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