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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: mpmont.h,v 1.4 1999/12/11 01:51:14 mdw Exp $ |
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4 | * |
5 | * Montgomery reduction |
6 | * |
7 | * (c) 1999 Straylight/Edgeware |
8 | */ |
9 | |
10 | /*----- Licensing notice --------------------------------------------------* |
11 | * |
12 | * This file is part of Catacomb. |
13 | * |
14 | * Catacomb is free software; you can redistribute it and/or modify |
15 | * it under the terms of the GNU Library General Public License as |
16 | * published by the Free Software Foundation; either version 2 of the |
17 | * License, or (at your option) any later version. |
18 | * |
19 | * Catacomb is distributed in the hope that it will be useful, |
20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
22 | * GNU Library General Public License for more details. |
23 | * |
24 | * You should have received a copy of the GNU Library General Public |
25 | * License along with Catacomb; if not, write to the Free |
26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
27 | * MA 02111-1307, USA. |
28 | */ |
29 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: mpmont.h,v $ |
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33 | * Revision 1.4 1999/12/11 01:51:14 mdw |
34 | * Use a Karatsuba-based reduction for large moduli. |
35 | * |
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36 | * Revision 1.3 1999/12/10 23:29:48 mdw |
37 | * Change header file guard names. |
38 | * |
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39 | * Revision 1.2 1999/11/19 13:17:43 mdw |
40 | * Add extra interface to exponentiation which returns a Montgomerized |
41 | * result. Add simultaneous exponentiation interface. |
42 | * |
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43 | * Revision 1.1 1999/11/17 18:02:16 mdw |
44 | * New multiprecision integer arithmetic suite. |
45 | * |
46 | */ |
47 | |
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48 | #ifndef CATACOMB_MPMONT_H |
49 | #define CATACOMB_MPMONT_H |
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50 | |
51 | #ifdef __cplusplus |
52 | extern "C" { |
53 | #endif |
54 | |
55 | /*----- Header files ------------------------------------------------------*/ |
56 | |
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57 | #ifndef CATACOMB_MP_H |
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58 | # include "mp.h" |
59 | #endif |
60 | |
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61 | /*----- Notes on Montgomery reduction -------------------------------------* |
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62 | * |
63 | * Given a little bit of precomputation, Montgomery reduction enables modular |
64 | * reductions of products to be calculated rather rapidly, without recourse |
65 | * to annoying things like division. |
66 | * |
67 | * Before starting, you need to do a little work. In particular, the |
68 | * following things need to be worked out: |
69 | * |
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70 | * * %$m$%, which is the modulus you'll be working with. This must be odd, |
71 | * otherwise the whole thing doesn't work. You're better off using |
72 | * Barrett reduction if your modulus might be even. |
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73 | * |
74 | * * %$b$%, the radix of the number system you're in (here, it's |
75 | * @MPW_MAX + 1@). |
76 | * |
77 | * * %$-m^{-1} \bmod b$%, a useful number for the reduction step. (This |
78 | * means that the modulus mustn't be even. This shouldn't be a problem.) |
79 | * |
80 | * * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%. |
81 | * |
82 | * * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing |
83 | * calculations such as exponentiation. |
84 | * |
85 | * The result of a Montgomery reduction of %$x$% is %$x R^{-1} \bmod m$%, |
86 | * which doesn't look ever-so useful. The trick is to initially apply a |
87 | * factor of %$R$% to all of your numbers so that when you multiply and |
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88 | * perform a Montgomery reduction you get %$(x R \cdot y R) R^{-1} \bmod m$%, |
89 | * which is just %$x y R \bmod m$%. Thanks to distributivity, even additions |
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90 | * and subtractions can be performed on numbers in this form -- the extra |
91 | * factor of %$R$% just runs through all the calculations until it's finally |
92 | * stripped out by a final reduction operation. |
93 | */ |
94 | |
95 | /*----- Data structures ---------------------------------------------------*/ |
96 | |
97 | /* --- A Montgomery reduction context --- */ |
98 | |
99 | typedef struct mpmont { |
100 | mp *m; /* Modulus */ |
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101 | mp *mi; /* %$-m^{-1} \bmod R$% */ |
102 | size_t n; /* %$\log_b R$% */ |
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103 | mp *r, *r2; /* %$R \bmod m$%, %$R^2 \bmod m$% */ |
104 | } mpmont; |
105 | |
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106 | /* --- A base/exponent pair for @mpmont_mexp@ --- */ |
107 | |
108 | typedef struct mpmont_factor { |
109 | mp *base; |
