Handbook of Elliptic and Hyperelliptic Curve Cryptography: 1st Edition (Hardback) book cover

Handbook of Elliptic and Hyperelliptic Curve Cryptography

1st Edition

Edited by Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren

Chapman and Hall/CRC

842 pages | 44 B/W Illus.

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Hardback: 9781584885184
pub: 2005-07-19
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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 explore their design and efficient implementations in smart cards. 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.


… very comprehensive coverage of this vast subject area … a useful and essential treatise for anyone involved in elliptic curve algorithms … this book offers the opportunity to grasp the ECC technology with a diversified and comprehensive perspective. … This book will remain on my shelf for a long time and will land on my desk on many occasions, if only because the coverage of the issues common to factoring and discrete log cryptosystems is excellent.

—IACR Book Reviews, June 2011

… the book is designed for people who are working in the area and want to learn more about a specific issue. The chapters are written to be relatively independent so that readers can focus on the part of interest for them. Such readers will be grateful for the excellent index and extensive bibliography. … the handbook covers a wide range of topics and will be a valuable reference for researchers in curve-based cryptography.

—Steven D. Galbraith, Mathematical Reviews, Issue 2007f

Table of Contents


Introduction to Public-Key Cryptography

Mathematical Background

Algebraic Background

Background on p-adic Numbers

Background on Curves and Jacobians

Varieties Over Special Fields

Background on Pairings

Background on Weil Descent

Cohomological Background on Point Counting

Elementary Arithmetic


Integer Arithmetic

Finite Field Arithmetic

Arithmetic of p-adic Numbers

Arithmetic of Curves

Arithmetic of Elliptic Curves

Arithmetic of Hyperelliptic Curves

Arithmetic of Special Curves

Implementation of Pairings

Point Counting

Point Counting on Elliptic and Hyperelliptic Curves

Complex Multiplication

Computation of Discrete Logarithms

Generic Algorithms for Computing Discrete Logarithms

Index Calculus

Index Calculus for Hyperelliptic Curves

Transfer of Discrete Logarithms


Algebraic Realizations of DL Systems

Pairing-Based Cryptography

Compositeness and Primality Testing-Factoring

Realizations of DL Systems

Fast Arithmetic Hardware

Smart Cards

Practical Attacks on Smart Cards

Mathematical Countermeasures Against Side-Channel Attacks

Random Numbers—Generation and Testing


About the Series

Discrete Mathematics and Its Applications

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Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Operating Systems / General
COMPUTERS / Security / General
MATHEMATICS / Combinatorics