1/* Copyright (c) 2015, Google Inc.
2 *
3 * Permission to use, copy, modify, and/or distribute this software for any
4 * purpose with or without fee is hereby granted, provided that the above
5 * copyright notice and this permission notice appear in all copies.
6 *
7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
14
15// A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
16//
17// Inspired by Daniel J. Bernstein's public domain nistp224 implementation
18// and Adam Langley's public domain 64-bit C implementation of curve25519.
19
20#include <openssl/base.h>
21
22#include <openssl/bn.h>
23#include <openssl/ec.h>
24#include <openssl/err.h>
25#include <openssl/mem.h>
26
27#include <string.h>
28
29#include "internal.h"
30#include "../delocate.h"
31#include "../../internal.h"
32
33
34#if defined(BORINGSSL_HAS_UINT128) && !defined(OPENSSL_SMALL)
35
36// Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
37// using 64-bit coefficients called 'limbs', and sometimes (for multiplication
38// results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 +
39// 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-p224_limb
40// representation is an 'p224_felem'; a 7-p224_widelimb representation is a
41// 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we
42// don't always reduce the representations: we ensure that inputs to each
43// p224_felem multiplication satisfy a_i < 2^60, so outputs satisfy b_i <
44// 4*2^60*2^60, and fit into a 128-bit word without overflow. The coefficients
45// are then again partially reduced to obtain an p224_felem satisfying a_i <
46// 2^57. We only reduce to the unique minimal representation at the end of the
47// computation.
48
49typedef uint64_t p224_limb;
50typedef uint128_t p224_widelimb;
51
52typedef p224_limb p224_felem[4];
53typedef p224_widelimb p224_widefelem[7];
54
55// Field element represented as a byte arrary. 28*8 = 224 bits is also the
56// group order size for the elliptic curve, and we also use this type for
57// scalars for point multiplication.
58typedef uint8_t p224_felem_bytearray[28];
59
60// Precomputed multiples of the standard generator
61// Points are given in coordinates (X, Y, Z) where Z normally is 1
62// (0 for the point at infinity).
63// For each field element, slice a_0 is word 0, etc.
64//
65// The table has 2 * 16 elements, starting with the following:
66// index | bits | point
67// ------+---------+------------------------------
68// 0 | 0 0 0 0 | 0G
69// 1 | 0 0 0 1 | 1G
70// 2 | 0 0 1 0 | 2^56G
71// 3 | 0 0 1 1 | (2^56 + 1)G
72// 4 | 0 1 0 0 | 2^112G
73// 5 | 0 1 0 1 | (2^112 + 1)G
74// 6 | 0 1 1 0 | (2^112 + 2^56)G
75// 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
76// 8 | 1 0 0 0 | 2^168G
77// 9 | 1 0 0 1 | (2^168 + 1)G
78// 10 | 1 0 1 0 | (2^168 + 2^56)G
79// 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
80// 12 | 1 1 0 0 | (2^168 + 2^112)G
81// 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
82// 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
83// 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
84// followed by a copy of this with each element multiplied by 2^28.
85//
86// The reason for this is so that we can clock bits into four different
87// locations when doing simple scalar multiplies against the base point,
88// and then another four locations using the second 16 elements.
89static const p224_felem g_p224_pre_comp[2][16][3] = {
90 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
91 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
92 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
93 {1, 0, 0, 0}},
94 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
95 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
96 {1, 0, 0, 0}},
97 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
98 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
99 {1, 0, 0, 0}},
100 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
101 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
102 {1, 0, 0, 0}},
103 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
104 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
105 {1, 0, 0, 0}},
106 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
107 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
108 {1, 0, 0, 0}},
109 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
110 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
111 {1, 0, 0, 0}},
112 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
113 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
114 {1, 0, 0, 0}},
115 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
116 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
117 {1, 0, 0, 0}},
118 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
119 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
120 {1, 0, 0, 0}},
121 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
122 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
123 {1, 0, 0, 0}},
124 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
125 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
126 {1, 0, 0, 0}},
127 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
128 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
129 {1, 0, 0, 0}},
130 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
131 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
132 {1, 0, 0, 0}},
133 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
134 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
135 {1, 0, 0, 0}}},
136 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
137 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
138 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
139 {1, 0, 0, 0}},
140 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
141 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
142 {1, 0, 0, 0}},
143 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
144 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
145 {1, 0, 0, 0}},
146 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
147 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
148 {1, 0, 0, 0}},
149 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
150 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
151 {1, 0, 0, 0}},
152 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
153 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
154 {1, 0, 0, 0}},
155 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
156 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
157 {1, 0, 0, 0}},
158 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
159 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
160 {1, 0, 0, 0}},
161 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
162 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
163 {1, 0, 0, 0}},
164 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
165 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
166 {1, 0, 0, 0}},
167 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
168 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
169 {1, 0, 0, 0}},
170 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
171 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
172 {1, 0, 0, 0}},
173 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
174 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
175 {1, 0, 0, 0}},
176 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
177 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
178 {1, 0, 0, 0}},
179 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
180 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
181 {1, 0, 0, 0}}}};
182
183static uint64_t p224_load_u64(const uint8_t in[8]) {
184 uint64_t ret;
185 OPENSSL_memcpy(&ret, in, sizeof(ret));
186 return ret;
187}
188
189// Helper functions to convert field elements to/from internal representation
190static void p224_bin28_to_felem(p224_felem out, const uint8_t in[28]) {
191 out[0] = p224_load_u64(in) & 0x00ffffffffffffff;
192 out[1] = p224_load_u64(in + 7) & 0x00ffffffffffffff;
193 out[2] = p224_load_u64(in + 14) & 0x00ffffffffffffff;
194 out[3] = p224_load_u64(in + 20) >> 8;
195}
196
197static void p224_felem_to_bin28(uint8_t out[28], const p224_felem in) {
198 for (size_t i = 0; i < 7; ++i) {
199 out[i] = in[0] >> (8 * i);
200 out[i + 7] = in[1] >> (8 * i);
201 out[i + 14] = in[2] >> (8 * i);
202 out[i + 21] = in[3] >> (8 * i);
203 }
204}
205
206static void p224_generic_to_felem(p224_felem out, const EC_FELEM *in) {
207 p224_bin28_to_felem(out, in->bytes);
208}
209
210// Requires 0 <= in < 2*p (always call p224_felem_reduce first)
211static void p224_felem_to_generic(EC_FELEM *out, const p224_felem in) {
212 // Reduce to unique minimal representation.
