{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:31:36Z","timestamp":1764981096772,"version":"3.46.0"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,8,17]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Dougherty et al. introduced the common information (CI) method as a method to produce non-Shannon inequalities satisfied by linear random variables, which are called linear rank inequalities. This method is based on the fact that linear random variables have CI. Dougerthy et al. asked whether this method is complete, in the sense that it can be used to produce all linear rank inequalities. We study this question, and we attack it using the theory of secret sharing schemes. To this end, we introduce the notions of Abelian secret sharing scheme and Abelian capacity. We prove that: If there exists an access structure whose Abelian capacity is smaller than its linear capacity, then the CI method is not complete. We investigate the existence of such an access structure.<\/jats:p>","DOI":"10.1515\/jmc-2022-0020","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T02:44:21Z","timestamp":1660704261000},"page":"233-250","source":"Crossref","is-referenced-by-count":0,"title":["Abelian sharing, common informations, and linear rank inequalities"],"prefix":"10.1515","volume":"16","author":[{"given":"Carolina","family":"Mejia","sequence":"first","affiliation":[{"name":"Mathematics Department, Universidad Nacional de Colombia , Bogot\u00e1 , Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan Andres","family":"Montoya","sequence":"additional","affiliation":[{"name":"Mathematics Department, Universidad Nacional de Colombia , Bogot\u00e1 , Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,8,17]]},"reference":[{"key":"2025120600292533132_j_jmc-2022-0020_ref_001","doi-asserted-by":"crossref","unstructured":"Beimel A. 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Linear secret sharing and the automatic search of linear rank inequalities. Appl Math Sci. 2015;9:5305\u201324.","DOI":"10.12988\/ams.2015.57478"},{"key":"2025120600292533132_j_jmc-2022-0020_ref_018","doi-asserted-by":"crossref","unstructured":"Mejia C, Andres Montoya J. On the information rates of homomorphic secret sharing schemes. J Inform Optimiz Sci. 2018;39(7):1463\u201382.","DOI":"10.1080\/02522667.2017.1367513"},{"key":"2025120600292533132_j_jmc-2022-0020_ref_019","doi-asserted-by":"crossref","unstructured":"Preneel B, Quisquater M, Vandewalle J. On the security of the threshold scheme based on the Chinese remainder theorem. Lecture Notes Comp Sci. 2002;2274:199\u2013210.","DOI":"10.1007\/3-540-45664-3_14"},{"key":"2025120600292533132_j_jmc-2022-0020_ref_020","unstructured":"Oxley J. Matroid theory. Oxford: The Clarendon Press, Oxford University Press; 1992."},{"key":"2025120600292533132_j_jmc-2022-0020_ref_021","doi-asserted-by":"crossref","unstructured":"Shamir A. 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