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0.30 release of LibTomMath
author Matt Johnston <matt@ucc.asn.au>
date Tue, 15 Jun 2004 14:42:57 +0000
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2:86e0b50a9b58 19:e1037a1e12e7
1 \BOOKMARK [0][-]{chapter.1}{Introduction}{}
2 \BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}
3 \BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1}
4 \BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1}
5 \BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}
6 \BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}
7 \BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}
8 \BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}
9 \BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3}
10 \BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3}
11 \BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3}
12 \BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3}
13 \BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}
14 \BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1}
15 \BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5}
16 \BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5}
17 \BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1}
18 \BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6}
19 \BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6}
20 \BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6}
21 \BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6}
22 \BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6}
23 \BOOKMARK [0][-]{chapter.2}{Getting Started}{}
24 \BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2}
25 \BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2}
26 \BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2}
27 \BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2}
28 \BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2}
29 \BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2}
30 \BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5}
31 \BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5}
32 \BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2}
33 \BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6}
34 \BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6}
35 \BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6}
36 \BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6}
37 \BOOKMARK [0][-]{chapter.3}{Basic Operations}{}
38 \BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3}
39 \BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3}
40 \BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2}
41 \BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2}
42 \BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3}
43 \BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3}
44 \BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4}
45 \BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4}
46 \BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3}
47 \BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5}
48 \BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5}
49 \BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3}
50 \BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6}
51 \BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6}
52 \BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{}
53 \BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}
54 \BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4}
55 \BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2}
56 \BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2}
57 \BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2}
58 \BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2}
59 \BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4}
60 \BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3}
61 \BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3}
62 \BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4}
63 \BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4}
64 \BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4}
65 \BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4}
66 \BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5}
67 \BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5}
68 \BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5}
69 \BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{}
70 \BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5}
71 \BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5}
72 \BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2}
73 \BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2}
74 \BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2}
75 \BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2}
76 \BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2}
77 \BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2}
78 \BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5}
79 \BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3}
80 \BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3}
81 \BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3}
82 \BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3}
83 \BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3}
84 \BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3}
85 \BOOKMARK [0][-]{chapter.6}{Modular Reduction}{}
86 \BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6}
87 \BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6}
88 \BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2}
89 \BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2}
90 \BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2}
91 \BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2}
92 \BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2}
93 \BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2}
94 \BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6}
95 \BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3}
96 \BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3}
97 \BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3}
98 \BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3}
99 \BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6}
100 \BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4}
101 \BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4}
102 \BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4}
103 \BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4}
104 \BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6}
105 \BOOKMARK [0][-]{chapter.7}{Exponentiation}{}
106 \BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7}
107 \BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1}
108 \BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7}
109 \BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2}
110 \BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2}
111 \BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7}
112 \BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3}
113 \BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7}
114 \BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{}
115 \BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8}
116 \BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1}
117 \BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1}
118 \BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1}
119 \BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8}
120 \BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2}
121 \BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2}
122 \BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2}
123 \BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2}
124 \BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8}
125 \BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8}
126 \BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4}
127 \BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4}
128 \BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{}
129 \BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9}
130 \BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1}
131 \BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9}
132 \BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9}
133 \BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3}
134 \BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9}
135 \BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4}
136 \BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9}
137 \BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5}
138 \BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5}
139 \BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5}