213 static const int64_t two56 = ((p224_limb)1) << 56;
214 // 0 <= in < 2*p, p = 2^224 - 2^96 + 1
215 // if in > p , reduce in = in - 2^224 + 2^96 - 1
216 int64_t tmp[4], a;
217 tmp[0] = in[0];
218 tmp[1] = in[1];
219 tmp[2] = in[2];
220 tmp[3] = in[3];
221 // Case 1: a = 1 iff in >= 2^224
222 a = (in[3] >> 56);
223 tmp[0] -= a;
224 tmp[1] += a << 40;
225 tmp[3] &= 0x00ffffffffffffff;
226 // Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and
227 // the lower part is non-zero
228 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
229 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
230 a &= 0x00ffffffffffffff;
231 // turn a into an all-one mask (if a = 0) or an all-zero mask
232 a = (a - 1) >> 63;
233 // subtract 2^224 - 2^96 + 1 if a is all-one
234 tmp[3] &= a ^ 0xffffffffffffffff;
235 tmp[2] &= a ^ 0xffffffffffffffff;
236 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
237 tmp[0] -= 1 & a;
238
239 // eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
240 // be non-zero, so we only need one step
241 a = tmp[0] >> 63;
242 tmp[0] += two56 & a;
243 tmp[1] -= 1 & a;
244
245 // carry 1 -> 2 -> 3
246 tmp[2] += tmp[1] >> 56;
247 tmp[1] &= 0x00ffffffffffffff;
248
249 tmp[3] += tmp[2] >> 56;
250 tmp[2] &= 0x00ffffffffffffff;
251
252 // Now 0 <= tmp < p
253 p224_felem tmp2;
254 tmp2[0] = tmp[0];
255 tmp2[1] = tmp[1];
256 tmp2[2] = tmp[2];
257 tmp2[3] = tmp[3];
258
259 p224_felem_to_bin28(out->bytes, tmp2);
260 // 224 is not a multiple of 64, so zero the remaining bytes.
261 OPENSSL_memset(out->bytes + 28, 0, 32 - 28);
262}
263
264
265// Field operations, using the internal representation of field elements.
266// NB! These operations are specific to our point multiplication and cannot be
267// expected to be correct in general - e.g., multiplication with a large scalar
268// will cause an overflow.
269
270static void p224_felem_assign(p224_felem out, const p224_felem in) {
271 out[0] = in[0];
272 out[1] = in[1];
273 out[2] = in[2];
274 out[3] = in[3];
275}
276
277// Sum two field elements: out += in
278static void p224_felem_sum(p224_felem out, const p224_felem in) {
279 out[0] += in[0];
280 out[1] += in[1];
281 out[2] += in[2];
282 out[3] += in[3];
283}
284
285// Subtract field elements: out -= in
286// Assumes in[i] < 2^57
287static void p224_felem_diff(p224_felem out, const p224_felem in) {
288 static const p224_limb two58p2 =
289 (((p224_limb)1) << 58) + (((p224_limb)1) << 2);
290 static const p224_limb two58m2 =
291 (((p224_limb)1) << 58) - (((p224_limb)1) << 2);
292 static const p224_limb two58m42m2 =
293 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2);
294
295 // Add 0 mod 2^224-2^96+1 to ensure out > in
296 out[0] += two58p2;
297 out[1] += two58m42m2;
298 out[2] += two58m2;
299 out[3] += two58m2;
300
301 out[0] -= in[0];
302 out[1] -= in[1];
303 out[2] -= in[2];
304 out[3] -= in[3];
305}
306
307// Subtract in unreduced 128-bit mode: out -= in
308// Assumes in[i] < 2^119
309static void p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) {
310 static const p224_widelimb two120 = ((p224_widelimb)1) << 120;
311 static const p224_widelimb two120m64 =
312 (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64);
313 static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) -
314 (((p224_widelimb)1) << 104) -
315 (((p224_widelimb)1) << 64);
316
317 // Add 0 mod 2^224-2^96+1 to ensure out > in
318 out[0] += two120;
319 out[1] += two120m64;
320 out[2] += two120m64;
321 out[3] += two120;
322 out[4] += two120m104m64;
323 out[5] += two120m64;
324 out[6] += two120m64;
325
326 out[0] -= in[0];
327 out[1] -= in[1];
328 out[2] -= in[2];
329 out[3] -= in[3];
330 out[4] -= in[4];
331 out[5] -= in[5];
332 out[6] -= in[6];
333}
334
335// Subtract in mixed mode: out128 -= in64
336// in[i] < 2^63
337static void p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) {
338 static const p224_widelimb two64p8 =
339 (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8);
340 static const p224_widelimb two64m8 =
341 (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8);
342 static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) -
343 (((p224_widelimb)1) << 48) -
344 (((p224_widelimb)1) << 8);
345
346 // Add 0 mod 2^224-2^96+1 to ensure out > in
347 out[0] += two64p8;
348 out[1] += two64m48m8;
349 out[2] += two64m8;
350 out[3] += two64m8;
351
352 out[0] -= in[0];
353 out[1] -= in[1];
354 out[2] -= in[2];
355 out[3] -= in[3];
356}
357
358// Multiply a field element by a scalar: out = out * scalar
359// The scalars we actually use are small, so results fit without overflow
360static void p224_felem_scalar(p224_felem out, const p224_limb scalar) {
361 out[0] *= scalar;
362 out[1] *= scalar;
363 out[2] *= scalar;
364 out[3] *= scalar;
365}
366
367// Multiply an unreduced field element by a scalar: out = out * scalar
368// The scalars we actually use are small, so results fit without overflow
369static void p224_widefelem_scalar(p224_widefelem out,
370 const p224_widelimb scalar) {
371 out[0] *= scalar;
372 out[1] *= scalar;
373 out[2] *= scalar;
374 out[3] *= scalar;
375 out[4] *= scalar;
376 out[5] *= scalar;
377 out[6] *= scalar;
378}
379
380// Square a field element: out = in^2
381static void p224_felem_square(p224_widefelem out, const p224_felem in) {
382 p224_limb tmp0, tmp1, tmp2;
383 tmp0 = 2 * in[0];
384 tmp1 = 2 * in[1];
385 tmp2 = 2 * in[2];
386 out[0] = ((p224_widelimb)in[0]) * in[0];
387 out[1] = ((p224_widelimb)in[0]) * tmp1;
388 out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1];
389 out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2;
390 out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2];
391 out[5] = ((p224_widelimb)in[3]) * tmp2;
392 out[6] = ((p224_widelimb)in[3]) * in[3];
393}
394
395// Multiply two field elements: out = in1 * in2
396static void p224_felem_mul(p224_widefelem out, const p224_felem in1,
397 const p224_felem in2) {
398 out[0] = ((p224_widelimb)in1[0]) * in2[0];
399 out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0];
400 out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] +
401 ((p224_widelimb)in1[2]) * in2[0];
402 out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] +
403 ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0];
404 out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] +
405 ((p224_widelimb)in1[3]) * in2[1];
406 out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2];
407 out[6] = ((p224_widelimb)in1[3]) * in2[3];
408}
409
410// Reduce seven 128-bit coefficients to four 64-bit coefficients.
