d3409d5e |
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 |