Elliptic Curves: A Computational Approach

377 Pages·2004·1.274 MB·English

by  Susanne Schmitt, Horst G. Zimmer

de Gruyter Studies in Mathematics 31
Editors: Carlos Kenig · Andrew Ranicki · Michael Röckner
de Gruyter Studies in Mathematics
1 Riemannian Geometry, 2nd rev. ed., Wilhelm P.A.Klingenberg
2 Semimartingales, Michel Me´tivier
3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup
4 Spaces of Measures, Corneliu Constantinescu
5 Knots, 2nd rev. and ext. ed., Gerhard Burde and Heiner Zieschang
6 Ergodic Theorems, Ulrich Krengel
7 Mathematical Theory of Statistics, Helmut Strasser
8 Transformation Groups, Tammo tom Dieck
9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii
10 Analyticity in Infinite Dimensional Spaces, Michel Herve´
11 Elementary Geometry in Hyperbolic Space, Werner Fenchel
12 Transcendental Numbers, Andrei B. Shidlovskii
13 Ordinary Differential Equations, Herbert Amann
14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and
Francis Hirsch
15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine
16 Rational Iteration, Norbert Steinmetz
17 Korovkin-typeApproximationTheoryanditsApplications,FrancescoAltomare
and Michele Campiti
18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev
19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima,
Yoichi Oshima and Masayoshi Takeda
20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R.Bloom
and Herbert Heyer
21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov
22 Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii,
Victor E. Shatalov and Boris Yu. Sternin
23 Probability Theory, Heinz Bauer
24 Variational Methods for Potential Operator Equations, Jan Chabrowski
25 The Structure of Compact Groups, Karl H. Hofmann and Sidney A. Morris
26 Measure and Integration Theory, Heinz Bauer
27 Stochastic Finance, Hans Föllmer and Alexander Schied
28 Painleve´ Differential Equations in the Complex Plane, Valerii I. Gromak, Ilpo
Laine and Shun Shimomura
29 Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel
and Jakob Nielsen
30 The Reidemeister Torsion of 3-Manifolds, Liviu I. Nicolaescu
Susanne Schmitt · Horst G. Zimmer
Elliptic Curves
A Computational Approach
With an Appendix by Attila Pethö
≥
Walter de Gruyter
Berlin · New York
Authors
SusanneSchmitt HorstGünterZimmer
Max-Planck-InstitutfürInformatik(MPII) FachbereichMathematik(Bau27.1)
AlgorithmsandComplexityGroup(AG1) UniversitätdesSaarlandes
Stuhlsatzenhausweg85 Postfach151150
66123Saarbrücken 66041Saarbrücken
Germany Germany
E-Mail:[email protected] E-Mail:[email protected]
SeriesEditors
CarlosE.Kenig AndrewRanicki MichaelRöckner
DepartmentofMathematics DepartmentofMathematics FakultätfürMathematik
UniversityofChicago UniversityofEdinburgh UniversitätBielefeld
5734UniversityAve MayfieldRoad Universitätsstraße25
Chicago,IL60637 EdinburghEH93JZ 33615Bielefeld
USA Scotland Germany
MathematicsSubjectClassification2000:11-01,14-01;11G05,11G07,11-04,11G20,11G40,11G50,
14-04,14H52
Keywords:ellipticcurves,Weierstrassfunction,height,torsion-group,rank,Mordell(cid:3)Weilgroup,algo-
rithms,cryptography
(cid:1)(cid:1) Printedonacid-freepaperwhichfallswithintheguidelinesoftheANSI
toensurepermanenceanddurability.
LibraryofCongress(cid:3)Cataloging-in-PublicationData
Schmitt,Susanne.
Ellipticcurves:acomputationalapproach/SusanneSchmitt,Horst
G.Zimmer;withanappendixbyAttilaPethö.
p. cm.(cid:3)(DeGruyterstudiesinmathematics;31)
Includesbibliographicalreferencesandindex.
ISBN3-11-016808-1(acid-freepaper)
1.Curves,Elliptic. I.Zimmer,HorstG. II.Title. III.Series.
