IEEE Consumer Electronics Magazine - September/October 2021 - 27

Figure 3. Comparison of the attacker's break-even point in both cases when g ¼ 2, 3, and 4.
Now, we consider the attack cost under the
combined mitigation. Let Sz denote the total
time to wait for z block confirmations. We first
consider whether Sz exceeds the deadline D.
The attacker's malicious behavior will not be
detected if Sz is within D, thus the attack cost
is consistent with Coriginal. The vendor will
refuse to deliver the commodity to the
attacker if Sz exceeds D.Inthiscase, thetime
when the attacker stops attacking is D as the
proposed attack is certain to fail, and the
attacker will lose the commodity and all the
block rewards. We assume the number of
blocks mined by the attacker within D is kD.
Moreover, in both cases, each identity of Sybil
nodes is charged a certain amount of the initial
entry fee and the participation fee. Since only
one block is created in each round of consensus
process, the number of rounds for charging
the participation fee is z.Weassume that
the total number of nodes in the Bitcoin network
is N, the ratio of Sybil nodes to the total
number of nodes is m, and each Sybil node has
g identities. Denote by " and s the initial entry
fee and the participation fee in each round of
consensus process charged for each identity,
respectively. Then, the total identity fee is
ð " þ zsÞgmN. According to the above analysis,
we can easily get the attack cost Ccombined in
two different cases as follows:
Ccombined ¼
ðÞ1 f v þ 12:5kzðÞþ " þ zsðÞgmN; Sz D
ðÞv þ 12:5kD þ " þ zsðÞgmN; Sz >D:
&
(2)
In order to find the attacker's break-even point
in the proposed attack, we also need to figure out
September/October 2021
the revenue of the attacker. The calculation of the
attacker's revenue is considered in two cases as
well. When Sz D, no matter the attacker succeeds
or fails, he can always get the value of the
commodity v, thus the attacker's revenue is v in
this case. When Sz >D, the attacker will lose the
revenue v since the vendor will refuse to hand
over the commodity to him. Let f2 denote the
probability that Sz D. By combining these two
cases, the attacker's revenue is f2v. As we know,
the attacker's break-even point is a situation
where the attack cost equals the attack revenue
(i.e., f2v ¼ Ccombined ). We select some specific
parameter values, e.g., z ¼ 6, " ¼ 0.01 BTC, s ¼
0.005 BTC, N ¼ 10000, and we use some probabilistic
models to do a calculation to get the
attacker's break-even point f2v or Ccombined.We
also calculate the attacker's break-even point in
the absence of any mitigations. We compare the
attacker's break-even point in both cases, and the
results are shown in Figure 3. From Figure 3, we
can clearly see that if we do not take any mitigations,
the more Sybil nodes deployed by the
attacker, and the more fake identities each Sybil
node owns, the lower the attacker's break-even
point, which means the attacker only pays a little
to complete the proposed attack in this case.
However, if we take the combined mitigation, the
attacker's break-even point always keeps the
trend of rising with the increase of m when g ¼ 2,
3, 4. Moreover, we can find that the changing
trend of the attacker's break-even point is similar
when g ¼ 2 and 3, whereas when g ¼ 4, the
attack's break-even point increases dramatically
with m changes from 0.04 to 0.05 since the attacker
needs to pay a huge cost for these fake identities.
Our findings demonstrate that the adoption of
this combined mitigation can effectively prevent
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IEEE Consumer Electronics Magazine - September/October 2021

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