1/*
2 * Copyright (c) 2007, 2017, Oracle and/or its affiliates. All rights reserved.
3 * Use is subject to license terms.
4 *
5 * This library is free software; you can redistribute it and/or
6 * modify it under the terms of the GNU Lesser General Public
7 * License as published by the Free Software Foundation; either
8 * version 2.1 of the License, or (at your option) any later version.
9 *
10 * This library is distributed in the hope that it will be useful,
11 * but WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 * Lesser General Public License for more details.
14 *
15 * You should have received a copy of the GNU Lesser General Public License
16 * along with this library; if not, write to the Free Software Foundation,
17 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
18 *
19 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
20 * or visit www.oracle.com if you need additional information or have any
21 * questions.
22 */
23
24/* *********************************************************************
25 *
26 * The Original Code is the elliptic curve math library for prime field curves.
27 *
28 * The Initial Developer of the Original Code is
29 * Sun Microsystems, Inc.
30 * Portions created by the Initial Developer are Copyright (C) 2003
31 * the Initial Developer. All Rights Reserved.
32 *
33 * Contributor(s):
34 * Stephen Fung <fungstep@hotmail.com>, Sun Microsystems Laboratories
35 *
36 * Last Modified Date from the Original Code: May 2017
37 *********************************************************************** */
38
39#include "ecp.h"
40#include "ecl-priv.h"
41#include "mplogic.h"
42#ifndef _KERNEL
43#include <stdlib.h>
44#endif
45
46#define MAX_SCRATCH 6
47
48/* Computes R = 2P. Elliptic curve points P and R can be identical. Uses
49 * Modified Jacobian coordinates.
50 *
51 * Assumes input is already field-encoded using field_enc, and returns
52 * output that is still field-encoded.
53 *
54 */
55mp_err
56ec_GFp_pt_dbl_jm(const mp_int *px, const mp_int *py, const mp_int *pz,
57 const mp_int *paz4, mp_int *rx, mp_int *ry, mp_int *rz,
58 mp_int *raz4, mp_int scratch[], const ECGroup *group)
59{
60 mp_err res = MP_OKAY;
61 mp_int *t0, *t1, *M, *S;
62
63 t0 = &scratch[0];
64 t1 = &scratch[1];
65 M = &scratch[2];
66 S = &scratch[3];
67
68#if MAX_SCRATCH < 4
69#error "Scratch array defined too small "
70#endif
71
72 /* Check for point at infinity */
73 if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
74 /* Set r = pt at infinity by setting rz = 0 */
75
76 MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
77 goto CLEANUP;
78 }
79
80 /* M = 3 (px^2) + a*(pz^4) */
81 MP_CHECKOK(group->meth->field_sqr(px, t0, group->meth));
82 MP_CHECKOK(group->meth->field_add(t0, t0, M, group->meth));
83 MP_CHECKOK(group->meth->field_add(t0, M, t0, group->meth));
84 MP_CHECKOK(group->meth->field_add(t0, paz4, M, group->meth));
85
86 /* rz = 2 * py * pz */
87 MP_CHECKOK(group->meth->field_mul(py, pz, S, group->meth));
88 MP_CHECKOK(group->meth->field_add(S, S, rz, group->meth));
89
90 /* t0 = 2y^2 , t1 = 8y^4 */
91 MP_CHECKOK(group->meth->field_sqr(py, t0, group->meth));
92 MP_CHECKOK(group->meth->field_add(t0, t0, t0, group->meth));
93 MP_CHECKOK(group->meth->field_sqr(t0, t1, group->meth));
94 MP_CHECKOK(group->meth->field_add(t1, t1, t1, group->meth));
95
96 /* S = 4 * px * py^2 = 2 * px * t0 */
97 MP_CHECKOK(group->meth->field_mul(px, t0, S, group->meth));
98 MP_CHECKOK(group->meth->field_add(S, S, S, group->meth));
99
100
101 /* rx = M^2 - 2S */
102 MP_CHECKOK(group->meth->field_sqr(M, rx, group->meth));
103 MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
104 MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
105
106 /* ry = M * (S - rx) - t1 */
107 MP_CHECKOK(group->meth->field_sub(S, rx, S, group->meth));
108 MP_CHECKOK(group->meth->field_mul(S, M, ry, group->meth));
109 MP_CHECKOK(group->meth->field_sub(ry, t1, ry, group->meth));
110
111 /* ra*z^4 = 2*t1*(apz4) */
112 MP_CHECKOK(group->meth->field_mul(paz4, t1, raz4, group->meth));
113 MP_CHECKOK(group->meth->field_add(raz4, raz4, raz4, group->meth));
114
115
116 CLEANUP:
117 return res;
118}
119
120/* Computes R = P + Q where R is (rx, ry, rz), P is (px, py, pz) and Q is
121 * (qx, qy, 1). Elliptic curve points P, Q, and R can all be identical.
