3 * $Id: mpmont.h,v 1.3 1999/12/10 23:29:48 mdw Exp $
7 * (c) 1999 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Revision history --------------------------------------------------*
33 * Revision 1.3 1999/12/10 23:29:48 mdw
34 * Change header file guard names.
36 * Revision 1.2 1999/11/19 13:17:43 mdw
37 * Add extra interface to exponentiation which returns a Montgomerized
38 * result. Add simultaneous exponentiation interface.
40 * Revision 1.1 1999/11/17 18:02:16 mdw
41 * New multiprecision integer arithmetic suite.
45 #ifndef CATACOMB_MPMONT_H
46 #define CATACOMB_MPMONT_H
52 /*----- Header files ------------------------------------------------------*/
58 /*----- Notes on Montgomery reduction -------------------------------------*
60 * Given a little bit of precomputation, Montgomery reduction enables modular
61 * reductions of products to be calculated rather rapidly, without recourse
62 * to annoying things like division.
64 * Before starting, you need to do a little work. In particular, the
65 * following things need to be worked out:
67 * * %$m$%, which is the modulus you'll be working with. This must be odd,
68 * otherwise the whole thing doesn't work. You're better off using
69 * Barrett reduction if your modulus might be even.
71 * * %$b$%, the radix of the number system you're in (here, it's
74 * * %$-m^{-1} \bmod b$%, a useful number for the reduction step. (This
75 * means that the modulus mustn't be even. This shouldn't be a problem.)
77 * * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%.
79 * * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing
80 * calculations such as exponentiation.
82 * The result of a Montgomery reduction of %$x$% is %$x R^{-1} \bmod m$%,
83 * which doesn't look ever-so useful. The trick is to initially apply a
84 * factor of %$R$% to all of your numbers so that when you multiply and
85 * perform a Montgomery reduction you get %$(x R \cdot y R) R^{-1} \bmod m$%,
86 * which is just %$x y R \bmod m$%. Thanks to distributivity, even additions
87 * and subtractions can be performed on numbers in this form -- the extra
88 * factor of %$R$% just runs through all the calculations until it's finally
89 * stripped out by a final reduction operation.
92 /*----- Data structures ---------------------------------------------------*/
94 /* --- A Montgomery reduction context --- */
96 typedef struct mpmont
{
98 mpw mi
; /* %$-m^{-1} \bmod b$% */
99 size_t shift
; /* %$\log_2 R$% */
100 mp
*r
, *r2
; /* %$R \bmod m$%, %$R^2 \bmod m$% */
103 /* --- A base/exponent pair for @mpmont_mexp@ --- */
105 typedef struct mpmont_factor
{
110 /*----- Functions provided ------------------------------------------------*/
112 /* --- @mpmont_create@ --- *
114 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
115 * @mp *m@ = modulus to use
119 * Use: Initializes a Montgomery reduction context ready for use.
120 * The argument @m@ must be a positive odd integer.
123 extern void mpmont_create(mpmont */
*mm*/
, mp */
*m*/
);
125 /* --- @mpmont_destroy@ --- *
127 * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context
131 * Use: Disposes of a context when it's no longer of any use to
135 extern void mpmont_destroy(mpmont */
*mm*/
);
137 /* --- @mpmont_reduce@ --- *
139 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
140 * @mp *d@ = destination
141 * @mp *a@ = source, assumed positive
143 * Returns: Result, %$a R^{-1} \bmod m$%.
146 extern mp
*mpmont_reduce(mpmont */
*mm*/
, mp */
*d*/
, mp */
*a*/
);
148 /* --- @mpmont_mul@ --- *
150 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
151 * @mp *d@ = destination
152 * @mp *a, *b@ = sources, assumed positive
154 * Returns: Result, %$a b R^{-1} \bmod m$%.
157 extern mp
*mpmont_mul(mpmont */
*mm*/
, mp */
*d*/
, mp */
*a*/
, mp */
*b*/
);
159 /* --- @mpmont_expr@ --- *
161 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
162 * @mp *d@ = fake destination
166 * Returns: Result, %$a^e R \bmod m$%. This is useful if further modular
167 * arithmetic is to be performed on the result.
170 extern mp
*mpmont_expr(mpmont */
*mm*/
, mp */
*d*/
, mp */
*a*/
, mp */
*e*/
);
172 /* --- @mpmont_exp@ --- *
174 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
175 * @mp *d@ = fake destination
179 * Returns: Result, %$a^e \bmod m$%.
182 extern mp
*mpmont_exp(mpmont */
*mm*/
, mp */
*d*/
, mp */
*a*/
, mp */
*e*/
);
184 /* --- @mpmont_mexpr@ --- *
186 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
187 * @mp *d@ = fake destination
188 * @mpmont_factor *f@ = pointer to array of factors
189 * @size_t n@ = number of factors supplied
191 * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the
192 * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result
195 * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} R \bmod m$%
198 extern mp
*mpmont_mexpr(mpmont */
*mm*/
, mp */
*d*/
,
199 mpmont_factor */
*f*/
, size_t /*n*/);
201 /* --- @mpmont_mexp@ --- *
203 * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context
204 * @mp *d@ = fake destination
205 * @mpmont_factor *f@ = pointer to array of factors
206 * @size_t n@ = number of factors supplied
208 * Returns: Product of bases raised to exponents, all mod @m@.
210 * Use: Convenient interface over @mpmont_mexpr@.
213 extern mp
*mpmont_mexp(mpmont */
*mm*/
, mp */
*d*/
,
214 mpmont_factor */
*f*/
, size_t /*n*/);
216 /*----- That's all, folks -------------------------------------------------*/