{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T22:39:49Z","timestamp":1648766389088},"reference-count":10,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":1127,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2007]]},"abstract":"<jats:p>Let <jats:italic>C<\/jats:italic> be an arbitrary smooth algebraic curve of genus <jats:italic>g<\/jats:italic> over a large finite field <jats:bold>F<\/jats:bold>. The authors of this paper revisit fast addition algorithms in the Jacobian of <jats:italic>C<\/jats:italic> due to Khuri-Makdisi [math.NT\/0409209, to appear in <jats:italic>Mathematics of Computation<\/jats:italic>]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |<jats:bold>K<\/jats:bold>| \u226b g and which asymptotically require <jats:italic>O<\/jats:italic>(<jats:italic>g<\/jats:italic><jats:sup>2.376<\/jats:sup>) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of <jats:italic>C<\/jats:italic><jats:sub>3,4<\/jats:sub> curves of genus 3. They show show that, typically, the addition of two distinct elements in the Jacobian of a <jats:italic>C<\/jats:italic><jats:sub>3,4<\/jats:sub> curve requires 117 multiplications and 2 inversions in <jats:bold>K<\/jats:bold>, and an element can be doubled using 129 multiplications and 2 inversions in <jats:bold>K<\/jats:bold>. This represents an improvement of approximately 20% over previous methods.<\/jats:p>","DOI":"10.1112\/s146115700000142x","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T07:42:31Z","timestamp":1375774951000},"page":"307-328","source":"Crossref","is-referenced-by-count":4,"title":["Fast Jacobian Group Operations for C<sub>3,4<\/sub> Curves over a Large Finite Field"],"prefix":"10.1112","volume":"10","author":[{"given":"Fatima K.","family":"Abu Salem","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kamal","family":"khuri-makdisi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S146115700000142X_ref010","article-title":"\u2018Asymptotically fast group operations on Jacobians of general curves\u2019","author":"Khuri-Makdisi","year":"2004","journal-title":"Math. Comp."},{"key":"S146115700000142X_ref008","unstructured":"8. Hess Florian , \u2018Zur Divisorenklassengruppenberechnung in globalen Funktionenk\u00f6rpern\u2019, PhD thesis, Technische Universitat Berlin, 1999, http:\/\/www.math.tu-berlin.de\/~kant\/publications\/diss\/hess.pdf."},{"key":"S146115700000142X_ref005","article-title":"\u2018Index calculus in class groups of non-hyperelliptic curves of genus three\u2019","author":"Diem","year":"2006","journal-title":"J. Cryptology"},{"key":"S146115700000142X_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/11792086_38"},{"key":"S146115700000142X_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24847-7_6"},{"key":"S146115700000142X_ref001","doi-asserted-by":"publisher","DOI":"10.1080\/00207160410001661311"},{"key":"S146115700000142X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24632-9_5"},{"key":"S146115700000142X_ref007","unstructured":"7. Flon St\u00e9phane and Oyono Roger and Ritzenthaler Christophe , \u2018Fast addition on non-hyperelliptic genus 3 curves\u2019, preprint, 2004, http:\/\/www.math.uwaterloo.ca\/~royono\/Quartic.html, http:\/\/www.exp-math.uni-essen.de\/~oyono\/Quartic.html."},{"key":"S146115700000142X_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-04-01699-0"},{"key":"S146115700000142X_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-03-01567-9"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S146115700000142X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,6]],"date-time":"2019-06-06T18:02:02Z","timestamp":1559844122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S146115700000142X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007]]},"references-count":10,"alternative-id":["S146115700000142X"],"URL":"https:\/\/doi.org\/10.1112\/s146115700000142x","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007]]}}}