## 5 Security Analysis
### Anonymity and Untraceability  
**Proposition 1**. : _The system achieves (preserves) \(\mathcal{P}\)'s anonymity and untraceability if for an adversary \(\mathcal{A}\) and \(\forall\{\alpha_{0},\alpha_{1}\}\), in a probabilistic polynomial time, \(\mathcal{A}\) has only at most an insignificant advantage over \(\mathcal{A}^{\tau}\) to win; that is, this is true if_  
\[ \begin{split}\mathit{Prob}[\delta^{{}^{\prime}}& \leftarrow\mathcal{A}(\mathit{SMP},\rho_{\mathcal{A}},\alpha_{0}, \alpha_{1})\colon\delta=\delta^{{}^{\prime}}]-\\ \mathit{Prob}[\delta^{{}^{\prime}}&\leftarrow \mathcal{A}^{\tau}(\rho_{\mathcal{A}},\alpha_{0},\alpha_{1})\colon\delta= \delta^{{}^{\prime}}]\leq\epsilon,\end{split}  \tag{4}\]  
_where \(\mathcal{A}^{\tau}\) is an adversary that reacts to all game difficulties with statistical inferences and \(\epsilon\) is the difference in probabilities for the adversary \(\mathcal{A}\), which is insignificant with respect to the security parameter \(\rho\)._  
Proof. : The proposed approach leverages the E-Bitmessage process during the course of communication in the SMP. Thus, the pair of session IDs is changed every time a session starts, therefore being used once and discarded. Hence, this makes it to almost impossible for \(\mathcal{A}\) to reply with a correct estimate \(\delta^{{}^{\prime}}\). As a result, \(\mathcal{A}\) has no or a less significant advantage over \(\mathcal{A}^{\tau}\) to win the game. Thus, changing the session IDs for each connection improves the security in terms of perfect forward secrecy [10]. This implies that even if the private pair of the session IDs gets stolen by any means, such as correlation of the public pair of the session IDs in the SMP, not only are past communications still secure, but all following communications will also be secure. This feature strengthens and ensures transactional data privacy in the system. All pairs of IDs only last for one trading session. However, encryption always needs the public key of the recipient, as well as decryption always needing the decryptor's private key. Unlike the public/private pairs of the session IDs, the original private/public key pairs are never dropped. Thus, even if the adversary is able to obtain one single ID out of the communication, it will neither be able to read messages from the past nor upcoming messages at a probabilistic polynomial time. **Theorem 1**. : _If anonymity and untraceability are achieved (preserved) for a given participant \(\mathcal{P}\), then it is presumed that the adversary \(\mathcal{A}\) cannot be able to launch any identity-related attacks._Proof. : (_By contradiction_) Assuming that \(\mathcal{A}\) is able to launch any of the aforementioned attacks even when the participants are anonymous and untraceable, this implies that for all adversaries \(\mathcal{A}\) and \(\forall\{\alpha_{0}\}\), \(\mathcal{A}\) has a significant advantage over \(\mathcal{A}^{\tau}\) to win the game at a probabilistic polynomial time, even though the system has satisfied transactional data privacy, anonymity and untraceability for the participant \(\mathcal{P}\). This statement contradicts Proposition 1. Hence, Theorem 1 holds. The participants' anonymity and untraceability offered by the proposed blockchain-based approach can be realized by measuring the degree to which the public pair of session IDs in the SMP can be linked to a particular address (public key) of a given participant. Proof. : This can be achieved by computing the estimate of \(\mathcal{A}\) using an \(n_{A}Xn_{A}\) matrix and \(E_{track}\), where \(E_{track}[i,j]=\{Prob_{i,j}\},i,j\in[1,n_{\mathcal{A}}]\). Then, the attacker \(\mathcal{A}\) calculates the probability \(Prob_{i,j}\) for which, in every ID \(\alpha_{i}\), a given \(\alpha_{i}\) belongs to the same participant as every \(\alpha_{i}\) in the SMP. However, \(E_{track}\) also consists of extra information that can be extracted from the SMP by \(\mathcal{A}\). This can be accomplished by means of statistical analysis or any other processes.