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 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \
23 !defined(OPENSSL_SMALL)
24
25 #include <openssl/bn.h>
26 #include <openssl/ec.h>
27 #include <openssl/err.h>
28 #include <openssl/mem.h>
29
30 #include <string.h>
31
32 #include "internal.h"
33 #include "../delocate.h"
34 #include "../../internal.h"
35
36
37 /* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
38 * using 64-bit coefficients called 'limbs', and sometimes (for multiplication
39 * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 +
40 * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-p224_limb
41 * representation is an 'p224_felem'; a 7-p224_widelimb representation is a
42 * 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we
43 * don't always reduce the representations: we ensure that inputs to each
44 * p224_felem multiplication satisfy a_i < 2^60, so outputs satisfy b_i <
45 * 4*2^60*2^60, and fit into a 128-bit word without overflow. The coefficients
46 * are then again partially reduced to obtain an p224_felem satisfying a_i <
47 * 2^57. We only reduce to the unique minimal representation at the end of the
48 * computation. */
49
50 typedef uint64_t p224_limb;
51 typedef uint128_t p224_widelimb;
52
53 typedef p224_limb p224_felem[4];
54 typedef p224_widelimb p224_widefelem[7];
55
56 /* Field element represented as a byte arrary. 28*8 = 224 bits is also the
57 * group order size for the elliptic curve, and we also use this type for
58 * scalars for point multiplication. */
59 typedef uint8_t p224_felem_bytearray[28];
60
61 /* Precomputed multiples of the standard generator
62 * Points are given in coordinates (X, Y, Z) where Z normally is 1
63 * (0 for the point at infinity).
64 * For each field element, slice a_0 is word 0, etc.
65 *
66 * The table has 2 * 16 elements, starting with the following:
67 * index | bits | point
68 * ------+---------+------------------------------
69 * 0 | 0 0 0 0 | 0G
70 * 1 | 0 0 0 1 | 1G
71 * 2 | 0 0 1 0 | 2^56G
72 * 3 | 0 0 1 1 | (2^56 + 1)G
73 * 4 | 0 1 0 0 | 2^112G
74 * 5 | 0 1 0 1 | (2^112 + 1)G
75 * 6 | 0 1 1 0 | (2^112 + 2^56)G
76 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
77 * 8 | 1 0 0 0 | 2^168G
78 * 9 | 1 0 0 1 | (2^168 + 1)G
79 * 10 | 1 0 1 0 | (2^168 + 2^56)G
80 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
81 * 12 | 1 1 0 0 | (2^168 + 2^112)G
82 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
83 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
84 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
85 * followed by a copy of this with each element multiplied by 2^28.
86 *
87 * The reason for this is so that we can clock bits into four different
88 * locations when doing simple scalar multiplies against the base point,
89 * and then another four locations using the second 16 elements. */
90 static const p224_felem g_p224_pre_comp[2][16][3] = {
91 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
92 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
93 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
94 {1, 0, 0, 0}},
95 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
96 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
97 {1, 0, 0, 0}},
98 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
99 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
100 {1, 0, 0, 0}},
101 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
102 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
103 {1, 0, 0, 0}},
104 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
105 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
106 {1, 0, 0, 0}},
107 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
108 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
109 {1, 0, 0, 0}},
110 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
111 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
112 {1, 0, 0, 0}},
113 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
114 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
115 {1, 0, 0, 0}},
116 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
117 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
118 {1, 0, 0, 0}},
119 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
120 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
121 {1, 0, 0, 0}},
122 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
123 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
124 {1, 0, 0, 0}},
125 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
126 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
127 {1, 0, 0, 0}},
128 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
129 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
