Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings PDF
Preview Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings
Lecture Notes in Computer Science 4076 CommencedPublicationin1973 FoundingandFormerSeriesEditors: GerhardGoos,JurisHartmanis,andJanvanLeeuwen EditorialBoard DavidHutchison LancasterUniversity,UK TakeoKanade CarnegieMellonUniversity,Pittsburgh,PA,USA JosefKittler UniversityofSurrey,Guildford,UK JonM.Kleinberg CornellUniversity,Ithaca,NY,USA FriedemannMattern ETHZurich,Switzerland JohnC.Mitchell StanfordUniversity,CA,USA MoniNaor WeizmannInstituteofScience,Rehovot,Israel OscarNierstrasz UniversityofBern,Switzerland C.PanduRangan IndianInstituteofTechnology,Madras,India BernhardSteffen UniversityofDortmund,Germany MadhuSudan MassachusettsInstituteofTechnology,MA,USA DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA DougTygar UniversityofCalifornia,Berkeley,CA,USA MosheY.Vardi RiceUniversity,Houston,TX,USA GerhardWeikum Max-PlanckInstituteofComputerScience,Saarbruecken,Germany Florian Hess Sebastian Pauli Michael Pohst (Eds.) Algorithmic Number Theory 7th International Symposium, ANTS-VII Berlin, Germany, July 23-28, 2006 Proceedings 1 3 VolumeEditors FlorianHess SebastianPauli MichaelPohst TechnischeUniversitätBerlin FakultätII,InstitutfürMathematikMA8-1 Strassedes17.Juni136,10623Berlin,Germany E-mail:{hess,pauli,pohst}@math.tu-berlin.de LibraryofCongressControlNumber:Appliedfor CRSubjectClassification(1998):F.2,G.2,E.3,I.1 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ISSN 0302-9743 ISBN-10 3-540-36075-1SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-36075-9SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable toprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2006 PrintedinGermany Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SPIN:11792086 06/3142 543210 Preface ThefirstAlgorithmicNumberTheorySymposium(ANTS)conferencewashosted byCornellUniversity,Ithaca,NewYork,USAin1994.Thegoaloftheconference was to bring together number theorists from around the world, and to advance theoreticalandpracticalresearchinthefield.ANTSIwassoonfollowedbycon- ferences in Bordeaux, France in 1996, Portland, Oregon, USA in 1998, Leiden, intheNetherlandsin2000,Sydney,Australiain2002,andBurlington,Vermont, USAin2004.TechnischeUniversit¨atBerlininGermanyhostedANTSVIIduring July, 23-28 2006. Five invited speakers attended ANTS VII. Thirty seven contributed papers were presented and a poster session was held. The invited speakers were Nigel BostonoftheUniversityofWisconsinatMadison,JohnCremonaoftheUniver- sityofNottingham,BasEdixhovenofUniversteitLeiden,Ju¨rgenKlu¨nersofUni- versita¨t Kassel, and Don Zagier from the Max-Planck-Institut fu¨r Mathematik, Bonn. Each submitted paper was reviewed by at least two experts external to the ProgramCommitteewhichdecidedaboutacceptanceorrejectiononthebasisof theirrecommendations.TheSelfridgeprizeincomputationalnumbertheorywas awardedtotheauthorsofthebestcontributedpaperpresentedattheconference. TheorganizersofANTSVIIexpresstheirgratitudeandthankstoformeror- ganizersDuncanBuell,JohnCannon,HenriCohen,DavidJoyner,BlairKellyIII and Peter Stevenhagen for their important and valuable advice. We also appre- ciate the sponsorships by the Deutsche Forschungsgemeinschaft and the Math- ematical Institute of Technische Unversita¨t Berlin. July 2006 Florian Hess Sebastian Pauli Michael Pohst The ANTS VII Organizers Organization Program Committee Karim Belabas, Universit´e Bordeaux 1 Johannes Buchmann, Universita¨t Darmstadt John Cannon, University of Sydney Gerhard Frey, Universita¨t Duisburg-Essen Istv´an Gaa´l, Debreceni Egyetem Franc¸ois Morain, E´cole Polytechnique Paris Ken Nakamula, Tokyo Metropolitan University Enric Nart, Universitat Auto`noma de Barcelona Takakazu Satoh, Tokyo Institute of Technology Peter Stevenhagen, Universiteit Leiden Fernando Villegas, University of Texas Hugh Williams, University of Calgary Conference Website The names of the winners of the Selfridge prize, material supplementing the contributed papers and errata for the proceedings, as well as the abstracts of the posters and the posters presented at ANTS VII, can be found under http://www.math.tu-berlin.de/~kant/ants. Table of Contents Invited Talks Computing Pro-P Galois Groups Nigel Boston, Harris Nover ..................................... 