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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 VercauterenContributors: Bertrand Byramjee, Jean-Christophe Courrège,
Sylvain Duquesne, Benoît Feix, Reynald Lercier, David Lubicz, Nicolas
Thériault, and Andrew WeiglDiscrete Mathematics and Its Applications 34ISBN: 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.

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

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.

**View:**

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.)

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

Preface

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

I MATHEMATICAL BACKGROUND

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)

II ELEMENTARY ARITHMETIC

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)

III ARITHMETIC OF CURVES

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)

IV POINT COUNTING

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

18. Complex Multiplication

V COMPUTATION OF DISCRETE LOGARITHMS

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)

VI APPLICATIONS

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)

VII REALIZATIONS OF DL SYSTEMS

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

References

Notation index

General Index

Book reviews

- MAA Online July 2006 by Fernando Q. Gouvêa.
- European Mathematical Society short review as recent book pp. 56-57.
- Zentralblatt MATH by Juan Tena Ayuso, ZBL 1082.94001.
- Mathematical Reviews by Steven D. Galbraith, MR 2162716. A cached pdf-file.
- mathematik.deRezension von Florian Hess im Computeralgebra-Rundbrief in Nr. 40 - März 2007.
- Amazon.com by Lee Carlson.

Thanks to Pierrick Gaudry for the favicon.