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BigNum Math: Implementing Cryptographic Multiple Precision
Bignum math is the backbone of modern computer security
algorithms. It is the ability to work with hundred-digit
numbers efficiently using techniques that are both elegant
and occasionally bizarre. This book introduces the reader to
the concept of bignum algorithms and proceeds to build an
entire library of functionality from
the ground up. Through
the use of theory, pseudo-code and actual fielded C source
code the book explains each and every algorithm that goes
into a modern bignum library. Excellent for the student as
a learning tool and practitioner as a reference alike BigNum
Math is for anyone with a background in computer science who
has taken introductory level mathematic courses. The text is
for students learning mathematics and cryptography as well
as the practioner who needs a reference for any of the
algorithms documented within.
* Complete coverage of Karatsuba Multiplication, the Barrett Algorithm, Toom-Cook 3-Way Multiplication, and More
* Tom St Denis is the developer of the industry standard cryptographic suite of tools called LibTom.
* This book provides step-by-step exercises to enforce concepts
Introduction 1
1.1 Multiple Precision Arithmetic
1.1.1 What is Multiple Precision Arithmetic?
1.1.2 The Need for Multiple Precision Arithmetic
1.1.3 Benefits of Multiple Precision Arithmetic
1.2 Purpose of This Text
1.3 Discussion and Notation
1.3.1 Notation
1.3.2 Precision Notation
1.3.3 Algorithm Inputs and Outputs
1.3.4 Mathematical Expressions
1.4 Exercises
1.5 Introduction to LibTomMath
1.5.1 What is LibTomMath?
1.5.2 Goals of LibTomMath
1.6 Choice of LibTomMath
1.6.1 Code Base
1.6.3 Optimizations
1.6.4 Portability and Stability
1.6.5 Choice
2 Getting Started 13
2.1 Library Basics
2.2 What is a Multiple Precision Integer?
2.2.1 The mp int Structure
2.3 Argument Passing
2.5 Initialization and Clearing
2.5.1 Initializing an mp int
2.5.2 Clearing an mp int
2.6 Maintenance Algorithms
2.6.1 Augmenting an mp int's Precision
2.6.2 Initializing Variable Precision mp ints
2.6.3 Multiple Integer Initializations and Clearings
2.6.4 Clamping Excess Digit
3 Basic Operations 35
3.1 Introduction
3.2 Assigning Values to mp int Structures
3.2.1 Copying an mp int
3.3 Zeroing an Integer .
3.4 Sign Manipulation
3.4.1 Absolute Value
3.4.2 Integer Negation
3.5.1 Setting Small Constants .
3.5.2 Setting Large Constants .
3.6 Comparisons .
3.6.1 Unsigned Comparisions
3.6.2 Signed Comparisons
4 Basic Arithmetic 53
4.1 Introduction
4.2.2 Low Level Subtraction
4.2.4 High Level Subtractio
4.3 Bit and Digit Shifting
4.3.2 Division by Two
4.4 Polynomial Basis Operation
4.4.1 Multiplication by x .
4.4.2 Division by x
4.5 Powers of Two
4.5.1 Multiplication by Power of Two
4.5.2 Division by Power of Two
4.5.3 Remainder of Division by Power of Two
5 Multiplication and Squaring 91
5.1 The Multipliers
5.2 Multiplication
5.2.1 The Baseline Multiplication
5.2.2 Faster Multiplication by the Comba Method
5.2.3 Polynomial Basis Multiplication
5.2.5 Toom-Cook 3-Way Multiplication
5.2.6 Signed Multiplication
5.3 Squaring
5.3.1 The Baseline Squaring Algorithm
5.3.2 Faster Squaring by the Comba Method
5.3.3 Polynomial Basis Squaring
5.3.4 Karatsuba Squaring
5.3.5 Toom-Cook Squaring
5.3.6 High Level Squaring
6.2 The Barrett Reduction
6.2.1 Fixed Point Arithmetic . .
6.2.5 The Barrett Algorithm
6.2.6 The Barrett Setup Algorithm
6.3 The Montgomery Reduction .
6.3.1 Digit Based Montgomery Reduction
6.3.2 Baseline Montgomery Reduction
6.3.3 Faster Comba Montgomery Reduction
6.3.4 Montgomery Setup
6.4 The Diminished Radix Algorithm
6.4.1 Choice of Moduli
6.4.2 Choice of k
6.4.3 Restricted Diminished Radix Reduction
6.4.4 Unrestricted Diminished Radix Reduction
6.5 Algorithm Comparison .
7 Exponentiation 187
7.1 Exponentiation Basics
7.1.1 Single Digit Exponentiation
7.2.1 Optimal Values of k
7.2.2 Sliding-Window Exponentiation
7.3 Modular Exponentiation
7.3.1 Barrett Modular Exponentiation
7.4 Quick Power of Two
8 Higher Level Algorithms 211
8.1 Integer Division with Remainde
8.1.1 Quotient Estimation
8.1.2 Normalized Integers
8.1.3 Radix- Division with Remainder
8.2 Single Digit Helpers
8.2.2 Single Digit Multiplication
8.2.3 Single Digit Division
8.2.4 Single Digit Root Extraction
8.3 Random Number Generation
8.4 Formatted Representations
8.4.1 Reading Radix-n Input
8.4.2 Generating Radix-n Output
9 Number Theoretic Algorithms
9.1 Greatest Common Divisor
9.2 Least Common Multiple
9.3 Jacobi Symbol Computation
9.3.1 Jacobi Symbol
9.4 Modular Inverse
9.4.1 General Case
9.5 Primality Tests
9.5.2 The Fermat Test
9.5.3 The Miller-Rabin Test
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