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When Should Selfish Miners Double-Spend?

Mustafa Doger, Sennur Ulukus

TL;DR

<3-5 sentence high-level summary> The paper investigates the profitability and risk of a combined double-spending and selfish mining attack in Nakamoto-style blockchains by introducing the $L$-stubborn mining framework, parameterized by adversarial share $\alpha$ and network influence $\gamma$. It derives closed-form revenue-ratio expressions $\rho_L$, identifies the maximal stubbornness $\Bar{L}$ for which the attack remains profitable, and shows that when $\Bar{L}>k$ under a $k$-confirmation rule, double-spending can occur at no cost to the attacker. A stealth variant, $S$-stealth mining, is proposed to conceal the attack and increase double-spend probability, with corresponding revenue and probability analyses. Numerical results demonstrate close alignment with epsilon-optimal MDP solutions and reveal actionable thresholds for Bitcoin-like settings, while connecting the analysis to Catalan numbers and Bertrand's ballot problem for exact combinatorial insight.

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, 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.

When Should Selfish Miners Double-Spend?

TL;DR

<3-5 sentence high-level summary> The paper investigates the profitability and risk of a combined double-spending and selfish mining attack in Nakamoto-style blockchains by introducing the -stubborn mining framework, parameterized by adversarial share and network influence . It derives closed-form revenue-ratio expressions , identifies the maximal stubbornness for which the attack remains profitable, and shows that when under a -confirmation rule, double-spending can occur at no cost to the attacker. A stealth variant, -stealth mining, is proposed to conceal the attack and increase double-spend probability, with corresponding revenue and probability analyses. Numerical results demonstrate close alignment with epsilon-optimal MDP solutions and reveal actionable thresholds for Bitcoin-like settings, while connecting the analysis to Catalan numbers and Bertrand's ballot problem for exact combinatorial insight.

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, 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 -confirmation rule. We show that, at each attack cycle, if the level of stubbornness is higher than , 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.
Paper Structure (35 sections, 17 theorems, 71 equations, 5 figures, 5 tables, 2 algorithms)

This paper contains 35 sections, 17 theorems, 71 equations, 5 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. System Model
  4. Propagation Delays
  5. Mining Strategies and Definitions
  6. $L$-Stubborn Mining
  7. An Example: $4$-Stubborn Mining
  8. Relation to the Existing Works
  9. Double-Spend Events
  10. $S$-Stealth Mining
  11. Definitions and Notations
  12. Analysis of $L$-Stubborn Mining
  13. Optimizing the Revenue Ratio
  14. $(k+1)$-Stubborn Mining and Double-Spending
  15. Combined Revenue Ratio
  16. ...and 20 more sections

Key Result

Lemma 1

Under the $L$-stubborn mining strategy, the probability that $A$-chain reaches length $L$ during an attack cycle while the maximum height of the honest block is $H=m<L$, is

Figures (5)

  • Figure 1: Sample paths of $4$-stubborn mining.
  • Figure 2: $L$-stubborn mining and $S$-stealth mining.
  • Figure 3: $(k+1)$-stubborn mining with $k=6$.
  • Figure 4: $(k+1)$-stealth mining with $k=6$.
  • Figure 5: $R^*_{6}$ for Bitcoin

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 6
  • Lemma 1
  • ...and 16 more