## 5 Spatio-Temporal Big Data Collaborative Storage Mechanism
### Aggregation Proof Protocol
**

\[\Lambda_{K} =\Lambda_{I}^{e_{INK}}\cdot\prod_{j\in I\setminus K}\chi_{j}^{v_{ j}} \tag{18}\] \[=\left(\prod_{j=1,j\
otin I}^{n}S_{j}^{v_{j}}\right)^{\frac{1}{ \Lambda_{I}}e_{I\cap K}}\cdot\prod_{j\in I\setminus K}\left(S_{j}^{\frac{1}{ \Lambda_{K}}}\right)^{v_{j}}\] \[=\left(\prod_{j=1,j\
otin I}^{n}S_{j}^{v_{j}}\right)^{\frac{1}{ \Lambda_{K}}}\cdot\left(\prod_{j\in I\setminus K}S_{j}^{v_{j}}\right)^{1/e_{ K}}\] \[=\left(\prod_{j=1,j\
otin K}^{n}S_{j}^{v_{j}}\right)^{1/e_{K}}\]

\(\rho_{I}\) and \(\sigma_{J}\) can be calculated as follows:

\[\rho_{I}:=\frac{\Lambda_{I}}{\Pi_{j\in I}\phi_{j}^{v_{j}}}=\left(\prod_{j=1,j \
otin I\cup J}^{n}S_{j}^{v_{j}}\right)^{1/e_{I}} \tag{19}\]\[\sigma_{l}:=\frac{\Lambda_{I}}{\prod_{j\in I}\psi_{j}^{v_{j}}}=\left( \prod_{j=1,j\
otin I\cup J}^{n}S_{j}^{v_{j}}\right)^{1/\epsilon_{j}} \tag{20}\] \[\Lambda_{K} :=ShamirTick(\rho_{l},\sigma_{l},\epsilon_{l},\epsilon_{l})\] \[=ShamirTick\left(\left(\prod_{j=1,j\
otin I\cup J}^{n}S_{j}^{v_{j} }\right)^{1/\epsilon_{l}},\left(\prod_{j=1,j\
otin I\cup J}^{n}S_{j}^{v_{j}} \right)^{1/\epsilon_{j}},\epsilon_{l},\epsilon_{l}\right)\] \[A_{K}\gets VC.Agg\big{(}(S_{I},S_{J}),(I,V_{I},A_{I}),(I,V_{J},A_{J}) \big{)} \tag{17}\]0

Theorem 4.1: _A malicious off-chain storage server that fails to convince the verifier to accept invalid outputs proves that the scheme is secure, i.e., the attacker succeeds if it persuades the verification algorithm to validate incorrect output values based on the input values. For any probability polynomial time adversary \(A\) has_

\[A_{K}\gets VC.Agg\big{(}(S_{I},S_{J}),(I,V_{I},A_{I}),(I,V_{J},A_{J}) \big{)} \tag{17}\]1

_where negl refers to the negligible function with respect to \(k\). Specifically, \(A\) has advantages in the following Algorithm 3:_

```
Input: security parameter \(k\) Output: accept or reject
1:\((pp,\delta_{0},st_{0})\leftarrow\text{Setup}(1^{k})\)
2:\(((op^{i},\Delta^{i},Y^{i}_{\Delta}),Q,V^{*}_{Q^{*}},\pi^{*})\gets A(pp, \delta_{0},st_{0})\)
3:for\(i\in[I]\)do
4:if\(op^{i}\in\{\text{mod},\text{add}\}\)then parse \(\Delta^{i}=(K,V^{i}_{k})\)
5:\(\forall i\in K\):\(V^{*}_{i}\gets V^{i}_{i};\forall i\in[|V|]\backslash K\):\(V^{*}_{i}\gets V_{i}\)
6:elseif\(op^{i}=\text{del}\)then parse \(\Delta^{i}=K\)
7:\(\forall i\in K\):\(V^{*}_{i}\gets V_{i}\);
8: run the update algorithm: input \((\delta_{i-1},op^{i},\Delta^{i},Y^{i}_{\Delta}),\text{output}(b_{i},\delta_{i})\);
9:endif
10:\(b\leftarrow\bigwedge b_{i}\)
11:endfor
12:\(b\gets b\wedge V^{*}_{i}\
eq V_{i}\wedge VC.Ver(pp,C,Q,V_{Q},\Lambda_{I})\);
13:return\(b\)
```

**Algorithm 3** Experiment

Proof: This approach can be regarded as sustaining and revising a vector commitment \(C\) and an RSA accumulator \(U\). \(U\) represents the RSA accumulator for all \(\epsilon_{i}\) and serves to authenticate \(S_{I}\). For every accumulator to maintain its integrity, the calculated cumulative value \(U\) must be derived truthfully. This is ensured in this scheme as we assume the existence of a valid history. It is possible to know through historical data whether \(U\) is of the correct form (\(U=g^{\prod_{i\in[i]}\epsilon_{i}}\)), and consequently it can be safely checked \(S_{I}\) is in the proper form (by checking \(S_{I}^{\prod_{i\in I}\epsilon_{i}}=U\)), this can be guaranteed by the security of the RSA accumulator. Upon verifying the validity of \(S_{I}\), it reduces to a positional binding issue for vector promises. Therefore, we give the theorem and proof of position binding of vector commitment below. **Theorem 3**. : _Consider Ggen as the generator of the hidden order group, where the low-order assumption is valid. Consequently, the previously defined SVC scheme possesses the location binding property._

Proof. : We begin by defining the game \(S_{0}\) as the genuine position-binding game stated in the theorem, with the goal of proving such that for any probabilistic polynomial-time algorithm A, \(Pr[\text{Game}_{0}=1]=\text{negl}(\lambda)\).