Conventional double-spending attack models ignore the revenue losses stemming from the orphan blocks. On the other hand, selfish mining literature usually ignores the chance of the attacker to double-spend at no-cost in each attack cycle. In this paper, we give a rigorous stochastic analysis of an attack where the goal of the adversary is to double-spend while mining selfishly. To do so, we first combine stubborn and selfish mining attacks, \textit{i.e.}, construct a strategy where the attacker acts stubborn until its private branch reaches a certain length and then switches to act selfish. We provide the optimal stubbornness for each parameter regime. Next, we provide the maximum stubbornness that is still more profitable than honest mining and argue a connection between the level of stubbornness and the $k$-confirmation rule. We show that, at each attack cycle, if the level of stubbornness is higher than $k$, the adversary gets a free shot at double-spending. At each cycle, for a given stubbornness level, we rigorously formulate how great the probability of double-spending is. We further modify the attack in the stubborn regime in order to conceal the attack and increase the double-spending probability.

翻译：传统双花攻击模型忽略了孤立区块带来的收益损失。另一方面，自私挖矿文献通常未考虑攻击者在每个攻击周期中无成本进行双花攻击的可能性。本文对攻击者以自私挖矿方式实施双花攻击的策略进行了严格的随机分析。为此，我们首先将顽固挖矿与自私挖矿攻击相结合，即构建一种策略：攻击者在其私有分支达到特定长度前保持顽固行为，之后转为自私行为。我们给出了不同参数区间的最优顽固程度。其次，我们确定了仍比诚实挖矿更有利可图的最高顽固程度，并论证了顽固程度与$k$确认规则之间的关联。研究表明，在每个攻击周期中，若顽固程度高于$k$，攻击者即可获得一次免费双花机会。针对给定顽固程度，我们严格推导了每个周期中双花攻击成功概率的表达式。此外，我们改进了顽固模式下的攻击策略，以隐藏攻击行为并提高双花攻击概率。