Hardcover ISBN:  9780821843116 
Product Code:  FIC/58 
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Electronic ISBN:  9781470417840 
Product Code:  FIC/58.E 
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Book DetailsFields Institute CommunicationsVolume: 58; 2010; 133 ppMSC: Primary 11; 14;
It is by now a wellknown paradigm that publickey 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 DenefVercauteren, 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.ReadershipGraduate students and research mathematicians interested in cryptography, applications of number theory and algebraic geometry.

Table of Contents

Chapters

V. Kumar Murty  An overview of algebraic curves and cryptography

Nicolas Thériault  Schoof’s point counting algorithm

Zubair Ashraf, Ali Juma and Pramathanath Sastry  Report on the DenefVercauteren/Kedlaya algorithm

Mohammed RadiBenjelloun  An introduction to Gröbner bases

Farzali Izadi  $C_{ab}$ curves and arithmetic on their Jacobians

Kenneth Shum  The zeta functions of two GarciaStichtenoth towers


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It is by now a wellknown paradigm that publickey 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 DenefVercauteren, 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.
Graduate students and research mathematicians interested in cryptography, applications of number theory and algebraic geometry.

Chapters

V. Kumar Murty  An overview of algebraic curves and cryptography

Nicolas Thériault  Schoof’s point counting algorithm

Zubair Ashraf, Ali Juma and Pramathanath Sastry  Report on the DenefVercauteren/Kedlaya algorithm

Mohammed RadiBenjelloun  An introduction to Gröbner bases

Farzali Izadi  $C_{ab}$ curves and arithmetic on their Jacobians

Kenneth Shum  The zeta functions of two GarciaStichtenoth towers