130 {1, 0, 0, 0}},
131 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
132 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
133 {1, 0, 0, 0}},
134 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
135 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
136 {1, 0, 0, 0}}},
137 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
138 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
139 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
140 {1, 0, 0, 0}},
141 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
142 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
143 {1, 0, 0, 0}},
144 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
145 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
146 {1, 0, 0, 0}},
147 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
148 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
149 {1, 0, 0, 0}},
150 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
151 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
152 {1, 0, 0, 0}},
153 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
154 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
155 {1, 0, 0, 0}},
156 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
157 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
158 {1, 0, 0, 0}},
159 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
160 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
161 {1, 0, 0, 0}},
162 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
163 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
164 {1, 0, 0, 0}},
165 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
166 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
167 {1, 0, 0, 0}},
168 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
169 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
170 {1, 0, 0, 0}},
171 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
172 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
173 {1, 0, 0, 0}},
174 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
175 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
176 {1, 0, 0, 0}},
177 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
178 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
179 {1, 0, 0, 0}},
180 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
181 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
182 {1, 0, 0, 0}}}};
183
p224_load_u64(const uint8_t in[8])184 static uint64_t p224_load_u64(const uint8_t in[8]) {
185 uint64_t ret;
186 OPENSSL_memcpy(&ret, in, sizeof(ret));
187 return ret;
188 }
189
190 /* Helper functions to convert field elements to/from internal representation */
p224_bin28_to_felem(p224_felem out,const uint8_t in[28])191 static void p224_bin28_to_felem(p224_felem out, const uint8_t in[28]) {
192 out[0] = p224_load_u64(in) & 0x00ffffffffffffff;
193 out[1] = p224_load_u64(in + 7) & 0x00ffffffffffffff;
194 out[2] = p224_load_u64(in + 14) & 0x00ffffffffffffff;
195 out[3] = p224_load_u64(in + 20) >> 8;
196 }
197
p224_felem_to_bin28(uint8_t out[28],const p224_felem in)198 static void p224_felem_to_bin28(uint8_t out[28], const p224_felem in) {
199 for (size_t i = 0; i < 7; ++i) {
200 out[i] = in[0] >> (8 * i);
201 out[i + 7] = in[1] >> (8 * i);
202 out[i + 14] = in[2] >> (8 * i);
203 out[i + 21] = in[3] >> (8 * i);
204 }
205 }
206
207 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
p224_flip_endian(uint8_t * out,const uint8_t * in,size_t len)208 static void p224_flip_endian(uint8_t *out, const uint8_t *in, size_t len) {
209 for (size_t i = 0; i < len; ++i) {
210 out[i] = in[len - 1 - i];
211 }
212 }
213
214 /* From OpenSSL BIGNUM to internal representation */
p224_BN_to_felem(p224_felem out,const BIGNUM * bn)215 static int p224_BN_to_felem(p224_felem out, const BIGNUM *bn) {
216 /* BN_bn2bin eats leading zeroes */
217 p224_felem_bytearray b_out;
218 OPENSSL_memset(b_out, 0, sizeof(b_out));
219 size_t num_bytes = BN_num_bytes(bn);
220 if (num_bytes > sizeof(b_out) ||
221 BN_is_negative(bn)) {
222 OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE);
223 return 0;
224 }
225
226 p224_felem_bytearray b_in;
227 num_bytes = BN_bn2bin(bn, b_in);
228 p224_flip_endian(b_out, b_in, num_bytes);
229 p224_bin28_to_felem(out, b_out);
230 return 1;
231 }
232
233 /* From internal representation to OpenSSL BIGNUM */
p224_felem_to_BN(BIGNUM * out,const p224_felem in)234 static BIGNUM *p224_felem_to_BN(BIGNUM *out, const p224_felem in) {
235 p224_felem_bytearray b_in, b_out;
236 p224_felem_to_bin28(b_in, in);
237 p224_flip_endian(b_out, b_in, sizeof(b_out));
238 return BN_bin2bn(b_out, sizeof(b_out), out);
239 }
240
241 /* Field operations, using the internal representation of field elements.
