Correction: Jang, J.; Lee, H.-N. Profitable Double-Spending Attacks. Appl. Sci. 2020, 10, 8477

Text Correction

1.

There was an error in the original publication [1]: “In a recentralized network, since most computing resources are concentrated on a small number of pools, it could be not difficult for them to conspire to alter the block content for their own benefits, if not aiming to double-spend, more probable.”

A correction has been made to this sentence located in Section 1, Introduction, in paragraph 4 on line 7:

“In a recentralized network, since most computing resources are concentrated on a small number of pools, it could be not difficult for them to conspire to alter the block content for their own benefits, if aiming to double-spend. ”

2.

There was an error in the original publication: “Definition 2. For a given cut-time $t_{cut}\in {ℝ}^{+}$, the success of DS attack is declared if and only if there exists a DSA time $T_{DSA}\in \left(0,t_{cut}\right)$ at which ${\mathcal{G}}^{\left(1\right)}$ and ${\mathcal{G}}^{\left(2\right)}$ in Definition 1 has been achieved.”

A correction has been made to Definition 2:

“Definition 2. For a given cut-time$t_{cut}\in {ℝ}^{+}$, the success of DS attack is declared if, and only if, there exists a DSA time$T_{DSA}\in \left(0,t_{cut}\right)$ atwhich ${\mathcal{G}}^{\left(1\right)}$and ${\mathcal{G}}^{\left(2\right)}$inDefinition1havebeenachieved.”

3.

There was an error in the original publication: “The sets ${\mathcal{D}}_j^{\left(1\right)}$ for each $j$ are mutually exclusive as each of them represents the first satisfaction of the block confirmation condition exactly at the $j$-th state.”

A correction has been made to this sentence located in Section 2, The Attack Model, Section 2.3, DS Attack Achieving Time, in the paragraph 5 on line 1:

“The sets ${\mathcal{D}}_j^{\left(1\right)}$ for all $j$ are mutually exclusive as each of them represents the first satisfaction of the block confirmation condition exactly at the $j$-th state.”

4.

There was an error in the original publication: “From Equation (13), the PDF of $T_{DSA}$ requires the probabilities of two random events; one is the state progression time $T_i$ in Equation (5); and the other is the event that a given state index $i$ satisfies $\omega \in {\mathcal{D}}_j^{\left(1\right)}\cap {\mathcal{D}}_{i,j}^{\left(2\right)}$.”

A correction has been made to this sentence located in Section 3, The Attack Probabilities, in the paragraph 2 on line 1:

“From Equation (13), the PDF of $T_{DSA}$ requires the probabilities of two random events: one is the state progression time $T_i$ in Equation (5); and the other is the event that a given state index $i$ satisfies $\omega \in {\mathcal{D}}_j^{\left(1\right)}\cap {\mathcal{D}}_{i,j}^{\left(2\right)}$.”

5.

There was an error in the original publication: “By taking infinite summations of $p_{DSA,j}$ in Lemma 1 for all indices $i\in \mathbb{N}$, we can compute the probability $P_{DSA}$ that a DS attack will ever achieve the necessary conditions in Definition 1.”

A correction has been made to this sentence located in Section 3, The Attack Probabilities, in the paragraph 4 on line 1:

“By taking infinite summations of $p_{DSA,i}$ in Lemma 1 for all indices $i\in \mathbb{N}$, we can compute the probability ${\mathbb{P}}_{DSA}$ that a DS attack will ever achieve the necessary conditions in Definition 1.”

6.

There was an error in the original publication: “If there does not exist such an index $i$, with the probability of $1-P_{DSA}$, then $T_{DSA}=\infty$.”

A correction has been made to this sentence located in Section 3, The Attack Probabilities, in the paragraph 5 on line 2:

“If there does not exist such an index $i$, with the probability of $1-{\mathbb{P}}_{DSA}$, $T_{DSA}=\infty$. ”

7.

There was an error to Equation (19) in the original publication:

$$“f_{T_{DSA}}\left(t\right)={\displaystyle \sum _{i=2N_{BC}+1}^{\infty }{p}_{DSA,i}{f}_{T_i}\left(t\right)}+\left(1-P_{DSA}\right)\delta \left(t-\infty \right),”$$

A correction has been made to Equation (19):

$$“f_{T_{DSA}}\left(t\right)={\displaystyle \sum _{i=2N_{BC}+1}^{\infty }{p}_{DSA,i}{f}_{T_i}\left(t\right)}+\left(1-{\mathbb{P}}_{DSA}\right)\delta \left(t-\infty \right),”$$

8.

There was an error in the original publication: “The previous probabilistic analyses in [1,6] show that the success of DS attacks is not guaranteed when $p_A<0.5$.”

A correction has been made to this sentence located in Section 4, Profitable DS Attacks, in the paragraph 1 on line 1:

“The previous probabilistic analyses in [1,6] have shown that the success of DS attacks is not guaranteed when $p_A<0.5$.”

9.

There was an error in the original publication: “With regards to $P$, if an attack succeeds, the revenue comes from $C$, as it is double-spent, and $R$ for the number of blocks minded during the time duration $T_{AS}$, i.e., $R\left({\lambda }_A,T_{AS}\right)$.”

