conference paper published yes Mirza Ahad Baig author 3EDE6DE4-AA5A-11E9-986D-341CE6697425 Christoph Ullrich Günther author ec98511c-eb8e-11eb-b029-edd25d7271a1 Krzysztof Z Pietrzak author 3E04A7AA-F248-11E8-B48F-1D18A9856A870000-0002-9139-1654 KrPi department AFT: Conference on Advances in Financial Technologies Security and Privacy by Design for Complex Systems project Security and Privacy by Design for Complex Systems project The blocks in the Bitcoin blockchain "record" the amount of work W that went into creating them through proofs of work. When honest parties control a majority of the work, consensus is achieved by picking the chain with the highest recorded weight. Resources other than work have been considered to secure such longest-chain blockchains. In Chia, blocks record the amount of disk-space S (via a proof of space) and sequential computational steps V (through a VDF). In this paper, we ask what weight functions Γ(S,V,W) (that assign a weight to a block as a function of the recorded space, speed, and work) are secure in the sense that whenever the weight of the resources controlled by honest parties is larger than the weight of adversarial parties, the blockchain is secure against private double-spending attacks. We completely classify such functions in an idealized "continuous" model: Γ(S,V,W) is secure against private double-spending attacks if and only if it is homogeneous of degree one in the "timed" resources V and W, i.e., αΓ(S,V,W) = Γ(S,α V, α W). This includes the Bitcoin rule Γ(S,V,W) = W and the Chia rule Γ(S,V,W) = S ⋅ V. In a more realistic model where blocks are created at discrete time-points, one additionally needs some mild assumptions on the dependency on S (basically, the weight should not grow too much if S is slightly increased, say linear as in Chia). Our classification is more general and allows various instantiations of the same resource. It provides a powerful tool for designing new longest-chain blockchains. E.g., consider combining different PoWs to counter centralization, say the Bitcoin PoW W₁ and a memory-hard PoW W₂. Previous work suggested to use W₁+W₂ as weight. Our results show that using e.g., √{W₁}⋅ √{W₂} or min{W₁,W₂} are also secure, and we argue that in practice these are much better choices. https://research-explorer.ista.ac.at/download/20587/20598/2025_LIPIcsAFT_Baig.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2025Pittsburgh, PA, United States eng 1868-8969 9783959774000 2508.0144810.4230/LIPIcs.AFT.2025.16 354 https://research-explorer.ista.ac.at/record/21651 Baig, Mirza Ahad, Christoph Ullrich Günther, and Krzysztof Z Pietrzak. “Nakamoto Consensus from Multiple Resources.” In <i>7th Conference on Advances in Financial Technologies</i>, Vol. 354. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href="https://doi.org/10.4230/LIPIcs.AFT.2025.16">https://doi.org/10.4230/LIPIcs.AFT.2025.16</a>. Baig, M. A., Günther, C. U., &#38; Pietrzak, K. Z. (2025). Nakamoto consensus from multiple resources. In <i>7th Conference on Advances in Financial Technologies</i> (Vol. 354). Pittsburgh, PA, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.AFT.2025.16">https://doi.org/10.4230/LIPIcs.AFT.2025.16</a> Baig, Mirza Ahad, et al. “Nakamoto Consensus from Multiple Resources.” <i>7th Conference on Advances in Financial Technologies</i>, vol. 354, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href="https://doi.org/10.4230/LIPIcs.AFT.2025.16">10.4230/LIPIcs.AFT.2025.16</a>. M. A. Baig, C. U. Günther, and K. Z. Pietrzak, “Nakamoto consensus from multiple resources,” in <i>7th Conference on Advances in Financial Technologies</i>, Pittsburgh, PA, United States, 2025, vol. 354. M.A. Baig, C.U. Günther, K.Z. Pietrzak, in:, 7th Conference on Advances in Financial Technologies, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. Baig MA, Günther CU, Pietrzak KZ. Nakamoto consensus from multiple resources. In: <i>7th Conference on Advances in Financial Technologies</i>. Vol 354. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href="https://doi.org/10.4230/LIPIcs.AFT.2025.16">10.4230/LIPIcs.AFT.2025.16</a> Baig MA, Günther CU, Pietrzak KZ. 2025. Nakamoto consensus from multiple resources. 7th Conference on Advances in Financial Technologies. AFT: Conference on Advances in Financial Technologies, LIPIcs, vol. 354, 16. 205872025-11-02T23:01:34Z2026-04-15T08:45:18Z