Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings

525 Pages·2002·3.28 MB·English

by  Manjul Bhargava (auth.)

Lecture Notes in Computer Science 2369
EditedbyG.Goos,J.Hartmanis,andJ.vanLeeuwen
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
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Claus Fieker David R. Kohel (Eds.)
Algorithmic
Number Theory
5th International Symposium,ANTS-V
Sydney,Australia, July 7-12, 2002
Proceedings
1 3
SeriesEditors
GerhardGoos,KarlsruheUniversity,Germany
JurisHartmanis,CornellUniversity,NY,USA
JanvanLeeuwen,UtrechtUniversity,TheNetherlands
VolumeEditors
ClausFieker
DavidR.Kohel
UniversityofSydney,SchoolofMathematicsandStatistics,F07
Sydney,NSW2006,Australia
E-mail:{claus,kohel}@maths.usyd.edu.au
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Algorithmicnumbertheory:5thinternationalsymposium;proceedings/
ANTS-V,Sydney,Australia,July7-12,2002.ClausFieker;DavidR.Kohel
(ed.).-Berlin;Heidelberg;NewYork;Barcelona;HongKong;London;
Milan;Paris;Tokyo:Springer,2002
(Lecturenotesincomputerscience;Vol.2369)
ISBN3-540-43863-7
CRSubjectClassification(1998):F.2,G.2,E.3
ISSN0302-9743
ISBN3-540-43863-7Springer-VerlagBerlinHeidelbergNewYork
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Preface
TheAlgorithmicNumberTheorySymposiabeganin1994atCornellUniversity
in Ithaca, New York to recognize the growing importance of algorithmic work
in the theory of numbers. The subject of the conference is broadly construed
to encompass a diverse body of mathematics, and to cover both the theoretical
and practical advances in the field. They have been held every two years since:
in Bordeaux (Universit´e Bordeaux I) in 1996, Portland (Reed College) in 1998,
Leiden (Universiteit Leiden) in 2000, and the present conference hosted by the
Magma Computational Algebra Group at the University of Sydney.
The conference program included invited talks by Manjul Bhargava (Prince-
ton),JohnCoates(Cambridge),AntoineJoux(DCSSICryptoLab),BjornPoo-
nen (Berkeley), and Takakazu Satoh (Saitama), as well as 34 contributed talks
invariousareasofnumbertheory.Inadditiontothemathematicalprogram,the
conferenceincludedaspecialdinnertohonourAlfvanderPoortenofMacquarie
University, on the occasion of his 60th birthday.
Each paper was reviewed by at least two experts external to the program
committee and the selection of papers was made on the basis of these recom-
mendations.Weexpressourappreciationtothe66expertrefereeswhoprovided
reportsonaverytightschedule.Refereeingofthesubmissionfromamemberof
the Magma group was organized by Joe Buhler.
Theprogramcommitteethanksthegenerousadvicefromorganizersofprevi-
ous ANTS conferences, particularly Joe Buhler, Wieb Bosma, Hendrik Lenstra,
and Bart de Smit. The conference was generously supported by the College of
Science and Technology, the School of Mathematics and Statistics (both at the
UniversityofSydney),theAustralianDefenceScienceTechnologyOrganisation,
and eSign.
April 2002 John Cannon
Claus Fieker
David Kohel
Table of Contents
Invited Talks
Gauss Composition and Generalizations .............................. 1
Manjul Bhargava
Elliptic Curves — The Crossroads of Theory and Computation .......... 9
John Coates
The Weil and Tate Pairings as Building Blocks
for Public Key Cryptosystems ....................................... 20
Antoine Joux
Using Elliptic Curves of Rank One towards the Undecidability
of Hilbert’s Tenth Problem over Rings of Algebraic Integers ............. 33
Bjorn Poonen
On p-adic Point Counting Algorithms for Elliptic Curves
over Finite Fields .................................................. 43
Takakazu Satoh
Number Theory
On Arithmetically Equivalent Number Fields of Small Degree ........... 67
Wieb Bosma, Bart de Smit
A Survey of Discriminant Counting .................................. 80
Henri Cohen, Francisco Diaz y Diaz, Michel Olivier
A Higher-Rank Mersenne Problem ................................... 95
Graham Everest, Peter Rogers, Thomas Ward
An Application of Siegel Modular Functions
to Kronecker’s Limit Formula........................................ 108
Takashi Fukuda, Keiichi Komatsu
Computational Aspects of NUCOMP................................. 120
Michael J. Jacobson, Jr., Alfred J. van der Poorten
Efficient Computation of Class Numbers of Real Abelian Number Fields .. 134
St´ephane R. Louboutin
An Accelerated Buchmann Algorithm for Regulator Computation
in Real Quadratic Fields ............................................ 148
Ulrich Vollmer
VIII Table of Contents
Arithmetic Geometry
Some Genus 3 Curves with Many Points ............................. 163
Roland Auer, Jaap Top
Trinomials ax7+bx+c and ax8+bx+c
with Galois Groups of Order 168 and 8·168........................... 172
Nils Bruin, Noam D. Elkies
Computations on Modular Jacobian Surfaces .......................... 189
Enrique Gonza´lez-Jim´enez, Josep Gonza´lez, Jordi Gua`rdia
Integral Points on Punctured Abelian Surfaces......................... 198
Andrew Kresch, Yuri Tschinkel
Genus 2 Curves with (3,3)-Split Jacobian
and Large Automorphism Group..................................... 205
Tony Shaska
Transportable Modular Symbols and the Intersection Pairing ............ 219
Helena A. Verrill
