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   1 /* -*-c-*-
   2  *
   3  * $Id: mpmont.h,v 1.1 1999/11/17 18:02:16 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 1.1  1999/11/17 18:02:16  mdw
  34  * New multiprecision integer arithmetic suite.
  35  *
  36  */
  37
  38 #ifndef MPMONT_H
  39 #define MPMONT_H
  40
  41 #ifdef __cplusplus
  42   extern "C" {
  43 #endif
  44
  45 /*----- Header files ------------------------------------------------------*/
  46
  47 #ifndef MP_H
  48 #  include "mp.h"
  49 #endif
  50
  51 /*----- What's going on here? ---------------------------------------------*
  52  *
  53  * Given a little bit of precomputation, Montgomery reduction enables modular
  54  * reductions of products to be calculated rather rapidly, without recourse
  55  * to annoying things like division.
  56  *
  57  * Before starting, you need to do a little work.  In particular, the
  58  * following things need to be worked out:
  59  *
  60  *   * %$m$%, which is the modulus you'll be working with.
  61  *
  62  *   * %$b$%, the radix of the number system you're in (here, it's
  63  *     @MPW_MAX + 1@).
  64  *
  65  *   * %$-m^{-1} \bmod b$%, a useful number for the reduction step.  (This
  66  *     means that the modulus mustn't be even.  This shouldn't be a problem.)
  67  *
  68  *   * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%.
  69  *
  70  *   * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing
  71  *     calculations such as exponentiation.
  72  *
  73  * The result of a Montgomery reduction of %$x$% is %$x R^{-1} \bmod m$%,
  74  * which doesn't look ever-so useful.  The trick is to initially apply a
  75  * factor of %$R$% to all of your numbers so that when you multiply and
  76  * perform a Montgomery reduction you get %$(xR \cdot yR)R^{-1} \bmod m$%,
  77  * which is just %$xyR \bmod m$%.  Thanks to distributivity, even additions
  78  * and subtractions can be performed on numbers in this form -- the extra
  79  * factor of %$R$% just runs through all the calculations until it's finally
  80  * stripped out by a final reduction operation.
  81  */
  82
  83 /*----- Data structures ---------------------------------------------------*/
  84
  85 /* --- A Montgomery reduction context --- */
  86
  87 typedef struct mpmont {
  88   mp *m;                                /* Modulus */
  89   mpw mi;                               /* %$-m^{-1} \bmod b$% */
  90   size_t shift;                         /* %$\log_2 R$% */
  91   mp *r, *r2;                           /* %$R \bmod m$%, %$R^2 \bmod m$% */
  92 } mpmont;
  93
  94 /*----- Functions provided ------------------------------------------------*/
  95
  96 /* --- @mpmont_create@ --- *
  97  *
  98  * Arguments:   @mpmont *mm@ = pointer to Montgomery reduction context
  99  *              @mp *m@ = modulus to use
 100  *
 101  * Returns:     ---
 102  *
 103  * Use:         Initializes a Montgomery reduction context ready for use.
 104  */
 105
 106 extern void mpmont_create(mpmont */*mm*/, mp */*m*/);
 107
 108 /*----- That's all, folks -------------------------------------------------*/
 109
 110 #ifdef __cplusplus
 111   }
 112 #endif
 113
 114 #endif