411// Requires in[i] < 2^126,
412// ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
413static void p224_felem_reduce(p224_felem out, const p224_widefelem in) {
414 static const p224_widelimb two127p15 =
415 (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15);
416 static const p224_widelimb two127m71 =
417 (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71);
418 static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) -
419 (((p224_widelimb)1) << 71) -
420 (((p224_widelimb)1) << 55);
421 p224_widelimb output[5];
422
423 // Add 0 mod 2^224-2^96+1 to ensure all differences are positive
424 output[0] = in[0] + two127p15;
425 output[1] = in[1] + two127m71m55;
426 output[2] = in[2] + two127m71;
427 output[3] = in[3];
428 output[4] = in[4];
429
430 // Eliminate in[4], in[5], in[6]
431 output[4] += in[6] >> 16;
432 output[3] += (in[6] & 0xffff) << 40;
433 output[2] -= in[6];
434
435 output[3] += in[5] >> 16;
436 output[2] += (in[5] & 0xffff) << 40;
437 output[1] -= in[5];
438
439 output[2] += output[4] >> 16;
440 output[1] += (output[4] & 0xffff) << 40;
441 output[0] -= output[4];
442
443 // Carry 2 -> 3 -> 4
444 output[3] += output[2] >> 56;
445 output[2] &= 0x00ffffffffffffff;
446
447 output[4] = output[3] >> 56;
448 output[3] &= 0x00ffffffffffffff;
449
450 // Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72
451
452 // Eliminate output[4]
453 output[2] += output[4] >> 16;
454 // output[2] < 2^56 + 2^56 = 2^57
455 output[1] += (output[4] & 0xffff) << 40;
456 output[0] -= output[4];
457
458 // Carry 0 -> 1 -> 2 -> 3
459 output[1] += output[0] >> 56;
460 out[0] = output[0] & 0x00ffffffffffffff;
461
462 output[2] += output[1] >> 56;
463 // output[2] < 2^57 + 2^72
464 out[1] = output[1] & 0x00ffffffffffffff;
465 output[3] += output[2] >> 56;
466 // output[3] <= 2^56 + 2^16
467 out[2] = output[2] & 0x00ffffffffffffff;
468
469 // out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
470 // out[3] <= 2^56 + 2^16 (due to final carry),
471 // so out < 2*p
472 out[3] = output[3];
473}
474
475// Get negative value: out = -in
476// Requires in[i] < 2^63,
477// ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
478static void p224_felem_neg(p224_felem out, const p224_felem in) {
479 p224_widefelem tmp = {0};
480 p224_felem_diff_128_64(tmp, in);
481 p224_felem_reduce(out, tmp);
482}
483
484// Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
485// elements are reduced to in < 2^225, so we only need to check three cases: 0,
486// 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
487static p224_limb p224_felem_is_zero(const p224_felem in) {
488 p224_limb zero = in[0] | in[1] | in[2] | in[3];
489 zero = (((int64_t)(zero)-1) >> 63) & 1;
490
491 p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) |
492 (in[2] ^ 0x00ffffffffffffff) |
493 (in[3] ^ 0x00ffffffffffffff);
494 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1;
495 p224_limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) |
496 (in[2] ^ 0x00ffffffffffffff) |
497 (in[3] ^ 0x01ffffffffffffff);
498 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1;
499 return (zero | two224m96p1 | two225m97p2);
500}
501
502// Invert a field element
503// Computation chain copied from djb's code
504static void p224_felem_inv(p224_felem out, const p224_felem in) {
505 p224_felem ftmp, ftmp2, ftmp3, ftmp4;
506 p224_widefelem tmp;
507
508 p224_felem_square(tmp, in);
509 p224_felem_reduce(ftmp, tmp); // 2
510 p224_felem_mul(tmp, in, ftmp);
511 p224_felem_reduce(ftmp, tmp); // 2^2 - 1
512 p224_felem_square(tmp, ftmp);
