## 5 Spatio-Temporal Big Data Collaborative Storage Mechanism
### Aggregation Proof Protocol
\begin{tabular}{l} \hline Game Game\({}_{0}\) \\ \hline
1: \(\mathit{crs}\leftarrow\text{VC.Setup}(1^{n},m)\) \\
2: \((C,I,\pi,\tau,\tau^{*})\gets A(\mathit{crs})\) \\
3: \(b\gets V\cup\mathit{V.Ver}(\mathit{crs},C,I,\vec{y},\pi)=1\wedge\vec{y} \
eq\vec{y}^{*}\wedge\mathit{VC.Ver}(\mathit{crs},C,I,\vec{y}^{*},\pi^{*})=1\); \\
4: **return**\(b\) \\ \hline \end{tabular} 

Now, let the game \(Game_{1}\) be the same as above, except the opponent outputs \(S_{I}\) and \(S_{I}^{\prime}\) making \(S_{I}=g^{e_{[n]}}I=S_{I}^{*}\). \begin{tabular}{l} \hline Game \(Game_{1}\) \\ \hline
1: \(\mathit{crs}\leftarrow\mathit{VC.Setup}(1^{\lambda},1,n)\); \\
2: \((C,I,\vec{y},(S_{I},A_{I}),\vec{y}^{*},(S_{I}^{*},A_{I}^{*}))\leftarrow\mathcal{ A}(\mathit{crs})\); \\
3: **if**\(S_{I}\
eq g^{e_{[n]}\lor t}\) **or**\(S_{I}^{*}\
eq g^{e_{[n]}\lor t}\) **then abort** \\
4: \(b\gets C=\Lambda_{I}^{e_{I}}\prod_{i\in I}\left(S_{I}^{e_{I}(i)}\right)^{y _{i}}\wedge\vec{y}=\vec{y}^{*}\wedge C=\Lambda_{I}^{*e_{I}}\prod_{i\in I} \left(S_{I}^{*e_{I}(i)}\right)^{\vec{y}^{*}}\); \\
5: **return**\(b\) \\ \hline \end{tabular} 

Then, \(Pr[\text{G}_{0}=1]<Pr[\text{G}_{1}=1]+\text{negl}(\lambda)\). In \(Game_{0}\), \(S_{\text{I}}^{\text{e}_{\text{I}}}=\text{U}_{n}=\text{g}^{e_{[n]}}\). Assume \(\text{S}_{\text{I}}\
eq\text{g}^{e_{[n]}\lor t}\) then \(\text{g}^{e_{[n]}\lor t}=\text{S}_{1}^{*}\); therefore, \(\text{S}_{\text{I}}^{\text{e}_{\text{I}}}=\text{S}_{\text{I}}^{*e_{\text{I} }}\), \(\left(\text{S}_{\text{I}}^{-1}\text{S}_{\text{I}}^{*}\right)^{\text{e}_{\text{I }}}=1\). Since \(S_{I}^{*}\) can be computed efficiently, and \(\text{e}_{\text{I}}<2^{\text{poly}(\lambda)}\) holds. This forms a low-order solution to the hidden order group Solution to the problem. Under low-level assumptions, it only occurs with negligible probability, and the same is true for \(S_{I}^{*}\). Let game \(Game_{2}\) be the same as above, except each opponent needs to receive \(e_{i}\gets PrimeGen(i)\) and \(S_{i}=g^{e_{[n]}\lor(i)}\), where \(i\in[n]\). \begin{tabular}{l} \hline Game \(Game_{2}\) \\ \hline
1: \((G,g,\text{PrimeGen})\gets VC.Setup(1^{\lambda},1,n)\); \\
2: \(e_{i}\leftarrow PrimeGen(i)\); \\
3: **for each**\(i\) **in**\([n]\)**do** \\
4: \(S_{I}=g^{\prod_{i\in[n]}(i)^{e_{i}}}\); \\
5: **end for** \\
6: \((C,I,\vec{y},A_{I},\vec{y}^{*},A_{I}^{*})\leftarrow\mathcal{A}(G,g,\text{ PrimeGen},\{S_{i}\}_{i\in[n]})\); \\
7: \(b\gets C=\Lambda_{I}^{e_{I}}\prod_{i\in I}S_{i}^{e_{i}}\wedge\vec{y}=\vec{y} ^{*}\wedge C=\Lambda_{I}^{*e_{I}}\prod_{i\in I}S_{i}^{\vec{y}^{*}}\); \\
8: **return**\(b\) \\ \hline \end{tabular} 

It can be directly concluded that \(Pr[\text{Game}_{1}=1]=Pr[\text{Game}_{2}=1]\), and \(Game_{2}\) Same as the position-bound game of the SVC scheme, based on the assumption \(Pr[\text{Game}_{2}=1]=\text{negl}(\lambda)\). Therefore, we conclude that the previously defined SVC scheme possesses the location binding property. ## 6 Performance of Our Scheme

In this scheme, the experimental analysis is divided into off-chain and on-chain parts based on whether the operation is executed on the blockchain. Firstly, we tested the communication overhead of the system, focusing mainly on the setup phase. Then, weevaluated the time cost of off-chain operations to assess the computational overhead of our protocol off-chain. We tested the time costs for querying, verification, and validation of the off-chain storage server with different numbers of data blocks. On-chain, we primarily conducted simulation analysis on the efficiency of smart contract execution. The proposed scheme was implemented using the pairing-based cryptography library (JPBC) and the off-chain storage database was deployed using MongoDB. The smart contract was deployed in the JavaScript VM environment of the Remix compiler.