# Supervised Blockchain Anonymous Transaction Model Based on Certificateless Signcryption
## 5 Concrete Scheme
### Specific Transaction Process
The transaction amount is encrypted using two methods: one is Paillier homomorphic encryption for transaction verification, where [PERSON] randomly selects an integer \(r_{1}\in Z_{q}^{*}\) and uses the regulatory public key \(PK_{R}=(n,g)\) to compute \(C_{A,1}=g^{\text{tx}}v_{1}^{\text{m}}\bmod n^{2}\); the other is certificateless signcryption for transaction updates, where [PERSON] randomly selects \(u_{A}\in Z_{q}^{*}\), computes \(U_{A}=u_{A}P\), \(T_{A}=u_{A}(X_{B}+G_{B}+h_{B,2}P_{pub})\), with \(h_{B,2}=H_{2}(EID_{B},X_{B},G_{B},P_{pub})\), computes \(C_{A,2}=v_{0}\oplus h_{A,3}\), with \(h_{A,3}=H_{3}(U_{A},T_{A},X_{B},G_{B})\), and computes \(S_{A}=u_{A}+h_{A,3}x_{A}+h_{A,4}d_{A}\), with \(h_{A,4}=H_{4}(EID_{B},C_{A,2},X_{B},U_{A},t_{A})\). The signature \(\sigma_{A}=(U_{A},S_{A},C_{A,1},C_{A,2})\) for \(v_{0}\) is generated, and the transaction \(tx_{send}=(EID_{A},PK_{A},\sigma_{A},t_{A})\) is sent on the blockchain. Additionally, [PERSON] makes a commitment and proves to the miner nodes through a commitment proof that the encrypted amount is always positive. ```
Input: PP, \(EID_{l_{1}}\), \(PK_{l_{1}}\), \(SK_{l_{1}}\), \(v_{0}\), \(EID_{l_{2}}\), \(PK_{l_{2}}\), \(PK_{R}\) Output: \(tx_{send}\)
1. Sender randomly selects \(r_{1}\in Z_{q}^{*}\). 2. Calculate \(C_{i_{1},1}=g^{h_{1}}g^{h_{1}}_{mod}\)\(n^{2}\). 3. Sender selects \(u_{i_{1}}\in Z_{q}^{*}\). 4. Calculate \(U_{l_{1}}=u_{i_{1}}P\). 5. Calculate \(h_{i_{2},2}=H_{2}(EID_{l_{2}},X_{l_{2}},G_{i_{2}},P_{pub})\). 6. Calculate \(T_{i_{1}}=u_{i_{1}}(X_{i_{2}}+G_{i_{2}}+h_{i_{2},2}P_{pub})\). 7. Calculate \(h_{i_{1},3}=H_{3}(U_{i_{1}},T_{i_{1}},X_{i_{2}},G_{i_{2}})\). 8. Calculate \(C_{i_{1},2}=v_{0}\oplus h_{i_{1},3}\). 9. Calculate \(h_{i_{1},4}=H_{4}(EID_{l_{2}},C_{i_{2}},X_{i_{2}},U_{i_{1}},t_{i_{1}})\). 10. Calculate \(S_{i_{1}}=u_{i_{1}}+h_{i_{1},3}x_{i_{1}}+h_{i_{1},4}d_{i_{1}}\). 11. Generate a signature \(\alpha_{i_{1}}=(U_{i_{1}},F_{i_{1}},C_{i_{1},2},C_{i_{1},2})\). 12. Send the transaction \(tx_{send}=(EID_{l_{1}},PK_{l_{1}},\sigma_{l_{1}},t_{i_{1}})\). ```  
**Algorithm 6**Transaction creation algorithm. (7) \(VerifyTx(tx_{send})\to b\): Algorithm 7 is invoked by the miner nodes and Alice. The miner nodes verify the correctness of \(C_{A,1}\) and the signature \(\alpha\). The correctness of \(\sigma\) is verified by checking whether the equation \(S_{A}P=U_{A}+h_{A,3}X_{A}+\)\(h_{A,4}(G_{A}+h_{A,1}P_{pub})\) holds. If it holds, \(tx_{send}\) is considered valid and legitimate, and the output is \(b=1\). The miner nodes then use a consensus algorithm to add the transaction record to the blockchain and broadcast it. Otherwise, the output is \(b=0\), and the transaction is sent to the regulatory authority for auditing. The correctness of \(C_{A,1}\) is proved as follows:  
**Lemma 1**. : _For \(\forall\theta\in{Z_{n^{2}}}^{*}\), the equation \(L(\theta^{\lambda}\bmod n^{2})=\lambda[\theta]_{n+1}\) holds._  
For any given base \(g\in{Z_{n^{2}}}^{*}\) and \(\theta\in{Z_{n^{2}}}^{*}\), \(L(\theta^{\lambda}\bmod n^{2})\) is invertible modulo n, and obtain  
\[\frac{L(\theta^{\lambda}\bmod n^{2})}{L(g^{\lambda}\bmod n^{2})}=\frac{ \lambda[\theta]_{1+n}}{\lambda[g]_{1+n}}=\frac{[\theta]_{1+n}}{[g]_{1+n}}=[ \theta]_{g}\bmod n \tag{6}\]  
In the above equation, given \(n,g,\theta\), computing \([\theta]_{g}\) is computationally hard. However, if \(\lambda\) is known, it is possible to compute any \([\theta]_{g}\). The correctness of \(\sigma\) is proved as follows:  
\[ \begin{array}{ll}S_{i}P&=(u_{i}+h_{i,3}x_{i}+h_{i,4}d_{i})\\ &=U_{i}+h_{i,3}X_{i}+h_{i,4}(g_{i}+sh_{i,1})P\\ &=U_{i}+h_{i,3}X_{i}+h_{i,4}(G_{i}+h_{i,1}P_{pub})\end{array}  \tag{7}\]  
```
Input: PP, \(tx_{send}\) Output: \(b\)
1. Calculate \(\frac{L(\theta^{\lambda}\bmod n^{2})}{L(g^{\lambda}\bmod n^{2})}\) to verify the correctness of \(C_{A,1}\). 2. Calculate \(S_{i_{1}}P=U_{i_{1}}+h_{i_{1},3}X_{i_{1}}+h_{i_{1},4}(G_{i}+h_{i_{1},1}P_{pub})\). ```  
**Algorithm 7**Transaction verification algorithm.