Isogenies of Elliptic Curves_ A Computational Approach椭圆曲线地同构计算方法.pdf

《Isogenies of Elliptic Curves_ A Computational Approach椭圆曲线地同构计算方法.pdf》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、IsogeniesofEllipticCurves:AComputationalApproachDanielShumowAthesispresentedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceUniversityofWashington2009ProgramAuthorizedtoOerDegree:MathematicsTABLEOFCONTENTSPageChapter1:Introduction............................1Chapter2:BasicTheory..

2、..........................42.1TheMultiplicationBymMap......................52.2Isogenies..................................152.2.1CoordinateMaps.........................172.2.2Separability............................212.2.3IsogeniesandDierentialForms.................242.2.4TheDualIsogeny...........

3、..............27Chapter3:Algorithms.............................293.1ComputingfromtheKernel.......................293.1.1Velu'sApproach:Computingfrompointsinthekernel....313.1.2Kohel'sApproach:Computingfromthekernelpolynomial..463.2Computingfromthekernelpolynomial:Generaldegreeisogenies...533.3Com

4、putingfromDomainandCodomain................593.3.1ANaiveApproach........................593.3.2Stark'sAlgorithm.........................60Bibliography....................................65AppendixA:AlgebraicComplexityTheoryandAlgorithms..........67A.1AlgebraicComplexity...........................67

5、A.2EcientPolynomialArithmetic.....................70A.2.1Solvingasystemofrstorderlineardierentialequations..71A.2.2Solvingasystemofrstordernonlineardierentialequations72A.2.3Polynomialreciprocalandexponentialfunctions........73iAppendixB:EllipticcurveisogeniesinSage..................76iiACKNOW

6、LEDGMENTSFirstandforemostImustthankReinierBr•okerforhisamazingamountofhelponthisproject.WithouthispatienceandapparentlylimitlessdepthofwillingnesstohelpIwouldnothavebeenabletolearnthismaterial.IamalsogratefulformyadvisorNealKoblitz,his exibilityandguidancehelpedmetogetthemostoutofmytimeintheMathema

7、ticsdepartment.KristinLauter,PeterMontgomery,JohnManferdelli,HenryCohn,JoshBenalohandtherestoftheMicrosoftResearchcryptographyteamhavebeenextraordinarilysupportiveandhelpfulinmypursuitofgraduateeducation,Ih