$BigNum Math : Implementing Cryptographic Multiple Precision Arithmetic - Tom St Denis$

BigNum Math

Implementing Cryptographic Multiple Precision Arithmetic

By: Tom St Denis

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Paperback | 1 June 2006

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316 Pages

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23.5 x 19.05 x 1.68

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Implementing cryptography requires integers of significant magnitude to resist cryptanalytic attacks. Modern programming languages only provide support for integers which are relatively small and single precision. The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.

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

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Non-Fiction Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Engineering & Technology Technology in General Computer Security Data Encryption Computer Science

Preface	p. xv
Introduction	p. 1
Multiple Precision Arithmetic	p. 1
What Is Multiple Precision Arithmetic?	p. 1
The Need for Multiple Precision Arithmetic	p. 2
Benefits of Multiple Precision Arithmetic	p. 3
Purpose of This Text	p. 4
Discussion and Notation	p. 5
Notation	p. 5
Precision Notation	p. 5
Algorithm Inputs and Outputs	p. 6
Mathematical Expressions	p. 6
Work Effort	p. 7
Exercises	p. 7
Introduction to LibTomMath	p. 9
What Is LibTomMath?	p. 9
Goals of LibTomMath	p. 9
Choice of LibTomMath	p. 10
Code Base	p. 10
API Simplicity	p. 11
Optimizations	p. 11
Portability and Stability	p. 12
Choice	p. 12
Getting Started	p. 13
Library Basics	p. 13
What Is a Multiple Precision Integer?	p. 14
The mp_int Structure	p. 15
Argument Passing	p. 17
Return Values	p. 18
Initialization and Clearing	p. 19
Initializing an mp_int	p. 19
Clearing an mp_int	p. 22
Maintenance Algorithms	p. 24
Augmenting an mp_int's Precision	p. 24
Initializing Variable Precision mp_ints	p. 27
Multiple Integer Initializations and Clearings	p. 29
Clamping Excess Digits	p. 31
Basic Operations	p. 35
Introduction	p. 35
Assigning Values to mp_int Structures	p. 35
Copying an mp_int	p. 35
Creating a Clone	p. 39
Zeroing an Integer	p. 41
Sign Manipulation	p. 42
Absolute Value	p. 42
Integer Negation	p. 43
Small Constants	p. 44
Setting Small Constants	p. 44
Setting Large Constants	p. 46
Comparisons	p. 47
Unsigned Comparisons	p. 47
Signed Comparisons	p. 50
Basic Arithmetic	p. 53
Introduction	p. 53
Addition and Subtraction	p. 54
Low Level Addition	p. 54
Low Level Subtraction	p. 59
High Level Addition	p. 63
High Level Subtraction	p. 66
Bit and Digit Shifting	p. 69
Multiplication by Two	p. 69
Division by Two	p. 72
Polynomial Basis Operations	p. 75
Multiplication by x	p. 75
Division by x	p. 78
Powers of Two	p. 81
Multiplication by Power of Two	p. 82
Division by Power of Two	p. 85
Remainder of Division by Power of Two	p. 88
Multiplication and Squaring	p. 91
The Multipliers	p. 91
Multiplication	p. 92
The Baseline Multiplication	p. 92
Faster Multiplication by the "Comba" Method	p. 97
Even Faster Multiplication	p. 104
Polynomial Basis Multiplication	p. 107
Karatsuba Multiplication	p. 109
Toom-Cook 3-Way Multiplication	p. 116
Signed Multiplication	p. 126
Squaring	p. 128
The Baseline Squaring Algorithm	p. 129
Faster Squaring by the "Comba" Method	p. 133
Even Faster Squaring	p. 137
Polynomial Basis Squaring	p. 138
Karatsuba Squaring	p. 138
Toom-Cook Squaring	p. 143
High Level Squaring	p. 144
Modular Reduction	p. 147
Basics of Modular Reduction	p. 147
The Barrett Reduction	p. 148
Fixed Point Arithmetic	p. 148
Choosing a Radix Point	p. 150
Trimming the Quotient	p. 151
Trimming the Residue	p. 152
The Barrett Algorithm	p. 153
The Barrett Setup Algorithm	p. 156
The Montgomery Reduction	p. 158
Digit Based Montgomery Reduction	p. 160
Baseline Montgomery Reduction	p. 162
Faster "Comba" Montgomery Reduction	p. 167
Montgomery Setup	p. 173
The Diminished Radix Algorithm	p. 175
Choice of Moduli	p. 177
Choice of k	p. 178
Restricted Diminished Radix Reduction	p. 178
Unrestricted Diminished Radix Reduction	p. 184
Algorithm Comparison	p. 189
Exponentiation	p. 191
Exponentiation Bascis	p. 191
Single Digit Exponentiation	p. 193
k-ary Exponentiation	p. 195
Optimal Values of k	p. 196
Sliding Window Exponentiation	p. 197
Modular Exponentiation	p. 198
Barrett Modular Exponentiation	p. 203
Quick Power of Two	p. 214
Higher Level Algorithms	p. 217
Integer Division with Remainder	p. 217
Quotient Estimation	p. 219
Normalized Integers	p. 220
Radix-&gb Division with Remainder	p. 221
Single Digit Helpers	p. 231
Single Digit Addition and Subtraction	p. 232
Single Digit Multiplication	p. 235
Single Digit Division	p. 237
Single Digit Root Extraction	p. 241
Random Number Generation	p. 245
Formatted Representations	p. 247
Reading Radix-n Input	p. 247
Generating Radix-n Output	p. 252
Number Theoretic Algorithms	p. 255
Greatest Common Divisor	p. 255
Complete Greatest Common Divisor	p. 258
Least Common Multiple	p. 263
Jacobi Symbol Computation	p. 265
Jacobi Symbol	p. 266
Modular Inverse	p. 271
General Case	p. 273
Primality Tests	p. 279
Trial Division	p. 279
The Fermat Test	p. 282
The Miller-Rabin Test	p. 284
Bibliography	p. 289
Index	p. 291
Table of Contents provided by Ingram. All Rights Reserved.

BigNum Math

Implementing Cryptographic Multiple Precision Arithmetic

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