{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T04:18:33Z","timestamp":1770956313225,"version":"3.50.1"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T00:00:00Z","timestamp":1648512000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T00:00:00Z","timestamp":1648512000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we present the <jats:italic>Multidimensional Byzantine Agreement (MBA) Protocol<\/jats:italic>, a <jats:italic>leaderless<\/jats:italic> Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant information regarding a set of observed events. The consensus process is carried out in parallel on each component, and the output is a vector whose components are either values with wide agreement in the network (even if no individual node agrees on every value) or a special value <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\bot $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u22a5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that signals irreconcilable disagreement. The MBA Protocol is probabilistic and its execution halts with probability 1, and the number of steps necessary to halt follows a Bernoulli-like distribution. The design combines a <jats:italic>Multidimensional Graded Consensus<\/jats:italic> and a <jats:italic>Multidimensional Binary Byzantine Agreement<\/jats:italic>, the generalization to the multidimensional case of two protocols presented by Micali et al. (SIAM J Comput 26(4):873\u2013933, 1997; Byzantine agreement, made trivial, 2016). We prove the correctness and security of the protocol assuming a synchronous network where less than a third of the nodes are malicious.<\/jats:p>","DOI":"10.1007\/s00200-022-00548-5","type":"journal-article","created":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T01:06:40Z","timestamp":1648602400000},"page":"233-251","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Multidimensional Byzantine agreement in a synchronous setting"],"prefix":"10.1007","volume":"35","author":[{"given":"Andrea","family":"Flamini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Riccardo","family":"Longo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessio","family":"Meneghetti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,3,29]]},"reference":[{"key":"548_CR1","unstructured":"Buterin, V., Griffith, V.: Casper the friendly finality gadget. arXiv preprint arXiv:1710.09437 (2017)"},{"key":"548_CR2","first-page":"173","volume":"99","author":"M Castro","year":"1999","unstructured":"Castro, M., Liskov, B., et al.: Practical byzantine fault tolerance. 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