Pool Block Withholding Attack with Rational Miners

Abstract

We revisit pool block withholding (PBW) attack in this work, while taking miners’ rationality into consideration. It has been shown that a malicious mining pool in Bitcoin may attack other pools by sending some dummy miners who do not reveal true solutions but obtain reward portions from attacked pools. This result assumes that the infiltrating miners are always loyal and behave as the malicious pool desires; but why? In this work, we study rational infiltrating miners who behave by maximizing their own utilities. We characterize the infiltrating miners’ optimal strategies for two kinds of betraying behaviors depending on whether they return to the original pool or not. Our characterizations show that the infiltrating miners’ optimal strategy is simple and binary: they either betray or stay loyal all together, which depends on the total amount of them sent out by the malicious pool. Accordingly, we further compute the pool’s optimal attacking strategy given the miners’ strategic behaviors. Our experiments also show that by launching PBW attacks, the benefit to a pool is actually incremental.

This work is supported by The Hong Kong Polytechnic University (Grant No. P0034420).

A Appendix

In the Appendix, we present the full experiment results.

1.1 A.1 Experiments in the BR Model

In the simulations, we do experiments to investigate the strategic behavior of pool 1. It illustrates how many fraction of the mining power is used for infiltrating, and how many benefits are brought by infiltrating. In all of the examples (see Table 3), suppose there are 5 mining pools, and set the total mining power $m=m_1+\cdots +m_5=1$. For different combinations of $m_i$, we compute the optimal strategy $\mathbf{x}^*$ of pool 1, and the corresponding optimal utility $u_1^*$. If pool 1 does not attack (i.e., $\mathbf{x}=\mathbf{0}$), clearly the utility is $u_1^{non}=\frac{m_1}{m}=m_1$. We use the ratio $\frac{u_1^*}{u_1^{non}}$ to measure the benefits that attacking can bring.

Table 3. Experiments for computing the optimal strategy in the BR Model.

Our experiment results show that, about 50% mining power of pool 1 is used for attacking in an optimal strategy, which will bring a less than 10% increase of utility.

1.2 A.2 Experiments in the BNR Model

In our experiments for the BNR model, arbitrarily fixing $m_1,\ldots ,m_n,x_3,\ldots ,x_n$,

$\alpha _3,\ldots ,\alpha _n$, we take $x_2$ as a variable, and increase the value of $x_2$. For each value of $x_2$, we compute the best reaction $\alpha _2$ of infiltrating miner 2. The experiments results show that such a threshold truly exists in every example, and thus support the conjecture. Moreover, on average, the threshold in the BNR model is at least 20% smaller than that in the BR model, which implies that the infiltrating miners in the BNR model are more likely to betray their original pool.

In Table 4, we fix $m_1,\ldots ,m_5,x_3,\ldots ,x_5,\alpha _3,\ldots ,\alpha _5$, and let $x_2$ be a variable increasing from 0.001 within a step length of 0.002. For each value of $x_2$, we compute a corresponding best reaction $\alpha _2^*$ of miner 2. A rough threshold t is given for each example: when $x_2<t$, $\alpha _2^*=x_2$; when $x_2>t$, $\alpha _2^*=0$. The threshold for these examples in the BR model is denoted by $t^{BR}$. We conclude that, for every example, the threshold in the BNR model is much smaller than that in the BR model, which implies that the pool in BNR model is less likely to attack other pools.

Table 4. Experiments for computing miner 2’s best reaction in the BNR model, where $\mathbf{x}=(x_2, x_3=0.1m_1, x_4=0.12 m_1, x_5=0.15 m_1$) and $\boldsymbol{\alpha }=(\alpha _2, \alpha _3=0, \alpha _4=0.5x_4, \alpha _5=x_5$).