Handbook of Elliptic and Hyperelliptic Curve Cryptography

Contents Authors Sample Chapter Table of Contents Links Errata

Handbook of Elliptic and Hyperelliptic Curve Cryptography

Scientific Editors: Henri Cohen, Gerhard Frey
Executive Editor: Christophe Doche

Authors: Roberto M. Avanzi, Henri Cohen, Christophe Doche, Gerhard Frey, Tanja Lange, Kim Nguyen, Frederik Vercauteren

Contributors: Bertrand Byramjee, Jean-Christophe Courrège, Sylvain Duquesne, Benoît Feix, Reynald Lercier, David Lubicz, Nicolas Thériault, and Andrew Weigl

Discrete Mathematics and Its Applications 34
ISBN: 1584885181
Publication Date: 7/19/2005
Number of Pages: 848

Chapman & Hall/CRC


From the official CRC flyer:

The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive. The main reason is that no subexponential algorithm for computing discrete logarithms on small genus curves is currently available, except in very special cases. Therefore curve-based cryptosystems require much smaller key sizes than RSA to attain the same security level. This makes them particularly attractive for implementations on memory-restricted devices like smart cards and in high-security applications.
The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography. After a very detailed exposition of the mathematical background, it provides ready-to-implement algorithms for the group operations and computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method and provides the algorithms in an explicit manner. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves. For some special curves the discrete logarithm problem can be transferred to an easier one; the consequences are explained and suggestions for good choices are given. The authors present applications to protocols for discrete- logarithm-based systems (including bilinear structures) and explain the use of elliptic and hyperelliptic curves in factorization and primality proving. Two chapters on smart cards explore their design and efficient implementations. Practical and theoretical aspects of side-channel attacks and countermeasures and a chapter devoted to (pseudo-)random number generation round off the exposition.
The broad coverage of all important areas makes this book a complete handbook of elliptic and hyperelliptic curve cryptography and an invaluable reference to anyone interested in this exciting field.

All chapters are listed here. For a complete Table of Contents see the section Sample Chapter.


Roberto M. Avanzi is currently Junior Professor at the Ruhr-University, Bochum. His research interests include arithmetic and algorithmic aspects of curve-based cryptography, integer recodings and addition chains, side-channel analysis, and Diophantine analysis.

Henri Cohen is Professor of Mathematics at the University of Bordeaux, France. His research interests are number theory in general, and computational number theory in particular.

Christophe Doche is lecturer at Macquarie University, Sydney, Australia. His research is focused on analytic and algorithmic number theory as well as cryptography.

Gerhard Frey holds a chair for number theory at the Institute for Experimental Mathematics at the University of Duisburg-Essen, Germany. His research interests are number theory and arithmetical geometry as well as applications to coding theory and cryptography.

Tanja Lange is Professor at Eindhoven Technical University. Her research covers mathematical aspects of public-key cryptography and computational number theory with focus on curve cryptography.

Kim Nguyen received a Ph.D. in number theory and cryptography in 2001 at the University of Essen. His first position outside academia was with the Cryptology Competence Center of Philips Semiconductors Hamburg. He currently works for the Bundesdruckerei GmbH in Berlin, Germany.

Frederik Vercauteren is a Post-Doc at the Katholieke Universiteit Leuven, Belgium. His research interests are computational algebraic geometry and number theory, with applications to cryptography.

Bertrand Byramjee, Jean-Christophe Courrège, Sylvain Duquesne, Benoît Feix, Reynald Lercier, David Lubicz, Nicolas Thériault, and Andrew Weigl

Sample Chapter

We are glad to provide access to one sample chapter. With permission of CRC you may view (but not save or print) the following table of contents, chapter and bibliography. The standard copyright notice from CRC Press applies to this electronic version. The first page of each file contains the full statement.


We display a different chapter each month, so stay tuned and check back - or buy the book. (This link points to the US page of CRC. Their catalogue number is C5181.)

Table of Contents

Short table of contents. The full version can be found in section Sample Chapter.

1. Introduction to Public-Key Cryptography (was online March 13 - April 21 2007)

2. Algebraic Background (was online June 01 - July 01, 2009)
3. Background on p-adic Numbers (was online September 29 - November 3, 2007)
4. Background on Curves and Jacobians (was online June 6 - July 6, 2007)
5. Varieties Over Special Fields (was online July 01 - September 03, 2009)
6. Background on Pairings (was online April 22 - June 5, 2007)
7. Background on Weil Descent (was online September 17, 2011)
8. Cohomological Background on Point Counting (was online June 8 -July 10, 2008)

9. Exponentiation (currently online)
10. Integer Arithmetic (was online January 8 - February 14, 2008)
11. Finite Field Arithmetic (was online February 12 - March 13, 2007)
12. Arithmetic of p-adic Numbers (was online September 03 - October 11, 2009)

13. Arithmetic of Elliptic Curves (was online September 10 - October 22, 2006)
14. Arithmetic of Hyperelliptic Curves (was online December 20, 2008 - May 31, 2009)
15. Arithmetic of Special Curves (was online February 14 - March 18, 2008)
16. Implementation of Pairings (was online Nov 24 2006 - Jan 10, 2007)

17. Point Counting on Elliptic and Hyperelliptic Curves (was online Jan 11 - Feb 12, 2007)
18. Complex Multiplication

19. Generic Algorithms for Computing Discrete Logarithms (was online September 14 - October 28, 2008)
20. Index Calculus (was online October 23 - November 23, 2006)
21. Index Calculus for Hyperelliptic Curves (was online July 11 - August 12, 2008)
22. Transfer of Discrete Logarithms (was online August 27 - September 29, 2007)

23. Algebraic Realizations of DL Systems (was online October 11 - February 28, 2009)
24. Pairing-Based Cryptography (was online August 13 - September 14, 2008)
25. Compositeness and Primality Testing-Factoring (was online November 4 - December 5, 2007)

26. Fast Arithmetic Hardware (was online March 19 - May 3, 2008)
27. Smart Cards (was online October 28 - December 20 2008)
28. Practical Attacks on Smart Cards (was online December 5, 2007 - January 7, 2008)
29. Mathematical Countermeasures Against Side-Channel Attacks (was online May 4 - June 8, 2008)
30. Random Numbers - Generation and Testing

Notation index
General Index

Book reviews
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