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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: ec.c,v 1.4.4.2 2004/03/20 00:13:31 mdw Exp $ |
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4 | * |
5 | * Elliptic curve definitions |
6 | * |
7 | * (c) 2001 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: ec.c,v $ |
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33 | * Revision 1.4.4.2 2004/03/20 00:13:31 mdw |
34 | * Projective coordinates for prime curves |
35 | * |
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36 | * Revision 1.4.4.1 2003/06/10 13:43:53 mdw |
37 | * Simple (non-projective) curves over prime fields now seem to work. |
38 | * |
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39 | * Revision 1.4 2003/05/15 23:25:59 mdw |
40 | * Make elliptic curve stuff build. |
41 | * |
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42 | * Revision 1.3 2002/01/13 13:48:44 mdw |
43 | * Further progress. |
44 | * |
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45 | * Revision 1.2 2001/05/07 17:29:44 mdw |
46 | * Treat projective coordinates as an internal representation. Various |
47 | * minor interface changes. |
48 | * |
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49 | * Revision 1.1 2001/04/29 18:12:33 mdw |
50 | * Prototype version. |
51 | * |
52 | */ |
53 | |
54 | /*----- Header files ------------------------------------------------------*/ |
55 | |
56 | #include "ec.h" |
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57 | #include "ec-exp.h" |
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58 | |
59 | /*----- Trivial wrappers --------------------------------------------------*/ |
60 | |
61 | /* --- @ec_create@ --- * |
62 | * |
63 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
64 | * |
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65 | * Returns: The argument @p@. |
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66 | * |
67 | * Use: Initializes a new point. The initial value is the additive |
68 | * identity (which is universal for all curves). |
69 | */ |
70 | |
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71 | ec *ec_create(ec *p) { EC_CREATE(p); return (p); } |
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72 | |
73 | /* --- @ec_destroy@ --- * |
74 | * |
75 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
76 | * |
77 | * Returns: --- |
78 | * |
79 | * Use: Destroys a point, making it invalid. |
80 | */ |
81 | |
82 | void ec_destroy(ec *p) { EC_DESTROY(p); } |
83 | |
84 | /* --- @ec_atinf@ --- * |
85 | * |
86 | * Arguments: @const ec *p@ = pointer to a point |
87 | * |
88 | * Returns: Nonzero if %$p = O$% is the point at infinity, zero |
89 | * otherwise. |
90 | */ |
91 | |
92 | int ec_atinf(const ec *p) { return (EC_ATINF(p)); } |
93 | |
94 | /* --- @ec_setinf@ --- * |
95 | * |
96 | * Arguments: @ec *p@ = pointer to a point |
97 | * |
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98 | * Returns: The argument @p@. |
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99 | * |
100 | * Use: Sets the given point to be the point %$O$% at infinity. |
101 | */ |
102 | |
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103 | ec *ec_setinf(ec *p) { EC_SETINF(p); return (p); } |
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104 | |
105 | /* --- @ec_copy@ --- * |
106 | * |
107 | * Arguments: @ec *d@ = pointer to destination point |
108 | * @const ec *p@ = pointer to source point |
109 | * |
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110 | * Returns: The destination @d@. |
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111 | * |
112 | * Use: Creates a copy of an elliptic curve point. |
113 | */ |
114 | |
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115 | ec *ec_copy(ec *d, const ec *p) { EC_COPY(d, p); return (d); } |
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116 | |
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117 | /*----- Standard curve operations -----------------------------------------*/ |
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118 | |
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119 | /* --- @ec_idin@, @ec_idout@, @ec_idfix@ --- * |
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120 | * |
121 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
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122 | * @ec *d@ = pointer to the destination |
123 | * @const ec *p@ = pointer to a source point |
