Implementing Elliptic Curve Cryptography

Michael Rosing

  • 出版商: Manning Publications
  • 出版日期: 1998-01-01
  • 定價: $1,630
  • 售價: 9.6$1,568
  • 貴賓價: 9.1$1,490
  • 語言: 英文
  • 頁數: 338
  • 裝訂: Paperback
  • ISBN: 1884777694
  • ISBN-13: 9781884777691

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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.

Table of Contents

  • 1 Introduction
          Why 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