## 5 Construction  
Table 1 outlines the descriptions for each symbol utilised within the system. In accordance with the scheme definition, the operational procedure is divided into eight phases: initial system communication, key generation, the first stage of encryption, the second stage of encryption, the verification of message source, proxy key generation, the first stage of decryption, and the second stage of decryption. The subsection provides a detailed explanation of each phase. 1. System Setup: \(Init(\lambda)\longmapsto(PK,MK_{i},\mathbb{U}_{mix})\)  
In this algorithm, each group's AA provides an attribute set specific to its group, where \(\mathbb{U}_{i}\) represents the attributes of each group. The attribute sets from all groups are then combined into \(\mathbb{U}_{mix}=\bigcup_{i\in L_{c}}\mathbb{U}_{i}\), where \(L_{c}\) denotes the groups participating in the communication. The AAs collaborate to establish the public parameters \(PK\) and the master key \(MK_{i}\). Figure 2 shows the sequence diagram of this process, providing a step-by-step depiction of parameter generation, sharing, and collaboration among the AAs. First, the Access Originator \(AA_{o}\) generates the parameters \(\mathbb{G},g,\;\mathbb{Z}_{p}\) and the hash function \(H_{1}\). Using the Joint-Shamir RSS [33], \(AA_{o}\) defines two secret polynomials and computes sub-secret value \(f_{a}(0)=\alpha_{o}\) and \(f_{\beta}(0)=\beta_{o}\), where \(\alpha_{o},\;\beta_{o}\in\mathbb{Z}_{p}\). It then computes the sub-share value \(\gamma_{o,i}=\left\{f_{a}(i)=\gamma_{o,i}^{\alpha},\;f_{\beta}(i)=\gamma_{o,i }^{\beta}\right\}\), with \(\gamma_{o,i}^{\alpha},\;\gamma_{o,i}^{\beta}\in\mathbb{Z}_{p}\). \(AA_{o}\) sends the attribute set \(\mathbb{U}_{o}\), \(L_{c}\), and these parameters to the other groups' \(AA_{i}\). Each \(AA_{i}\) receives the parameters \(\left(G_{0},g,\;\mathbb{Z}_{p},\;\gamma_{o,i},\;\mathbb{U}_{o},\;L_{c},\;H_{1}\right)\). \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(AA\) & Attribute authority \\ \(CN\) & Compute node \\ \(CR\) & Controller \\ \(CT_{1}\) & A ciphertext generated by CR in the encryption phase \\ \(CT_{2}\) & A ciphertext generated by CN in the encryption phase \\ \(SK_{CR_{j}}\) & Attribute key of the \(CR_{j}\) \\ \(tk_{CR_{j}}\) & One – time proxy key of the \(CR_{j}\) \\ \(IDM\) & Intermediate decryption message \\ \(key\) & Encrypting and decrypting data with an asymmetric key \\ \(M\) & Plaintext \\ \(\mathcal{T}\) & An access tree \\ \(PK\) & Public parameter \\ \(MK_{i}\) & Master keys for group \(i\) \\ \(gpk_{i}\) & Public parameter for group \(i\) \\ \(H\) & One-way hash function \\ \(HMAC\) & Hash-based message authentication code \\ \(t,r\) & Transmitter and receiver, respectively \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation and description. After transmitting the parameters, each group's AA defines two new secret polynomials and computes sub-secret values, \(\alpha_{i}\) and \(\beta_{i}\) in \(\mathbb{Z}_{p}\), and sub-shares \(\gamma_{ij}=\left\{\gamma_{i,j}^{\alpha},\ \gamma_{ij,j}^{\beta}\right\}\). Based on \(L_{c}\), \(AA_{i}\) sends \(\left(\gamma_{i,j},\mathbb{U}_{i}\right)\), to the other groups, denoted as \(AA_{j}\), including \(AA_{o}\). Each \(AA\) in \(L_{c}\) receives sub-share values \(\gamma_{ij}\) and attributes \(\mathbb{U}_{i}\) from other groups. Using Equation (1), the master-share value \(\gamma_{j}=\left\{\gamma_{j}^{\alpha},\gamma_{j}^{\beta}\right\}\) is computed, where \(\gamma_{j}=\sum_{t=1}^{n}\gamma_{i,j}\ mod\ p\), and the combined attribute set \(\mathbb{U}_{mix}=\bigcup_{i\in L_{c}}\mathbb{U}_{i}\). The master-share value \(\gamma_{j}\) is then distributed to each AA, enabling the calculation of the master-secret value (\(\alpha\), \(\beta\)). Finally, each AA selects a random number \(u_{i}\in\mathbb{Z}_{p}\) and calculates \(gpk_{i}=g^{u_{i}}\) as the group public key.