All-out Attack: Optimal Block Withholding Under Pay-Per-Share Scheme

Abstract

Classical Block Withholding (BWH) attacks have been extensively studied in block-dependent reward schemes, where pool members are compensated upon a block discovery within the pool. However, most contemporary mining pools operate under share-based schemes, wherein participants are paid immediately upon submission of valid shares. In this paper, we analyze BWH under Pay-Per-Share (PPS) and Full-PPS (FPPS) schemes for Nakamoto-style blockchains and prove that these mechanisms are not incentive compatible -- contrary to claims in prior literature. Under PPS/FPPS, the optimal strategy for a BWH attacker is the All-out Attack (AoA): the adversary allocates its entire hashpower toward the victim pool, submitting only partial Proof-of-Work shares (pPoW) while withholding all valid blocks, i.e., full Proof-of-Work (fPoW). Prior to the first difficulty adjustment, the adversary incurs negligible loss from withheld fPoWs. After the adjustment reduces block difficulty, the adversary either generates more pPoWs per unit time or, if pPoW difficulty is held fixed, earns a higher reward per share, in both cases achieving a relative gain of $\fracα{1-α}$ over pre-adjustment rates, where $α$ is the adversarial hashpower fraction. Honest miners benefit at the same rate as the adversary per unit hashpower, while the victim pool operator bears all losses, paying out-of-pocket for pPoW submissions without receiving fPoW compensation in return. Finally, advanced BWH variants such as Fork After Withholding (FAW) yield no additional profit under PPS/FPPS.