122 * Uses mixed Modified_Jacobian-affine coordinates. Assumes input is
123 * already field-encoded using field_enc, and returns output that is still
124 * field-encoded. */
125mp_err
126ec_GFp_pt_add_jm_aff(const mp_int *px, const mp_int *py, const mp_int *pz,
127 const mp_int *paz4, const mp_int *qx,
128 const mp_int *qy, mp_int *rx, mp_int *ry, mp_int *rz,
129 mp_int *raz4, mp_int scratch[], const ECGroup *group)
130{
131 mp_err res = MP_OKAY;
132 mp_int *A, *B, *C, *D, *C2, *C3;
133
134 A = &scratch[0];
135 B = &scratch[1];
136 C = &scratch[2];
137 D = &scratch[3];
138 C2 = &scratch[4];
139 C3 = &scratch[5];
140
141#if MAX_SCRATCH < 6
142#error "Scratch array defined too small "
143#endif
144
145 /* If either P or Q is the point at infinity, then return the other
146 * point */
147 if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
148 MP_CHECKOK(ec_GFp_pt_aff2jac(qx, qy, rx, ry, rz, group));
149 MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
150 MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
151 MP_CHECKOK(group->meth->
152 field_mul(raz4, &group->curvea, raz4, group->meth));
153 goto CLEANUP;
154 }
155 if (ec_GFp_pt_is_inf_aff(qx, qy) == MP_YES) {
156 MP_CHECKOK(mp_copy(px, rx));
157 MP_CHECKOK(mp_copy(py, ry));
158 MP_CHECKOK(mp_copy(pz, rz));
159 MP_CHECKOK(mp_copy(paz4, raz4));
160 goto CLEANUP;
161 }
162
163 /* A = qx * pz^2, B = qy * pz^3 */
164 MP_CHECKOK(group->meth->field_sqr(pz, A, group->meth));
165 MP_CHECKOK(group->meth->field_mul(A, pz, B, group->meth));
166 MP_CHECKOK(group->meth->field_mul(A, qx, A, group->meth));
167 MP_CHECKOK(group->meth->field_mul(B, qy, B, group->meth));
168
169 /*
170 * Additional checks for point equality and point at infinity
171 */
172 if (mp_cmp(px, A) == 0 && mp_cmp(py, B) == 0) {
173 /* POINT_DOUBLE(P) */
174 MP_CHECKOK(ec_GFp_pt_dbl_jm(px, py, pz, paz4, rx, ry, rz, raz4,
175 scratch, group));
176 goto CLEANUP;
177 }
178
179 /* C = A - px, D = B - py */
180 MP_CHECKOK(group->meth->field_sub(A, px, C, group->meth));
181 MP_CHECKOK(group->meth->field_sub(B, py, D, group->meth));
182
183 /* C2 = C^2, C3 = C^3 */
184 MP_CHECKOK(group->meth->field_sqr(C, C2, group->meth));
185 MP_CHECKOK(group->meth->field_mul(C, C2, C3, group->meth));
186
187 /* rz = pz * C */
188 MP_CHECKOK(group->meth->field_mul(pz, C, rz, group->meth));
189
190 /* C = px * C^2 */
191 MP_CHECKOK(group->meth->field_mul(px, C2, C, group->meth));
192 /* A = D^2 */
193 MP_CHECKOK(group->meth->field_sqr(D, A, group->meth));
194
195 /* rx = D^2 - (C^3 + 2 * (px * C^2)) */
196 MP_CHECKOK(group->meth->field_add(C, C, rx, group->meth));
197 MP_CHECKOK(group->meth->field_add(C3, rx, rx, group->meth));
198 MP_CHECKOK(group->meth->field_sub(A, rx, rx, group->meth));
199
200 /* C3 = py * C^3 */
201 MP_CHECKOK(group->meth->field_mul(py, C3, C3, group->meth));
202
203 /* ry = D * (px * C^2 - rx) - py * C^3 */
