BigNum Math - Impl. Crypo. Mult. Precicion Arithmetic

315 Pages·2006·12.41 MB·English

by  T. St. Denis

BigNum Math
Implementing Cryptographic Multiple Precision Arithmetic
Tom St Denis Greg Rose
LibTom Projects QUALCOMM Australia
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003 829KM8NJH2
004 HJ9899923N
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BigNum Math: ImplementingCryptographic Multiple Precision Arithmetic
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Contents
Preface xv
1 Introduction 1
1.1 Multiple Precision Arithmetic . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 What Is Multiple Precision Arithmetic? . . . . . . . . . . . 1
1.1.2 The Need for Multiple Precision Arithmetic . . . . . . . . . 2
1.1.3 Benefits of Multiple Precision Arithmetic . . . . . . . . . . 3
1.2 Purpose of This Text . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Discussion and Notation . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Precision Notation . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.3 Algorithm Inputs and Outputs . . . . . . . . . . . . . . . . 6
1.3.4 Mathematical Expressions . . . . . . . . . . . . . . . . . . . 6
1.3.5 Work Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Introduction to LibTomMath . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 What Is LibTomMath? . . . . . . . . . . . . . . . . . . . . 9
1.5.2 Goals of LibTomMath . . . . . . . . . . . . . . . . . . . . . 9
1.6 Choice of LibTomMath . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.1 Code Base. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.2 API Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.3 Optimizations. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.4 Portability and Stability . . . . . . . . . . . . . . . . . . . . 12
1.6.5 Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
v
2 Getting Started 13
2.1 Library Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 What Is a Multiple Precision Integer? . . . . . . . . . . . . . . . . 14
2.2.1 The mp int Structure . . . . . . . . . . . . . . . . . . . . . 15
2.3 Argument Passing . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Return Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Initialization and Clearing . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1 Initializing an mp int . . . . . . . . . . . . . . . . . . . . . 19
2.5.2 Clearing an mp int . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Maintenance Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.1 Augmenting an mp int’s Precision . . . . . . . . . . . . . . 24
2.6.2 Initializing Variable Precision mp ints . . . . . . . . . . . . 27
2.6.3 Multiple Integer Initializations and Clearings . . . . . . . . 29
2.6.4 Clamping Excess Digits . . . . . . . . . . . . . . . . . . . . 31
3 Basic Operations 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Assigning Values to mp int Structures . . . . . . . . . . . . . . . . 35
3.2.1 Copying an mp int . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Creating a Clone . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Zeroing an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Sign Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Integer Negation . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Small Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 Setting Small Constants . . . . . . . . . . . . . . . . . . . . 44
3.5.2 Setting Large Constants . . . . . . . . . . . . . . . . . . . . 46
3.6 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.1 Unsigned Comparisons . . . . . . . . . . . . . . . . . . . . . 47
3.6.2 Signed Comparisons . . . . . . . . . . . . . . . . . . . . . . 50
4 Basic Arithmetic 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Low Level Addition . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Low Level Subtraction . . . . . . . . . . . . . . . . . . . . . 59
4.2.3 High Level Addition . . . . . . . . . . . . . . . . . . . . . . 63
4.2.4 High Level Subtraction . . . . . . . . . . . . . . . . . . . . 66
4.3 Bit and Digit Shifting . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Multiplication by Two . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 Division by Two . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Polynomial Basis Operations . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 Multiplication by x . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.2 Division by x . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Powers of Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.1 Multiplication by Power of Two. . . . . . . . . . . . . . . . 82
4.5.2 Division by Power of Two . . . . . . . . . . . . . . . . . . . 85
4.5.3 Remainder of Division by Power of Two . . . . . . . . . . . 88
5 Multiplication and Squaring 91
