] and the proposed GAKA protocol enable communicating entities to belong to multiple groups, and so are effective for hypergraphs. The final comparison involved whether the protocol provided the anonymity of users. The GAKA protocols [3; 7; 8; 13; 14] reveal users' identities, and fail to protect user privacy. Only the proposed GAKA protocol did not reveal users' identities, and so protected users' anonymity. Table 3 lists the simulation environment, including used hardware/software specifications and algorithms. Figure 3 illustrates simulation results for the response time of related protocols and the proposed one for \(n=5\), \(10\), \(15\), \(\ldots\), \(30\). Due to the use of extended Chebyshev chaotic map operations, the proposed protocol required less response time than related protocols.