204 MP_CHECKOK(group->meth->field_sub(C, rx, ry, group->meth));
205 MP_CHECKOK(group->meth->field_mul(D, ry, ry, group->meth));
206 MP_CHECKOK(group->meth->field_sub(ry, C3, ry, group->meth));
207
208 /* raz4 = a * rz^4 */
209 MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
210 MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
211 MP_CHECKOK(group->meth->
212 field_mul(raz4, &group->curvea, raz4, group->meth));
213CLEANUP:
214 return res;
215}
216
217/* Computes R = nP where R is (rx, ry) and P is the base point. Elliptic
218 * curve points P and R can be identical. Uses mixed Modified-Jacobian
219 * co-ordinates for doubling and Chudnovsky Jacobian coordinates for
220 * additions. Assumes input is already field-encoded using field_enc, and
221 * returns output that is still field-encoded. Uses 5-bit window NAF
222 * method (algorithm 11) for scalar-point multiplication from Brown,
223 * Hankerson, Lopez, Menezes. Software Implementation of the NIST Elliptic
224 * Curves Over Prime Fields. */
225mp_err
226ec_GFp_pt_mul_jm_wNAF(const mp_int *n, const mp_int *px, const mp_int *py,
227 mp_int *rx, mp_int *ry, const ECGroup *group,
228 int timing)
229{
230 mp_err res = MP_OKAY;
231 mp_int precomp[16][2], rz, tpx, tpy, tpz;
232 mp_int raz4, tpaz4;
233 mp_int scratch[MAX_SCRATCH];
234 signed char *naf = NULL;
235 int i, orderBitSize;
236 int numDoubles, numAdds, extraDoubles, extraAdds;
237
238 MP_DIGITS(&rz) = 0;
239 MP_DIGITS(&raz4) = 0;
240 MP_DIGITS(&tpx) = 0;
241 MP_DIGITS(&tpy) = 0;
242 MP_DIGITS(&tpz) = 0;
243 MP_DIGITS(&tpaz4) = 0;
244 for (i = 0; i < 16; i++) {
245 MP_DIGITS(&precomp[i][0]) = 0;
246 MP_DIGITS(&precomp[i][1]) = 0;
247 }
248 for (i = 0; i < MAX_SCRATCH; i++) {
249 MP_DIGITS(&scratch[i]) = 0;
250 }
251
252 ARGCHK(group != NULL, MP_BADARG);
253 ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);
254
255 /* initialize precomputation table */
256 MP_CHECKOK(mp_init(&tpx, FLAG(n)));
257 MP_CHECKOK(mp_init(&tpy, FLAG(n)));
258 MP_CHECKOK(mp_init(&tpz, FLAG(n)));
259 MP_CHECKOK(mp_init(&tpaz4, FLAG(n)));
260 MP_CHECKOK(mp_init(&rz, FLAG(n)));
261 MP_CHECKOK(mp_init(&raz4, FLAG(n)));
262
263 for (i = 0; i < 16; i++) {
264 MP_CHECKOK(mp_init(&precomp[i][0], FLAG(n)));
265 MP_CHECKOK(mp_init(&precomp[i][1], FLAG(n)));
266 }
267 for (i = 0; i < MAX_SCRATCH; i++) {
268 MP_CHECKOK(mp_init(&scratch[i], FLAG(n)));
269 }
270
271 /* Set out[8] = P */
272 MP_CHECKOK(mp_copy(px, &precomp[8][0]));
273 MP_CHECKOK(mp_copy(py, &precomp[8][1]));
274
275 /* Set (tpx, tpy) = 2P */
276 MP_CHECKOK(group->
277 point_dbl(&precomp[8][0], &precomp[8][1], &tpx, &tpy,
278 group));
279
280 /* Set 3P, 5P, ..., 15P */
281 for (i = 8; i < 15; i++) {
282 MP_CHECKOK(group->
283 point_add(&precomp[i][0], &precomp[i][1], &tpx, &tpy,
284 &precomp[i + 1][0], &precomp[i + 1][1],
285 group));
286 }
287
288 /* Set -15P, -13P, ..., -P */
289 for (i = 0; i < 8; i++) {
