| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: mpmont.h,v 1.4 1999/12/11 01:51:14 mdw Exp $ |
| 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 $ |
| 33 | * Revision 1.4 1999/12/11 01:51:14 mdw |
| 34 | * Use a Karatsuba-based reduction for large moduli. |
| 35 | * |
| 36 | * Revision 1.3 1999/12/10 23:29:48 mdw |
| 37 | * Change header file guard names. |
| 38 | * |
| 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 | * |
| 43 | * Revision 1.1 1999/11/17 18:02:16 mdw |
| 44 | * New multiprecision integer arithmetic suite. |
| 45 | * |
| 46 | */ |
| 47 | |
| 48 | #ifndef CATACOMB_MPMONT_H |
| 49 | #define CATACOMB_MPMONT_H |
| 50 | |
| 51 | #ifdef __cplusplus |
| 52 | extern "C" { |
| 53 | #endif |
| 54 | |
| 55 | /*----- Header files ------------------------------------------------------*/ |
| 56 | |
| 57 | #ifndef CATACOMB_MP_H |
| 58 | # include "mp.h" |
| 59 | #endif |
| 60 | |
| 61 | /*----- Notes on Montgomery reduction -------------------------------------* |
| 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 | * |
| 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. |
| 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 |
| 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 |
| 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 */ |
| 101 | mp *mi; /* %$-m^{-1} \bmod R$% */ |
| 102 | size_t n; /* %$\log_b R$% */ |
| 103 | mp *r, *r2; /* %$R \bmod m$%, %$R^2 \bmod m$% */ |
| 104 | } mpmont; |
| 105 | |
| 106 | /* --- A base/exponent pair for @mpmont_mexp@ --- */ |
| 107 | |
| 108 | typedef struct mpmont_factor { |
| 109 | mp *base; |
| 110 | mp *exp; |
| 111 | } mpmont_factor; |
| 112 | |
| 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. |
| 123 | * The argument @m@ must be a positive odd integer. |
| 124 | */ |
| 125 | |
| 126 | extern void mpmont_create(mpmont */*mm*/, mp */*m*/); |
| 127 | |
| 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 |
| 144 | * @mp *a@ = source, assumed positive |
| 145 | * |
| 146 | * Returns: Result, %$a R^{-1} \bmod m$%. |
| 147 | */ |
| 148 | |
| 149 | extern mp *mpmont_reduce(mpmont */*mm*/, mp */*d*/, mp */*a*/); |
| 150 | |
| 151 | /* --- @mpmont_mul@ --- * |
| 152 | * |
| 153 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 154 | * @mp *d@ = destination |
| 155 | * @mp *a, *b@ = sources, assumed positive |
| 156 | * |
| 157 | * Returns: Result, %$a b R^{-1} \bmod m$%. |
| 158 | */ |
| 159 | |
| 160 | extern mp *mpmont_mul(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/); |
| 161 | |
| 162 | /* --- @mpmont_expr@ --- * |
| 163 | * |
| 164 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 165 | * @mp *d@ = fake destination |
| 166 | * @mp *a@ = base |
| 167 | * @mp *e@ = exponent |
| 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 | |
| 173 | extern mp *mpmont_expr(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
| 174 | |
| 175 | /* --- @mpmont_exp@ --- * |
| 176 | * |
| 177 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 178 | * @mp *d@ = fake destination |
| 179 | * @mp *a@ = base |
| 180 | * @mp *e@ = exponent |
| 181 | * |
| 182 | * Returns: Result, %$a^e \bmod m$%. |
| 183 | */ |
| 184 | |
| 185 | extern mp *mpmont_exp(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
| 186 | |
| 187 | /* --- @mpmont_mexpr@ --- * |
| 188 | * |
| 189 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 190 | * @mp *d@ = fake destination |
| 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 | |
| 201 | extern mp *mpmont_mexpr(mpmont */*mm*/, mp */*d*/, |
| 202 | mpmont_factor */*f*/, size_t /*n*/); |
| 203 | |
| 204 | /* --- @mpmont_mexp@ --- * |
| 205 | * |
| 206 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 207 | * @mp *d@ = fake destination |
| 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 | |
| 216 | extern mp *mpmont_mexp(mpmont */*mm*/, mp */*d*/, |
| 217 | mpmont_factor */*f*/, size_t /*n*/); |
| 218 | |
| 219 | /*----- That's all, folks -------------------------------------------------*/ |
| 220 | |
| 221 | #ifdef __cplusplus |
| 222 | } |
| 223 | #endif |
| 224 | |
| 225 | #endif |