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d3409d5e | 1 | /* -*-c-*- |
d3409d5e | 2 | * |
3 | * Montgomery reduction | |
4 | * | |
5 | * (c) 1999 Straylight/Edgeware | |
6 | */ | |
7 | ||
45c0fd36 | 8 | /*----- Licensing notice --------------------------------------------------* |
d3409d5e | 9 | * |
10 | * This file is part of Catacomb. | |
11 | * | |
12 | * Catacomb is free software; you can redistribute it and/or modify | |
13 | * it under the terms of the GNU Library General Public License as | |
14 | * published by the Free Software Foundation; either version 2 of the | |
15 | * License, or (at your option) any later version. | |
45c0fd36 | 16 | * |
d3409d5e | 17 | * Catacomb is distributed in the hope that it will be useful, |
18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
20 | * GNU Library General Public License for more details. | |
45c0fd36 | 21 | * |
d3409d5e | 22 | * You should have received a copy of the GNU Library General Public |
23 | * License along with Catacomb; if not, write to the Free | |
24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
25 | * MA 02111-1307, USA. | |
26 | */ | |
27 | ||
b3f05084 | 28 | #ifndef CATACOMB_MPMONT_H |
29 | #define CATACOMB_MPMONT_H | |
d3409d5e | 30 | |
31 | #ifdef __cplusplus | |
32 | extern "C" { | |
33 | #endif | |
34 | ||
35 | /*----- Header files ------------------------------------------------------*/ | |
36 | ||
b3f05084 | 37 | #ifndef CATACOMB_MP_H |
d3409d5e | 38 | # include "mp.h" |
39 | #endif | |
40 | ||
b3f05084 | 41 | /*----- Notes on Montgomery reduction -------------------------------------* |
d3409d5e | 42 | * |
43 | * Given a little bit of precomputation, Montgomery reduction enables modular | |
44 | * reductions of products to be calculated rather rapidly, without recourse | |
45 | * to annoying things like division. | |
46 | * | |
47 | * Before starting, you need to do a little work. In particular, the | |
48 | * following things need to be worked out: | |
49 | * | |
b3f05084 | 50 | * * %$m$%, which is the modulus you'll be working with. This must be odd, |
51 | * otherwise the whole thing doesn't work. You're better off using | |
52 | * Barrett reduction if your modulus might be even. | |
d3409d5e | 53 | * |
54 | * * %$b$%, the radix of the number system you're in (here, it's | |
55 | * @MPW_MAX + 1@). | |
56 | * | |
fa17e5dc MW |
57 | * * %$m' = -m^{-1} \bmod b$%, a useful number for the reduction step. |
58 | * (This means that the modulus mustn't be even. This shouldn't be a | |
59 | * problem.) | |
d3409d5e | 60 | * |
61 | * * %$R = b^n > m > b^{n - 1}$%, or at least %$\log_2 R$%. | |
62 | * | |
63 | * * %$R \bmod m$% and %$R^2 \bmod m$%, which are useful when doing | |
64 | * calculations such as exponentiation. | |
65 | * | |
fa17e5dc MW |
66 | * Suppose that %$0 \le a_i \le (b^n + b^i - 1) m$% with %$a_i \equiv {}$% |
67 | * %$0 \pmod{b^i}$%. Let %$w_i = m' a_i/b^i \bmod b$%, and set %$a_{i+1} = | |
68 | * a_i + b^i w_i m$%. Then obviously %$a_{i+1} \equiv {} $% %$a_i | |
69 | * \pmod{m}$%, and less obviously %$a_{i+1}/b^i \equiv a_i/b^i + {}$% %$m m' | |
70 | * a_i/b^i \equiv 0 \pmod{b}$% so %$a_{i+1} \equiv 0 \pmod{b^{i+1}}$%. | |
71 | * Finally, we can bound %$a_{i+1} \le {}$% %$a_i + b^i (b - 1) m = {}$% | |
72 | * %$a_i + (b^{i+1} - b^i) m \le (b^n + b^{i+1} - 1) m$%. As a result, if | |
73 | * we're given some %a_0%, we can calculate %$a_n \equiv 0 \pmod{R}$%, with | |
74 | * $%a_n \equiv a_0 \pmod{n}$%, i.e., %$a_n/R \equiv a_0 R^{-1} \pmod{m}$%; | |
75 | * furthermore, if %$0 \le a_0 < m + b^n%$ then %$0 \le a_n/R < 2 m$%, so a | |
76 | * fully reduced result can be obtained with a single conditional | |
77 | * subtraction. | |
78 | * | |
79 | * The result of reduing %$a$% is then %$a R^{-1}$% \bmod m$%. This is | |
80 | * actually rather useful for reducing products, if we run an extra factor of | |
81 | * %$R$% through the calculation: the result of reducing the product of | |
82 | * %$(x R)(y R) = x y R^2$% is then %$x y R \bmod m$%, preserving the running | |
83 | * factor. Thanks to distributivity, additions and subtractions can be | |
84 | * performed on numbers in this form -- the extra factor of %$R$% just runs | |
85 | * through all the calculations until it's finally stripped out by a final | |
86 | * reduction operation. | |
d3409d5e | 87 | */ |
88 | ||
89 | /*----- Data structures ---------------------------------------------------*/ | |
90 | ||
91 | /* --- A Montgomery reduction context --- */ | |
92 | ||
93 | typedef struct mpmont { | |
94 | mp *m; /* Modulus */ | |
f5f35081 | 95 | mp *mi; /* %$-m^{-1} \bmod R$% */ |