Elliptic Curves and CM
Action of Modular Correspondences around CM Points ................. 234
Jean-Marc Couveignes, Thierry Henocq
Curves Dy2 =x3−x of Odd Analytic Rank........................... 244
Noam D. Elkies
Comparing Invariants for Class Fields of Imaginary Quadratic Fields ..... 252
Andreas Enge, Fran¸cois Morain
A Database of Elliptic Curves – First Report .......................... 267
William A. Stein, Mark Watkins
Point Counting
Isogeny Volcanoes and the SEA Algorithm ............................ 276
Mireille Fouquet, Franc¸ois Morain
Fast Elliptic Curve Point Counting Using Gaussian Normal Basis ........ 292
Hae Young Kim, Jung Youl Park, Jung Hee Cheon, Je Hong Park,
Jae Heon Kim, Sang Geun Hahn
An Extension of Kedlaya’s Algorithm to Artin-Schreier Curves
in Characteristic 2 ................................................. 308
Jan Denef, Frederik Vercauteren
Table of Contents IX
Cryptography
Implementing the Tate Pairing ...................................... 324
Steven D. Galbraith, Keith Harrison, David Soldera
Smooth Orders and Cryptographic Applications ....................... 338
Carl Pomerance, Igor E. Shparlinski
Chinese Remaindering for Algebraic Numbers in a Hidden Field ......... 349
Igor E. Shparlinski, Ron Steinfeld
Function Fields
An Algorithm for Computing Weierstrass Points ....................... 357
Florian Hess
New Optimal Tame Towers of Function Fields over Small Finite Fields ... 372
Wen-Ching W. Li, Hiren Maharaj, Henning Stichtenoth,
Noam D. Elkies
Periodic Continued Fractions in Elliptic Function Fields ................ 390
Alfred J. van der Poorten, Xuan Chuong Tran
Discrete Logarithms and Factoring
Fixed Points and Two-Cycles of the Discrete Logarithm ................ 405
Joshua Holden
Random Cayley Digraphs and the Discrete Logarithm .................. 416
Jeremy Horwitz, Ramarathnam Venkatesan
The Function Field Sieve Is Quite Special ............................. 431
Antoine Joux, Reynald Lercier
MPQS with Three Large Primes ..................................... 446
PaulLeyland,ArjenLenstra,BruceDodson,AlecMuffett,SamWagstaff
An Improved Baby Step Giant Step Algorithm
for Point Counting of Hyperelliptic Curves over Finite Fields ............ 461
Kazuto Matsuo, Jinhui Chao, Shigeo Tsujii
Factoring N =pq2 with the Elliptic Curve Method..................... 475
Peter Ebinger, Edlyn Teske
Gr¨obner Bases
A New Scheme for Computing with Algebraically Closed Fields.......... 491
Allan Steel
X Table of Contents
Complexity
Additive Complexity and Roots of Polynomials
over Number Fields and p-adic Fields................................. 506
J. Maurice Rojas
Author Index................................................... 517
Gauss Composition and Generalizations
Manjul Bhargava(cid:1)
Clay Mathematics Institute and Princeton University
Abstract. We discuss several higher analogues of Gauss composition
and consider their potential algorithmic applications.
1 Introduction
The class groups of quadratic fields have long held a special place in the annals
of algorithmic algebraic number theory. This special place has been due in large
part to the close relationship between ideal class groups of quadratic fields and
integral binary quadratic forms, which allows one to reduce the study and com-
putation of ideal classes in quadratic orders to the study of lattice points in a
certain fixed three-dimensional real vector space—namely the space of binary
quadratic forms over R.
Thisfundamentalcorrespondence,knownclassicallyas“Gausscomposition”,
was discovered by Gauss almost exactly 200 years ago in his celebrated work
Disquisitiones Arithmeticae of 1801. Even after two centuries, there is still no
faster way known for computing the ideal class groups of quadratic fields than
by Gauss composition.
The key feature of Gauss composition, which makes it so useful, is that one
has a bijective correspondence between the arithmetic objects of interest (ideal
classes of quadratic orders) with the integer points in a vector space—rather
than, say, with the integer points on a high codimension variety in an affine
space. The principle here is that one can readily locate all the integer points
in a codimension zero region in a vector space, whereas searching for integer
points on higher codimension subvarieties is extremely difficult in general, both
computationally and theoretically.
Thus situations where one has a direct bijection between arithmetic objects
of study and the integer points in a vector space (modulo, say, the action of a
reductive group over Z) are clearly of intrinsic interest, both from a theoretical
and an algorithmic standpoint; and the question naturally arises as to whether
there exist any spaces in addition to Gauss’s space of binary quadratic forms
that might share this remarkable property.
(cid:1) IamverygratefultoProfessorsAndrewWilesandPeterSarnakforalltheirenthusi-
asmandencouragement,andtoJonathanHanke,KiranKedlaya,andLennyNgfor
helpful comments on an earlier draft of this paper. This work was supported by
the Hertz Foundation and the Clay Mathematics Institute, and was conducted at
Princeton University.
C.FiekerandD.R.Kohel(Eds.):ANTS2002,LNCS2369,pp.1–8,2002.
(cid:1)c Springer-VerlagBerlinHeidelberg2002