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1 | /* -*-c-*- |
2 | * |
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3 | * $Id$ |
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4 | * |
5 | * Montgomery reduction |
6 | * |
7 | * (c) 1999 Straylight/Edgeware |
8 | */ |
9 | |
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10 | /*----- Licensing notice --------------------------------------------------* |
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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. |
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18 | * |
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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. |
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23 | * |
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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 | |
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30 | #ifndef CATACOMB_MPMONT_H |
31 | #define CATACOMB_MPMONT_H |
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32 | |
33 | #ifdef __cplusplus |
34 | extern "C" { |
35 | #endif |
36 | |
37 | /*----- Header files ------------------------------------------------------*/ |
38 | |
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39 | #ifndef CATACOMB_MP_H |
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40 | # include "mp.h" |
41 | #endif |
42 | |
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43 | /*----- Notes on Montgomery reduction -------------------------------------* |
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44 | * |
45 | * Given a little bit of precomputation, Montgomery reduction enables modular |
46 | * reductions of products to be calculated rather rapidly, without recourse |
47 | * to annoying things like division. |
48 | * |
49 | * Before starting, you need to do a little work. In particular, the |
50 | * following things need to be worked out: |
51 | * |
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52 | * * %$m$%, which is the modulus you'll be working with. This must be odd, |
53 | * otherwise the whole thing doesn't work. You're better off using |
54 | * Barrett reduction if your modulus might be even. |
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55 | * |
56 | * * %$b$%, the radix of the number system you're in (here, it's |
57 | * @MPW_MAX + 1@). |
58 | * |
59 | * * %$-m^{-1} \bmod b$%, a useful number for the reduction step. (This |
60 | * means that the modulus mustn't be even. This shouldn't be a problem.) |
61 | * |
62 | * * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%. |
63 | * |
64 | * * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing |
65 | * calculations such as exponentiation. |
66 | * |
67 | * The result of a Montgomery reduction of %$x$% is %$x R^{-1} \bmod m$%, |
68 | * which doesn't look ever-so useful. The trick is to initially apply a |
69 | * factor of %$R$% to all of your numbers so that when you multiply and |
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70 | * perform a Montgomery reduction you get %$(x R \cdot y R) R^{-1} \bmod m$%, |
71 | * which is just %$x y R \bmod m$%. Thanks to distributivity, even additions |
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72 | * and subtractions can be performed on numbers in this form -- the extra |
73 | * factor of %$R$% just runs through all the calculations until it's finally |
74 | * stripped out by a final reduction operation. |
75 | */ |
76 | |
77 | /*----- Data structures ---------------------------------------------------*/ |
78 | |
79 | /* --- A Montgomery reduction context --- */ |
80 | |
81 | typedef struct mpmont { |
82 | mp *m; /* Modulus */ |
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83 | mp *mi; /* %$-m^{-1} \bmod R$% */ |
84 | size_t n; /* %$\log_b R$% */ |
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85 | mp *r, *r2; /* %$R \bmod m$%, %$R^2 \bmod m$% */ |
86 | } mpmont; |
87 | |
88 | /*----- Functions provided ------------------------------------------------*/ |
89 | |
90 | /* --- @mpmont_create@ --- * |
91 | * |
92 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
93 | * @mp *m@ = modulus to use |
94 | * |
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95 | * Returns: Zero on success, nonzero on error. |
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96 | * |
97 | * Use: Initializes a Montgomery reduction context ready for use. |
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98 | * The argument @m@ must be a positive odd integer. |
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99 | */ |
100 | |
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101 | extern int mpmont_create(mpmont */*mm*/, mp */*m*/); |
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102 | |
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103 | /* --- @mpmont_destroy@ --- * |
104 | * |
105 | * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context |
106 | * |
107 | * Returns: --- |
108 | * |
109 | * Use: Disposes of a context when it's no longer of any use to |
110 | * anyone. |
111 | */ |
112 | |
113 | extern void mpmont_destroy(mpmont */*mm*/); |
114 | |
115 | /* --- @mpmont_reduce@ --- * |
116 | * |
117 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
118 | * @mp *d@ = destination |
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119 | * @mp *a@ = source, assumed positive |
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120 | * |
121 | * Returns: Result, %$a R^{-1} \bmod m$%. |
122 | */ |
123 | |
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124 | extern mp *mpmont_reduce(mpmont */*mm*/, mp */*d*/, mp */*a*/); |
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125 | |
126 | /* --- @mpmont_mul@ --- * |
127 | * |
128 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
129 | * @mp *d@ = destination |
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130 | * @mp *a, *b@ = sources, assumed positive |
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131 | * |
132 | * Returns: Result, %$a b R^{-1} \bmod m$%. |
133 | */ |
134 | |
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135 | extern mp *mpmont_mul(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/); |
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136 | |
137 | /* --- @mpmont_expr@ --- * |
138 | * |
139 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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140 | * @mp *d@ = fake destination |
141 | * @mp *a@ = base |
142 | * @mp *e@ = exponent |
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143 | * |
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144 | * Returns: Result, %$(a R^{-1})^e R \bmod m$%. This is useful if |
145 | * further modular arithmetic is to be performed on the result. |
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146 | */ |
147 | |
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148 | extern mp *mpmont_expr(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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149 | |
150 | /* --- @mpmont_exp@ --- * |
151 | * |
152 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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153 | * @mp *d@ = fake destination |
154 | * @mp *a@ = base |
155 | * @mp *e@ = exponent |
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156 | * |
157 | * Returns: Result, %$a^e \bmod m$%. |
158 | */ |
159 | |
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160 | extern mp *mpmont_exp(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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161 | |
162 | /* --- @mpmont_mexpr@ --- * |
163 | * |
164 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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165 | * @mp *d@ = fake destination |
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166 | * @const mp_expfactor *f@ = pointer to array of factors |
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167 | * @size_t n@ = number of factors supplied |
168 | * |
169 | * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the |
170 | * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result |
171 | * is: |
172 | * |
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173 | * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$% |
174 | * |
175 | * |
176 | * except that the %$g_i$% and result are in Montgomery form. |
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177 | */ |
178 | |
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179 | extern mp *mpmont_mexpr(mpmont */*mm*/, mp */*d*/, |
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180 | const mp_expfactor */*f*/, size_t /*n*/); |
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181 | |
182 | /* --- @mpmont_mexp@ --- * |
183 | * |
184 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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185 | * @mp *d@ = fake destination |
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186 | * @const mp_expfactor *f@ = pointer to array of factors |
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187 | * @size_t n@ = number of factors supplied |
188 | * |
189 | * Returns: Product of bases raised to exponents, all mod @m@. |
190 | * |
191 | * Use: Convenient interface over @mpmont_mexpr@. |
192 | */ |
193 | |
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194 | extern mp *mpmont_mexp(mpmont */*mm*/, mp */*d*/, |
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195 | const mp_expfactor */*f*/, size_t /*n*/); |
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196 | |
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197 | /*----- That's all, folks -------------------------------------------------*/ |
198 | |
199 | #ifdef __cplusplus |
200 | } |
201 | #endif |
202 | |
203 | #endif |