110 | mp *exp; |
111 | } mpmont_factor; |
112 | |
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113 | /*----- Functions provided ------------------------------------------------*/ |
114 | |
115 | /* --- @mpmont_create@ --- * |
116 | * |
117 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
118 | * @mp *m@ = modulus to use |
119 | * |
120 | * Returns: --- |
121 | * |
122 | * Use: Initializes a Montgomery reduction context ready for use. |
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123 | * The argument @m@ must be a positive odd integer. |
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124 | */ |
125 | |
126 | extern void mpmont_create(mpmont */*mm*/, mp */*m*/); |
127 | |
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128 | /* --- @mpmont_destroy@ --- * |
129 | * |
130 | * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context |
131 | * |
132 | * Returns: --- |
133 | * |
134 | * Use: Disposes of a context when it's no longer of any use to |
135 | * anyone. |
136 | */ |
137 | |
138 | extern void mpmont_destroy(mpmont */*mm*/); |
139 | |
140 | /* --- @mpmont_reduce@ --- * |
141 | * |
142 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
143 | * @mp *d@ = destination |
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144 | * @mp *a@ = source, assumed positive |
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145 | * |
146 | * Returns: Result, %$a R^{-1} \bmod m$%. |
147 | */ |
148 | |
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149 | extern mp *mpmont_reduce(mpmont */*mm*/, mp */*d*/, mp */*a*/); |
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150 | |
151 | /* --- @mpmont_mul@ --- * |
152 | * |
153 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
154 | * @mp *d@ = destination |
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155 | * @mp *a, *b@ = sources, assumed positive |
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156 | * |
157 | * Returns: Result, %$a b R^{-1} \bmod m$%. |
158 | */ |
159 | |
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160 | extern mp *mpmont_mul(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/); |
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161 | |
162 | /* --- @mpmont_expr@ --- * |
163 | * |
164 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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165 | * @mp *d@ = fake destination |
166 | * @mp *a@ = base |
167 | * @mp *e@ = exponent |
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168 | * |
169 | * Returns: Result, %$a^e R \bmod m$%. This is useful if further modular |
170 | * arithmetic is to be performed on the result. |
171 | */ |
172 | |
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173 | extern mp *mpmont_expr(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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174 | |
175 | /* --- @mpmont_exp@ --- * |
176 | * |
177 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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178 | * @mp *d@ = fake destination |
179 | * @mp *a@ = base |
180 | * @mp *e@ = exponent |
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181 | * |
182 | * Returns: Result, %$a^e \bmod m$%. |
183 | */ |
184 | |
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185 | extern mp *mpmont_exp(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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186 | |
187 | /* --- @mpmont_mexpr@ --- * |
188 | * |
189 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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190 | * @mp *d@ = fake destination |
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191 | * @mpmont_factor *f@ = pointer to array of factors |
192 | * @size_t n@ = number of factors supplied |
193 | * |
194 | * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the |
195 | * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result |
196 | * is: |
197 | * |
198 | * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} R \bmod m$% |
199 | */ |
200 | |
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201 | extern mp *mpmont_mexpr(mpmont */*mm*/, mp */*d*/, |
202 | mpmont_factor */*f*/, size_t /*n*/); |
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203 | |
204 | /* --- @mpmont_mexp@ --- * |
205 | * |
206 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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207 | * @mp *d@ = fake destination |
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208 | * @mpmont_factor *f@ = pointer to array of factors |
209 | * @size_t n@ = number of factors supplied |
210 | * |
211 | * Returns: Product of bases raised to exponents, all mod @m@. |
212 | * |
213 | * Use: Convenient interface over @mpmont_mexpr@. |
214 | */ |
215 | |
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216 | extern mp *mpmont_mexp(mpmont */*mm*/, mp */*d*/, |
217 | mpmont_factor */*f*/, size_t /*n*/); |
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218 | |
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219 | /*----- That's all, folks -------------------------------------------------*/ |
220 | |
221 | #ifdef __cplusplus |
222 | } |
223 | #endif |
224 | |
225 | #endif |