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Time-Restricted Double-Spending Attack on PoW-based Blockchains

Yiming Jiang, Jiangfan Zhang

TL;DR

Problem addressed: double-spending attacks on PoW blockchains for applications with finite task windows; approach: define a time-restricted DSA (TR-DSA) and derive a closed-form success probability $P_s^{(TR)}$ by mapping the catch-up event to a two-dimensional boundary-hitting random walk, with $b$ following a negative-binomial distribution and $Q(l,m,n)$ expressed via Catalan-number structures; contributions: (i) full closed-form for $P_s^{(TR)}$, (ii) proof that $P_s^{(TR)} ≤ P_s^{(TU)}$ with strict inequality for $0<I<1$, and (iii) numerical validation showing monotone increase of $P_s^{(TR)}$ with horizon $L$ and nonzero risk even if $I≥0.5$; practical impact: offers a quantitative risk instrument to set secure confirmation depths $Z$ and to evaluate attacker feasibility under limited resources.

Abstract

Numerous blockchain applications are designed with tasks that naturally have finite durations, and hence, a double-spending attack (DSA) on such blockchain applications leans towards being conducted within a finite timeframe, specifically before the completion of their tasks. Furthermore, existing research suggests that practical attackers typically favor executing a DSA within a finite timeframe due to their limited computational resources. These observations serve as the impetus for this paper to investigate a time-restricted DSA (TR-DSA) model on Proof-of-Work based blockchains. In this TR-DSA model, an attacker only mines its branch within a finite timeframe, and the TR-DSA is considered unsuccessful if the attacker's branch fails to surpass the honest miners' branch when the honest miners' branch has grown by a specific number of blocks. First, we developed a general closed-form expression for the success probability of a TR-DSA. This developed probability not only can assist in evaluating the risk of a DSA on blockchain applications with timely tasks, but also can enable practical attackers with limited computational resources to assess the feasibility and expected reward of launching a TR-DSA. In addition, we provide rigorous proof that the success probability of a TR-DSA is no greater than that of a time-unrestricted DSA where the attacker indefinitely mines its branch. This result implies that blockchain applications with timely tasks are less vulnerable to DSAs than blockchain applications that provide attackers with an unlimited timeframe for their attacks. Furthermore, we show that the success probability of a TR-DSA is always smaller than one even though the attacker controls more than half of the hash rate in the network. This result alerts attackers that there is still a risk of failure in launching a TR-DSA even if they amass a majority of the hash rate in the network.

Time-Restricted Double-Spending Attack on PoW-based Blockchains

TL;DR

Problem addressed: double-spending attacks on PoW blockchains for applications with finite task windows; approach: define a time-restricted DSA (TR-DSA) and derive a closed-form success probability by mapping the catch-up event to a two-dimensional boundary-hitting random walk, with following a negative-binomial distribution and expressed via Catalan-number structures; contributions: (i) full closed-form for , (ii) proof that with strict inequality for , and (iii) numerical validation showing monotone increase of with horizon and nonzero risk even if ; practical impact: offers a quantitative risk instrument to set secure confirmation depths and to evaluate attacker feasibility under limited resources.

Abstract

Numerous blockchain applications are designed with tasks that naturally have finite durations, and hence, a double-spending attack (DSA) on such blockchain applications leans towards being conducted within a finite timeframe, specifically before the completion of their tasks. Furthermore, existing research suggests that practical attackers typically favor executing a DSA within a finite timeframe due to their limited computational resources. These observations serve as the impetus for this paper to investigate a time-restricted DSA (TR-DSA) model on Proof-of-Work based blockchains. In this TR-DSA model, an attacker only mines its branch within a finite timeframe, and the TR-DSA is considered unsuccessful if the attacker's branch fails to surpass the honest miners' branch when the honest miners' branch has grown by a specific number of blocks. First, we developed a general closed-form expression for the success probability of a TR-DSA. This developed probability not only can assist in evaluating the risk of a DSA on blockchain applications with timely tasks, but also can enable practical attackers with limited computational resources to assess the feasibility and expected reward of launching a TR-DSA. In addition, we provide rigorous proof that the success probability of a TR-DSA is no greater than that of a time-unrestricted DSA where the attacker indefinitely mines its branch. This result implies that blockchain applications with timely tasks are less vulnerable to DSAs than blockchain applications that provide attackers with an unlimited timeframe for their attacks. Furthermore, we show that the success probability of a TR-DSA is always smaller than one even though the attacker controls more than half of the hash rate in the network. This result alerts attackers that there is still a risk of failure in launching a TR-DSA even if they amass a majority of the hash rate in the network.
Paper Structure (9 sections, 3 theorems, 54 equations, 8 figures)

This paper contains 9 sections, 3 theorems, 54 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Summary of Results and Main Contributions
  3. Related Work
  4. Probability of Success in Launching A Time-Restricted Double-Spending Attack
  5. Time-Restricted Double-Spending Attack Model
  6. The Success Probability of a Time-Restricted Double-Spending Attack
  7. Comparison of the Success Probability of A TR-DSA and the Success Probability of A TU-DSA
  8. Numerical Results
  9. Conclusions

Key Result

Theorem 1

The probability $Q(l,m,n)$ that the attacker finally wins the competition in which at some time instant, the attacker's branch lags by $m$ blocks behind the honest branch and the honest branch has already grown by $n$ blocks can be expressed as, $m =-1,0,...$, and $\forall n \in \{0,1,\cdots, l\}$, where the coefficient $a_{i,m}$ is defined as and the constant $C_i$ is the $i$-th Catalan number

Figures (8)

  • Figure 1: Illustration of a time-restricted double-spending attack.
  • Figure 2: Two-dimensional random walk illustration.
  • Figure 3: The relationship between $Q(l,m,n)$ and $m$ for different $I$.
  • Figure 4: The relationship between $Q(l,m,n)$ and $m$ for different $(l-n)$.
  • Figure 5: The comparison between $Q(l,m,n)$ obtained from Theorem \ref{['Theorem_DSA']} and Theorem \ref{['Theorem_asymptotic']}.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof : Proof:
  • Theorem 2
  • proof : Proof:
  • Corollary 1
  • proof