git.distorted.org.uk Git - u/mdw/catacomb/blob - mpmont.h

   1 /* -*-c-*-
   2  *
   3  * $Id$
   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 #ifndef CATACOMB_MPMONT_H
  31 #define CATACOMB_MPMONT_H
  32
  33 #ifdef __cplusplus
  34   extern "C" {
  35 #endif
  36
  37 /*----- Header files ------------------------------------------------------*/
  38
  39 #ifndef CATACOMB_MP_H
  40 #  include "mp.h"
  41 #endif
  42
  43 /*----- Notes on Montgomery reduction -------------------------------------*
  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  *
  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.
  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
  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
  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 */
  83   mp *mi;                               /* %$-m^{-1} \bmod R$% */
  84   size_t n;                             /* %$\log_b R$% */
  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  *
  95  * Returns:     Zero on success, nonzero on error.
  96  *
  97  * Use:         Initializes a Montgomery reduction context ready for use.
  98  *              The argument @m@ must be a positive odd integer.
  99  */
 100
 101 extern int mpmont_create(mpmont */*mm*/, mp */*m*/);
 102
 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
 119  *              @mp *a@ = source, assumed positive
 120  *
 121  * Returns:     Result, %$a R^{-1} \bmod m$%.
 122  */
 123
 124 extern mp *mpmont_reduce(mpmont */*mm*/, mp */*d*/, mp */*a*/);
 125
 126 /* --- @mpmont_mul@ --- *
 127  *
 128  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
 129  *              @mp *d@ = destination
 130  *              @mp *a, *b@ = sources, assumed positive
 131  *
 132  * Returns:     Result, %$a b R^{-1} \bmod m$%.
 133  */
 134
 135 extern mp *mpmont_mul(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/);
 136
 137 /* --- @mpmont_expr@ --- *
 138  *
 139  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
 140  *              @mp *d@ = fake destination
 141  *              @mp *a@ = base
 142  *              @mp *e@ = exponent
 143  *
 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.
 146  */
 147
 148 extern mp *mpmont_expr(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/);
 149
 150 /* --- @mpmont_exp@ --- *
 151  *
 152  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
 153  *              @mp *d@ = fake destination
 154  *              @mp *a@ = base
 155  *              @mp *e@ = exponent
 156  *
 157  * Returns:     Result, %$a^e \bmod m$%.
 158  */
 159
 160 extern mp *mpmont_exp(mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/);
 161
 162 /* --- @mpmont_mexpr@ --- *
 163  *
 164  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
 165  *              @mp *d@ = fake destination
 166  *              @const mp_expfactor *f@ = pointer to array of factors
 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  *
 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.
 177  */
 178
 179 extern mp *mpmont_mexpr(mpmont */*mm*/, mp */*d*/,
 180                         const mp_expfactor */*f*/, size_t /*n*/);
 181
 182 /* --- @mpmont_mexp@ --- *
 183  *
 184  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
 185  *              @mp *d@ = fake destination
 186  *              @const mp_expfactor *f@ = pointer to array of factors
 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
 194 extern mp *mpmont_mexp(mpmont */*mm*/, mp */*d*/,
 195                        const mp_expfactor */*f*/, size_t /*n*/);
 196
 197 /*----- That's all, folks -------------------------------------------------*/
 198
 199 #ifdef __cplusplus
 200   }
 201 #endif
 202
 203 #endif