# Profitable Double-Spending Attacks
## 4 Profitable DS Attacks
: Substituting \(x_{1}=x_{2}\), \(x_{3}=x_{4}\), \(r_{1}=r_{2}\), and \(r_{3}=r_{4}\) into Equation (35) results in Equation (36). Theorem 1 shows that not only superior attackers with \(p_{A}\in(0.5,1)\) but also inferior attackers with \(p_{A}\in(0,0.5)\) are able to expect profitable DS attacks once a high enough value \(C\) greater than \(C_{\text{Req. }}\) of the target transaction is designed. The condition \(C_{\text{Req. }}\) in Equation (36) can be pre-computed before carrying out an attack, as it stochastically estimates the future expected cost, for a given position \(p_{A}\in(0,1)\) and a cut-time \(t_{cut}\) of an attacker, and a given set of network environment parameters \(\gamma\) and \(\beta\). Tables 1 and 2 list the resources including \(C_{\text{Req}}\), \(\mathbb{E}_{X}\), and \(\mathbb{E}_{T_{AS}}\) required for profitable DS attacks respectively using \(p_{A}=0.35\) and \(p_{A}=0.4\), when \(t_{cut}=cN_{BC}\lambda_{H}^{-1}\) with \(c=4\). Note that the expectation of the time spent for the block confirmation equals \(N_{BC}\lambda_{H}^{-1}\), and we let \(t_{cut}\) linear to it. In other words, as normal traders wait for \(N_{BC}\lambda_{H}^{-1}\) seconds on the average, attackers shall be tolerable as well and wait for the same scale of time duration. Note that the \(\mathbb{P}_{AS}\) for \(N_{BC}=1\) is smaller than that for \(N_{BC}=3\) due to not long enough \(t_{cut}\). We scaled the results by parameters \(\lambda_{H}\) and \(\gamma\), which we will explain how to obtain from the internet in the next subsection. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Block Confirmation Number \((N_{BC})\)** & **1** & **3** & **5** & **7** & **9** \\ \hline Attack success probability \((\mathbb{P}_{AS})\) & 0.315 & 0.279 & 0.218 & 0.170 & 0.132 \\ \hline Expected attack success time \(\left(\mathbb{E}_{T_{\infty}}\right)\)(Scaled by \(\lambda_{H}\)\({}^{-1}\)) & 2.004 & 5.518 & 8.681 & 11.694 & 14.607 \\ \hline Expected OPEX \((\mathbb{E}_{X})\)(Scaled by \(\gamma\)) & 1.815 & 5.487 & 9.440 & 13.588 & 17.859 \\ \hline \hline \multirow{3}{*}{Required value of target transaction (\(C_{Suf.}\)) (Scaled by \(\gamma\))} & 1.079 & 2.971 & 4.675 & 6.297 & 7.866 \\  & \(\cdot(1-\mu)\) & \(\cdot(1-\mu)\) & \(\cdot(1-\mu)\) & \(\cdot(1-\mu)\) & \(\cdot(1-\mu)\) \\ \cline{1-1}  & \(+\) 4.680 & \(+\) 16.68 & \(+\) 38.62 & \(+\) 73.84 & \(+\) 127.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical computations of required resources for profitable double-spending (DS) attacks with \(p_{A}=0.35\) when \(t_{cut}=cN_{BC}\lambda_{H}^{-1}\) with \(c=4\). The following Theorem 2 is for the inferior attackers with \(p_{A}\in(0,0.5)\) and shows the importance of setting a finite \(t_{cut}\). Suppose \(x_{1}=x_{2}\) and \(x_{3}=x_{4}\) in Equation (29), and \(r_{1}=r_{2}\) and \(r_{3}=r_{4}\) in Equation (30). Then, a DS attack \(\mathsf{DS}(C,p_{A},t_{cut};N_{\text{BC}})\) with \(p_{A}\in(0,0.5)\) is profitable only if \(t_{cut}<\infty\). Proof. : For any \(p_{A}\in(0,0.5)\), it always holds that \(\mathbb{P}_{AS}<1\). In this case, if \(t_{cut}\to\infty\) then \(C_{\text{Req. }}\to\infty\) from Equation (36); i.e., infinite value \(C\) of fraudulent transaction is required for a DS attack \(\mathsf{DS}(C,p_{A},t_{cut};N_{\text{BC}})\) to be profitable. Thus, for a DS attack with \(p_{A}\in(0,0.5)\) to be profitable, a finite cut-time \(t_{cut}<\infty\) must be set. Theorem 2 shows that for \(p_{A}\in(0,0.5)\), setting \(t_{cut}=\infty\) is expected to incur infinite deficit. On the contrary, for \(p_{A}\in(0.5,1)\), what we have numerically checked out but omitted due to space limitation is the result that \(\mathbb{E}_{P}\) is an increasing function of \(t_{cut}\); i.e., setting \(t_{cut}=\infty\) is the optimal choice in the superior attack regime. Applying \(p_{A}\in(0.5,1)\) and \(t_{cut}=\infty\) into Equation (36) leads to \(\mathbb{P}_{AS}=1\), and thus \(C_{\text{Req.