1 The Elliptic Curve Database for Conductors to 130000 John Cremona................................................. 11 On the Computation of the Coefficients of a Modular Form Bas Edixhoven ................................................ 30 Cohen–Lenstra Heuristics of Quadratic Number Fields E´tienne Fouvry, Ju¨rgen Klu¨ners ................................. 40 Algebraic Number Theory An Algorithm for Computing p-Class Groups of Abelian Number Fields Miho Aoki, Takashi Fukuda ..................................... 56 Computation of Locally Free Class Groups Werner Bley, Robert Boltje ..................................... 72 Numerical Results on Class Groups of Imaginary Quadratic Fields Michael J. Jacobson Jr., Shantha Ramachandran, Hugh C. Williams.............................................. 87 Cyclic Polynomials Arising from Kummer Theory of Norm Algebraic Tori Masanari Kida ................................................ 102 The Totally Real Primitive Number Fields of Discriminant at Most 109 Gunter Malle.................................................. 114 A Modular Method for Computing the Splitting Field of a Polynomial Gu´ena¨el Renault, Kazuhiro Yokoyama ............................ 124 Analytic and Elementary Number Theory On the Density of Sums of Three Cubes Jean-Marc Deshouillers, Franc¸ois Hennecart, Bernard Landreau ..... 141 VIII Table of Contents The Mertens Conjecture Revisited Tadej Kotnik, Herman te Riele .................................. 156 Fast Bounds on the Distribution of Smooth Numbers Scott T. Parsell, Jonathan P. Sorenson ........................... 168 Use of Extended Euclidean Algorithm in Solving a System of Linear Diophantine Equations with Bounded Variables Parthasarathy Ramachandran ................................... 182 The PseudosquaresPrime Sieve Jonathan P. Sorenson .......................................... 193 Doubly-Focused Enumeration of Pseudosquares and Pseudocubes Kjell Wooding, Hugh C. Williams ................................ 208 Lattices Practical Lattice Basis Sampling Reduction Johannes Buchmann, Christoph Ludwig .......................... 222 LLL on the Average Phong Q. Nguyen, Damien Stehl´e ................................ 238 On the Randomness of Bits Generated by Sufficiently Smooth Functions Damien Stehl´e................................................. 257 Curves and Varieties over Fields of Characteristic Zero Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q John Cremona, Samir Siksek .................................... 275 Points of Low Height on Elliptic Curves and Surfaces I: Elliptic Surfaces over P1 with Small d Noam D. Elkies................................................ 287 Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples Noam D. Elkies................................................ 302 Table of Contents IX The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds Andreas-Stephan Elsenhans, J¨org Jahnel .......................... 317 Testing Equivalence of Ternary Cubics Tom Fisher ................................................... 333 Classification of Genus 3 Curves in Special Strata of the Moduli Space Martine Girard, David R. Kohel ................................. 346 Heegner Point Computations Via Numerical p-Adic Integration Matthew Greenberg............................................. 361 Symmetric Powersof Elliptic Curve L-Functions Phil Martin, Mark Watkins ..................................... 377 Determined Sequences, Continued Fractions, and Hyperelliptic Curves Alfred J. van der Poorten ....................................... 393 Computing CM Points on Shimura Curves Arising from Cocompact Arithmetic Triangle Groups John Voight ................................................... 406 Curves over Finite Fields and Applications Arithmetic of Generalized Jacobians Isabelle D´ech`ene ............................................... 421 Hidden Pairings and Trapdoor DDH Groups Alexander W. Dent, Steven D. Galbraith .......................... 