513 p224_felem_reduce(ftmp, tmp); // 2^3 - 2
514 p224_felem_mul(tmp, in, ftmp);
515 p224_felem_reduce(ftmp, tmp); // 2^3 - 1
516 p224_felem_square(tmp, ftmp);
517 p224_felem_reduce(ftmp2, tmp); // 2^4 - 2
518 p224_felem_square(tmp, ftmp2);
519 p224_felem_reduce(ftmp2, tmp); // 2^5 - 4
520 p224_felem_square(tmp, ftmp2);
521 p224_felem_reduce(ftmp2, tmp); // 2^6 - 8
522 p224_felem_mul(tmp, ftmp2, ftmp);
523 p224_felem_reduce(ftmp, tmp); // 2^6 - 1
524 p224_felem_square(tmp, ftmp);
525 p224_felem_reduce(ftmp2, tmp); // 2^7 - 2
526 for (size_t i = 0; i < 5; ++i) { // 2^12 - 2^6
527 p224_felem_square(tmp, ftmp2);
528 p224_felem_reduce(ftmp2, tmp);
529 }
530 p224_felem_mul(tmp, ftmp2, ftmp);
531 p224_felem_reduce(ftmp2, tmp); // 2^12 - 1
532 p224_felem_square(tmp, ftmp2);
533 p224_felem_reduce(ftmp3, tmp); // 2^13 - 2
534 for (size_t i = 0; i < 11; ++i) { // 2^24 - 2^12
535 p224_felem_square(tmp, ftmp3);
536 p224_felem_reduce(ftmp3, tmp);
537 }
538 p224_felem_mul(tmp, ftmp3, ftmp2);
539 p224_felem_reduce(ftmp2, tmp); // 2^24 - 1
540 p224_felem_square(tmp, ftmp2);
541 p224_felem_reduce(ftmp3, tmp); // 2^25 - 2
542 for (size_t i = 0; i < 23; ++i) { // 2^48 - 2^24
543 p224_felem_square(tmp, ftmp3);
544 p224_felem_reduce(ftmp3, tmp);
545 }
546 p224_felem_mul(tmp, ftmp3, ftmp2);
547 p224_felem_reduce(ftmp3, tmp); // 2^48 - 1
548 p224_felem_square(tmp, ftmp3);
549 p224_felem_reduce(ftmp4, tmp); // 2^49 - 2
550 for (size_t i = 0; i < 47; ++i) { // 2^96 - 2^48
551 p224_felem_square(tmp, ftmp4);
552 p224_felem_reduce(ftmp4, tmp);
553 }
554 p224_felem_mul(tmp, ftmp3, ftmp4);
555 p224_felem_reduce(ftmp3, tmp); // 2^96 - 1
556 p224_felem_square(tmp, ftmp3);
557 p224_felem_reduce(ftmp4, tmp); // 2^97 - 2
558 for (size_t i = 0; i < 23; ++i) { // 2^120 - 2^24
559 p224_felem_square(tmp, ftmp4);
560 p224_felem_reduce(ftmp4, tmp);
561 }
562 p224_felem_mul(tmp, ftmp2, ftmp4);
563 p224_felem_reduce(ftmp2, tmp); // 2^120 - 1
564 for (size_t i = 0; i < 6; ++i) { // 2^126 - 2^6
565 p224_felem_square(tmp, ftmp2);
566 p224_felem_reduce(ftmp2, tmp);
567 }
568 p224_felem_mul(tmp, ftmp2, ftmp);
569 p224_felem_reduce(ftmp, tmp); // 2^126 - 1
570 p224_felem_square(tmp, ftmp);
571 p224_felem_reduce(ftmp, tmp); // 2^127 - 2
572 p224_felem_mul(tmp, ftmp, in);
573 p224_felem_reduce(ftmp, tmp); // 2^127 - 1
574 for (size_t i = 0; i < 97; ++i) { // 2^224 - 2^97
575 p224_felem_square(tmp, ftmp);
576 p224_felem_reduce(ftmp, tmp);
577 }
578 p224_felem_mul(tmp, ftmp, ftmp3);
579 p224_felem_reduce(out, tmp); // 2^224 - 2^96 - 1
580}
581
582// Copy in constant time:
583// if icopy == 1, copy in to out,
584// if icopy == 0, copy out to itself.
585static void p224_copy_conditional(p224_felem out, const p224_felem in,
586 p224_limb icopy) {
587 // icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
588 const p224_limb copy = -icopy;
589 for (size_t i = 0; i < 4; ++i) {
590 const p224_limb tmp = copy & (in[i] ^ out[i]);
591 out[i] ^= tmp;
592 }
593}
594
595// ELLIPTIC CURVE POINT OPERATIONS
596//
597// Points are represented in Jacobian projective coordinates:
598// (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
599// or to the point at infinity if Z == 0.
600
601// Double an elliptic curve point:
602// (X', Y', Z') = 2 * (X, Y, Z), where
603// X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
604// Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
605// Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
606// Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
607// while x_out == y_in is not (maybe this works, but it's not tested).
