# Profitable Double-Spending Attacks
## 4 Profitable DS Attacks  
The previous probabilistic analyses in [1; 6] have shown that the success of DS attacks is not guaranteed when \(p_{A}<0.5\). However, DS attacks with \(p_{A}<0.5\) can be vigorously pursued as long as they bring profit. We analyze the profitability of DS attacks and to this end, we define a profit function \(P\) of a DS attack \(\text{DS}(C,p_{A},t_{cut};N_{BC})\), where \(C\) is the value of a fraudulent transaction, in terms of revenue and operating expense (OPEX) of the computing power. The OPEX \(X\) (e.g., the rental fee for the computing power) and the block mining reward \(R\) tend to increase with respect to \(\lambda_{A}\) and the time \(t\) consumed during the attack. Thus, \(X\) and \(R\) are expressed as functions of \(\lambda_{A}\) and \(t\), and they can be any increasing function; e.g., linear, exponential, or logarithm. We define \(X\) and \(R\), respectively, as follows:  
\[X(\lambda_{A},t):=\gamma\lambda_{A}t\Big{(}\log_{x_{1}}x_{2}\Big{)}^{\lambda_{ A}}\Big{(}\log_{x_{3}}x_{4}\Big{)}^{t} \tag{29}\]  
for real constants \(\gamma>0\), \(x_{1},x_{2}>1\), and \(x_{3},x_{4}>1\), and  
\[R(\lambda_{A},t):=\beta\lambda_{A}t\Big{(}\log_{\tau_{2}}r_{2}\Big{)}^{\lambda _{A}}\Big{(}\log_{\tau_{3}}r_{4}\Big{)}^{t} \tag{30}\]  
for real constants \(\beta>0\), \(r_{1},r_{2}>1\), and \(r_{3},r_{4}>1\). We denote the ratio of \(\gamma\) and \(\beta\) by  
\[\mu:=\beta\gamma^{-1}. \tag{31}\]  
With regards to \(P\), if an attack succeeds, the revenue comes from \(C\), as it is double-spent, added to \(R\) for the number of blocks mined during the time duration \(T_{AS}\), i.e., \(R(\lambda_{A},T_{AS})\). In this case, the cost is the OPEX for the time duration \(T_{AS}\), i.e., \(X(\lambda_{A},T_{AS})\). If the attack fails, the cost is the OPEX \(X(\lambda_{A},t_{cut})\) for the time duration \(t_{cut}\), and there is no revenue. Hence, for a DS attack \(\text{DS}(C,p_{A},t_{cut};N_{BC})\), we define \(P\) as follows,  
\[P:= \begin{cases}C+R(\lambda_{A},T_{AS})-X(\lambda_{A},T_{AS}),&\text{if}\quad T _{DSA}<t_{cut},\\ -X(\lambda_{A},t_{cut}),&\text{otherwise}.\end{cases}  \tag{32}\]  
Subsequently, the expected profit function is  
\[ \begin{array}{ll}\mathbb{E}_{P}&=\mathbb{P}_{AS}(t_{cut})\cdot(C+\mathbb{E}[ R(\lambda_{A},T_{AS})]-\mathbb{E}[X(\lambda_{A},T_{AS})])-(1-\mathbb{P}_{AS}(t_{ cut}))X(\lambda_{A},t_{cut})\\ &=\mathbb{P}_{AS}(t_{cut})\cdot(C+\mathbb{E}[R(\lambda_{A},T_{AS})])-\mathbb{E} _{X},\end{array}  \tag{33}\]  
where \(\mathbb{E}_{X}\) is the expected OPEX defined as  
\[\mathbb{E}_{X}:=\mathbb{P}_{AS}(t_{cut})\mathbb{E}[X(\lambda_{A},T_{AS})]+(1- \mathbb{P}_{AS}(t_{cut}))X(\lambda_{A},t_{cut}). \tag{34}\]  
A DS attack \(\text{DS}(C,p_{A},t_{cut};N_{BC})\) is said to be profitable if and only if the expected profit \(\mathbb{E}_{P}>0\), where \(\mathbb{E}_{P}\) is defined in Equation (33). The key factor in determining the profitability of DS attacks is the value \(C\) of the fraudulent transaction. Thus, attackers would be interested in the minimum value required for profitable DSattacks [25]. Definition 5 implies that a DS attack \(\text{DS}(C,p_{A},t_{cut};N_{BC})\) is profitable if and only if \(C>C_{\text{Req. }}\), where the required value of target transaction \(C_{\text{Req. }}\) is  
\[C_{\text{Req.}}=\frac{\mathbb{E}_{X}}{\mathbb{P}_{AS}}-\mathbb{E}[R(\lambda_{ A},T_{AS})]. \tag{35}\]  
The following results in Theorem 1 and Theorem 2 focus on the case where both \(X(\lambda_{A},t)\) and \(R(\lambda_{A},t)\) are linearly increasing functions of \(\lambda_{A}\) and \(t\). **Theorem 1**. : _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 \(\text{DS}(C,p_{A},t_{cut};N_{BC})\) for any \(p_{A}\in(0,1)\) and for any \(t_{cut}\in(0,\infty]\) is profitable if and only if \(C>C_{\text{Req}}\), where_  
\[C_{\text{Req.}}=\frac{(1-\mathbb{P}_{AS}(t_{cut}))}{\mathbb{P}_{AS}(t_{cut}) }\gamma\lambda_{A}t_{cut}-(\mu-1)\gamma\lambda_{A}\mathbb{E}_{T_{AS}}(t_{cut}). \tag{36}\]  
Proof.