Abstract
In this paper we analyze permissionless blockchain protocols, whose distributed consensus algorithm lies on a Proof-of-Work composed of \(k > 1\) consecutive hash-puzzles, that have to be solved sequentially. Our contribution is twofold. First, under common assumptions in the literature, we provide a closed-form expression for the mining probability of a miner, that is, the probability that the miner completes the Proof-of-Work of the next block to be added to the blockchain before every other miner does. Second, we show that, contrary to single-stage Proof-of-Works (i.e., \(k=1\)), in multi-stage Proof-of-Works, the mining probability might not be strictly related to the miner hash rate. This feature could be exploited by a smart miner, and could cause fairness and centralization issues in mining, which make the design of practical multi-stage Proof-of-Work blockchain protocols not trivial.
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Notes
- 1.
Mathematically, it is the inverse of the hash puzzle target value.
- 2.
- 3.
The hash rate indicates the number of trials in the PoW game a miner performs per second, and is measured in hash/s [11]. By global we identify the number of trials performed per second by the entire network.
- 4.
Application specific integrated circuit.
- 5.
In a homogeneous Poisson point process the average number of successes in t units of time is \(\lambda \, t\) [19]. If we let t be a unit of time, i.e. \(t = 1\), then the statement holds.
- 6.
The merge of the N processes by cumulating their respective points.
- 7.
Expression computable in a finite number of standard operations.
- 8.
The benchmarking was performed on a HP Pavilion Laptop 15-cs2023nl, equipped with a quad-core CPU Intel Core™ i7-8565U 1.80 GHz CPU, 8 MB cache, and 16 GB (\(2\times 8\)) SO-DIMM SDRAM 2400MHz DDR4.
- 9.
Our source code, its comprehensive documentation, and the information regarding the benchmark results are available on GitHub: https://github.com/FraMog/MiningProbabilityMultiStageProof-of-Work.
- 10.
The ratio of the global hash rate the first miner possesses is \((h_{1,0} + h_{1,1}) / (h_{1,0} + h_{1,1} + h_{2,0} + h_{2,1})\).
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Acknowledgements
Francesco Mogavero’s work was partially funded by Sapienza’s Progetto di Ateneo 2020: La disintermediazione della Pubblica Amministrazione: il ruolo della tecnologia blockchain e le sue implicazioni nei processi e nei ruoli della PA.
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D’Arco, P., Mogavero, F. (2022). On (Multi-stage) Proof-of-Works. In: Prieto, J., Partida, A., Leitão, P., Pinto, A. (eds) Blockchain and Applications. BLOCKCHAIN 2021. Lecture Notes in Networks and Systems, vol 320. Springer, Cham. https://doi.org/10.1007/978-3-030-86162-9_10
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