About the Group Law for the -hec-Jacobi Variety

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1、Beitr¨agezurAlgebraundGeometrieContributionstoAlgebraandGeometryVolume46(2005),No.1,125-130.AbouttheGroupLawfortheJacobiVarietyofaHyperellipticCurveFrankLeitenbergerFachbereichMathematik,Universit¨atRostockRostock,D-18051,Germanye-mail:frank.leitenberger@math

2、ematik.uni-rostock.deAbstract.WegeneralizethegrouplawofcurvesofdegreethreebychordsandtangentstotheJacobivarietyofahyperellipticcurve.Inthecaseofgenus2weaccomplishtheconstructionbyacubicparabola.WederiveexplicitrationalformulasfortheadditiononadensesetintheJac

3、obian.1.IntroductionTheintentionofthisremarkisanexplicitdescriptionofthegrouplawofhyperellipticcurves.Itappearsthatitispossibletogeneralizethechordandtangentmethodforcurvesofdegreethreeinaverynaivewaybyreplacingpointsbypointgroupsofgpointsandbyreplacinglinesb

4、ycertaininterpolationfunctions.Explicitdescriptionsofthegrouplawplayalessimportantroleinthehistoryofthesubject.Theyappearfirstinthenewliterature.Casselsremarked1983:“IcannotevenfindintheliteratureanexplicitsetofequationsfortheJacobianofacurveofgenus2togetherwit

5、hexplicitexpressionsforthegroupoperationinaformamenabletocalculation...”(cf.[2,3]).Mazurremarked1986:“...anaiveattempttogeneralizethisgroupstructure[ofdegree3curves]tocurvesofhigherdegree(evenquartics)willnotwork.”(cf.[8],p.230).Withthedevelopmentofcryptograp

6、hyarosealgorithmsforthegrouplaw.In1987Cantordescribedthegrouplawofahyperellipticcurveinthecontextofcryptography(cf.[1,6]).Latergrouplawsofmoregeneralclassesofcurvesweredescribedin[4,11].Thesegrouplawsworkstepbystepanddonotallowavisualization.Inthisremarkweder

7、iveexplicitformulasforthegrouplawfortheJacobivarietyofacurveofgenus2startingfromaninterpolatingcubicparabola.Astheabovealgorithms0138-4821/93$2.50c2005HeldermannVerlag126F.Leitenberger:AbouttheGroupLawfortheJacobiVarietyofaHyperellipticCurveperformthereductio

8、ninseveralstepsweexecutethereductioninonlyonestep.Thecaseg>2canbeperformedbyrationalinterpolationfunctionsanalogously.TheseinterpolationfunctionswerefirstconsideredbyJacobiinconnectionwith