608static void p224_point_double(p224_felem x_out, p224_felem y_out,
609 p224_felem z_out, const p224_felem x_in,
610 const p224_felem y_in, const p224_felem z_in) {
611 p224_widefelem tmp, tmp2;
612 p224_felem delta, gamma, beta, alpha, ftmp, ftmp2;
613
614 p224_felem_assign(ftmp, x_in);
615 p224_felem_assign(ftmp2, x_in);
616
617 // delta = z^2
618 p224_felem_square(tmp, z_in);
619 p224_felem_reduce(delta, tmp);
620
621 // gamma = y^2
622 p224_felem_square(tmp, y_in);
623 p224_felem_reduce(gamma, tmp);
624
625 // beta = x*gamma
626 p224_felem_mul(tmp, x_in, gamma);
627 p224_felem_reduce(beta, tmp);
628
629 // alpha = 3*(x-delta)*(x+delta)
630 p224_felem_diff(ftmp, delta);
631 // ftmp[i] < 2^57 + 2^58 + 2 < 2^59
632 p224_felem_sum(ftmp2, delta);
633 // ftmp2[i] < 2^57 + 2^57 = 2^58
634 p224_felem_scalar(ftmp2, 3);
635 // ftmp2[i] < 3 * 2^58 < 2^60
636 p224_felem_mul(tmp, ftmp, ftmp2);
637 // tmp[i] < 2^60 * 2^59 * 4 = 2^121
638 p224_felem_reduce(alpha, tmp);
639
640 // x' = alpha^2 - 8*beta
641 p224_felem_square(tmp, alpha);
642 // tmp[i] < 4 * 2^57 * 2^57 = 2^116
643 p224_felem_assign(ftmp, beta);
644 p224_felem_scalar(ftmp, 8);
645 // ftmp[i] < 8 * 2^57 = 2^60
646 p224_felem_diff_128_64(tmp, ftmp);
647 // tmp[i] < 2^116 + 2^64 + 8 < 2^117
648 p224_felem_reduce(x_out, tmp);
649
650 // z' = (y + z)^2 - gamma - delta
651 p224_felem_sum(delta, gamma);
652 // delta[i] < 2^57 + 2^57 = 2^58
653 p224_felem_assign(ftmp, y_in);
654 p224_felem_sum(ftmp, z_in);
655 // ftmp[i] < 2^57 + 2^57 = 2^58
656 p224_felem_square(tmp, ftmp);
657 // tmp[i] < 4 * 2^58 * 2^58 = 2^118
658 p224_felem_diff_128_64(tmp, delta);
659 // tmp[i] < 2^118 + 2^64 + 8 < 2^119
660 p224_felem_reduce(z_out, tmp);
661
662 // y' = alpha*(4*beta - x') - 8*gamma^2
663 p224_felem_scalar(beta, 4);
664 // beta[i] < 4 * 2^57 = 2^59
665 p224_felem_diff(beta, x_out);
666 // beta[i] < 2^59 + 2^58 + 2 < 2^60
667 p224_felem_mul(tmp, alpha, beta);
668 // tmp[i] < 4 * 2^57 * 2^60 = 2^119
669 p224_felem_square(tmp2, gamma);
670 // tmp2[i] < 4 * 2^57 * 2^57 = 2^116
671 p224_widefelem_scalar(tmp2, 8);
672 // tmp2[i] < 8 * 2^116 = 2^119
673 p224_widefelem_diff(tmp, tmp2);
674 // tmp[i] < 2^119 + 2^120 < 2^121
675 p224_felem_reduce(y_out, tmp);
676}
677
678// Add two elliptic curve points:
679// (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
680// X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
681// 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
682// Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 *
683// X_1)^2 - X_3) -
684// Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
685// Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
686//
687// This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
688
689// This function is not entirely constant-time: it includes a branch for
690// checking whether the two input points are equal, (while not equal to the
691// point at infinity). This case never happens during single point
692// multiplication, so there is no timing leak for ECDH or ECDSA signing.
693static void p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3,
694 const p224_felem x1, const p224_felem y1,
695 const p224_felem z1, const int mixed,
696 const p224_felem x2, const p224_felem y2,
697 const p224_felem z2) {
698 p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
699 p224_widefelem tmp, tmp2;
700 p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal;
701
702 if (!mixed) {
703 // ftmp2 = z2^2
704 p224_felem_square(tmp, z2);
705 p224_felem_reduce(ftmp2, tmp);
706
707 // ftmp4 = z2^3
708 p224_felem_mul(tmp, ftmp2, z2);
709 p224_felem_reduce(ftmp4, tmp);
710
711 // ftmp4 = z2^3*y1
712 p224_felem_mul(tmp2, ftmp4, y1);
713 p224_felem_reduce(ftmp4, tmp2);
714
715 // ftmp2 = z2^2*x1
716 p224_felem_mul(tmp2, ftmp2, x1);
717 p224_felem_reduce(ftmp2, tmp2);
718 } else {
719 // We'll assume z2 = 1 (special case z2 = 0 is handled later)
720
721 // ftmp4 = z2^3*y1
722 p224_felem_assign(ftmp4, y1);
723
724 // ftmp2 = z2^2*x1
725 p224_felem_assign(ftmp2, x1);
726 }
727
728 // ftmp = z1^2
729 p224_felem_square(tmp, z1);
730 p224_felem_reduce(ftmp, tmp);
731
732 // ftmp3 = z1^3
733 p224_felem_mul(tmp, ftmp, z1);
734 p224_felem_reduce(ftmp3, tmp);
735
736 // tmp = z1^3*y2
737 p224_felem_mul(tmp, ftmp3, y2);
738 // tmp[i] < 4 * 2^57 * 2^57 = 2^116
739
740 // ftmp3 = z1^3*y2 - z2^3*y1
741 p224_felem_diff_128_64(tmp, ftmp4);
742 // tmp[i] < 2^116 + 2^64 + 8 < 2^117
743 p224_felem_reduce(ftmp3, tmp);
744
745 // tmp = z1^2*x2
746 p224_felem_mul(tmp, ftmp, x2);
747 // tmp[i] < 4 * 2^57 * 2^57 = 2^116
748
749 // ftmp = z1^2*x2 - z2^2*x1
750 p224_felem_diff_128_64(tmp, ftmp2);
751 // tmp[i] < 2^116 + 2^64 + 8 < 2^117
752 p224_felem_reduce(ftmp, tmp);
753
754 // the formulae are incorrect if the points are equal
755 // so we check for this and do doubling if this happens
756 x_equal = p224_felem_is_zero(ftmp);
757 y_equal = p224_felem_is_zero(ftmp3);
758 z1_is_zero = p224_felem_is_zero(z1);
759 z2_is_zero = p224_felem_is_zero(z2);
760 // In affine coordinates, (X_1, Y_1) == (X_2, Y_2)
761 p224_limb is_nontrivial_double =
762 x_equal & y_equal & (1 - z1_is_zero) & (1 - z2_is_zero);
763 if (is_nontrivial_double) {
764 p224_point_double(x3, y3, z3, x1, y1, z1);