242 * NB! These operations are specific to our point multiplication and cannot be
243 * expected to be correct in general - e.g., multiplication with a large scalar
244 * will cause an overflow. */
245
p224_felem_assign(p224_felem out,const p224_felem in)246 static void p224_felem_assign(p224_felem out, const p224_felem in) {
247 out[0] = in[0];
248 out[1] = in[1];
249 out[2] = in[2];
250 out[3] = in[3];
251 }
252
253 /* Sum two field elements: out += in */
p224_felem_sum(p224_felem out,const p224_felem in)254 static void p224_felem_sum(p224_felem out, const p224_felem in) {
255 out[0] += in[0];
256 out[1] += in[1];
257 out[2] += in[2];
258 out[3] += in[3];
259 }
260
261 /* Get negative value: out = -in */
262 /* Assumes in[i] < 2^57 */
p224_felem_neg(p224_felem out,const p224_felem in)263 static void p224_felem_neg(p224_felem out, const p224_felem in) {
264 static const p224_limb two58p2 =
265 (((p224_limb)1) << 58) + (((p224_limb)1) << 2);
266 static const p224_limb two58m2 =
267 (((p224_limb)1) << 58) - (((p224_limb)1) << 2);
268 static const p224_limb two58m42m2 =
269 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2);
270
271 /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
272 out[0] = two58p2 - in[0];
273 out[1] = two58m42m2 - in[1];
274 out[2] = two58m2 - in[2];
275 out[3] = two58m2 - in[3];
276 }
277
278 /* Subtract field elements: out -= in */
279 /* Assumes in[i] < 2^57 */
p224_felem_diff(p224_felem out,const p224_felem in)280 static void p224_felem_diff(p224_felem out, const p224_felem in) {
281 static const p224_limb two58p2 =
282 (((p224_limb)1) << 58) + (((p224_limb)1) << 2);
283 static const p224_limb two58m2 =
284 (((p224_limb)1) << 58) - (((p224_limb)1) << 2);
285 static const p224_limb two58m42m2 =
286 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2);
287
288 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
289 out[0] += two58p2;
290 out[1] += two58m42m2;
291 out[2] += two58m2;
292 out[3] += two58m2;
293
294 out[0] -= in[0];
295 out[1] -= in[1];
296 out[2] -= in[2];
297 out[3] -= in[3];
298 }
299
300 /* Subtract in unreduced 128-bit mode: out -= in */
301 /* Assumes in[i] < 2^119 */
p224_widefelem_diff(p224_widefelem out,const p224_widefelem in)302 static void p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) {
303 static const p224_widelimb two120 = ((p224_widelimb)1) << 120;
304 static const p224_widelimb two120m64 =
305 (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64);
306 static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) -
307 (((p224_widelimb)1) << 104) -
308 (((p224_widelimb)1) << 64);
309
310 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
311 out[0] += two120;
312 out[1] += two120m64;
313 out[2] += two120m64;
314 out[3] += two120;
315 out[4] += two120m104m64;
316 out[5] += two120m64;
317 out[6] += two120m64;
318
319 out[0] -= in[0];
320 out[1] -= in[1];
321 out[2] -= in[2];
322 out[3] -= in[3];
323 out[4] -= in[4];
324 out[5] -= in[5];
325 out[6] -= in[6];
326 }
327
328 /* Subtract in mixed mode: out128 -= in64 */
329 /* in[i] < 2^63 */
p224_felem_diff_128_64(p224_widefelem out,const p224_felem in)330 static void p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) {
331 static const p224_widelimb two64p8 =
332 (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8);
333 static const p224_widelimb two64m8 =
334 (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8);
335 static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) -
336 (((p224_widelimb)1) << 48) -
337 (((p224_widelimb)1) << 8);
338
339 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
340 out[0] += two64p8;
341 out[1] += two64m48m8;
342 out[2] += two64m8;
343 out[3] += two64m8;
344
345 out[0] -= in[0];
346 out[1] -= in[1];
347 out[2] -= in[2];
348 out[3] -= in[3];
349 }
350
351 /* Multiply a field element by a scalar: out = out * scalar
352 * The scalars we actually use are small, so results fit without overflow */
p224_felem_scalar(p224_felem out,const p224_limb scalar)353 static void p224_felem_scalar(p224_felem out, const p224_limb scalar) {
354 out[0] *= scalar;
355 out[1] *= scalar;
356 out[2] *= scalar;
357 out[3] *= scalar;
