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Implementing Elliptic Curve Cryptography Michael Rosing 1998 | 330 pages ISBN: 1884777694 |
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$47.95 | Softbound print book | |
Table of Contents
1 IntroductionWhy Elliptic Curves?
Why C?
Orders of Magnitude
Structure of Book
Comments on Style
Acknowledgements
2 Basics of Number Theory
Large Integer Math Header
Large Integer Math Routines
Multiplication
Division
Large Integer Code Example
Back to Number Theory
Greatest Common Factor
Modular Arithmetic
Fermat's Theorem
Finite Fields
Generators
3 Polynomial Math Over Finite Fields
Polynomial Basis Header Files
Polynomial Addition
Polynomial Multiplication
Polynomial Division
Modular Polynomial Arithmetic
Inversion Over Prime Polynomials
Polynomial Greatest Common Divisor
Prime Polynomials
Summary
4 Normal Basis Mathematics
Squaring Normal Basis Numbers
Multiplication in Theory
Type I Optimal Normal Basis
Type II Optimal Normal Basis
Multiplication in Practice
Inversion over Optimal Normal Basis
5 Elliptic Curves
Mathematics of Elliptic Curves Over Real Numbers
Mathematics of Elliptic Curves Over Prime Fields
Mathematics of Elliptic Curves Over Galois Fields
Polynomial Basis Elliptic Curve Subroutines
Optimal Normal Basis Elliptic curve Subroutines
Multiplication Over Elliptic Curves
Balanced Integer Conversion Code
Following the Balanced Representation
6 Cryptography
Fundamentals of Elliptic Curve Cryptography
Choosing an Elliptic Curve
Non-supersingular Curves
Embedding Data on a Curve
Solving Quadratic Equations in Binary Fields
The Trace Function
Solving Quadratic Equations in Normal Basis
Solving Quadratic Equations in Polynomial Basis
Quadratic Polynomials, the Code
Using the T Matrix
Embedding Data Using Polynomial Basis
Summary of Quadratic Solving
7 Simple Protocols
Random Bit Generator
Choosing Random Curves
Diffie-Hellman
ElGamal Protocol
ElGamal Using Optimal Normal Basis
ElGamal, Polynomial Basis
Menezes-Qu-Vanstone Key Agreement Scheme
MQV the Code, Simple Version
8 Elliptic Curve Encryption
Mask Generation Function
Hash Fucntion SHA-1
Mask Generation, the Code
ECES, the Code
Polynomial Basis
Normal Basis
9 Advanced Protocols, Key Exchange
Polynomial solution to g^3 = g + 1
Massey-Omura Protocol
Massey-Omura, the Code
MQV, the Standard
MQV, Normal Basis Version
MQV, Polynomial Basis Version
10 Advanced Protocols, Signatures
Message Hash
Nyberg-Rueppel Signature Scheme
Nyberg-Rueppel signature, Normal Basis
Nyberg-Rueppel, Polynomial Basis
Elliptic Curve DSA
DSA in Normal Basis
DSA, Polynomial Basis
11 State of the Art
High Speed Inversion for ON
Faster Inversion, Preliminary Subroutines
Faster Inversion, the Code
Security from Cryptography
Counting Points
Polynomials Base p
Hyper-Elliptic Curves
References
DESCRIPTION
Implementing Elliptic Curve Cryptography proceeds step-by-step to explain basic number theory, polynomial mathematics, normal basis mathematics and elliptic curve mathematics. With these in place, applications to cryptography are introduced. The book is filled with C code to illustrate how mathematics is put into a computer, and the last several chapters show how to implement several cryptographic protocols. The most important is a description of P1363, an IEEE draft standard for public key cryptography.
The main purpose of Implementing Elliptic Curve Cryptography is to help "crypto engineers" implement functioning, state-of-the-art cryptographic algorithms in the minimum time. With detailed descriptions of the mathematics, the reader can expand on the code given in the book and develop optimal hardware or software for their own applications.
Implementing Elliptic Curve Cryptography assumes the reader has at least a high school background in algebra, but it explains, in stepwise fashion, what has been considered to be a topic only for graduate-level students.
WHAT THE EXPERTS SAY ABOUT THIS BOOK...
"The book provides all the theory and working programs needed to create real applications based on the latest IEEE P1363 standard."
--Reviewed in Cryptologia
ABOUT THE AUTHOR...
With a background that includes nuclear engineering, hardware computer design, and telephony, Michael Rosing has seen how cryptography plays an important role in military and commercial applications. At present he is employed building DSP hardware for brain research at the University of Wisconsin in Madison.