765 return;
766 }
767
768 // ftmp5 = z1*z2
769 if (!mixed) {
770 p224_felem_mul(tmp, z1, z2);
771 p224_felem_reduce(ftmp5, tmp);
772 } else {
773 // special case z2 = 0 is handled later
774 p224_felem_assign(ftmp5, z1);
775 }
776
777 // z_out = (z1^2*x2 - z2^2*x1)*(z1*z2)
778 p224_felem_mul(tmp, ftmp, ftmp5);
779 p224_felem_reduce(z_out, tmp);
780
781 // ftmp = (z1^2*x2 - z2^2*x1)^2
782 p224_felem_assign(ftmp5, ftmp);
783 p224_felem_square(tmp, ftmp);
784 p224_felem_reduce(ftmp, tmp);
785
786 // ftmp5 = (z1^2*x2 - z2^2*x1)^3
787 p224_felem_mul(tmp, ftmp, ftmp5);
788 p224_felem_reduce(ftmp5, tmp);
789
790 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2
791 p224_felem_mul(tmp, ftmp2, ftmp);
792 p224_felem_reduce(ftmp2, tmp);
793
794 // tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3
795 p224_felem_mul(tmp, ftmp4, ftmp5);
796 // tmp[i] < 4 * 2^57 * 2^57 = 2^116
797
798 // tmp2 = (z1^3*y2 - z2^3*y1)^2
799 p224_felem_square(tmp2, ftmp3);
800 // tmp2[i] < 4 * 2^57 * 2^57 < 2^116
801
802 // tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3
803 p224_felem_diff_128_64(tmp2, ftmp5);
804 // tmp2[i] < 2^116 + 2^64 + 8 < 2^117
805
806 // ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
807 p224_felem_assign(ftmp5, ftmp2);
808 p224_felem_scalar(ftmp5, 2);
809 // ftmp5[i] < 2 * 2^57 = 2^58
810
811 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
812 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
813 p224_felem_diff_128_64(tmp2, ftmp5);
814 // tmp2[i] < 2^117 + 2^64 + 8 < 2^118
815 p224_felem_reduce(x_out, tmp2);
816
817 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out
818 p224_felem_diff(ftmp2, x_out);
819 // ftmp2[i] < 2^57 + 2^58 + 2 < 2^59
820
821 // tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
822 p224_felem_mul(tmp2, ftmp3, ftmp2);
823 // tmp2[i] < 4 * 2^57 * 2^59 = 2^118
824
825 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
826 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
827 p224_widefelem_diff(tmp2, tmp);
828 // tmp2[i] < 2^118 + 2^120 < 2^121
829 p224_felem_reduce(y_out, tmp2);
830
831 // the result (x_out, y_out, z_out) is incorrect if one of the inputs is
832 // the point at infinity, so we need to check for this separately
833
834 // if point 1 is at infinity, copy point 2 to output, and vice versa
835 p224_copy_conditional(x_out, x2, z1_is_zero);
836 p224_copy_conditional(x_out, x1, z2_is_zero);
837 p224_copy_conditional(y_out, y2, z1_is_zero);
838 p224_copy_conditional(y_out, y1, z2_is_zero);
839 p224_copy_conditional(z_out, z2, z1_is_zero);
840 p224_copy_conditional(z_out, z1, z2_is_zero);
841 p224_felem_assign(x3, x_out);
842 p224_felem_assign(y3, y_out);
843 p224_felem_assign(z3, z_out);
844}
845
846// p224_select_point selects the |idx|th point from a precomputation table and
847// copies it to out.
848static void p224_select_point(const uint64_t idx, size_t size,
849 const p224_felem pre_comp[/*size*/][3],
850 p224_felem out[3]) {
851 p224_limb *outlimbs = &out[0][0];
852 OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem));
853
854 for (size_t i = 0; i < size; i++) {
855 const p224_limb *inlimbs = &pre_comp[i][0][0];
856 uint64_t mask = i ^ idx;
857 mask |= mask >> 4;
858 mask |= mask >> 2;
859 mask |= mask >> 1;
860 mask &= 1;
861 mask--;
862 for (size_t j = 0; j < 4 * 3; j++) {
863 outlimbs[j] |= inlimbs[j] & mask;
864 }
865 }
866}
867
868// p224_get_bit returns the |i|th bit in |in|
869static char p224_get_bit(const p224_felem_bytearray in, size_t i) {
870 if (i >= 224) {
871 return 0;
872 }
873 return (in[i >> 3] >> (i & 7)) & 1;
874}
875
876// Takes the Jacobian coordinates (X, Y, Z) of a point and returns
877// (X', Y') = (X/Z^2, Y/Z^3)
878static int ec_GFp_nistp224_point_get_affine_coordinates(
879 const EC_GROUP *group, const EC_RAW_POINT *point, EC_FELEM *x,
880 EC_FELEM *y) {
881 if (ec_GFp_simple_is_at_infinity(group, point)) {
882 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
883 return 0;
884 }
885
886 p224_felem z1, z2;
887 p224_widefelem tmp;
888 p224_generic_to_felem(z1, &point->Z);
889 p224_felem_inv(z2, z1);
890 p224_felem_square(tmp, z2);
891 p224_felem_reduce(z1, tmp);
892
893 if (x != NULL) {
894 p224_felem x_in, x_out;
895 p224_generic_to_felem(x_in, &point->X);
896 p224_felem_mul(tmp, x_in, z1);
897 p224_felem_reduce(x_out, tmp);
898 p224_felem_to_generic(x, x_out);
899 }
900
901 if (y != NULL) {
902 p224_felem y_in, y_out;
903 p224_generic_to_felem(y_in, &point->Y);
904 p224_felem_mul(tmp, z1, z2);
905 p224_felem_reduce(z1, tmp);
906 p224_felem_mul(tmp, y_in, z1);
907 p224_felem_reduce(y_out, tmp);
908 p224_felem_to_generic(y, y_out);
909 }
910
911 return 1;
912}
913
914static void ec_GFp_nistp224_add(const EC_GROUP *group, EC_RAW_POINT *r,
915 const EC_RAW_POINT *a, const EC_RAW_POINT *b) {
916 p224_felem x1, y1, z1, x2, y2, z2;
917 p224_generic_to_felem(x1, &a->X);
918 p224_generic_to_felem(y1, &a->Y);
919 p224_generic_to_felem(z1, &a->Z);
920 p224_generic_to_felem(x2, &b->X);
921 p224_generic_to_felem(y2, &b->Y);
922 p224_generic_to_felem(z2, &b->Z);
923 p224_point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2, z2);
924 // The outputs are already reduced, but still need to be contracted.