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124 | * |
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125 | * Returns: The destination @d@. |
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126 | * |
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127 | * Use: An identity operation if your curve has no internal |
128 | * representation. (The field internal representation is still |
129 | * used.) |
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130 | */ |
131 | |
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132 | ec *ec_idin(ec_curve *c, ec *d, const ec *p) |
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133 | { |
134 | if (EC_ATINF(p)) |
135 | EC_SETINF(d); |
136 | else { |
137 | field *f = c->f; |
138 | d->x = F_IN(f, d->x, p->x); |
139 | d->y = F_IN(f, d->y, p->y); |
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140 | mp_drop(d->z); d->z = 0; |
141 | } |
142 | return (d); |
143 | } |
144 | |
145 | ec *ec_idout(ec_curve *c, ec *d, const ec *p) |
146 | { |
147 | if (EC_ATINF(p)) |
148 | EC_SETINF(d); |
149 | else { |
150 | field *f = c->f; |
151 | d->x = F_OUT(f, d->x, p->x); |
152 | d->y = F_OUT(f, d->y, p->y); |
153 | mp_drop(d->z); d->z = 0; |
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154 | } |
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155 | return (d); |
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156 | } |
157 | |
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158 | ec *ec_idfix(ec_curve *c, ec *d, const ec *p) |
159 | { |
160 | EC_COPY(d, p); |
161 | return (d); |
162 | } |
163 | |
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164 | /* --- @ec_projin@, @ec_projout@ --- * |
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165 | * |
166 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
41a324a7 |
167 | * @ec *d@ = pointer to the destination |
168 | * @const ec *p@ = pointer to a source point |
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169 | * |
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170 | * Returns: The destination @d@. |
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171 | * |
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172 | * Use: Conversion functions if your curve operations use a |
173 | * projective representation. |
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174 | */ |
175 | |
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176 | ec *ec_projin(ec_curve *c, ec *d, const ec *p) |
177 | { |
178 | if (EC_ATINF(p)) |
179 | EC_SETINF(d); |
180 | else { |
181 | field *f = c->f; |
182 | d->x = F_IN(f, d->x, p->x); |
183 | d->y = F_IN(f, d->y, p->y); |
184 | mp_drop(d->z); d->z = MP_COPY(f->one); |
185 | } |
186 | return (d); |
187 | } |
188 | |
189 | ec *ec_projout(ec_curve *c, ec *d, const ec *p) |
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190 | { |
191 | if (EC_ATINF(p)) |
192 | EC_SETINF(d); |
193 | else { |
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194 | mp *x, *y, *z, *zz; |
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195 | field *f = c->f; |
196 | z = F_INV(f, MP_NEW, p->z); |
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197 | zz = F_SQR(f, MP_NEW, z); |
198 | z = F_MUL(f, z, zz, z); |
199 | x = F_MUL(f, d->x, p->x, zz); |
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200 | y = F_MUL(f, d->y, p->y, z); |
201 | mp_drop(z); |
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202 | mp_drop(zz); |
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203 | mp_drop(d->z); |
204 | d->x = F_OUT(f, x, x); |
205 | d->y = F_OUT(f, y, y); |
206 | d->z = 0; |
207 | } |
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208 | return (d); |
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209 | } |
210 | |
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211 | ec *ec_projfix(ec_curve *c, ec *d, const ec *p) |
212 | { |
213 | if (EC_ATINF(p)) |
214 | EC_SETINF(d); |
215 | else if (d->z == c->f->one) |
216 | EC_COPY(d, p); |
217 | else { |
218 | mp *z, *zz; |
219 | field *f = c->f; |
220 | z = F_INV(f, MP_NEW, p->z); |
221 | zz = F_SQR(f, MP_NEW, z); |
222 | z = F_MUL(f, z, zz, z); |
223 | d->x = F_MUL(f, d->x, p->x, zz); |
224 | d->y = F_MUL(f, d->y, p->y, z); |
225 | mp_drop(z); |
226 | mp_drop(zz); |
227 | mp_drop(d->z); |
228 | d->z = MP_COPY(f->one); |
229 | } |
230 | return (d); |
231 | } |
232 | |
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233 | /* --- @ec_stdsub@ --- * |
