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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: mpmont.h,v 1.7 2004/04/01 12:50:09 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 1.7 2004/04/01 12:50:09 mdw |
34 | * Add cyclic group abstraction, with test code. Separate off exponentation |
35 | * functions for better static linking. Fix a buttload of bugs on the way. |
36 | * Generally ensure that negative exponents do inversion correctly. Add |
37 | * table of standard prime-field subgroups. (Binary field subgroups are |
38 | * currently unimplemented but easy to add if anyone ever finds a good one.) |
39 | * |
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40 | * Revision 1.6 2002/01/13 13:49:25 mdw |
41 | * Make @const@-correct. |
42 | * |
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43 | * Revision 1.5 2001/06/16 13:00:04 mdw |
44 | * Moved @mpmont_factor@ to <mp.h>. Documented interface change to |
45 | * @mpmont_expr@ and @mpmont_mexpr@ -- the arguments are now in Montgomery |
46 | * form. |
47 | * |
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48 | * Revision 1.4 1999/12/11 01:51:14 mdw |
49 | * Use a Karatsuba-based reduction for large moduli. |
50 | * |
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51 | * Revision 1.3 1999/12/10 23:29:48 mdw |
52 | * Change header file guard names. |
53 | * |
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54 | * Revision 1.2 1999/11/19 13:17:43 mdw |
55 | * Add extra interface to exponentiation which returns a Montgomerized |
56 | * result. Add simultaneous exponentiation interface. |
57 | * |
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58 | * Revision 1.1 1999/11/17 18:02:16 mdw |
59 | * New multiprecision integer arithmetic suite. |
60 | * |
61 | */ |
62 | |
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63 | #ifndef CATACOMB_MPMONT_H |
64 | #define CATACOMB_MPMONT_H |
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65 | |
66 | #ifdef __cplusplus |
67 | extern "C" { |
68 | #endif |
69 | |
70 | /*----- Header files ------------------------------------------------------*/ |
71 | |
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72 | #ifndef CATACOMB_MP_H |
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73 | # include "mp.h" |
74 | #endif |
75 | |
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76 | /*----- Notes on Montgomery reduction -------------------------------------* |
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77 | * |
78 | * Given a little bit of precomputation, Montgomery reduction enables modular |
79 | * reductions of products to be calculated rather rapidly, without recourse |
80 | * to annoying things like division. |
81 | * |
82 | * Before starting, you need to do a little work. In particular, the |
83 | * following things need to be worked out: |
84 | * |
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85 | * * %$m$%, which is the modulus you'll be working with. This must be odd, |
86 | * otherwise the whole thing doesn't work. You're better off using |
87 | * Barrett reduction if your modulus might be even. |
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88 | * |
89 | * * %$b$%, the radix of the number system you're in (here, it's |
90 | * @MPW_MAX + 1@). |
91 | * |
92 | * * %$-m^{-1} \bmod b$%, a useful number for the reduction step. (This |
93 | * means that the modulus mustn't be even. This shouldn't be a problem.) |
94 | * |
95 | * * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%. |
96 | * |
97 | * * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing |
98 | * calculations such as exponentiation. |
99 | * |
100 | * The result of a Montgomery reduction of %$x$% is %$x R^{-1} \bmod m$%, |
101 | * which doesn't look ever-so useful. The trick is to initially apply a |
102 | * factor of %$R$% to all of your numbers so that when you multiply and |
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103 | * perform a Montgomery reduction you get %$(x R \cdot y R) R^{-1} \bmod m$%, |
104 | * which is just %$x y R \bmod m$%. Thanks to distributivity, even additions |
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105 | * and subtractions can be performed on numbers in this form -- the extra |
106 | * factor of %$R$% just runs through all the calculations until it's finally |
107 | * stripped out by a final reduction operation. |
108 | */ |
109 | |
110 | /*----- Data structures ---------------------------------------------------*/ |
111 | |
112 | /* --- A Montgomery reduction context --- */ |
113 | |
114 | typedef struct mpmont { |
115 | mp *m; /* Modulus */ |
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116 | mp *mi; /* %$-m^{-1} \bmod R$% */ |
117 | size_t n; /* %$\log_b R$% */ |
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118 | mp *r, *r2; /* %$R \bmod m$%, %$R^2 \bmod m$% */ |
119 | } mpmont; |
120 | |
121 | /*----- Functions provided ------------------------------------------------*/ |
122 | |
123 | /* --- @mpmont_create@ --- * |
124 | * |
125 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
126 | * @mp *m@ = modulus to use |
127 | * |
128 | * Returns: --- |
129 | * |
130 | * Use: Initializes a Montgomery reduction context ready for use. |