290 MP_CHECKOK(mp_copy(&precomp[15 - i][0], &precomp[i][0]));
291 MP_CHECKOK(group->meth->
292 field_neg(&precomp[15 - i][1], &precomp[i][1],
293 group->meth));
294 }
295
296 /* R = inf */
297 MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
298
299 orderBitSize = mpl_significant_bits(&group->order);
300
301 /* Allocate memory for NAF */
302#ifdef _KERNEL
303 naf = (signed char *) kmem_alloc((orderBitSize + 1), FLAG(n));
304#else
305 naf = (signed char *) malloc(sizeof(signed char) * (orderBitSize + 1));
306 if (naf == NULL) {
307 res = MP_MEM;
308 goto CLEANUP;
309 }
310#endif
311
312 /* Compute 5NAF */
313 ec_compute_wNAF(naf, orderBitSize, n, 5);
314
315 numAdds = 0;
316 numDoubles = orderBitSize;
317 /* wNAF method */
318 for (i = orderBitSize; i >= 0; i--) {
319
320 if (ec_GFp_pt_is_inf_jac(rx, ry, &rz) == MP_YES) {
321 numDoubles--;
322 }
323
324 /* R = 2R */
325 ec_GFp_pt_dbl_jm(rx, ry, &rz, &raz4, rx, ry, &rz,
326 &raz4, scratch, group);
327
328 if (naf[i] != 0) {
329 ec_GFp_pt_add_jm_aff(rx, ry, &rz, &raz4,
330 &precomp[(naf[i] + 15) / 2][0],
331 &precomp[(naf[i] + 15) / 2][1], rx, ry,
332 &rz, &raz4, scratch, group);
333 numAdds++;
334 }
335 }
336
337 /* extra operations to make timing less dependent on secrets */
338 if (timing) {
339 /* low-order bit of timing argument contains no entropy */
340 timing >>= 1;
341
342 MP_CHECKOK(ec_GFp_pt_set_inf_jac(&tpx, &tpy, &tpz));
343 mp_zero(&tpaz4);
344
345 /* Set the temp value to a non-infinite point */
346 ec_GFp_pt_add_jm_aff(&tpx, &tpy, &tpz, &tpaz4,
347 &precomp[8][0],
348 &precomp[8][1], &tpx, &tpy,
349 &tpz, &tpaz4, scratch, group);
350
351 /* two bits of extra adds */
352 extraAdds = timing & 0x3;
353 timing >>= 2;
354 /* Window size is 5, so the maximum number of additions is ceil(orderBitSize/5) */
355 /* This is the same as (orderBitSize + 4) / 5 */
356 for(i = numAdds; i <= (orderBitSize + 4) / 5 + extraAdds; i++) {
357 ec_GFp_pt_add_jm_aff(&tpx, &tpy, &tpz, &tpaz4,
358 &precomp[9 + (i % 3)][0],
359 &precomp[9 + (i % 3)][1], &tpx, &tpy,
360 &tpz, &tpaz4, scratch, group);
361 }
362
363 /* two bits of extra doubles */
364 extraDoubles = timing & 0x3;
365 timing >>= 2;
366 for(i = numDoubles; i <= orderBitSize + extraDoubles; i++) {
367 ec_GFp_pt_dbl_jm(&tpx, &tpy, &tpz, &tpaz4, &tpx, &tpy, &tpz,
368 &tpaz4, scratch, group);
369 }
370
371 }
372
373 /* convert result S to affine coordinates */
374 MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));
375
376 CLEANUP:
377 for (i = 0; i < MAX_SCRATCH; i++) {
378 mp_clear(&scratch[i]);
379 }
380 for (i = 0; i < 16; i++) {
381 mp_clear(&precomp[i][0]);
382 mp_clear(&precomp[i][1]);
383 }
384 mp_clear(&tpx);
385 mp_clear(&tpy);
386 mp_clear(&tpz);
387 mp_clear(&tpaz4);
388 mp_clear(&rz);
389 mp_clear(&raz4);
390#ifdef _KERNEL
391 kmem_free(naf, (orderBitSize + 1));
392#else
393 free(naf);
394#endif
395 return res;
396}
397