Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings

608 Pages·2006·6.6 MB·English

by  Nigel Boston

Lecture Notes in Computer Science 4076
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CarnegieMellonUniversity,Pittsburgh,PA,USA
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CornellUniversity,Ithaca,NY,USA
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ETHZurich,Switzerland
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UniversityofBern,Switzerland
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IndianInstituteofTechnology,Madras,India
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UniversityofCalifornia,LosAngeles,CA,USA
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UniversityofCalifornia,Berkeley,CA,USA
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RiceUniversity,Houston,TX,USA
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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
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LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues
ISSN 0302-9743
ISBN-10 3-540-36075-1SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-36075-9SpringerBerlinHeidelbergNewYork
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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