The 2-adic CM method for genus 2 curves with application to cryptography
Identifieur interne : 004A86 ( Hal/Corpus ); précédent : 004A85; suivant : 004A87The 2-adic CM method for genus 2 curves with application to cryptography
Auteurs : Pierrick Gaudry ; Thomas Houtmann ; David Kohel ; Christophe Ritzenthaler ; Annegret WengSource :
Abstract
The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i sqrt(75 + 12 sqrt(17))), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre's algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.
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X</orgName><desc><address><country key="FR"></country></address></desc></org></tutelle><tutelle name="UMR7161" active="#struct-441569" type="direct"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno><idno type="IdRef">02636817X</idno><orgName>Centre National de la Recherche Scientifique</orgName><orgName type="acronym">CNRS</orgName><date type="start">1939-10-19</date><desc><address><country key="FR"></country></address><ref type="url">http://www.cnrs.fr/</ref></desc></org></tutelle><tutelle active="#struct-118511" type="direct"><org type="laboratory" xml:id="struct-118511" status="VALID"><idno type="RNSR">200818248E</idno><orgName>INRIA Saclay - Ile de France</orgName><desc><address><addrLine>1 rue Honoré d'Estienne d'OrvesBâtiment Alan TuringCampus de l'École Polytechnique91120 Palaiseau</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre/saclay</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle></tutelles></hal:affiliation></affiliation></author></analytic></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i sqrt(75 + 12 sqrt(17))), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre's algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">The 2-adic CM method for genus 2 curves with application to cryptography</title><author role="aut"><persName><forename type="first">Pierrick</forename><surname>Gaudry</surname></persName><email>gaudry@loria.fr</email><idno type="idhal">pierrickgaudry</idno><idno type="halauthor">66001</idno><affiliation ref="#struct-2364"></affiliation></author><author role="aut"><persName><forename type="first">Thomas</forename><surname>Houtmann</surname></persName><email></email><idno type="halauthor">134713</idno><affiliation ref="#struct-56056"></affiliation></author><author role="aut"><persName><forename type="first">David</forename><surname>Kohel</surname></persName><email></email><idno type="idhal">drkohel</idno><idno type="halauthor">134715</idno><affiliation ref="#struct-75152"></affiliation></author><author role="aut"><persName><forename type="first">Christophe</forename><surname>Ritzenthaler</surname></persName><email></email><idno type="idhal">christophe-ritzenthaler</idno><idno type="halauthor">68040</idno><affiliation ref="#struct-892"></affiliation></author><author role="aut"><persName><forename type="first">Annegret</forename><surname>Weng</surname></persName><email></email><idno type="halauthor">134714</idno><affiliation ref="#struct-56056"></affiliation></author><editor role="depositor"><persName><forename>Pierrick</forename><surname>Gaudry</surname></persName><email>pierrick.gaudry@loria.fr</email></editor></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2006-10-04 13:44:40</date><date type="whenModified">2016-05-18 08:55:44</date><date type="whenReleased">2006-10-04 14:40:14</date><date type="whenProduced">2006-12</date><date type="whenEndEmbargoed">2006-10-04</date><ref type="file" target="https://hal.inria.fr/inria-00103435/document"><date notBefore="2006-10-04"></date></ref><ref type="file" n="1" target="https://hal.inria.fr/inria-00103435/file/42840114.pdf"><date notBefore="2006-10-04"></date></ref></edition><respStmt><resp>contributor</resp><name key="103933"><persName><forename>Pierrick</forename><surname>Gaudry</surname></persName><email>pierrick.gaudry@loria.fr</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">inria-00103435</idno><idno type="halUri">https://hal.inria.fr/inria-00103435</idno><idno type="halBibtex">gaudry:inria-00103435</idno><idno type="halRefHtml">X. Lay and K. Chen. Asiacrypt 2006, Dec 2006, Shangai, China. Springer-Verlag, 4284, pp.114-129, 2006, Lecture notes in computer science</idno><idno type="halRef">X. Lay and K. Chen. Asiacrypt 2006, Dec 2006, Shangai, China. Springer-Verlag, 4284, pp.114-129, 2006, 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="LIX">Laboratoire d'informatique de l'école polytechnique</idno><idno type="stamp" n="X-LIX" p="X">Laboratoire d'informatique de l'X (LIX)</idno><idno type="stamp" n="X-DEP-INFO" p="X-DEP">Département d'informatique</idno><idno type="stamp" n="X-DEP">Polytechnique</idno><idno type="stamp" n="X" p="PARISTECH">Ecole Polytechnique</idno><idno type="stamp" n="PARISTECH">ParisTech</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="IML" p="I2M">Institut de Mathématiques de Luminy</idno><idno type="stamp" n="I2M">Institut de Mathématiques de Marseille</idno><idno type="stamp" n="INRIA-SACLAY">INRIA Saclay - Ile de France</idno><idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno><idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno><idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="UNIV-AMU">Aix Marseille Université</idno><idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno></seriesStmt><notesStmt><note type="audience" n="1">Not set</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">The 2-adic CM method for genus 2 curves with application to cryptography</title><author role="aut"><persName><forename type="first">Pierrick</forename><surname>Gaudry</surname></persName><email>gaudry@loria.fr</email><idno type="idHal">pierrickgaudry</idno><idno type="halAuthorId">66001</idno><affiliation ref="#struct-2364"></affiliation></author><author role="aut"><persName><forename type="first">Thomas</forename><surname>Houtmann</surname></persName><idno type="halAuthorId">134713</idno><affiliation ref="#struct-56056"></affiliation></author><author role="aut"><persName><forename type="first">David</forename><surname>Kohel</surname></persName><idno type="idHal">drkohel</idno><idno type="halAuthorId">134715</idno><affiliation ref="#struct-75152"></affiliation></author><author role="aut"><persName><forename type="first">Christophe</forename><surname>Ritzenthaler</surname></persName><idno type="idHal">christophe-ritzenthaler</idno><idno type="halAuthorId">68040</idno><affiliation ref="#struct-892"></affiliation></author><author role="aut"><persName><forename type="first">Annegret</forename><surname>Weng</surname></persName><idno type="halAuthorId">134714</idno><affiliation ref="#struct-56056"></affiliation></author></analytic><monogr><meeting><title>Asiacrypt 2006</title><date type="start">2006-12</date><settlement>Shangai</settlement><country key="CN">China</country></meeting><editor>X. Lay and K. Chen</editor><imprint><publisher>Springer-Verlag</publisher><biblScope unit="serie">Lecture notes in computer science</biblScope><biblScope unit="volume">4284</biblScope><biblScope unit="pp">114-129</biblScope><date type="datePub">2006-12</date></imprint></monogr></biblStruct></sourceDesc><profileDesc><langUsage><language ident="en">English</language></langUsage><textClass><classCode scheme="halDomain" n="info.info-cr">Computer Science [cs]/Cryptography and Security [cs.CR]</classCode><classCode scheme="halTypology" n="COMM">Conference papers</classCode></textClass><abstract xml:lang="en">The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i sqrt(75 + 12 sqrt(17))), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre's algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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