This paper studies the probability of a double-spend attack in an Ouroboros-like Proof-of-Stake (PoS) setting when confirmation decisions must be made for a finite number of blocks. Existing security analyses of Ouroboros-family protocols are mainly asymptotic and therefore do not directly provide the attack probability for a fixed confirmation depth. We consider an analytically tractable model that allows empty slots and multiple slot leaders, and assumes fixed stake distribution within an epoch, one-block growth of the public longest chain in any slot containing at least one honest leader, and next-slot block visibility. These assumptions hold when the time slot length is much greater than the network delay, and are applicable to practical deployment scenarios such as Cardano. Under these assumptions, for the first time, an exact closed-form solution for the success probability of a double-spend attack considering a realistic model with multiple leaders and empty time slots. Numerical examples illustrate how the required confirmation depth depends on the adversarial stake ratio and the active slot coefficient. The results apply to the stated analytical model and do not yet cover delayed fork resolution or the full protocol-level fork-choice and finality mechanisms of Ouroboros Praos.
Kovalchuk et al. (Fri,) studied this question.
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