Elliptic Curves, Modular Forms and Cryptography: Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory

339 Pages·2003·25.32 MB·English

by  Ashwani K. Bhandari

Elliptic Curves, Modular
Forms and Cryptography
Proceedings of the Advanced Instructional
Workshop on Aigebraic Number Theory
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Elliptic Curves, Modular
Forms and Cryptography
Proceedings of the Advanced Instructional
Workshop on Aigebraic Number Theory
Edited by
A.K. Bhandari
D.S. Nagaraj
B. Ramakrishnan
T.N. Venkataramana
~HINDUSTAN
U LQj UBOOKAGENCY
Editors:
Ashwani K. Bhandari D.S. Nagaraj
Centre for Advanced Study in Institute of Mathematical Sciences
Mathematics C I T Campus, Taramani
Panjab University Chennai 600 113, India
Chandigarh 160 014, India e-mail: [e-mail]
e-mail: [e-mail]
T.N. Venkataramana
B. Ramakrishnan School of Mathematics
Harish-Chandra Research Institute Tata Institute of Fundamental
Chhatnag Road Research
Jhusi Dr. Homi Bhabha Road
Allahabad 211 019, India Mumbai 400 005, India.
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ISBN 978-81-85931-42-5 ISBN 978-93-86279-15-6 (eBook)
DOI 10.1007/978-93-86279-15-6
Preface
This volume represents the proceedings of the Advanced Instruc
tional Workshop on Algebraic Number Theory held at Harish-Chandra
Research Institute (HRI), (Formerly, Mehta Research Institute), Alla
habad, during November 2000.
The theme of the workshop was Aigebraic N umber Theory with spe
cial emphasis on Elliptic Curves. The theory of Elliptic Curves has
been the source ofnew approaches to classical problems in Number The
ory. It has also found applications in Cryptography. The workshop also
covered some aspects of Cryptography. During the workshop, several
Mathematicians from India gathered at HRI and gave lectures on vari
ous topics in these fields. This volume consists of articles prepared by
some of the speakers of the workshop and we thank all the contributors
to this volume.
The volume is in three parts, the first part contains articles in the
field of Elliptic Curves, the second contains articles on Modular Forms.
Some basics as well as some advanced topics on Cryptography are pre
sented in the third and final part of these proceedings.
Each part contains an introduction, which, in some sense, gives the
overall picture of the contents in that part. Most of the articles are
presented in a self-contained style and they give a different flavour to
the subject. Though some of the contents of a few articles are already
contained in some text books, they are presented here (with due refer
ences) in order to make this volume complete to some extent. We hope
that the graduate students who want to pursue their research career in
Number Theory will benefit from this volume.
This workshop was followed by an International Conference on N um
ber Theory, the proceedings ofwhich had already been published through
Hindustan Book Agency, New Delhi, who also brings out the present one.
We thank Mr. Jainendra K. Jain for agreeing to publish this volume
and also for his patience in spite of the unexpected delay in finalising
the proceedings.
VI Preface
It is our pleasure to express our sincere thanks to Professor S. D.
Adhikari whose encouragement as weIl as help virtually brought out
this volume.
We thank Prof. H. S. Mani, the then director of HRI, and Prof. Ravi
S. Kulkarni, Director of HRI for their constant support and encourage
ment in our endeavour. We thank Professor S. A. Katre for his help
in organising the workshop. Thanks are also due to the administrative
staff of HRI for the hard work they put in to make the workshop a grand
success.
FinaIly, we acknowledge the generous financial support from HRI
and the Department of Science and Technology, Govt. of India.
April 2003 A. K. Bhandari
D. S. Nagaraj
B. Ramakrishnan
T. N. Venkataramana
Contents
Preface v
Part I. Elliptic Curves 1
An Overview
D. S. NAGARAJ 3
A Quick Introduction to Algebraic Geometry and Elliptic Curves
D.S. NAGARAJ AND B. SURY 5
Elliptic Curves over Finite Fields
B. SURY 33
The Nagell-Lutz Theorem
RAJAT TANDON 49
Weak Mordell-Weil Theorem
C. S. RAJAN 63
The Mordell-Weil Theorem
D.S. NAGARAJ AND B. SURY 73
Complex Multiplication
EKNATH GHATE 85
The Main Theorem of Complex Multiplication
DIPENDRA PRASAD 109
Approximations of Algebraic Numbers by Rationals:
A Theorem of Thue
T. N. SHOREY 119
viii Contents
Siegel's Theorem: Finiteness of Integral Points
S. D. ADHIKARI AND D. S. RAMANA 139
p-adic Theta Functions and Tate Curves
ALEXANDER F. BROWN 151
t-adic Representation Attached to an Elliptic Curve
over a Nu mber Field
D. S. NAGARAJ 167
Arithmetic on Curves
CHANDAN SINGH DALAWAT 193
Part 11. Modular Forms 201
Introduction
B. RAMAKRISHNAN 203
Elliptic Functions
PARVATI SHASTRI 205
An Introduction to Modular Forms and Hecke Operators
M. MANICKAM AND B. RAMAKRISHNAN 223
L-Functions of Modular Forms
C. S. YOGANANDA 247
On the Eichler-Shimura Congruence Relation
T. N. VENKATARAMANA 255
Part 111. Cryptography 261
Cryptography
ASHWANI K. BHANDARI 269
Classical Cryptosystems
R. THANGADURAI 275
The Public Key Cryptography
ASHWANI K. BHANDARI 287
Primality and Factoring
AMORA N ONGKYNRIH 303
Elliptic Curves and Cryptography
R. BALASUBRAMANIAN 325
Part I
ELLIPTIC CURVES