96 | size_t n; /* %$\log_b R$% */ | |
d3409d5e | 97 | mp *r, *r2; /* %$R \bmod m$%, %$R^2 \bmod m$% */ |
98 | } mpmont; | |
99 | ||
100 | /*----- Functions provided ------------------------------------------------*/ | |
101 | ||
102 | /* --- @mpmont_create@ --- * | |
103 | * | |
104 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context | |
105 | * @mp *m@ = modulus to use | |
106 | * | |
f4535c64 | 107 | * Returns: Zero on success, nonzero on error. |
d3409d5e | 108 | * |
109 | * Use: Initializes a Montgomery reduction context ready for use. | |
b3f05084 | 110 | * The argument @m@ must be a positive odd integer. |
d3409d5e | 111 | */ |
112 | ||
f4535c64 | 113 | extern int mpmont_create(mpmont */*mm*/, mp */*m*/); |
d3409d5e | 114 | |
2af1930e | 115 | /* --- @mpmont_destroy@ --- * |
116 | * | |
117 | * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context | |
118 | * | |
119 | * Returns: --- | |
120 | * | |
121 | * Use: Disposes of a context when it's no longer of any use to | |
122 | * anyone. | |
123 | */ | |
124 | ||
125 | extern void mpmont_destroy(mpmont */*mm*/); | |
126 | ||
127 | /* --- @mpmont_reduce@ --- * | |
128 | * | |
295f4f90 | 129 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
2af1930e | 130 | * @mp *d@ = destination |
b3f05084 | 131 | * @mp *a@ = source, assumed positive |
2af1930e | 132 | * |
133 | * Returns: Result, %$a R^{-1} \bmod m$%. | |
134 | */ | |
135 | ||
295f4f90 | 136 | extern mp *mpmont_reduce(const mpmont */*mm*/, mp */*d*/, mp */*a*/); |
2af1930e | 137 | |
138 | /* --- @mpmont_mul@ --- * | |
139 | * | |
295f4f90 | 140 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
2af1930e | 141 | * @mp *d@ = destination |
b3f05084 | 142 | * @mp *a, *b@ = sources, assumed positive |
2af1930e | 143 | * |
144 | * Returns: Result, %$a b R^{-1} \bmod m$%. | |
145 | */ | |
146 | ||
295f4f90 | 147 | extern mp *mpmont_mul(const mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*b*/); |
2af1930e | 148 | |
149 | /* --- @mpmont_expr@ --- * | |
150 | * | |
295f4f90 | 151 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
b3f05084 | 152 | * @mp *d@ = fake destination |
153 | * @mp *a@ = base | |
154 | * @mp *e@ = exponent | |
2af1930e | 155 | * |
932f6ca7 | 156 | * Returns: Result, %$(a R^{-1})^e R \bmod m$%. This is useful if |
157 | * further modular arithmetic is to be performed on the result. | |
2af1930e | 158 | */ |
159 | ||
295f4f90 MW |
160 | extern mp *mpmont_expr(const mpmont */*mm*/, mp */*d*/, |
161 | mp */*a*/, mp */*e*/); | |
2af1930e | 162 | |
163 | /* --- @mpmont_exp@ --- * | |
164 | * | |
295f4f90 | 165 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
b3f05084 | 166 | * @mp *d@ = fake destination |
167 | * @mp *a@ = base | |
168 | * @mp *e@ = exponent | |
2af1930e | 169 | * |
170 | * Returns: Result, %$a^e \bmod m$%. | |
171 | */ | |
172 | ||
295f4f90 | 173 | extern mp *mpmont_exp(const mpmont */*mm*/, mp */*d*/, mp */*a*/, mp */*e*/); |
2af1930e | 174 | |
175 | /* --- @mpmont_mexpr@ --- * | |
176 | * | |
295f4f90 | 177 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
b3f05084 | 178 | * @mp *d@ = fake destination |
34e4f738 | 179 | * @const mp_expfactor *f@ = pointer to array of factors |
2af1930e | 180 | * @size_t n@ = number of factors supplied |
181 | * | |
182 | * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the | |
183 | * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result | |
184 | * is: | |
185 | * | |
932f6ca7 | 186 | * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$% |
187 | * | |
188 | * | |
189 | * except that the %$g_i$% and result are in Montgomery form. | |
2af1930e | 190 | */ |
191 | ||
295f4f90 | 192 | extern mp *mpmont_mexpr(const mpmont */*mm*/, mp */*d*/, |
34e4f738 | 193 | const mp_expfactor */*f*/, size_t /*n*/); |
2af1930e | 194 | |
195 | /* --- @mpmont_mexp@ --- * | |
196 | * | |
295f4f90 | 197 | * Arguments: @const mpmont *mm@ = pointer to Montgomery reduction context |
b3f05084 | 198 | * @mp *d@ = fake destination |
3beded37 | 199 | * @const mp_expfactor *f@ = pointer to array of factors |
2af1930e | 200 | * @size_t n@ = number of factors supplied |
201 | * | |
202 | * Returns: Product of bases raised to exponents, all mod @m@. | |
203 | * | |
204 | * Use: Convenient interface over @mpmont_mexpr@. | |
205 | */ | |
206 | ||
295f4f90 | 207 | extern mp *mpmont_mexp(const mpmont */*mm*/, mp */*d*/, |
3beded37 | 208 | const mp_expfactor */*f*/, size_t /*n*/); |
2af1930e | 209 | |
d3409d5e | 210 | /*----- That's all, folks -------------------------------------------------*/ |
211 | ||
212 | #ifdef __cplusplus | |
213 | } | |
214 | #endif | |
215 | ||
216 | #endif |