An analytic evaluation for the impact of uncle blocks by selfish and stubborn mining in an imperfect Ethereum network

Besides supporting smart contract, another Ethereum distinct feature lies in the improvement of running efficiency achieved by the uncle block mechanism. To mitigate network congestion, an orphan block has a chance to be referenced by a regular block as an uncle block rather than being abandoned directly. In this paper, we use a Markov state machine to model selfish and stubborn mining in an imperfect Ethereum network with natural forks. The security threshold against selfish mining, which indicates how difficult an attacker could gain more than he deserves, decreases to 23.8% in an uncle block rate of 11%. Two advanced selfish mining strategies, trail stubborn mining and equal fork combined with trail stubborn mining, are also analyzed with different uncle block rates in this paper. In different feasible regions, strategies that bring about the optimal reward vary from each other. All evaluations in this paper are carried out in an analytic method, which is more flexible and scalable than a numeric method if it combines multiple attacks or extends to several attackers.