## 6 Computation Overhead Analysis
### Proposed RCA Method
Figure 13 shows the computation cost per message, including the signature generation time and verification time for the three proposed authentication methods and the six previous methods [19; 20; 21; 22; 23; 24], for the single-zone scenario. For the single-zone, the signature generation time per message of [PERSON] et al. [24] is as long as 9.7275 ms, as one signature generation requires 3 bilinear pairings and five scalar multiplication over group \(G_{1}\). It is 16 times longer than the proposed three authentication methods, which consume only 0.614 ms for signature generation. For signature verification, the previous methods [19; 20; 21; 22; 23; 24] incur as long as 16 ms (for the case of [22]), due to their excessive use of bilinear pairing in verification. It is 4 times longer than the three proposed authentication methods that incur only 5.035 ms for verification. In a multi-zone scenario, the previous methods repeatedly transmit the same message with different signatures to multiple destination zones. However, the three proposed authentication methods send a single signed message to multiple receivers in different zones. Each receiver in multiple zones individually verifies the same signature by just one verification step, which provides a constant verification time regardless of \(N\). As shown in Figure 14, we compare the signature generation time in multi-zone scenarios for the three proposed authentication methods (TCA, SCA, RCA) and three previous methods, [PERSON] et al. [22], [PERSON] et al. [23], and [PERSON] et al. [24]. TCA for multi-zone scenarios introduces a fixed computation time of 0.614 ms for a signature generation, regardless of the number of destination zones. SCA introduces 0.614n ms, which linearly increases with the number of destination zones. While RCA introduces a little increase in singing time, due to aggregation of \(sk_{aggr}\) and \(pk_{aggr}\). Figure 14: Comparison of signature generation time for the three proposed authentication methods and the two previous methods for multi-zone scenarios. Figure 13: A comparison of signature generation time and verification time for the proposed TCA method and the six previous methods in a single-zone scenario. For \(N=5\), [PERSON] et al. [22], [PERSON] et al. [23], [PERSON] et al. [24], SCA, TCA, and RCA consume signature generation time of 37.3 ms, 12.745 ms, 48.6 ms, 3.07 ms, 0.614 ms, and 0.67 ms, respectively. Therefore, the proposed methods significantly reduce the signature generation time by 16 times-80 times, as compared to the two previous methods [22; 24]. As shown in Figure 15, we compare the signature verification time in multi-zone scenarios for the three proposed authentication methods (TCA, SCA, RCA) and two previous methods, [PERSON] et al. [22] and [PERSON] et al. [24]. TCA and SCA for multi-zone scenarios introduce a fixed computation time of 5.035 ms, for a signature verification, regardless of the number of destination zones. For \(N=5\), [PERSON] et al. [22], [PERSON] et al. [24], SCA, TCA, and RCA consume signature generation times of 80.66 ms, 50.27 ms, 5.035 ms, 5.035 ms, and 5.079 ms, respectively. Therefore, the proposed methods significantly reduce the signature verification time by 10-16 times, as compared to the two previous methods. In summary, the reduction ratio in communication and computation overhead that the proposed methods provide, tends to rapidly increase for a large-scale network with a large number of neighboring zones. In our future work, we intend to improve the proposed authentication methods by integrating them into different 5G applications, as recommended by the authors of [38]. We also intend to implement the proposed authentication methods using hardware devices and compare the performance with the results of our previous decentralized hash-chain-based protocol [39].