{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T17:27:32Z","timestamp":1790530052632,"version":"4.1.0"},"reference-count":13,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:p>\n                    Lattice basis reduction is an important problem in geometry of numbers with applications in combinatorial optimization, computer algebra, and cryptography. The well-known sequential LLL algorithm finds a short vector in O(n\n                    <jats:sup>4<\/jats:sup>\n                    log B) arithmetic operations on integers having binary length O(n log B), where n denotes the dimension of the lattice and B denotes the maximum L\n                    <jats:sub>2<\/jats:sub>\n                    norm of the initial basis vectors. In this paper a new analysis of the parallel algorithm of Roch and Villard is presented. It is shown that on an n x n mesh it needs O(n\n                    <jats:sup>2<\/jats:sup>\n                    log B) arithmetic operations on integers having binary length O(n log B). This improves the previous analysis and shows that an asymptotical speedup of n\n                    <jats:sup>2<\/jats:sup>\n                    is possible using n\n                    <jats:sup>2<\/jats:sup>\n                    processors.\n                  <\/jats:p>","DOI":"10.1137\/s0097539795295626","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1295-1302","source":"Crossref","is-referenced-by-count":10,"title":["Complexity Analysis of a Parallel Lattice Basis Reduction Algorithm"],"prefix":"10.1137","volume":"27","author":[{"given":"C.","family":"Heckler","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Thiele","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,28]]},"reference":[{"key":"R1","unstructured":"G. Golub and C. van Loan,\n                      Matrix Computations\n                      , The Johns Hopkins University Press, Baltimore, MD, 1983."},{"key":"R2","unstructured":"C. Heckler,\n                      Automatische Parallelisierung und parallele Gitterbasisreduktion\n                      , dissertation, Institut f\u00fcr Mikroelektronik, Universit\u00e4t des Saarlandes, Saarbr\u00fccken, Germany, March 1995."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"C. Heckler and L. Thiele,\n                      A parallel lattice basis reduction for mesh\u2010connected processor arrays and parallel complexity\n                      , in Proc. SPDP\u201993, Dallas, TX, December 1\u20134, 1993, pp. 400\u2013407.","DOI":"10.1109\/SPDP.1993.395505"},{"key":"R4","unstructured":"A. Joux,\n                      A fast parallel lattice reduction algorithm\n                      , in Proc. Second Gauss Symposium, Munich, 1993."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"Erich Kaltofen, On the complexity of finding short vectors in integer lattices, Lecture Notes in Comput. Sci., Vol. 162, Springer, Berlin, 1983, 236\u201324486m:11101","DOI":"10.1007\/3-540-12868-9_107"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"Ravi Kannan, Algorithmic geometry of numbers, Annual Reviews, Palo Alto, CA, 1987, 231\u201326789a:11131","DOI":"10.1146\/annurev.cs.02.060187.001311"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457454"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1287\/moor.8.4.538"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"J. Roch and G. Villard,\n                      Parallel gcd and lattice basis reduction\n                      , in Proc. CONPAR92, Lyon, LNCS 632, Springer\u2010Verlag, Berlin, New York, 1992, pp. 557\u2013564.","DOI":"10.1007\/3-540-55895-0_455"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(88)90004-1"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581144"},{"key":"R12","unstructured":"A. Skhrei\u02d8ver, Teoriya lineinogo i tselochislennogo programmirovaniya. 2, \u201cMir\u201d, 1991, 361\u2013704, Translated from the English by S. A. Tarasov, M. A. Frumkin and V. I. Shlyk; Translation edited by L. G. Khachiyan94g:90005"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"G. Villard,\n                      Parallel lattice basis reduction\n                      , in Proc. International Symposium on Symbolic and Algebraic Computation, Berkeley, CA, ACM Press, New York, 1992, pp. 269\u2013277.","DOI":"10.1145\/143242.143327"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539795295626","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:14:10Z","timestamp":1787336050000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539795295626"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":13,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["10.1137\/S0097539795295626"],"URL":"https:\/\/doi.org\/10.1137\/s0097539795295626","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}