{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:35:39Z","timestamp":1764981339848,"version":"3.46.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2012,10,1]],"date-time":"2012-10-01T00:00:00Z","timestamp":1349049600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,10,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n                  <jats:p>\n                    Bergman's ring\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2012-0014_5ffe37bc14f60252855a8ffa12319619.png\"\/>\n                    , parameterized by a prime number\n                    <jats:italic>p<\/jats:italic>\n                    ,\nis a ring with\n                    <jats:italic>\n                      p\n                      <jats:sup>5<\/jats:sup>\n                    <\/jats:italic>\n                    elements that cannot be embedded in a ring of matrices over any commutative ring.\nThis ring was discovered in 1974.\nIn 2011, Climent, Navarro and Tortosa described an efficient implementation of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2012-0014_5ffe37bc14f60252855a8ffa12319619.png\"\/>\n                    using simple modular arithmetic, and suggested that this ring may be a useful source\nfor intractable cryptographic problems.\nWe present a deterministic polynomial time reduction of the discrete logarithm problem in\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2012-0014_5ffe37bc14f60252855a8ffa12319619.png\"\/>\n                    to the classical discrete logarithm problem in\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2012-0014_7d9668eed00aff84dc48384be9314c3f.png\"\/>\n                    , the\n                    <jats:italic>p<\/jats:italic>\n                    -element field.\nIn particular, the discrete logarithm problem in\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2012-0014_5ffe37bc14f60252855a8ffa12319619.png\"\/>\n                    can be solved, by conventional computers,\nin sub-exponential time.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2012-0014","type":"journal-article","created":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T12:21:08Z","timestamp":1347452468000},"page":"171-182","source":"Crossref","is-referenced-by-count":2,"title":["The discrete logarithm problem in Bergman's non-representable ring"],"prefix":"10.1515","volume":"6","author":[{"given":"Matan","family":"Banin","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel"}]},{"given":"Boaz","family":"Tsaban","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel"}]}],"member":"374","published-online":{"date-parts":[[2012,10,5]]},"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2012-0014\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2012-0014\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:30:55Z","timestamp":1764981055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2012-0014\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,1]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,10,5]]},"published-print":{"date-parts":[[2012,10,1]]}},"alternative-id":["10.1515\/jmc-2012-0014"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2012-0014","relation":{},"ISSN":["1862-2984","1862-2976"],"issn-type":[{"type":"electronic","value":"1862-2984"},{"type":"print","value":"1862-2976"}],"subject":[],"published":{"date-parts":[[2012,10,1]]}}}