QA567.2.E44S35 2003
516.3152(cid:3)dc21 2002041531
ISBN3-11-016808-1
BibliographicinformationpublishedbyDieDeutscheBibliothek
DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataisavailableintheInternetat(cid:1)http://dnb.ddb.de(cid:2).
(cid:1)Copyright2003byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany.
Allrightsreserved,includingthoseoftranslationintoforeignlanguages.Nopartofthisbookmaybe
reproducedinanyformorbyanymeans,electronicormechanical,includingphotocopy,recording,or
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Preface
Thepurposeofthepresenttextistogiveanelementaryintroductiontothearithmetic
ofellipticcurvesovernumberfieldsfromacomputationalpointofview. Thetextis
thereforeequippedwiththecorrespondingalgorithms. Manyexamplesand,ofcourse,
someexercisesareadded. Inallthoseregardswemakeextensiveuseoftheavailability
ofcomputers.
Thearithmeticofellipticcurvescanbecomealmostarbitrarilysophisticated. That
iswhyanelementaryintroductionrequiresacertainrestrictionontheselectionoftopics
treated. However,themostimportanttopicsaredealtwithinthebook. Theyinclude
thedeterminationoftorsiongroups,computationsconcerningtheMordell–Weilgroup,
especiallyconcerningtherankandabasisofthatgroup,heightcalculations,andthe
determinationofintegraland,moregenerally,S-integralpoints. Toavoidoverlapping
we occasionally, instead of giving proofs, cite the books [204], [207] of Silverman.
For some more exercises, we also refer to the books of Silverman which contain a
lotofexercisesrelatedtothepresenttextaswell. Weshouldalsomentionthatthere
aresomesurveyarticlesonthistopic,forinstancethoseofFrey[69],Stroeker[216],
Zagier[245]andthesecondauthor[254],[255]
Elliptic curves admit many applications, in pure mathematics and in computer
science. TheyplayanimportantroleintheproofbyFaltingsofMordell’sconjecture
andintheproofbyWilesofFermat’slasttheoremontheonehand,andtheyareused
e.g. forfactoringintegers,derivingpropertiesofprimenumbers,andincryptography
on the other (see Jacobson, Menezes, Stein [104]). Thus cryptography enters the
pictureheretoo.
Oneaimofthebookisalsotoinformaboutresultsobtainedintheresearchgroup
of the second author. Specifically, the research group was developing the computer
algebra system SIMATH.This system will be further developed from the year 2002
onbyKenNakamulaandhisgroupinTokyoandbyothers.
TheSIMATHpackagefocusesonellipticcurves. Thereforeitplaysanimportant
part in the book: The algorithms are implemented in SIMATH and the examples
are produced by SIMATH. Of course, among other things, SIMATH contains all
the elementary algorithms used in the book. More about SIMATH can be found on
http://diana.math.uni-sb.de/˜simath/.
Ellipticcurvesareaspecialcaseofdiophantineequations. Itisthereforeinorder
to briefly consider in an appendix how to solve general diophantine equations. The
LLL-algorithm plays an essential role here. In addition, lower bounds for linear
forms in logarithms are important. Finally, the p-adic methods used are outlined in
AppendixA.
Wehopethatthetextwillbereadbymathematiciansaswellascomputerscientists,
andscientistsfromindustry. Inviewoftheapplicationsincryptography,itisespecially
vi Preface
importantforcomputerscientiststoknowsomeelementsofthearithmeticofelliptic
curves.
Thebibliographydoesnotcompriseallthebooksorpapersrelevanttothefieldof
ellipticcurvesbecauseitwasnearlyimpossibleinviewofitsextend.
TheauthorswishtoexpresstheirheartfeltthankstoF.LemmermeyerfromtheUni-
versity of San Marcos, and M. Kida from the University of Electro-Communication
Chofu, Tokyo, for their detailed and helpful reading of the manuscript. F. Lem-
mermeyer and M. Kida also made a number of suggestions for improvements and
extensions. Manyofthemcouldbeincorporatedinthetext.
A. Petho˝ has not only writtenAppendixA but also essentially contributed to the
manuscript. Therefore,thanksareduetohimtoo.
Last but not least we thank Dr. M. Karbe from the de Gruyter Verlag for his
permanentengagementforthemanuscriptandforhispatiencewiththeauthors.
