Two double-spend attack strategies on a proof-of-stake consensus are considered. For each strategy, the probability of its success is obtained, which depends on the network parameters and the number of confirmation blocks. These results can be used to define how many confirmation blocks a vendor should wait after a correspondent transaction before sending goods or services.

The transition from centralized power systems to decentralized infrastructures with a high share of renewable energy sources calls for reliable settlement in P2P electricity trading across “smart” regions. Blockchain platforms can enhance transparency and facilitate automated settlement; however, double-spend attacks still pose a threat to transaction finality and, consequently, undermine trust in the payment layer. This paper quantifies this risk through a probabilistic analysis of classical double-spend scenarios for Proof-of-Work (PoW) and Proof-of-Stake (PoS) blockchains augmented with periodic checkpoints, which render the chain history prior to the latest checkpoint effectively irreversible. We develop attack models for both consensus mechanisms and derive explicit formulas for the attacker’s success probability as a function of the adversarial share, the spacing between checkpoints, and the number of confirmation blocks. On this basis, we compute the minimum confirmation depth needed to satisfy a predefined risk threshold. Numerical evaluation using the derived expressions shows that checkpoints consistently reduce double-spend probability relative to checkpoint-free baselines; in the evaluated settings, the reduction reaches up to 44% and becomes more pronounced as the adversarial share increases. Finally, the analysis yields practical guidance for energy trading applications: accept a payment after the computed number of confirmations when it fits within a single checkpoint interval; otherwise, treat finality as reaching the next checkpoint.

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.