925 p224_felem_to_generic(&r->X, x1);
926 p224_felem_to_generic(&r->Y, y1);
927 p224_felem_to_generic(&r->Z, z1);
928}
929
930static void ec_GFp_nistp224_dbl(const EC_GROUP *group, EC_RAW_POINT *r,
931 const EC_RAW_POINT *a) {
932 p224_felem x, y, z;
933 p224_generic_to_felem(x, &a->X);
934 p224_generic_to_felem(y, &a->Y);
935 p224_generic_to_felem(z, &a->Z);
936 p224_point_double(x, y, z, x, y, z);
937 // The outputs are already reduced, but still need to be contracted.
938 p224_felem_to_generic(&r->X, x);
939 p224_felem_to_generic(&r->Y, y);
940 p224_felem_to_generic(&r->Z, z);
941}
942
943static void ec_GFp_nistp224_make_precomp(p224_felem out[17][3],
944 const EC_RAW_POINT *p) {
945 OPENSSL_memset(out[0], 0, sizeof(p224_felem) * 3);
946
947 p224_generic_to_felem(out[1][0], &p->X);
948 p224_generic_to_felem(out[1][1], &p->Y);
949 p224_generic_to_felem(out[1][2], &p->Z);
950
951 for (size_t j = 2; j <= 16; ++j) {
952 if (j & 1) {
953 p224_point_add(out[j][0], out[j][1], out[j][2], out[1][0], out[1][1],
954 out[1][2], 0, out[j - 1][0], out[j - 1][1], out[j - 1][2]);
955 } else {
956 p224_point_double(out[j][0], out[j][1], out[j][2], out[j / 2][0],
957 out[j / 2][1], out[j / 2][2]);
958 }
959 }
960}
961
962static void ec_GFp_nistp224_point_mul(const EC_GROUP *group, EC_RAW_POINT *r,
963 const EC_RAW_POINT *p,
964 const EC_SCALAR *scalar) {
965 p224_felem p_pre_comp[17][3];
966 ec_GFp_nistp224_make_precomp(p_pre_comp, p);
967
968 // Set nq to the point at infinity.
969 p224_felem nq[3], tmp[4];
970 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));
971
972 int skip = 1; // Save two point operations in the first round.
973 for (size_t i = 220; i < 221; i--) {
974 if (!skip) {
975 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
976 }
977
978 // Add every 5 doublings.
979 if (i % 5 == 0) {
980 uint64_t bits = p224_get_bit(scalar->bytes, i + 4) << 5;
981 bits |= p224_get_bit(scalar->bytes, i + 3) << 4;
982 bits |= p224_get_bit(scalar->bytes, i + 2) << 3;
983 bits |= p224_get_bit(scalar->bytes, i + 1) << 2;
984 bits |= p224_get_bit(scalar->bytes, i) << 1;
985 bits |= p224_get_bit(scalar->bytes, i - 1);
986 uint8_t sign, digit;
987 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
988
989 // Select the point to add or subtract.
990 p224_select_point(digit, 17, (const p224_felem(*)[3])p_pre_comp, tmp);
991 p224_felem_neg(tmp[3], tmp[1]); // (X, -Y, Z) is the negative point
992 p224_copy_conditional(tmp[1], tmp[3], sign);
993
994 if (!skip) {
995 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */,
996 tmp[0], tmp[1], tmp[2]);
997 } else {
998 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));
999 skip = 0;
1000 }
1001 }
1002 }
1003
1004 // Reduce the output to its unique minimal representation.
1005 p224_felem_to_generic(&r->X, nq[0]);
1006 p224_felem_to_generic(&r->Y, nq[1]);
1007 p224_felem_to_generic(&r->Z, nq[2]);
1008}
1009
1010static void ec_GFp_nistp224_point_mul_base(const EC_GROUP *group,
1011 EC_RAW_POINT *r,
1012 const EC_SCALAR *scalar) {
1013 // Set nq to the point at infinity.
1014 p224_felem nq[3], tmp[3];
1015 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));
1016
1017 int skip = 1; // Save two point operations in the first round.
1018 for (size_t i = 27; i < 28; i--) {
1019 // double
1020 if (!skip) {
1021 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1022 }
1023
1024 // First, look 28 bits upwards.
1025 uint64_t bits = p224_get_bit(scalar->bytes, i + 196) << 3;
1026 bits |= p224_get_bit(scalar->bytes, i + 140) << 2;
1027 bits |= p224_get_bit(scalar->bytes, i + 84) << 1;
1028 bits |= p224_get_bit(scalar->bytes, i + 28);
1029 // Select the point to add, in constant time.
1030 p224_select_point(bits, 16, g_p224_pre_comp[1], tmp);
1031
1032 if (!skip) {
1033 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
1034 tmp[0], tmp[1], tmp[2]);
1035 } else {
1036 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));
1037 skip = 0;
1038 }
1039
1040 // Second, look at the current position/
1041 bits = p224_get_bit(scalar->bytes, i + 168) << 3;
1042 bits |= p224_get_bit(scalar->bytes, i + 112) << 2;
1043 bits |= p224_get_bit(scalar->bytes, i + 56) << 1;
1044 bits |= p224_get_bit(scalar->bytes, i);
1045 // Select the point to add, in constant time.
1046 p224_select_point(bits, 16, g_p224_pre_comp[0], tmp);
1047 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
1048 tmp[0], tmp[1], tmp[2]);
1049 }
1050
1051 // Reduce the output to its unique minimal representation.
1052 p224_felem_to_generic(&r->X, nq[0]);
1053 p224_felem_to_generic(&r->Y, nq[1]);
1054 p224_felem_to_generic(&r->Z, nq[2]);
1055}
1056
1057static void ec_GFp_nistp224_point_mul_public(const EC_GROUP *group,
1058 EC_RAW_POINT *r,
1059 const EC_SCALAR *g_scalar,
1060 const EC_RAW_POINT *p,
1061 const EC_SCALAR *p_scalar) {
1062 // TODO(davidben): If P-224 ECDSA verify performance ever matters, using
1063 // |ec_compute_wNAF| for |p_scalar| would likely be an easy improvement.