Saarbrücken,September2003 SusanneSchmitt
Horst-GünterZimmer
Contents
Preface v
1 Ellipticcurves 1
1.1 Normalforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Theadditionlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Multiplicationformulas . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Factorizationandprimalitytest . . . . . . . . . . . . . . . . . . . . . 24
1.5 Isogeniesandendomorphismsofellipticcurves . . . . . . . . . . . . 27
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Ellipticcurvesoverthecomplexnumbers 33
2.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Weierstraß℘-function. . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Periodsofellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Complexmultiplication . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Ellipticcurvesoverfinitefields 63
3.1 Frobeniusendomorphismandsupersingularcurves . . . . . . . . . . 63
3.2 Computingthenumberofpoints . . . . . . . . . . . . . . . . . . . . 65
3.3 Constructionofellipticcurveswithgivengrouporder . . . . . . . . . 74
3.4 Ellipticcurvesincryptography . . . . . . . . . . . . . . . . . . . . . 79
3.5 Thediscretelogarithmproblemonellipticcurves . . . . . . . . . . . 83
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Ellipticcurvesoverlocalfields 87
4.1 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Thefiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 ThetheoremofNagell,Lutz,andCassels . . . . . . . . . . . . . . . 98
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 TheMordell–Weiltheoremandheights 103
5.1 TheoremofMordellandWeil. . . . . . . . . . . . . . . . . . . . . . 103
5.2 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Computationoftheheights . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Pointsofboundedheight . . . . . . . . . . . . . . . . . . . . . . . . 133
5.5 Thedifferencesbetweentheheights . . . . . . . . . . . . . . . . . . 136
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
viii Contents
6 Torsiongroup 147
6.1 Structureofthetorsiongroup . . . . . . . . . . . . . . . . . . . . . . 147
6.2 Ellipticcurveswithintegralj-invariant . . . . . . . . . . . . . . . . 151
6.3 ThetheoremofNagell,Lutz,andCassels . . . . . . . . . . . . . . . 177
6.4 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.5 Computationofthetorsiongroup . . . . . . . . . . . . . . . . . . . . 185
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7 Therank 198
7.1 L-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.2 ThecoefficientsoftheL-series . . . . . . . . . . . . . . . . . . . . . 202
7.3 ContinuationoftheL-series . . . . . . . . . . . . . . . . . . . . . . 207
7.4 Conjecturesconcerningtherank . . . . . . . . . . . . . . . . . . . . 214
7.5 TheSelmerandtheTate–Shafarevichgroup . . . . . . . . . . . . . . 216
7.6 2-descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.7 Therankinfieldextensions . . . . . . . . . . . . . . . . . . . . . . . 233
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8 Basis 242
8.1 Linearlyindependentpoints . . . . . . . . . . . . . . . . . . . . . . 242
8.2 Computationofabasis . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.4 Heegnerpointmethod. . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9 S-integralpoints 263
9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.2 Ellipticlogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3 S-integralpointsoverQ. . . . . . . . . . . . . . . . . . . . . . . . . 272
9.4 Proofofthetheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 278
9.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
A Algorithmictheoryofdiophantineequations 294
A.1 Hilbert’s10th problem . . . . . . . . . . . . . . . . . . . . . . . . . 294
A.2 IntroductiontoBaker’smethod . . . . . . . . . . . . . . . . . . . . . 295
A.3 S-unitequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
A.4 Thueequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
A.5 Smallcollectionofotherresults . . . . . . . . . . . . . . . . . . . . 305
A.6 Lowerboundsforlinearformsinlogarithms . . . . . . . . . . . . . . 308
A.7 LLL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
A.8 Reductionofthelargebound . . . . . . . . . . . . . . . . . . . . . . 311
Contents ix
B Multiquadraticnumberfields 316
B.1 MultiquadraticfieldsandGaloisgroups . . . . . . . . . . . . . . . . 316
B.2 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
B.3 IntegralBases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
B.4 DecompositionLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
B.5 Biquadraticnumberfields . . . . . . . . . . . . . . . . . . . . . . . . 330
B.6 Totallyrealandtotallycomplexbiquadraticfields . . . . . . . . . . . 341
B.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Bibliography 351
Index 365