## 5 Performance Evaluation
### Computation Overhead  
(1). Spatial aggregation  
Let C\({}_{ma}\) and C\({}_{ppf}\) denote respectively the cost of Modular addition operation and keys generation operation with PRF, respectively let C\({}_{add}\) and C\({}_{mul}\) denote the cost of addition and multiplication operation respectively, and C\({}_{enc}\) and C\({}_{dec}\) denote the cost of homomorphic encryption and decryption operation respectively. In our spatial aggregation scheme, for every node, partitioning individual data into \(k\) partitions costs one C\({}_{ma}\); generating \(k\) pairwise keys costs \(k\)-C\({}_{ppf}\) receiving \(k\) encrypted messages and adding them up cost \(k\)-C\({}_{add}\), then the computation overhead per node is C\({}_{ma}\)\(+\)\(k\)-C\({}_{ppf}\)\(+\)\(k\)-C\({}_{add}\) and the total computation overhead per aggregator is (\(n\)-1)-C\({}_{add}\) for aggregating data from \(n\) nodes. In [PERSON] et al. 's scheme [23], at the \(d^{\text{th}}\) time step, every hash function cost is \(C_{\text{hash}}\), \(k\) masking random keys cost is \(k\)-C\({}_{ppf}\) and computing total masking keys cost is \(2k\)-C\({}_{add}\), and then encrypting individual data cost is \(C_{enc}\), so the total computation overhead is \(C_{hash}\)\(+\)\(k\)-C\({}_{ppf}\)\(+\)\(2k\)-C\({}_{add}\)\(+\)\(C_{enc}\). In [PERSON] et al. 's scheme [24], at the \(d^{\text{th}}\) time step, the additive secret sharing cost is \(C_{\text{ss}}\), \(k\) hash functions cost is \(k\)-C\({}_{hash}\), and then \(k\)-order polynomial operation is x\({}^{k}\) and \(k\) matrix multiplication operations cost is \((k^{2}+2k)\)-C\({}_{mul}\), so the total computation overhead is: \(C_{ss}\)\(+\)\(k\)-C\({}_{lnsh}\)\(+\)\((k^{2}+2k)\)-C\({}_{mul}\). We provide the individual spatial computation overhead comparison in Table 3.  
As described in the related work, the scheme in Reference [23] sets all nodes as communication nodes instead of selecting a limited number of communication nodes as in ours and [22]; however, for convenient comparison, we assume that \(k\) communication nodes are selected, which is on the same experiment platform as ours and the scheme in [23]. Even under such relaxation, we can still prove ours is superior in terms of computation and communication cost through the following performance evaluation. The Figure 4 plots the comparison of spatial computation overhead between our scheme and the schemes in References [23; 24] with the value of \(k\) increasing. The Figure 4 shows that the three schemes' computation overheads all increase with the value of \(k\) increasing, the computation overhead in Reference [23] and ours are lower compared with the scheme in References [24], in which polynomial \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notations** & **Descriptions** & **Time Cost** \\ \hline C\({}_{add}\) & Addition & \(\approx\)0.038 ms \\ C\({}_{hash}\) & Hash (100 randoms) & \(\approx\)0.85 ms \\ C\({}_{mul}{mul}{}\) & Multiplication & \(\approx\)0.013 ms \\ C\({}_{\text{henc}}\) & Homomorphic encryption & \(\approx\)2.7 ms \\ C\({}_{\text{hdec}}\) & Homomorphic decryption & \(\approx\)0.61 ms \\ C\({}_{\text{ma}}\) & Hash/Modular addition & \(\approx\)0.0023 ms \\ C\({}_{prf}\) & Pseudorandom function & \(\approx\)0.074 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average time for functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Scheme** & **Computation Overhead Per Smart Meter** \\ \hline Scheme in [23] & \(C_{hash}\)\(+\)\(k\)-C\({}_{prf}\)\(+\)\(2k\)-C\({}_{add}\)\(+\)\(C_{enc}\) \\ Scheme in [24] & \(C_{ss}\)\(+\)\(k\)-C\({}_{hash}\)\(+\)\((k^{2}+2k)\)-C\({}_{mul}\) \\ Our scheme & \(C_{ma}\)\(+\)\(k\)-C\({}_{prf}\)\(+\)\(k\)-C\({}_{add}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Individual spatial computation overhead comparison (msec). operation \(x^{k}\) and \(k\) matrix multiplication operations generate too much computation overhead with \(k\) growing, it has more cost significantly than ours and Erkin et al.