1064 p224_felem p_pre_comp[17][3];
1065 ec_GFp_nistp224_make_precomp(p_pre_comp, p);
1066
1067 // Set nq to the point at infinity.
1068 p224_felem nq[3], tmp[3];
1069 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));
1070
1071 // Loop over both scalars msb-to-lsb, interleaving additions of multiples of
1072 // the generator (two in each of the last 28 rounds) and additions of p (every
1073 // 5th round).
1074 int skip = 1; // Save two point operations in the first round.
1075 for (size_t i = 220; i < 221; i--) {
1076 if (!skip) {
1077 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1078 }
1079
1080 // Add multiples of the generator.
1081 if (i <= 27) {
1082 // First, look 28 bits upwards.
1083 uint64_t bits = p224_get_bit(g_scalar->bytes, i + 196) << 3;
1084 bits |= p224_get_bit(g_scalar->bytes, i + 140) << 2;
1085 bits |= p224_get_bit(g_scalar->bytes, i + 84) << 1;
1086 bits |= p224_get_bit(g_scalar->bytes, i + 28);
1087
1088 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
1089 g_p224_pre_comp[1][bits][0], g_p224_pre_comp[1][bits][1],
1090 g_p224_pre_comp[1][bits][2]);
1091 assert(!skip);
1092
1093 // Second, look at the current position.
1094 bits = p224_get_bit(g_scalar->bytes, i + 168) << 3;
1095 bits |= p224_get_bit(g_scalar->bytes, i + 112) << 2;
1096 bits |= p224_get_bit(g_scalar->bytes, i + 56) << 1;
1097 bits |= p224_get_bit(g_scalar->bytes, i);
1098 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
1099 g_p224_pre_comp[0][bits][0], g_p224_pre_comp[0][bits][1],
1100 g_p224_pre_comp[0][bits][2]);
1101 }
1102
1103 // Incorporate |p_scalar| every 5 doublings.
1104 if (i % 5 == 0) {
1105 uint64_t bits = p224_get_bit(p_scalar->bytes, i + 4) << 5;
1106 bits |= p224_get_bit(p_scalar->bytes, i + 3) << 4;
1107 bits |= p224_get_bit(p_scalar->bytes, i + 2) << 3;
1108 bits |= p224_get_bit(p_scalar->bytes, i + 1) << 2;
1109 bits |= p224_get_bit(p_scalar->bytes, i) << 1;
1110 bits |= p224_get_bit(p_scalar->bytes, i - 1);
1111 uint8_t sign, digit;
1112 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1113
1114 // Select the point to add or subtract.
1115 OPENSSL_memcpy(tmp, p_pre_comp[digit], 3 * sizeof(p224_felem));
1116 if (sign) {
1117 p224_felem_neg(tmp[1], tmp[1]); // (X, -Y, Z) is the negative point
1118 }
1119
1120 if (!skip) {
1121 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */,
1122 tmp[0], tmp[1], tmp[2]);
1123 } else {
1124 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));
1125 skip = 0;
1126 }
1127 }
1128 }
1129
1130 // Reduce the output to its unique minimal representation.
1131 p224_felem_to_generic(&r->X, nq[0]);
1132 p224_felem_to_generic(&r->Y, nq[1]);
1133 p224_felem_to_generic(&r->Z, nq[2]);
1134}
1135
1136static void ec_GFp_nistp224_felem_mul(const EC_GROUP *group, EC_FELEM *r,
1137 const EC_FELEM *a, const EC_FELEM *b) {
1138 p224_felem felem1, felem2;
1139 p224_widefelem wide;
1140 p224_generic_to_felem(felem1, a);
1141 p224_generic_to_felem(felem2, b);
1142 p224_felem_mul(wide, felem1, felem2);
1143 p224_felem_reduce(felem1, wide);
1144 p224_felem_to_generic(r, felem1);
1145}
1146
1147static void ec_GFp_nistp224_felem_sqr(const EC_GROUP *group, EC_FELEM *r,
1148 const EC_FELEM *a) {
1149 p224_felem felem;
1150 p224_generic_to_felem(felem, a);
1151 p224_widefelem wide;
1152 p224_felem_square(wide, felem);
1153 p224_felem_reduce(felem, wide);
1154 p224_felem_to_generic(r, felem);
1155}
1156
1157static int ec_GFp_nistp224_bignum_to_felem(const EC_GROUP *group, EC_FELEM *out,
1158 const BIGNUM *in) {
1159 return bn_copy_words(out->words, group->field.width, in);
1160}
1161
1162static int ec_GFp_nistp224_felem_to_bignum(const EC_GROUP *group, BIGNUM *out,
1163 const EC_FELEM *in) {
1164 return bn_set_words(out, in->words, group->field.width);
1165}
1166
1167DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) {
1168 out->group_init = ec_GFp_simple_group_init;
1169 out->group_finish = ec_GFp_simple_group_finish;
1170 out->group_set_curve = ec_GFp_simple_group_set_curve;
1171 out->point_get_affine_coordinates =
1172 ec_GFp_nistp224_point_get_affine_coordinates;
1173 out->add = ec_GFp_nistp224_add;
1174 out->dbl = ec_GFp_nistp224_dbl;
1175 out->mul = ec_GFp_nistp224_point_mul;
1176 out->mul_base = ec_GFp_nistp224_point_mul_base;
1177 out->mul_public = ec_GFp_nistp224_point_mul_public;
1178 out->felem_mul = ec_GFp_nistp224_felem_mul;
1179 out->felem_sqr = ec_GFp_nistp224_felem_sqr;
1180 out->bignum_to_felem = ec_GFp_nistp224_bignum_to_felem;
1181 out->felem_to_bignum = ec_GFp_nistp224_felem_to_bignum;
1182 out->scalar_inv_montgomery = ec_simple_scalar_inv_montgomery;
1183 out->scalar_inv_montgomery_vartime = ec_GFp_simple_mont_inv_mod_ord_vartime;
1184 out->cmp_x_coordinate = ec_GFp_simple_cmp_x_coordinate;
1185}
1186
1187#endif // BORINGSSL_HAS_UINT128 && !SMALL
1188