436 Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10 David Freeman ................................................ 452 FastBilinearMapsfromthe Tate-LichtenbaumPairingonHyperelliptic Curves Gerhard Frey, Tanja Lange ..................................... 466 High Security Pairing-BasedCryptography Revisited Robert Granger, Dan Page, Nigel P. Smart ....................... 480 Efficiently Computable Endomorphisms for Hyperelliptic Curves David R. Kohel, Benjamin A. Smith ............................. 495 X Table of Contents Construction of Rational Points on Elliptic Curves over Finite Fields Andrew Shallue, Christiaan E. van de Woestijne ................... 510 20 Years of ECM Paul Zimmermann, Bruce Dodson ............................... 525 Discrete Logarithms An Index Calculus Algorithm for Plane Curves of Small Degree Claus Diem ................................................... 543 Signature Calculus and Discrete Logarithm Problems Ming-Deh Huang, Wayne Raskind ............................... 558 Spectral Analysis of PollardRho Collisions Stephen D. Miller, Ramarathnam Venkatesan...................... 573 Hard Instances of the Constrained Discrete Logarithm Problem Ilya Mironov, Anton Mityagin, Kobbi Nissim ...................... 582 Author Index................................................... 599 Computing Pro-P Galois Groups(cid:2) Nigel Boston and Harris Nover Department of Mathematics, University of Wisconsin, Madison, WI 53706 {boston, nover}@math.wisc.edu Abstract. We describe methods for explicit computation of Galois groups of certain tamely ramified p-extensions. In the finite case this yieldsashortlistofcandidatesfortheGaloisgroup.Intheinfinitecaseit produces a family or few families of likely candidates. 1 Introduction Throughout this paper K will denote a number field, p a rational prime, and S a finite set of primes, none of which lies above p. Furthermore, Kun,p will denote the maximal everywhere unramified p-extension of K. Our aim is to computetheGaloisgroupGofthemaximalp-extensionofK unramifiedoutside S. Whereas much is known about p-extensions ramified at p [16],[23], those unramified at p are poorly understood. Wingberg [26] calls them among the most mysterious objects in number theory. They are, however, important test cases for the Fontaine-Mazur conjecture [13], which here implies that G has no infinite p-adic analytic quotient. The first author’s work on this conjecture [5], [6]suggeststhatinfactGshouldhavenontrivialactionsonlocallyfinite,rooted trees,providingglimpsesofatheoryofarborealGaloisrepresentationsinparallel to the well-developed theory of p-adic Galois representations. Wewillfocuson2-extensions.Thesortofinformationavailableistheabelian- ization of low index (usually 1,2,4,and 8) subgroups,computed as quotients of ray class groups thanks to class field theory, and exact values of, or at least bounds on, the generator and relation ranks of G. In addition, in the cases that Gisinfinite,classfieldtheorygivesthefurtherinformationthateverysubgroup of finite index has finite abelianization (such a group is called FAb). In certain cases, such as K =Q, something is known about the form of the relations. If a finite index subgroup H has cyclic abelianization, then Burnside’s basis theoremforcesH(cid:2) ={1}andsoGisfinite.Moreover,afinite index subgroupH withabelianizationthe Klein4-groupforcesGto be finite sinceby anoldresult of Taussky H has a cyclic subgroup of index 2. This allowed Boston and Perry [8]tofindtheGaloisgroupsofseveral2-extensionsofQ.Ontheotherhand,the method is limited since in most cases these conditions do not hold at low index. BostonandLeedham-Green[7]nextintroduceda new methodfor computing Ginmoregeneralcircumstances.TheideaistosearchforGinO’Brien’stree[20], (cid:2) TheauthorswouldliketothankRafeJones,JeremyRouse,RobRhoadesandJayce Getzforusefuldiscussions.NigelBostonwaspartiallysupportedbytheNSF.Harris NoverwassupportedbytheOfficeofNavalResearchthroughanNDSEGfellowship. F.Hess,S.Pauli,andM.Pohst(Eds.):ANTS2006,LNCS4076,pp.1–10,2006. (cid:3)c Springer-VerlagBerlinHeidelberg2006
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