A correction has been made to this sentence located in Section 4, Profitable DS Attacks, in the paragraph 4 on line 1:

”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., $R\left({\lambda }_A,T_{AS}\right)$.”

10.

There was an error to Equation (35) in the original publication:

$$“C_{\mathrm{Req}.}==\frac{{\mathbb{E}}_X}{{\mathbb{P}}_{AS}}-\mathbb{E}\left[R\left({\lambda }_A,T_{AS}\right)\right].”$$

A correction has been made to Equation (35):

$$“C_{\mathrm{Req}.}=\frac{{\mathbb{E}}_X}{{\mathbb{P}}_{AS}}-\mathbb{E}\left[R\left({\lambda }_A,T_{AS}\right)\right].”$$

11.

There was an error in the original publication: “Specifically, under, it is required for the honest chain to have added $N_{BC}$ blocks, while under their condition, it was the fraudulent chain to do so (see Section 3 of [16]).”

A correction has been made to a sentence located in Section 6, Related Works, in the paragraph 10 on line 6:

“Specifically, under our definition, it is required for the honest chain to have added $N_{BC}$ blocks, while under their condition it was fraudulent for the chain to do so (see Section 3 of [16]). ”

12.

There was an error in the original publication: “Note that ${\mathcal{D}}_j^{\left(1\right)}\cap \left({\mathcal{D}}_{i,j}^{\left(2\right)}\right)=\varphi$ for $i\le 2N_{BC}$, since the minimum number of states for an attack success is $2N_{BC}+1$; $N_{BC}$ number of $+1$’s state transitions for the block confirmation; and $N_{BC}+1$ number of $-1$’s state transitions for the success of PoW competition. Thus, $\mathrm{p}{p}_{DSA,i}=0$ for $i\le 2N_{BC}$. □”

A correction has been made to the paragraph 2 of Appendix A, Proof of Lemma 1:

“Note that for ${\mathcal{D}}_j^{\left(1\right)}\cap \left({\mathcal{D}}_{i,j}^{\left(2\right)}\right)=\varphi$ for $i\le 2N_{BC}$, since the minimum number of states for an attack success is $2N_{BC}+1$: $N_{BC}$ number of +1’s state transitions for the block confirmation; and $N_{BC}+1$ number of −1’s state transitions for the success of PoW competition. Thus, $p_{DSA,i}=0$ for $i\le 2N_{BC}$.”

13.

There was an error in the original publication: “Finally, substituting Equation (A3) and Equation (A4) into Equation (A2) results in Equation (15).”

A correction has been made to the last paragraph of Appendix A, Proof of Lemma 1:

“Finally, substituting Equations (A3) and (A4) into Equation (A2) results in Equation (15). □ ”

14.

There was an error in the original publication: “From Equation (19) and Equation (26), when $t_{cut}=\infty$, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{T_{AS}}=& \frac{{\displaystyle \underset{t_{cut}\to {\infty }^{-}}{lim}}{\int }_0^{t_{cut}}t{f}_{T_{DSA}}\left(t\right)dt}{{\mathbb{P}}_{AS}\left(t_{cut}\right)}=\frac{{\displaystyle \sum _{i=2N_{BC}+1}^{\infty }}\mathbb{E}\left[T_i\right]p_{DSA,i}}{{\mathbb{P}}_{DSA}}\hfill \\ \hfill =& \frac{{\displaystyle \sum _{i=2N_{BC}+1}^{\infty }}\frac{i}{{\lambda }_T}{p}_{DSA,i}}{{\mathbb{P}}_{DSA}}\hfill \end{array}$$

(A17)

where $\mathbb{E}\left[T_i\right]=i\lambda T{1}^{-1}$ [22].”

A correction has been made to this sentence located in Appendix B, Proof of Proposition 2, in the paragraph 1 on line 1:

“From Equations (19) and (26), when $t_{cut}=\infty$, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{T_{AS}}=& \frac{{\displaystyle \underset{t_{cut}\to {\infty }^{-}}{lim}}{\int }_0^{t_{cut}}t{f}_{T_{DS\mathrm{A}}}\left(t\right)dt}{{\mathbb{P}}_{AS}\left(t_{cut}\right)}=\frac{{\displaystyle \sum _{i=2N_{BC}+1}^{\infty }}\mathbb{E}\left[T_i\right]p_{DSA,i}}{{\mathbb{P}}_{DSA}}\hfill \\ \hfill =& \frac{{\displaystyle \sum _{i=2N_{BC}+1}^{\infty }}\frac{i}{{\lambda }_T}{p}_{DSA,i}}{{\mathbb{P}}_{DSA}}\hfill \end{array}$$

(A17)

where $\mathbb{E}\left[T_i\right]=i{{\lambda }_T}^{-1}$ [22]. ”

15.

There was an error in the original publication: “Attack success probability ($P_{AS}$)”

A correction has been made to the second item on the first column of Table 1:

“Attack success probability (${\mathbb{P}}_{AS}$)”

16.

There was an error in the original publication: “Attack success probability ($P_{AS}$)”

A correction has been made to the second item on the first column of Table 2:

“Attack success probability (${\mathbb{P}}_{AS}$)”

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.