When Should Selfish Miners Double-Spend?

A new research paper by Mustafa Doger and Sennur Ulukus, titled 'When Should Selfish Miners Double-Spend?', provides a rigorous stochastic analysis of a novel blockchain attack. The work bridges a gap in existing security models: conventional double-spending attacks ignore revenue losses from orphaned blocks, while selfish mining literature often overlooks the attacker's chance to double-spend at no cost within an attack cycle. The authors propose and analyze a hybrid 'stubborn-selfish' mining strategy. In this attack, the adversary initially acts stubborn—persistently mining on its private chain—until that chain reaches a predetermined length threshold, at which point it switches to a classic selfish mining tactic.

The core contribution is a mathematical framework that determines the optimal level of 'stubbornness' (the length to mine privately before switching) for various network parameters. Crucially, the model establishes a direct connection between this stubbornness level and a blockchain's k-confirmation rule. It proves that if an attacker's stubbornness exceeds k, they gain a 'free shot' at double-spending in every attack cycle, as their private chain has a chance to overtake the public one after the victim considers a transaction final. The paper further modifies the stubborn regime to conceal the attack and boost the double-spending probability, presenting a more sophisticated and potentially more profitable threat model than previously analyzed.