5.1 The Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 The Baseline Multiplication . . . . . . . . . . . . . . . . . . 92
5.2.2 Faster Multiplication by the “Comba” Method . . . . . . . 97
5.2.3 Even Faster Multiplication . . . . . . . . . . . . . . . . . . 104
5.2.4 Polynomial Basis Multiplication . . . . . . . . . . . . . . . 107
5.2.5 Karatsuba Multiplication . . . . . . . . . . . . . . . . . . . 109
5.2.6 Toom-Cook 3-Way Multiplication. . . . . . . . . . . . . . . 116
5.2.7 Signed Multiplication . . . . . . . . . . . . . . . . . . . . . 126
5.3 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.1 The Baseline Squaring Algorithm . . . . . . . . . . . . . . . 129
5.3.2 Faster Squaring by the “Comba” Method . . . . . . . . . . 133
5.3.3 Even Faster Squaring . . . . . . . . . . . . . . . . . . . . . 137
5.3.4 Polynomial Basis Squaring . . . . . . . . . . . . . . . . . . 138
5.3.5 Karatsuba Squaring . . . . . . . . . . . . . . . . . . . . . . 138
5.3.6 Toom-Cook Squaring. . . . . . . . . . . . . . . . . . . . . . 143
5.3.7 High Level Squaring . . . . . . . . . . . . . . . . . . . . . . 144
6 Modular Reduction 147
6.1 Basics of Modular Reduction . . . . . . . . . . . . . . . . . . . . . 147
6.2 The Barrett Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2.1 Fixed Point Arithmetic . . . . . . . . . . . . . . . . . . . . 148
6.2.2 Choosing a Radix Point . . . . . . . . . . . . . . . . . . . . 150
6.2.3 Trimming the Quotient . . . . . . . . . . . . . . . . . . . . 151
6.2.4 Trimming the Residue . . . . . . . . . . . . . . . . . . . . . 152
6.2.5 The Barrett Algorithm. . . . . . . . . . . . . . . . . . . . . 153
6.2.6 The Barrett Setup Algorithm . . . . . . . . . . . . . . . . . 156
6.3 The Montgomery Reduction . . . . . . . . . . . . . . . . . . . . . . 158
6.3.1 Digit Based Montgomery Reduction . . . . . . . . . . . . . 160
6.3.2 Baseline Montgomery Reduction . . . . . . . . . . . . . . . 162
6.3.3 Faster “Comba” Montgomery Reduction . . . . . . . . . . . 167
6.3.4 Montgomery Setup . . . . . . . . . . . . . . . . . . . . . . . 173
6.4 The Diminished Radix Algorithm . . . . . . . . . . . . . . . . . . . 175
6.4.1 Choice of Moduli . . . . . . . . . . . . . . . . . . . . . . . . 177
6.4.2 Choice of k . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.3 Restricted Diminished Radix Reduction . . . . . . . . . . . 178
6.4.4 Unrestricted Diminished Radix Reduction . . . . . . . . . . 184
6.5 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 Exponentiation 191
7.1 Exponentiation Basics . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.1.1 Single Digit Exponentiation . . . . . . . . . . . . . . . . . . 193
7.2 k-ary Exponentiation. . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.2.1 Optimal Values of k . . . . . . . . . . . . . . . . . . . . . . 196
7.2.2 Sliding Window Exponentiation. . . . . . . . . . . . . . . . 197
7.3 Modular Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . 198
7.3.1 Barrett Modular Exponentiation . . . . . . . . . . . . . . . 203
7.4 Quick Power of Two . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8 Higher Level Algorithms 217
8.1 Integer Division with Remainder . . . . . . . . . . . . . . . . . . . 217
8.1.1 Quotient Estimation . . . . . . . . . . . . . . . . . . . . . . 219
8.1.2 Normalized Integers . . . . . . . . . . . . . . . . . . . . . . 220
8.1.3 Radix-β Division with Remainder . . . . . . . . . . . . . . 221
8.2 Single Digit Helpers . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8.2.1 Single Digit Addition and Subtraction . . . . . . . . . . . . 232
8.2.2 Single Digit Multiplication . . . . . . . . . . . . . . . . . . 235
8.2.3 Single Digit Division . . . . . . . . . . . . . . . . . . . . . . 237
8.2.4 Single Digit Root Extraction . . . . . . . . . . . . . . . . . 241
8.3 Random Number Generation . . . . . . . . . . . . . . . . . . . . . 245
8.4 Formatted Representations . . . . . . . . . . . . . . . . . . . . . . 247
8.4.1 Reading Radix-n Input . . . . . . . . . . . . . . . . . . . . 247
8.4.2 Generating Radix-n Output . . . . . . . . . . . . . . . . . . 252
9 Number Theoretic Algorithms 255
9.1 Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . 255
9.1.1 Complete Greatest Common Divisor . . . . . . . . . . . . . 258
9.2 Least Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . 263
9.3 Jacobi Symbol Computation. . . . . . . . . . . . . . . . . . . . . . 265
9.3.1 Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.4 Modular Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.5 Primality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.1 Trial Division . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.2 The Fermat Test . . . . . . . . . . . . . . . . . . . . . . . . 282
9.5.3 The Miller-Rabin Test . . . . . . . . . . . . . . . . . . . . . 284
Bibliography 289
Index 291