358 }
359
360 /* Multiply an unreduced field element by a scalar: out = out * scalar
361 * The scalars we actually use are small, so results fit without overflow */
p224_widefelem_scalar(p224_widefelem out,const p224_widelimb scalar)362 static void p224_widefelem_scalar(p224_widefelem out,
363 const p224_widelimb scalar) {
364 out[0] *= scalar;
365 out[1] *= scalar;
366 out[2] *= scalar;
367 out[3] *= scalar;
368 out[4] *= scalar;
369 out[5] *= scalar;
370 out[6] *= scalar;
371 }
372
373 /* Square a field element: out = in^2 */
p224_felem_square(p224_widefelem out,const p224_felem in)374 static void p224_felem_square(p224_widefelem out, const p224_felem in) {
375 p224_limb tmp0, tmp1, tmp2;
376 tmp0 = 2 * in[0];
377 tmp1 = 2 * in[1];
378 tmp2 = 2 * in[2];
379 out[0] = ((p224_widelimb)in[0]) * in[0];
380 out[1] = ((p224_widelimb)in[0]) * tmp1;
381 out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1];
382 out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2;
383 out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2];
384 out[5] = ((p224_widelimb)in[3]) * tmp2;
385 out[6] = ((p224_widelimb)in[3]) * in[3];
386 }
387
388 /* Multiply two field elements: out = in1 * in2 */
p224_felem_mul(p224_widefelem out,const p224_felem in1,const p224_felem in2)389 static void p224_felem_mul(p224_widefelem out, const p224_felem in1,
390 const p224_felem in2) {
391 out[0] = ((p224_widelimb)in1[0]) * in2[0];
392 out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0];
393 out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] +
394 ((p224_widelimb)in1[2]) * in2[0];
395 out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] +
396 ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0];
397 out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] +
398 ((p224_widelimb)in1[3]) * in2[1];
399 out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2];
400 out[6] = ((p224_widelimb)in1[3]) * in2[3];
401 }
402
403 /* Reduce seven 128-bit coefficients to four 64-bit coefficients.
404 * Requires in[i] < 2^126,
405 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
p224_felem_reduce(p224_felem out,const p224_widefelem in)406 static void p224_felem_reduce(p224_felem out, const p224_widefelem in) {
407 static const p224_widelimb two127p15 =
408 (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15);
409 static const p224_widelimb two127m71 =
410 (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71);
411 static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) -
412 (((p224_widelimb)1) << 71) -
413 (((p224_widelimb)1) << 55);
414 p224_widelimb output[5];
415
416 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
417 output[0] = in[0] + two127p15;
418 output[1] = in[1] + two127m71m55;
419 output[2] = in[2] + two127m71;
420 output[3] = in[3];
421 output[4] = in[4];
422
423 /* Eliminate in[4], in[5], in[6] */
424 output[4] += in[6] >> 16;
425 output[3] += (in[6] & 0xffff) << 40;
426 output[2] -= in[6];
427
428 output[3] += in[5] >> 16;
429 output[2] += (in[5] & 0xffff) << 40;
430 output[1] -= in[5];
431
432 output[2] += output[4] >> 16;
433 output[1] += (output[4] & 0xffff) << 40;
434 output[0] -= output[4];
435
436 /* Carry 2 -> 3 -> 4 */
437 output[3] += output[2] >> 56;
438 output[2] &= 0x00ffffffffffffff;
439
440 output[4] = output[3] >> 56;
441 output[3] &= 0x00ffffffffffffff;
442
443 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
444
445 /* Eliminate output[4] */
446 output[2] += output[4] >> 16;
447 /* output[2] < 2^56 + 2^56 = 2^57 */
448 output[1] += (output[4] & 0xffff) << 40;
449 output[0] -= output[4];
450
451 /* Carry 0 -> 1 -> 2 -> 3 */
452 output[1] += output[0] >> 56;
453 out[0] = output[0] & 0x00ffffffffffffff;
454
455 output[2] += output[1] >> 56;
456 /* output[2] < 2^57 + 2^72 */
457 out[1] = output[1] & 0x00ffffffffffffff;
458 output[3] += output[2] >> 56;
459 /* output[3] <= 2^56 + 2^16 */
460 out[2] = output[2] & 0x00ffffffffffffff;
461
462 /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
463 * out[3] <= 2^56 + 2^16 (due to final carry),
464 * so out < 2*p */
465 out[3] = output[3];
466 }
467
468 /* Reduce to unique minimal representation.
