The 2-adic CM method for genus 2 curves with application to cryptography
Identifieur interne : 004F27 ( Main/Exploration ); précédent : 004F26; suivant : 004F28The 2-adic CM method for genus 2 curves with application to cryptography
Auteurs : Pierrick Gaudry [France] ; Thomas Houtmann [France] ; David Kohel [Australie] ; Christophe Ritzenthaler [France] ; Annegret Weng [France]Source :
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.
Url:
Affiliations:
- Australie, France
- Grand Est, Lorraine (région)
- Nancy
- Institut national polytechnique de Lorraine, Université Nancy 2, Université de Lorraine
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Le document en format XML
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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><country>France</country></affiliation></author><author><name sortKey="Kohel, David" sort="Kohel, David" uniqKey="Kohel D" first="David" last="Kohel">David Kohel</name><affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-75152" status="VALID"><orgName>School of Mathematics and statistics [Sydney]</orgName><desc><address><addrLine>School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia</addrLine><country key="AU"></country></address><ref type="url">http://www.maths.usyd.edu.au/</ref></desc><listRelation><relation active="#struct-374701" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-374701" type="direct"><org type="institution" xml:id="struct-374701" status="VALID"><orgName>The University of Sydney [Sydney]</orgName><desc><address><addrLine>New South Wales 2006</addrLine><country key="AU"></country></address><ref type="url">http://sydney.edu.au/</ref></desc></org></tutelle></tutelles></hal:affiliation><country>Australie</country></affiliation></author><author><name sortKey="Ritzenthaler, Christophe" sort="Ritzenthaler, Christophe" uniqKey="Ritzenthaler C" first="Christophe" last="Ritzenthaler">Christophe Ritzenthaler</name><affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-892" status="OLD"><orgName>Institut de mathématiques de Luminy</orgName><orgName type="acronym">IML</orgName><desc><address><addrLine>case 907 163 Av de Luminy 13288 MARSEILLE CEDEX 09</addrLine><country key="FR"></country></address><ref type="url">http://iml.univ-mrs.fr/</ref></desc><listRelation><relation active="#struct-5033" type="direct"></relation><relation name="UMR6206" active="#struct-441569" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-5033" type="direct"><org type="institution" xml:id="struct-5033" status="OLD"><idno type="IdRef">026402882</idno><orgName>Université de la Méditerranée - Aix-Marseille 2</orgName><date type="start">1969</date><date type="end">2011</date><desc><address><addrLine>58, boulevard Charles Livon - 13284 Marseille cedex 07</addrLine><country key="FR"></country></address><ref type="url">http://www.univmed.fr/</ref></desc></org></tutelle><tutelle name="UMR6206" 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></tutelles></hal:affiliation><country>France</country></affiliation></author><author><name sortKey="Weng, Annegret" sort="Weng, Annegret" uniqKey="Weng A" first="Annegret" last="Weng">Annegret Weng</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-56056" status="OLD"><idno type="RNSR">200318378T</idno><orgName>Algorithmic number theory for cryptology</orgName><orgName type="acronym">TANC</orgName><desc><address><addrLine>LIX</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/equipes/tanc</ref></desc><listRelation><relation active="#struct-2071" type="direct"></relation><relation active="#struct-300340" type="direct"></relation><relation name="UMR7161" active="#struct-441569" type="direct"></relation><relation active="#struct-118511" type="direct"></relation><relation active="#struct-300009" type="indirect"></relation></listRelation><tutelles><tutelle active="#struct-2071" type="direct"><org type="laboratory" xml:id="struct-2071" status="VALID"><orgName>Laboratoire d'informatique de l'École polytechnique [Palaiseau]</orgName><orgName type="acronym">LIX</orgName><desc><address><addrLine>Route de Saclay 91128 PALAISEAU CEDEX</addrLine><country key="FR"></country></address><ref type="url">http://www.lix.polytechnique.fr/</ref></desc><listRelation><relation active="#struct-300340" type="direct"></relation><relation name="UMR7161" active="#struct-441569" type="direct"></relation></listRelation></org></tutelle><tutelle active="#struct-300340" type="direct"><org type="institution" xml:id="struct-300340" status="VALID"><orgName>Polytechnique - 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><country>France</country></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><affiliations><list><country><li>Australie</li><li>France</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Nancy</li></settlement><orgName><li>Institut national polytechnique de Lorraine</li><li>Université Nancy 2</li><li>Université de Lorraine</li></orgName></list><tree><country name="France"><region name="Grand Est"><name sortKey="Gaudry, Pierrick" sort="Gaudry, Pierrick" uniqKey="Gaudry P" first="Pierrick" last="Gaudry">Pierrick Gaudry</name></region><name sortKey="Houtmann, Thomas" sort="Houtmann, Thomas" uniqKey="Houtmann T" first="Thomas" last="Houtmann">Thomas Houtmann</name><name sortKey="Ritzenthaler, Christophe" sort="Ritzenthaler, Christophe" uniqKey="Ritzenthaler C" first="Christophe" last="Ritzenthaler">Christophe Ritzenthaler</name><name sortKey="Weng, Annegret" sort="Weng, Annegret" uniqKey="Weng A" first="Annegret" last="Weng">Annegret Weng</name></country><country name="Australie"><noRegion><name sortKey="Kohel, David" sort="Kohel, David" uniqKey="Kohel D" first="David" last="Kohel">David Kohel</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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