**Fields Institute Communications**

Volume: 58;
2010;
133 pp;
Hardcover

MSC: Primary 11; 14;

Print ISBN: 978-0-8218-4311-6

Product Code: FIC/58

List Price: $83.00

Individual Member Price: $66.40

**Electronic ISBN: 978-1-4704-1784-0
Product Code: FIC/58.E**

List Price: $83.00

Individual Member Price: $66.40

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# Algebraic Curves and Cryptography

Share this page *Edited by *
*V. Kumar Murty*

A co-publication of the AMS and Fields Institute

It is by now a well-known paradigm that
public-key cryptosystems can be built using finite Abelian groups and
that algebraic geometry provides a supply of such groups through
Abelian varieties over finite fields. Of special interest are the
Abelian varieties that are Jacobians of algebraic curves. All of the
articles in this volume are centered on the theme of point counting
and explicit arithmetic on the Jacobians of curves over finite
fields. The topics covered include Schoof's \(\ell\)-adic point
counting algorithm, the \(p\)-adic algorithms of Kedlaya and
Denef-Vercauteren, explicit arithmetic on the Jacobians of
\(C_{ab}\) curves and zeta functions.

This volume is based on seminars on algebraic curves and cryptography held
at the GANITA Lab of the University of Toronto during 2001–2008. The
articles are mostly suitable for independent study by graduate students who
wish to enter the field, both in terms of introducing basic material as well
as guiding them in the literature. The literature in cryptography seems to
be growing at an exponential rate. For a new entrant into the subject,
navigating through this ocean can seem quite daunting. In this volume, the
reader is steered toward a discussion of a few key ideas of the subject,
together with some brief guidance for further reading. It is hoped that this
approach may render the subject more approachable.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in cryptography, applications of number theory and algebraic geometry.

# Table of Contents

## Algebraic Curves and Cryptography

- Cover Cover11
- Title page i2
- Contents iii4
- An overview of algebraic curves and cryptography 16
- Schoof’s point counting algorithm 1116
- Report on the Denef-Vercauteren/Kedlaya algorithm 6570
- An introduction to Gröbner bases 8388
- 𝐶_{𝑎𝑏} curves and arithmetic on their Jacobians 99104
- The zeta functions of two Garcia-Stichtenoth towers 119124
- Bibliography 129134
- Index 133138
- Back Cover Back Cover1142