# Supervised Blockchain Anonymous Transaction Model Based on Certificateless Signcryption
## 4 System and Security Models
### Security Model
The challenger \(C\) interacts with adversary \(A_{II}\) as follows: (1) Initialization: \(C\) executes the system initialization algorithm to generate the system master key \(s\) and public parameters PP. \(C\) sends both PP and \(s\) to \(A_{II}\). (2) Query Phase: \(A_{II}\) adaptively issues queries to the oracles except private key queries and public key replacement queries, under the constraints:
* \(A_{II}\) cannot request a private key query for the receiver \(EID_{j}\). * \(A_{II}\) cannot request an unsigncryption query for the challenge ciphertext \(C^{*}\). (3) Challenge Phase: Same as in Game 1. (4) Guess Phase: \(A_{II}\) outputs a guess \(b^{\prime}\). If \(b=b^{\prime}\), \(A_{II}\) wins the game. The advantage of \(A_{II}\) is defined analogously as: \[Adv_{A_{2}}^{IND-CLSC-CCA2}=|Pr[b^{\prime}=b]-\frac{1}{2}|\] (3)  
**Definition 2** (Unforgeability). : _Under the Random Oracle Model (ROM), if no adversary can achieve a non-negligible advantage in winning Game 3 and Game 4 within probabilistic polynomial time (PPT), then the CLSC_SAT model satisfies existential unforgeability of certifateless signcryption under chosen message attack (EUF--CLSC--CMA)._  
**Game 3** (EUF--CLSC--CMA-G3): The challenger \(C\) interacts with adversary \(A_{I}\) as follows: (1) Initialization: \(A_{I}\) adaptively executes the same initialization phase as in Game 1. (2) Query Phase: \(A_{I}\) adaptively performs the above oracle query if the following conditions are met. * \(A_{I}\) cannot request a signcryption query for the tuple \(\{m,EID_{i},EID_{j}\}\). * If the sender \(EID_{i}\)'s public key has been replaced, A1 cannot request a partial private key query for \(EID_{i}\). * \(A_{I}\) cannot request a private key query for the sender \(EID_{i}\). (3) Forgery Phase: \(A_{I}\) outputs a forged signcryption \(C^{*}\) for message \(m\). If there is no output \(\bot\), the signature is legal and \(A_{I}\) wins this game. The advantage of \(A_{I}\) is defined as: \[Adv_{A_{1}}^{EUF-CLSC-CMA}(k)=Pr[Succ_{A_{1}}(k)]\] (4) where \(Succ_{A_{1}}(k)\) denotes the probability that \(A_{I}\) successfully forges a valid signcryption under security parameter \(k\). **Game 4** (EUF--CLSC--CMA-G4): The challenger \(C\) interacts with adversary \(A_{II}\) as follows: (1) Initialization: \(A_{II}\) adaptively executes the same initialization phase as in Game 2. 2. Query Phase: \(A_{II}\) adaptively performs the above oracle query if the following conditions are met. * \(A_{II}\) cannot request a signcryption query for the tuple \(\{m,EID_{i},EID_{j}\}\). * \(A_{II}\) cannot request a private key query for the sender \(EID_{i}\). * 3. Forgery Phase: \(A_{II}\) outputs a forged signcryption C\({}^{*}\) for message \(m\). If there is no output \(\bot\), the signature is legal and \(A_{II}\) wins this game. The advantage of \(A_{II}\) is defined as: \[Adv_{A_{2}}^{EUF-CLSC-CMA}(k)=Pr[Succ_{A_{2}}(k)]\] (5) where \(Succ_{A_{2}}(k)\) denotes the probability that \(A_{II}\) successfully forges a valid signcryption under security parameter \(k\).