Low-Dimensional Lattice Reduction Revisited (Extended Abstract)
Identifieur interne : 002F92 ( Hal/Corpus ); précédent : 002F91; suivant : 002F93Low-Dimensional Lattice Reduction Revisited (Extended Abstract)
Auteurs : Phong Q. Nguyen ; Damien StehléSource :
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Abstract
Most of the interesting algorithmic problems in the geometry of numbers are NP-hard as the lattice dimension increases. This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which generalizes in a natural way the well-known two-dimensional Gaussian algorithm and a recent three-dimensional algorithm by Semaev. Using this algorithm, we show that up to dimension four, one can compute a Minkowski-reduced basis, a Korkine-Zolotarev-reduced basis, and a closest vector in quadratic time without fast integer arithmetic: all other algorithms known in dimension four have a bit-complexity which is at least cubic. Our analysis, based on geometric properties of low-dimensional lattices and in particular Voronoï cells, arguably simplifies Semaev's analysis in dimension three, but breaks down in dimension five. More precisely, in dimension five or higher, the core algorithm does not necessarily output a Minkowski-reduced basis, and it is unknown whether or not the algorithm is still polynomial.
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This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which generalizes in a natural way the well-known two-dimensional Gaussian algorithm and a recent three-dimensional algorithm by Semaev. Using this algorithm, we show that up to dimension four, one can compute a Minkowski-reduced basis, a Korkine-Zolotarev-reduced basis, and a closest vector in quadratic time without fast integer arithmetic: all other algorithms known in dimension four have a bit-complexity which is at least cubic. Our analysis, based on geometric properties of low-dimensional lattices and in particular Voronoï cells, arguably simplifies Semaev's analysis in dimension three, but breaks down in dimension five. More precisely, in dimension five or higher, the core algorithm does not necessarily output a Minkowski-reduced basis, and it is unknown whether or not the algorithm is still polynomial.</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">Low-Dimensional Lattice Reduction Revisited (Extended Abstract)</title><author role="aut"><persName><forename type="first">Phong Q.</forename><surname>Nguyen</surname></persName><email></email><idno type="halauthor">131412</idno><orgName ref="#struct-441569"></orgName></author><author role="aut"><persName><forename type="first">Damien</forename><surname>Stehlé</surname></persName><email>damien.stehle@loria.fr</email><idno type="halauthor">130769</idno><orgName ref="#struct-364475"></orgName><affiliation ref="#struct-2364"></affiliation></author><editor role="depositor"><persName><forename>Publications</forename><surname>Loria</surname></persName><email>publications@loria.fr</email></editor></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2006-09-26 10:15:09</date><date type="whenModified">2016-05-19 01:09:06</date><date type="whenReleased">2006-09-28 15:22:47</date><date type="whenProduced">2004</date></edition><respStmt><resp>contributor</resp><name key="108626"><persName><forename>Publications</forename><surname>Loria</surname></persName><email>publications@loria.fr</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">inria-00100176</idno><idno type="halUri">https://hal.inria.fr/inria-00100176</idno><idno type="halBibtex">nguyen:inria-00100176</idno><idno type="halRefHtml">Duncan Buell. 6th International Symposium on Algorithmic Number Theory - ANTS VI, 2004, Burligton, United States. Springer, 3076, pp.338--357, 2004, Lecture notes in Computer Science</idno><idno type="halRef">Duncan Buell. 6th International Symposium on Algorithmic Number Theory - ANTS VI, 2004, Burligton, United States. Springer, 3076, pp.338--357, 2004, Lecture notes in Computer Science</idno></publicationStmt><seriesStmt><idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno><idno type="stamp" n="INPL">Institut National Polytechnique de Lorraine</idno><idno type="stamp" n="LORIA2">Publications du LORIA</idno><idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno><idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno><idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno><idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno><idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno></seriesStmt><notesStmt><note type="commentary">Colloque avec actes et comité de lecture. internationale.</note><note type="audience" n="2">International</note><note type="invited" n="0">No</note><note type="popular" n="0">No</note><note type="peer" n="1">Yes</note><note type="proceedings" n="1">Yes</note></notesStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">Low-Dimensional Lattice Reduction Revisited (Extended Abstract)</title><author role="aut"><persName><forename type="first">Phong Q.</forename><surname>Nguyen</surname></persName><idno type="halAuthorId">131412</idno><orgName ref="#struct-441569"></orgName></author><author role="aut"><persName><forename type="first">Damien</forename><surname>Stehlé</surname></persName><email>damien.stehle@loria.fr</email><idno type="halAuthorId">130769</idno><orgName ref="#struct-364475"></orgName><affiliation ref="#struct-2364"></affiliation></author></analytic><monogr><idno type="localRef">A04-R-015 || nguyen04a</idno><meeting><title>6th International Symposium on Algorithmic Number Theory - ANTS VI</title><date type="start">2004</date><settlement>Burligton</settlement><country key="US">United States</country></meeting><editor>Duncan Buell</editor><imprint><publisher>Springer</publisher><biblScope unit="serie">Lecture notes in Computer Science</biblScope><biblScope unit="volume">3076</biblScope><biblScope unit="pp">338--357</biblScope><date type="datePub">2004</date></imprint></monogr></biblStruct></sourceDesc><profileDesc><langUsage><language ident="en">English</language></langUsage><textClass><keywords scheme="author"><term xml:lang="fr">lattice reduction</term><term xml:lang="fr">réduction de réseaux</term><term xml:lang="fr">semaev algorithm</term><term xml:lang="fr">algorithme de semaev</term><term xml:lang="fr">gauss algorithm</term><term xml:lang="fr">algorithme de gauss</term></keywords><classCode scheme="halDomain" n="info.info-oh">Computer Science [cs]/Other [cs.OH]</classCode><classCode scheme="halTypology" n="COMM">Conference papers</classCode></textClass><abstract xml:lang="en">Most of the interesting algorithmic problems in the geometry of numbers are NP-hard as the lattice dimension increases. This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which generalizes in a natural way the well-known two-dimensional Gaussian algorithm and a recent three-dimensional algorithm by Semaev. Using this algorithm, we show that up to dimension four, one can compute a Minkowski-reduced basis, a Korkine-Zolotarev-reduced basis, and a closest vector in quadratic time without fast integer arithmetic: all other algorithms known in dimension four have a bit-complexity which is at least cubic. Our analysis, based on geometric properties of low-dimensional lattices and in particular Voronoï cells, arguably simplifies Semaev's analysis in dimension three, but breaks down in dimension five. More precisely, in dimension five or higher, the core algorithm does not necessarily output a Minkowski-reduced basis, and it is unknown whether or not the algorithm is still polynomial.</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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