# Privacy-Preserving Data Aggregation with Dynamic Billing in Fog-Based Smart Grid
## 6 Comparison and Evaluation
### Communication Overhead  
In this section, we also take a sub-region (with \(n\) SMs) as an example to analyse and compare the communication overhead of the scheme [23; 27] and PPDB. For the sake of fairness, we also focus only on the Reports Generation, Data Aggregation and Data Reading phases, phases, which are present in all three schemes. According to the construction of the schemes, we divide the communication process into five parts: (1) SM-to-FN communication; (2) FN-to-AC communication; (3) FN-to-SP communication; (4) AC-to-SP communication; and (5) SM-to-SP communication. We select \(|N|=1024\) bits for the Paillier cryptosystem and the length of the elements in \(G\) is 512 bits. In addition, we assume that the sum of the length of the identity and timestamp is 64 bits. The communication overhead of each part is given in Table 4.  
In our proposed PPDB, each SM sends the report \(\mathcal{T}_{ij,m}=(ID_{SM_{ij}},u_{ij,m},v_{ij,m},T_{ij,m},\sigma_{ij,m})\) to the corresponding FN, and its length is \(L_{SM}=64+512+512+512=1600\) bits. Next, we analyse the communication overhead from FN to AC. After generating a signature, FN sends the report \(\mathcal{T}_{ij,m}=(ID_{SM_{ij}},u_{ij,m},v_{ij,m},T_{ij,m},\sigma_{ij,m})\) to AC. Therefore, the communication overhead from FN to AC should be \(L_{FN}=64+512+512+512=1600\) bits. In the Data Reading phase, AC sends \(\mathcal{T}_{AC_{i,m}}=(ID_{AC},u_{i,m},C^{\prime}_{i,m},\mathcal{T}_{AC_{i,m }},\sigma_{AC_{i,m}})\) to the SP, the size of which is \(L_{AC}=64+512+512+512=1600\) bits. The FN collects reports for \(n\) SMs in the coverage area, AC receives one report from the FN and the SP also receives only one report from AC. Therefore, the total communication overhead of the proposed PPDB in the Reports Generation, Data Aggregation and Data Reading phases is \((nL_{SM}+L_{FN}+L_{AC})\) bits.  
In the scheme [23], each SM sends \(\{l_{i},ID_{SM},TS,\sigma_{i}\}\) to the FN, where \(l_{i}\in Z_{N^{2}}\). Thus, the communication overhead from the SM to the FN is \(L_{SM^{\prime}}=64+2048+512=2624\) bits. The report sent from the FN to AC is in the form of \(\{C,ID_{GW},TS,\sigma\}\), and its length is \(L_{FN^{\prime}}=64+2048+512=2624\) bits. Therefore, the total communication overhead of the scheme [23] is \((nL_{SM^{\prime}}+L_{FN^{\prime}})\) bits.  
In the scheme [27], each SM sends \(\left\{ID_{i},C^{a}_{i},C^{a}_{i},T_{i},\sigma_{i}\right\}\) to the FN with the size of \(L_{SM^{\prime\prime}}=64+512+512+512=1600\) bits. In the Data Reading phase, the SM also needs to send a report to the SP in the form of \(\left\{ID_{i},D_{i},T^{d}_{i},\sigma^{d}_{i}\right\}\), and its length is \(L_{SM^{\prime\prime}}=64+512+512=1088\) bits. After generating a signature, the FN sends the report \(\left\{ID_{GW},C^{a},C^{b},T_{GW},\sigma_{GW}\right\}\) to the SP. Therefore, the communication overhead from the FN to the SP should be \(L_{FN^{\prime\prime}}=64+512+512+512=1600\) bits. Therefore, the total communication overhead of the scheme [27] is \((nL_{SM^{\prime\prime}}+nL_{SM^{\prime\prime}}+L_{FN^{\prime\prime}})\) bits.  
Comparison of total communication overhead between our proposed PPDB and schemes [23; 27] is shown in Figure 9. It can be observed that compared with schemes [23; 27], our PPDB has the lowest communication overhead. When \(n=100\), the communication overhead of scheme [23] and scheme [27] is about 1.8 times that of PPDB.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **SM-to-FN** & **FN-to-AC** & **FN-to-SP** & **AC-to-SP** & **SM-to-SP** \\ \hline [PERSON] et al.’s scheme [23] & 2624 bits & 2624 bits & / & / & / \\ [PERSON] et al.’s scheme [27] & 1600 bits & / & 1600 bits & / & 1600 bits \\ Our scheme & 1600 bits & 1600 bits & / & 1600 bits & / \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of communication overhead.