After 18 years of research and development, elliptic curve cryptography has gained widespread exposure and acceptance in the commercial world. Industry, banking, and government standards have been drafted to facilitate widespread deployment.
This book is the first to provide a comprehensive treatment of the practical aspects of elliptic curve cryptography. It explains the basic mathematics, provides detailed descriptions of state-in-the-art in methods for efficient implementation, and presents standardized protocols for public-key encryption, digital signatures, and key establishment. Also considered are some issues that arise in software and hardware implementation, side-channel attacks and countermeasures, and efficient parameter generation.
Written by professors who have extensive experience studying the security of elliptic curve cryptography, its implementation, and industrial applications, this book is an important resource for security professionals, developers, and implementers.
A broad overview and guide to one of the most widely used cryptosystems in wireless devices and smartcards today. The authors are some of the key researchers and developers of the EEC standards technology. Essential reference for any practitioners or professionals incomputer science, computer engineering, network design, and network/data security.

Contents
List of Algorithms
List of Tables
List of Figures
Acronyms
Preface 1 Introduction and Overview
1.1 Cryptography basics
1.2 Public-key cryptography
1.2.1 RSAsystems
1.2.2 Discrete logarithmsystems
1.2.3 Elliptic curve systems
1.3 Why elliptic curve cryptography?
1.4 Roadmap
1.5 Notes and further references 2 Finite Field Arithmetic
2.1 Introduction to finite fields
2.2 Primefieldarithmetic
2.2.1 Addition and subtraction
2.2.2 Integer multiplication
2.2.3 Integer squaring
2.2.4 Reduction
2.2.5 Inversion
2.2.6 NISTprimes
2.3 Binary field arithmetic
2.3.1 Addition
2.3.2 Multiplication
2.3.3 Polynomial multiplication
2.3.4 Polynomial squaring
2.3.5 Reduction
2.3.6 Inversion and division
2.4 Optimal extension field arithmetic
2.4.1 Addition and subtraction
2.4.2 Multiplication and reduction
2.4.3 Inversion
2.5 Notes andfurther references 3 Elliptic Curve Arithmetic
3.1 Introduction to elliptic curves
3.1.1 Simplified Weierstrass equations
3.1.2 Group law
3.1.3 Group order
3.1.4 Group structure
3.1.5 Isomorphism classes
3.2 Point representation and the group law
3.2.1 Projective coordinates
3.2.2 The elliptic curve y2 = x3 +ax +b
3.2.3 The elliptic curve y2 +xy = x3 +ax2 +b
3.3 Point multiplication
3.3.1 Unknown point
3.3.2 Fixed point
3.3.3 Multiple point multiplication
3.4 Koblitz curves
3.4.1 The Frobenius map and the ring Z[ô]
3.4.2 Point multiplication
3.5 Curves with efficiently computable endomorphisms
3.6 Point multiplication using halving
3.6.1 Point halving
3.6.2 Performing point halving efficiently
3.6.3 Point multiplication
3.7 Point multiplication costs
3.8 Notes andfurther references 4 Cryptographic Protocols
4.1 The elliptic curve discrete logarithm problem
4.1.1 Pohlig-Hellmanattack
4.1.2 Pollard¿s rho attack
4.1.3 Index-calculus attacks
4.1.4 Isomorphism attacks
4.1.5 Related problems
4.2 Domain parameters
4.2.1 Domain parameter generation and validation
4.2.2 Generating elliptic curves verifiably at random
4.2.3 Determining the number of points on an elliptic curve
4.3 Keypairs
4.4 Signature schemes
4.4.1 ECDSA
4.4.2 EC-KCDSA
4.5 Public-key encryption
4.5.1 ECIES
4.5.2 PSEC
4.6 Keyestablishment
4.6.1 Station-to-station
4.6.2 ECMQV
4.7 Notes and further references 5 Implementation Issues
5.1 Software implementation
5.1.1 Integer arithmetic
5.1.2 Floating-point arithmetic
5.1.3 SIMD and field arithmetic
5.1.4 Platform miscellany
5.1.5 Timings
5.2 Hardware implementation
5.2.1 Design criteria
5.2.2 Field arithmetic processors
5.3 Secure implementation
5.3.1 Power analysis attacks
5.3.2 Electromagnetic analysis attacks
5.3.3 Error message analysis
5.3.4 Fault analysis attacks
5.3.5 Timing attacks
5.4 Notes and further references A Sample Parameters
A.1 Irreducible polynomials
A.2 Elliptic curves
A.2.1 Random elliptic curves over Fp
A.2.2 Random elliptic curves over F2m
A.2.3 Koblitz elliptic curves over F2m
B ECC Standards
C Software Tools
C.1 General-purpose tools
C.2 Libraries
Bibliography
Index