The blockchain data structure maintained via the longest-chain rule—popularized by Bitcoin—is a powerful algorithmic tool for consensus algorithms. Such algorithms achieve consistency for blocks in the chain as a function of their depth from the end of the chain. While the analysis of Bitcoin guarantees consistency with error 2k for blocks of depth O(k), the state-of-the-art of proof-of-stake (PoS) blockchains suffers from a quadratic dependence on k: these protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that depth Θ(k2) is sufficient. Whether this quadratic gap is an intrinsic limitation of PoS—due to issues such as the nothing-at-stake problem—has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctnes.
We give an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longest-chain rule and achieve, in broad generality, Θ(k) dependence on depth in order to achieve consistency error 2k In particular, for the first time we show that PoS protocols can match proof-of-work protocols for linear consistency.
We analyze the associated stochastic process, give a recursive relation for the critical functionals of this process, and derive tail bounds in both i.i.d. and martingale settings via associated generating functions.

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cover image Proceedings
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1135 - 1154
Editor: Shuchi Chawla
ISBN (Online): 978-1-611975-99-4


Published online: 23 December 2019




Erica Blum's work was partly supported by financial assistance award 70NANB19H126 from U.S. Department of Commerce, National Institute of Standards and Technology. Aggelos Kiayias' research was partly supported by H2020 Grant #780477, PRIViLEDGE. Cristopher Moore's research was partly supported by NSF grant BIGDATA-1838251. Alexander Russell's work was partly supported by NSF Grant #1717432.

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