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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}% 2
\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1}% 3
\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1}% 4
\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}% 5
\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}% 7
\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}% 8
\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3}% 9
\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3}% 10
\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3}% 11
\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3}% 12
\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}% 13
\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1}% 14
\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5}% 15
\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5}% 16
\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1}% 17
\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6}% 18
\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6}% 19
\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6}% 20
\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6}% 21
\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6}% 22
\BOOKMARK [0][-]{chapter.2}{Getting Started}{}% 23
\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2}% 24
\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2}% 25
\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2}% 26
\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2}% 27
\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2}% 28
\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2}% 29
\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5}% 30
\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5}% 31
\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2}% 32
\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6}% 33
\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6}% 34
\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6}% 35
\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6}% 36
\BOOKMARK [0][-]{chapter.3}{Basic Operations}{}% 37
\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3}% 38
\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3}% 39
\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2}% 40
\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2}% 41
\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3}% 42
\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3}% 43
\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4}% 44
\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4}% 45
\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3}% 46
\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5}% 47
\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5}% 48
\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3}% 49
\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6}% 50
\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6}% 51
\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{}% 52
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 53
\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4}% 54
\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2}% 55
\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2}% 56
\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2}% 57
\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2}% 58
\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4}% 59
\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3}% 60
\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3}% 61
\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4}% 62
\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4}% 63
\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4}% 64
\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4}% 65
\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5}% 66
\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5}% 67
\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5}% 68
\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{}% 69
\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5}% 70
\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5}% 71
\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2}% 72
\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2}% 73
\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2}% 74
\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2}% 75
\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2}% 76
\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2}% 77
\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5}% 78
\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3}% 79
\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3}% 80
\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3}% 81
\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3}% 82
\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3}% 83
\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3}% 84
\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{}% 85
\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6}% 86
\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6}% 87
\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2}% 88
\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2}% 89
\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2}% 90
\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2}% 91
\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2}% 92
\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2}% 93
\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6}% 94
\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3}% 95
\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3}% 96
\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3}% 97
\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3}% 98
\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6}% 99
\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4}% 100
\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4}% 101
\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4}% 102
\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4}% 103
\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6}% 104
\BOOKMARK [0][-]{chapter.7}{Exponentiation}{}% 105
\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7}% 106
\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1}% 107
\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7}% 108
\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2}% 109
\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2}% 110
\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7}% 111
\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3}% 112
\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7}% 113
\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{}% 114
\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8}% 115
\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1}% 116
\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1}% 117
\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1}% 118
\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8}% 119
\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2}% 120
\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2}% 121
\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2}% 122
\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2}% 123
\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8}% 124
\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8}% 125
\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4}% 126
\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4}% 127
\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{}% 128
\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9}% 129
\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1}% 130
\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9}% 131
\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9}% 132
\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3}% 133
\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9}% 134
\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4}% 135
\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9}% 136
\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5}% 137
\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5}% 138
\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5}% 139
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