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, 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：传统的双花攻击模型忽略了由孤块带来的收益损失。另一方面，自私挖矿的相关研究通常忽略了攻击者在每个攻击周期内无成本进行双花的机会。本文对一种攻击进行了严格的随机分析，其中对手的目标是在自私挖矿的同时进行双花。为此，我们首先结合了顽固挖矿与自私挖矿攻击，即构建一种策略，使攻击者在私有分支达到特定长度前采取顽固行为，随后切换为自私行为。我们给出了各参数区间下的最优顽固度。接着，我们提供了仍比诚实挖矿更有利可图的最大顽固度，并论证了顽固度水平与 $k$ 次确认规则之间的关联。我们证明，在每个攻击周期中，若顽固度水平高于 $k$，对手将获得一次免费的双花机会。在每个周期中，对于给定的顽固度水平，我们严格推导了双花概率的大小。此外，我们进一步修改了顽固阶段中的攻击策略，以隐藏攻击并提高双花概率。