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Handbook of Elliptic and Hyperelliptic Curve Cryptography

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

Print publication date:  July  2005
Online publication date:  July  2005

Print ISBN: 9781584885184
eBook ISBN: 9781420034981
Adobe ISBN:

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Book description

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.

Table of contents

Chapter  1:  Introduction to Public-Key Cryptography Download PDF
Chapter  2:  Algebraic Background Download PDF
Chapter  3:  Background on p-adic Numbers Download PDF
Chapter  4:  Background on Curves and Jacobians Download PDF
Chapter  5:  Varieties over Special Fields Download PDF
Chapter  6:  Background on Pairings Download PDF
Chapter  7:  Background on Weil Descent Download PDF
Chapter  8:  Cohomological Background on Point Counting Download PDF
Chapter  9:  Exponentiation Download PDF
Chapter  10:  Integer Arithmetic Download PDF
Chapter  11:  Finite Field Arithmetic Download PDF
Chapter  12:  Arithmetic of p-adic Numbers Download PDF
Chapter  13:  Arithmetic of Elliptic Curves Download PDF
Chapter  14:  Arithmetic of Hyperelliptic Curves Download PDF
Chapter  15:  Arithmetic of Special Curves Download PDF
Chapter  16:  Implementation of Pairings Download PDF
Chapter  17:  Point Counting on Elliptic and Hyperelliptic Curves Download PDF
Chapter  18:  Complex Multiplication Download PDF
Chapter  19:  Generic Algorithms for Computing Discrete Logarithms Download PDF
Chapter  20:  Index Calculus Download PDF
Chapter  21:  Index Calculus for Hyperelliptic Curves Download PDF
Chapter  22:  Transfer of Discrete Logarithms Download PDF
Chapter  23:  Algebraic Realizations of DL Systems Download PDF
Chapter  24:  Pairing-Based Cryptography Download PDF
Chapter  25:  Compositeness and Primality Testing Factoring Download PDF
Chapter  26:  Fast Arithmetic in Hardware Download PDF
Chapter  27:  Smart Cards Download PDF
Chapter  28:  Practical Attacks on Smart Cards Download PDF
Chapter  29:  Mathematical Countermeasures against Side-Channel Attacks Download PDF
Chapter  30:  Random Numbers Generation and Testing Download PDF
prelims Download PDF
References Download PDF
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