Block withholding resilience
Cyril Grunspan, Ricardo Perez-Marco
TL;DR
The paper addresses block withholding in Nakamoto consensus by introducing a generalized difficulty adjustment algorithm (DAA) that accounts for orphan blocks. Using a martingale-based analysis of a Poisson-block model, it proves that, under the generalized DAA, honest mining is optimal and adversarial strategies become non-profitable, regardless of attacker connectivity. It further extends the framework to a broader DAA form with a difficulty function $D$, showing that when orphan blocks are properly reported (or rewarded within limits), the profitability ratio $\Gamma$ remains bounded by the attacker's hash power $q$, preserving network security. The work also discusses practical considerations, such as the potential inflation from orphan rewards and how to mitigate it (e.g., adjusting halving), providing a rigorous defense of Nakamoto consensus against selfish mining under more realistic reward schemes.
Abstract
It has been known for some time that the Nakamoto consensus as implemented in the Bitcoin protocol is not totally aligned with the individual interests of the participants. More precisely, it has been shown that block withholding mining strategies can exploit the difficulty adjustment algorithm of the protocol and obtain an unfair advantage. However, we show that a modification of the difficulty adjustment formula taking into account orphan blocks makes honest mining the only optimal strategy. Surprinsingly, this is still true when orphan blocks are rewarded with an amount smaller to the official block reward. This gives an incentive to signal orphan blocks. The results are independent of the connectivity of the attacker.