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131 | * The argument @m@ must be a positive odd integer. |
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132 | */ |
133 | |
134 | extern void mpmont_create(mpmont */*mm*/, mp */*m*/); |
135 | |
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136 | /* --- @mpmont_destroy@ --- * |
137 | * |
138 | * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context |
139 | * |
140 | * Returns: --- |
141 | * |
142 | * Use: Disposes of a context when it's no longer of any use to |
143 | * anyone. |
144 | */ |
145 | |
146 | extern void mpmont_destroy(mpmont */*mm*/); |
147 | |
148 | /* --- @mpmont_reduce@ --- * |
149 | * |
150 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
151 | * @mp *d@ = destination |
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152 | * @mp *a@ = source, assumed positive |
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153 | * |
154 | * Returns: Result, %$a R^{-1} \bmod m$%. |
155 | */ |
156 | |
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157 | extern mp *mpmont_reduce(mpmont */*mm*/, mp */*d*/, mp */*a*/); |
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158 | |
159 | /* --- @mpmont_mul@ --- * |
160 | * |
161 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
162 | * @mp *d@ = destination |
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163 | * @mp *a, *b@ = sources, assumed positive |
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164 | * |
165 | * Returns: Result, %$a b R^{-1} \bmod m$%. |
166 | */ |
167 | |
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168 | extern mp *mpmont_mul(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/); |
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169 | |
170 | /* --- @mpmont_expr@ --- * |
171 | * |
172 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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173 | * @mp *d@ = fake destination |
174 | * @mp *a@ = base |
175 | * @mp *e@ = exponent |
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176 | * |
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177 | * Returns: Result, %$(a R^{-1})^e R \bmod m$%. This is useful if |
178 | * further modular arithmetic is to be performed on the result. |
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179 | */ |
180 | |
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181 | extern mp *mpmont_expr(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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182 | |
183 | /* --- @mpmont_exp@ --- * |
184 | * |
185 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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186 | * @mp *d@ = fake destination |
187 | * @mp *a@ = base |
188 | * @mp *e@ = exponent |
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189 | * |
190 | * Returns: Result, %$a^e \bmod m$%. |
191 | */ |
192 | |
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193 | extern mp *mpmont_exp(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
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194 | |
195 | /* --- @mpmont_mexpr@ --- * |
196 | * |
197 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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198 | * @mp *d@ = fake destination |
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199 | * @const mp_expfactor *f@ = pointer to array of factors |
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200 | * @size_t n@ = number of factors supplied |
201 | * |
202 | * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the |
203 | * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result |
204 | * is: |
205 | * |
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206 | * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$% |
207 | * |
208 | * |
209 | * except that the %$g_i$% and result are in Montgomery form. |
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210 | */ |
211 | |
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212 | extern mp *mpmont_mexpr(mpmont */*mm*/, mp */*d*/, |
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213 | const mp_expfactor */*f*/, size_t /*n*/); |
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214 | |
215 | /* --- @mpmont_mexp@ --- * |
216 | * |
217 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
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218 | * @mp *d@ = fake destination |
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219 | * @const mp_expfactor *f@ = pointer to array of factors |
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220 | * @size_t n@ = number of factors supplied |
221 | * |
222 | * Returns: Product of bases raised to exponents, all mod @m@. |
223 | * |
224 | * Use: Convenient interface over @mpmont_mexpr@. |
225 | */ |
226 | |
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227 | extern mp *mpmont_mexp(mpmont */*mm*/, mp */*d*/, |
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228 | const mp_expfactor */*f*/, size_t /*n*/); |
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229 | |
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230 | /*----- That's all, folks -------------------------------------------------*/ |
231 | |
232 | #ifdef __cplusplus |
233 | } |
234 | #endif |
235 | |
236 | #endif |