# Blockchain-Based Distributed Computing Consistency Verification for IoT Mobile Applications
## 7 Security Analysis  
In this section, we conduct a series of theoretical analyses to prove the security of BDCA. We first provide a security proof of the advanced gamma multi-signature scheme (AGMS) and then prove that BDCA can achieve the design goals defined in Section 5.3. The detailed security proofs are shown as follows:  
**Theorem 1**: _The security of AGMS can be assured under the assumption that the computation of discrete logarithms is a computationally infeasible problem._  
Proof. : The idea behind the proof lies in the notion that if it were computationally feasible to fabricate the signature of AGMS, it would imply that adversary \(\mathcal{A}\) has the ability to circumvent the discrete logarithm (DL) assumption [39]. Otherwise, if the DL assumption holds, then the security of AGMS can be ensured, thereby preventing potential cross-chain tampering attacks and privacy leakage attacks. We assume that adversary \(\mathcal{A}\) has the ability to resolve the DL assumption in \(\mathcal{G}\), which can forge a signature \((c^{*},\text{sig}^{*},hv^{*})\) to pass the verification. Thus, we can deduce Equation (5):  
\[ \begin{split} g^{sig}&=V_{\text{leader}}^{c}\, \cdot\,PK_{\text{leader}}^{-hv}\,=V_{\text{leader}}^{c}\,\prod_{i\in[1,\omega]}pk _{i}^{-hv}\\ g^{\text{sig}}&=V_{\text{leader}}^{c^{*}}\,\cdot\,PK_ {\text{leader}}^{-hv^{*}}\,=V_{\text{leader}}^{c^{*}}\,\prod_{i\in[1,\omega]}pk _{i}^{*-hv^{*}},\end{split}  \tag{5}\]  
where \(\left\{pk_{i}\right\}_{i\in[1,\omega]}\) and \(\left\{pk_{i}^{*}\right\}_{i\in[1,\omega]}\) are two public key sets aiming to forge the signature. We can conclude that the signature can be forged successfully only if \(V_{\text{leader}}^{c}\,=V_{\text{leader}}^{c^{*}}\), \(c^{-1}hv=c^{*-1}hv^{*}\), and \(\left\{pk_{i}^{-hv}\right\}_{i\in[1,\omega]}=\left\{pk_{i}^{*-hv^{*}}\right\} _{i\in[1,\omega]}\). Hence, we can deduce Equation (6). \[ \begin{split} V_{\text{leader}}&=g^{\text{sig}-1} \prod_{i\in[1,\omega]}pk_{i}^{c^{-1}}hvs.\\ V_{\text{leader}}&=g^{\text{sig}^{*}c^{*-1}}\prod _{i\in[1,\omega]}pk_{i}^{c^{*-1}hv^{*}}\end{split}  \tag{6}\]  
Based on Equation (6), we can obtain Equation (7) and it can be asserted that \(\mathcal{A}\) has the capability to generate all possible private keys with the exception of its own if \(c^{*}=c\) and \(e^{*}=e\) from Equation (7). According to [40], it can be concluded that the likelihood of the adversary \(\mathcal{A}\) succeeding in generating two distinct outputs that satisfy the verification criteria is exceedingly small. As a consequence, the robustness of the AGMS scheme can effectively mitigate the potential risks of cross-chain tampering attacks and privacy leakage attacks. \[g^{\text{sig}^{-1}-sig^{*}c^{*-1}}=\prod_{i\in[1,\omega]}pk_{i}^{c^{*-1}hv^{* }-c^{-1}hv} \tag{7}\]  
**Theorem 2**: _If all entities of the BDCA have duly complied with the prescribed procedures, it can be asserted that the consistency of dynamic cross-chain data can be reliably ensured._  
Proof. : The idea behind the proof lies in the notion that if all entities involved in the BDCA abide by the prescribed procedures in an honest and diligent manner, certain associated parameters can be computed with integrity and accuracy. Under such circumstances, \(AC\) is able to ensure the auditing consistency between the source chain (\(SC\)) and target chain (\(TC\)), thereby promoting the reliability and trustworthiness of cross-chain transactions. In the process of data auditing, \(AC\) generates verification information \(\Omega^{(s_{1})}\) based on the \(proof_{SC}\)and checks the validity of \(proof_{TC}\) based on Equation (4). The correctness and soundness of Equation (4) can be deduced as follows:  
\[