234 | * |
235 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
236 | * @ec *d@ = pointer to the destination |
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237 | * @const ec *p, *q@ = the operand points |
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238 | * |
239 | * Returns: The destination @d@. |
240 | * |
241 | * Use: Standard point subtraction operation, in terms of negation |
242 | * and addition. This isn't as efficient as a ready-made |
243 | * subtraction operator. |
244 | */ |
245 | |
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246 | ec *ec_stdsub(ec_curve *c, ec *d, const ec *p, const ec *q) |
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247 | { |
248 | ec t = EC_INIT; |
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249 | EC_NEG(c, &t, q); |
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250 | EC_FIX(c, &t, &t); |
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251 | EC_ADD(c, d, p, &t); |
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252 | EC_DESTROY(&t); |
253 | return (d); |
254 | } |
255 | |
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256 | /*----- Creating curves ---------------------------------------------------*/ |
257 | |
258 | /* --- @ec_destroycurve@ --- * |
259 | * |
260 | * Arguments: @ec_curve *c@ = pointer to an ellptic curve |
261 | * |
262 | * Returns: --- |
263 | * |
264 | * Use: Destroys a description of an elliptic curve. |
265 | */ |
266 | |
267 | void ec_destroycurve(ec_curve *c) { c->ops->destroy(c); } |
268 | |
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269 | /*----- Real arithmetic ---------------------------------------------------*/ |
270 | |
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271 | /* --- @ec_find@ --- * |
272 | * |
273 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
274 | * @ec *d@ = pointer to the destination point |
275 | * @mp *x@ = a possible x-coordinate |
276 | * |
277 | * Returns: Zero if OK, nonzero if there isn't a point there. |
278 | * |
279 | * Use: Finds a point on an elliptic curve with a given x-coordinate. |
280 | */ |
281 | |
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282 | ec *ec_find(ec_curve *c, ec *d, mp *x) |
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283 | { |
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284 | x = F_IN(c->f, MP_NEW, x); |
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285 | if ((d = EC_FIND(c, d, x)) != 0) |
286 | EC_OUT(c, d, d); |
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287 | MP_DROP(x); |
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288 | return (d); |
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289 | } |
290 | |
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291 | /* --- @ec_neg@ --- * |
292 | * |
293 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
294 | * @ec *d@ = pointer to the destination point |
295 | * @const ec *p@ = pointer to the operand point |
296 | * |
297 | * Returns: The destination point. |
298 | * |
299 | * Use: Computes the negation of the given point. |
300 | */ |
301 | |
302 | ec *ec_neg(ec_curve *c, ec *d, const ec *p) |
303 | { |
304 | EC_IN(c, d, p); |
305 | EC_NEG(c, d, d); |
306 | return (EC_OUT(c, d, d)); |
307 | } |
308 | |
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309 | /* --- @ec_add@ --- * |
310 | * |
311 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
312 | * @ec *d@ = pointer to the destination point |
313 | * @const ec *p, *q@ = pointers to the operand points |
314 | * |
315 | * Returns: --- |
316 | * |
317 | * Use: Adds two points on an elliptic curve. |
318 | */ |
319 | |
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320 | ec *ec_add(ec_curve *c, ec *d, const ec *p, const ec *q) |
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321 | { |
322 | ec pp = EC_INIT, qq = EC_INIT; |
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323 | EC_IN(c, &pp, p); |
324 | EC_IN(c, &qq, q); |
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325 | EC_ADD(c, d, &pp, &qq); |
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326 | EC_OUT(c, d, d); |
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327 | EC_DESTROY(&pp); |
328 | EC_DESTROY(&qq); |
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329 | return (d); |
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330 | } |
331 | |
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332 | /* --- @ec_sub@ --- * |
