When Should Selfish Miners Double-Spend?

TL;DR

This paper provides a detailed stochastic analysis of selfish mining combined with stubbornness, revealing conditions under which double-spending can occur at no cost, and explores optimal strategies and their profitability.

Contribution

It introduces a combined stubborn and selfish mining attack model, deriving optimal stubbornness levels and analyzing double-spending probabilities under various conditions.

Findings

Higher stubbornness than k enables free double-spending attempts.

Optimal stubbornness depends on network parameters and confirmation rules.

Modified attack strategies can conceal double-spending and increase success probability.

Abstract

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.