Honest majority is a vital assumption of Bitcoin-style blockchains. However, recent 51% attacks render this assumption unrealistic in practice. In this paper, we analyse two possible 51% attacks launched by a rational miner, who is profit driven. The first attack considers a rational miner, who moves his mining power from a stronger blockchain to a weaker blockchain to launch 51% attacks, provided that (1) the mining power is compatible in both blockchains, and (2) the transferred mining power dominates the weaker blockchain. We say a blockchain is stronger if the total mining power of this blockchain is higher than other (weaker) blockchains. The second attack considers a rational miner, who rents cloud mining power to launch 51% attacks. The former attack is new, and we name it mining power migration attack; the latter is called cloud mining attack, and it was initially covered by the bribery attack (FC’ 16).
We formalise the two attacks by using Markov Decision Process. We then test the feasibility of launching both attacks on leading blockchains in the wild by using our model. We show that both attacks are feasible and profitable. For example, our result shows that with 12.5% mining power of Bitcoin, a rational miner can gain approximately 6% (18,946.5 USD) extra profit than honest mining, by launching mining power migration attack to double spend a transaction of 3000 BCH (equivalent to $378,930) on BitcoinCash. We also investigate the 51% attack on Ethereum Classic happened in Jan. 2019, by applying our model into this attack to provide some insights to understand more about it.