# A Secure and Efficient Group Key Management Scheme for Clusters of String Inverters
## 5 Performance Evaluation
### Latency for Group Key Distribution
In Figure 6, we present the (mean) latency (in seconds) of the prototype as a function of the group size and type/length of the key, specifically: RSA vs. EC keys with 128 bits security level. Confidence intervals are not shown due to the scale on the y-axis. The comparison experimental results in Figure 6 show that typical PV plants' timing constraints were definitely met. In general, the (mean) latency  
Figure 6: Latency (in seconds) for group key generation and distribution in a leave operation. Comparison experimental results as a function of the cardinality of set of remaining inverters and type/length of the keys: RSA vs. Elliptic Curve (EC) keys with 128 bits security level. Figure 7: Latency (in seconds) for group key generation and distribution in a leave operation. Comparison experimental results as a function of the cardinality of set of remaining inverters and RSA-keys security levels: 112-bit vs. 128-bit. increased linearly with the group size. We demonstrated that we expected such behavior in our discussion in Section 6.2. Furthermore, because of ECDSA's faster processing times and the lower demands on bandwidth with respect to RSA [81], latency in the case of EC keys is less than half the latency required in the case of RSA keys. Due to the architecture of Power-One Italy S.p.A. developed inverters and under development, we then specifically focused on RSA keys only. In Figure 7, we show the mean latency as a function of the group size and RSA key security levels considered: 112-bit vs. 128-bit. As one might expect, the (mean) latency was approximately 83-87.5% higher in the case of 128-bit strength RSA keys. Though, in this case, the scheme only required 5.7 s to distribute a new group key to 50 inverters. This accounted only for 9.5% of the maximum time allowed for group key distribution, namely 60 s. Therefore, we evaluated the efficiency of the scheme. Specifically, we provided a first definition of efficiency according to the following equation:  
\[efficiency_{latency}=1-\frac{\textit{time to distribute the group key to all group members}}{\textit{maximum time allowed for group key distribution}} \tag{1}\]  
It turns out that \(efficiency_{latency}=1-\frac{5.7}{60}=90.5\%\). It is worthwhile to notice that if the PKI hierarchy chain consisted of intermediate issuing CAs, rather than the root CA only, it would be necessary to verify the digital signatures on the certificate chain up to the root CA. This would increase the latency. Last but not least, it is important to highlight that we evaluated the (mean) latency, reported in Figures 6 and 7, for group sizes greater than 10 members by concurrently running more processes on inverters. As inverters were equipped with a single core CPU, the mean values shown above need to be considered as higher estimates. In Table 4 we present the main results regarding latency for group key distribution in a leave operation. As repeated throughout the paper, to the best of author's knowledge, we are the first addressing the issue of secure group communication in clusters of smart string inverters. Consequently, the results presented below cannot be compared neither with the states of the arts nor with other peoples' work.