333 | * |
334 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
335 | * @ec *d@ = pointer to the destination point |
336 | * @const ec *p, *q@ = pointers to the operand points |
337 | * |
338 | * Returns: The destination @d@. |
339 | * |
340 | * Use: Subtracts one point from another on an elliptic curve. |
341 | */ |
342 | |
343 | ec *ec_sub(ec_curve *c, ec *d, const ec *p, const ec *q) |
344 | { |
345 | ec pp, qq; |
346 | EC_IN(c, &pp, p); |
347 | EC_IN(c, &qq, q); |
348 | EC_SUB(c, d, &qq, &qq); |
349 | EC_OUT(c, d, d); |
350 | EC_DESTROY(&pp); |
351 | EC_DESTROY(&qq); |
352 | return (d); |
353 | } |
354 | |
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355 | /* --- @ec_dbl@ --- * |
356 | * |
357 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
358 | * @ec *d@ = pointer to the destination point |
359 | * @const ec *p@ = pointer to the operand point |
360 | * |
361 | * Returns: --- |
362 | * |
363 | * Use: Doubles a point on an elliptic curve. |
364 | */ |
365 | |
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366 | ec *ec_dbl(ec_curve *c, ec *d, const ec *p) |
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367 | { |
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368 | EC_IN(c, d, p); |
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369 | EC_DBL(c, d, d); |
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370 | return (EC_OUT(c, d, d)); |
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371 | } |
372 | |
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373 | /* --- @ec_check@ --- * |
374 | * |
375 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
376 | * @const ec *p@ = pointer to the point |
377 | * |
378 | * Returns: Zero if OK, nonzero if this is an invalid point. |
379 | * |
380 | * Use: Checks that a point is actually on an elliptic curve. |
381 | */ |
382 | |
383 | int ec_check(ec_curve *c, const ec *p) |
384 | { |
385 | ec t = EC_INIT; |
386 | int rc; |
387 | |
388 | if (EC_ATINF(p)) |
389 | return (0); |
390 | EC_IN(c, &t, p); |
391 | rc = EC_CHECK(c, &t); |
392 | EC_DESTROY(&t); |
393 | return (rc); |
394 | } |
395 | |
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396 | /* --- @ec_imul@, @ec_mul@ --- * |
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397 | * |
398 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
399 | * @ec *d@ = pointer to the destination point |
400 | * @const ec *p@ = pointer to the generator point |
401 | * @mp *n@ = integer multiplier |
402 | * |
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403 | * Returns: The destination @d@. |
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404 | * |
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405 | * Use: Multiplies a point by a scalar, returning %$n p$%. The |
406 | * @imul@ variant uses internal representations for argument |
407 | * and result. |
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408 | */ |
409 | |
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410 | ec *ec_imul(ec_curve *c, ec *d, const ec *p, mp *n) |
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411 | { |
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412 | ec t = EC_INIT; |
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413 | |
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414 | EC_COPY(&t, p); |
415 | if (t.x && (n->f & MP_BURN)) |
416 | t.x->f |= MP_BURN; |
417 | MP_SHRINK(n); |
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418 | EC_SETINF(d); |
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419 | if (MP_LEN(n) == 0) |
420 | ; |
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421 | else { |
422 | if (n->f & MP_NEG) |
423 | EC_NEG(c, &t, &t); |
424 | if (MP_LEN(n) < EXP_THRESH) |
425 | EXP_SIMPLE(*d, t, n); |
426 | else |
427 | EXP_WINDOW(*d, t, n); |
428 | } |
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429 | EC_DESTROY(&t); |
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430 | return (d); |
431 | } |
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432 | |
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433 | ec *ec_mul(ec_curve *c, ec *d, const ec *p, mp *n) |
434 | { |
435 | EC_IN(c, d, p); |
436 | ec_imul(c, d, d, n); |
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437 | return (EC_OUT(c, d, d)); |
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438 | } |
439 | |
440 | /*----- That's all, folks -------------------------------------------------*/ |