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Advances in Elliptic Curve Cryptography 2nd Edition by Ian F Blake ISBN 052160415X 9780521604154

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Author(s): Ian F. Blake
ISBN(s): 9780521604154, 052160415X
Edition: 2nd
File Details: PDF, 4.66 MB
Year: 2005
Language: english
SKU: EB-2008730 Category: Tag:

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ISBN 10: 052160415X 
ISBN 13: 9780521604154
Author: Ian F Blake

In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

Advances in Elliptic Curve Cryptography 2nd Table of contents:

CHAPTER I Introduction
I.1. Cryptography Based on Groups
I.2. What Types of Group are Used
I.3. What it Means in Practice
CHAPTER II Finite Field Arithmetic
II.1 Fields of Odd Characteristic
II.2. Fields of Characteristic Two
CHAPTER III Arithmetic on an Elliptic Curve
III.1. General Elliptic Curves
III.2. The Group Law
III.3. Elliptic Curves over Finite Fields
III.4. The Division Polynomials
III.5. The Weil Pairing
III.6. Isogenies, Endomorphisms and Torsion
III.7. Various Functions and q-Expansions
III.8. Modular Polynomials and Variants
CHAPTER IV Efficient Implementation of Elliptic Curves
IV.1. Point Addition
IV.2. Point Multiplication
IV.3. Frobenius Expansions
IV.4. Point Compression
CHAPTER V The Elliptic Curve Discrete Logarithm Problem
V.1. The Simplification of Pohlig and Hellman
V.2. The MOV Attack
V.3. The Anomalous Attack
V.4. Baby Step/Giant Step
V.5. Methods based on Random Walks
V.6. Index Calculus Methods
V.7. Summary
CHAPTER VI Determining the Group Order
VI.1. Main Approaches
VI.2. Checking the Group Order
VI.3. The Method of Shanks and Mestre
VI.5. Searching for Good Curves
CHAPTER VII Schoof’s Algorithm and Extensions
VII.1. Schoof’s Algorithm
VII.2. Beyond Schoof
VII.3. More on the Modular Polynomials
VIl.4. Finding Factors of Division Polynomials through Isogenies: Odd Characteristic
VII.5. Finding Factors of Division Polynomials through Isogenies: Characteristic Two
VII.6. Determining the Trace Modulo a Prime Power
VII.7. The Elkies Procedure
VII.8. The Atkin Procedure
VII.9. Combining the Information from Elkies and Atkin Primes
VII.10. Examples
VII.11. Further Discussion
CHAPTER VIII Generating Curves using Complex Multiplication
VIlI.1. The Theory of Complex Multiplication
VIII.2. Generating Curves over Large Prime Fields using CM
VIII.3. Weber Polynomials
VIll.4. Further Discussion
CHAPTER IX Other Applications of Elliptic Curves
IX.1. Factoring Using Elliptic Curves
IX.2. The Pocklington-Lehmer Primality Test
IX.3. The ECPP Algorithm
IX.4. Equivalence between DLP and DHP
CHAPTER X Hyperelliptic Cryptosystems
X.1. Arithmetic of Hyperelliptic Curves
X.2. Generating Suitable Curves
X.3. The Hyperelliptic Discrete Logarithm Problem
APPENDIX A Curve Examples
A.1. Odd Characteristic
A.2. Characteristic Two

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Ian F Blake,Elliptic Curve,Cryptography