469 * Requires 0 <= in < 2*p (always call p224_felem_reduce first) */
p224_felem_contract(p224_felem out,const p224_felem in)470 static void p224_felem_contract(p224_felem out, const p224_felem in) {
471 static const int64_t two56 = ((p224_limb)1) << 56;
472 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
473 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
474 int64_t tmp[4], a;
475 tmp[0] = in[0];
476 tmp[1] = in[1];
477 tmp[2] = in[2];
478 tmp[3] = in[3];
479 /* Case 1: a = 1 iff in >= 2^224 */
480 a = (in[3] >> 56);
481 tmp[0] -= a;
482 tmp[1] += a << 40;
483 tmp[3] &= 0x00ffffffffffffff;
484 /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and
485 * the lower part is non-zero */
486 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
487 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
488 a &= 0x00ffffffffffffff;
489 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
490 a = (a - 1) >> 63;
491 /* subtract 2^224 - 2^96 + 1 if a is all-one */
492 tmp[3] &= a ^ 0xffffffffffffffff;
493 tmp[2] &= a ^ 0xffffffffffffffff;
494 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
495 tmp[0] -= 1 & a;
496
497 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
498 * be non-zero, so we only need one step */
499 a = tmp[0] >> 63;
500 tmp[0] += two56 & a;
501 tmp[1] -= 1 & a;
502
503 /* carry 1 -> 2 -> 3 */
504 tmp[2] += tmp[1] >> 56;
505 tmp[1] &= 0x00ffffffffffffff;
506
507 tmp[3] += tmp[2] >> 56;
508 tmp[2] &= 0x00ffffffffffffff;
509
510 /* Now 0 <= out < p */
511 out[0] = tmp[0];
512 out[1] = tmp[1];
513 out[2] = tmp[2];
514 out[3] = tmp[3];
515 }
516
517 /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
518 * elements are reduced to in < 2^225, so we only need to check three cases: 0,
519 * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */
p224_felem_is_zero(const p224_felem in)520 static p224_limb p224_felem_is_zero(const p224_felem in) {
521 p224_limb zero = in[0] | in[1] | in[2] | in[3];
522 zero = (((int64_t)(zero)-1) >> 63) & 1;
523
524 p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) |
525 (in[2] ^ 0x00ffffffffffffff) |
526 (in[3] ^ 0x00ffffffffffffff);
527 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1;
528 p224_limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) |
529 (in[2] ^ 0x00ffffffffffffff) |
530 (in[3] ^ 0x01ffffffffffffff);
531 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1;
532 return (zero | two224m96p1 | two225m97p2);
533 }
534
535 /* Invert a field element */
536 /* Computation chain copied from djb's code */
p224_felem_inv(p224_felem out,const p224_felem in)537 static void p224_felem_inv(p224_felem out, const p224_felem in) {
538 p224_felem ftmp, ftmp2, ftmp3, ftmp4;
539 p224_widefelem tmp;
540
541 p224_felem_square(tmp, in);
542 p224_felem_reduce(ftmp, tmp); /* 2 */
543 p224_felem_mul(tmp, in, ftmp);
544 p224_felem_reduce(ftmp, tmp); /* 2^2 - 1 */
545 p224_felem_square(tmp, ftmp);
546 p224_felem_reduce(ftmp, tmp); /* 2^3 - 2 */
547 p224_felem_mul(tmp, in, ftmp);
548 p224_felem_reduce(ftmp, tmp); /* 2^3 - 1 */
549 p224_felem_square(tmp, ftmp);
550 p224_felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
551 p224_felem_square(tmp, ftmp2);
552 p224_felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
553 p224_felem_square(tmp, ftmp2);
554 p224_felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
555 p224_felem_mul(tmp, ftmp2, ftmp);
556 p224_felem_reduce(ftmp, tmp); /* 2^6 - 1 */
557 p224_felem_square(tmp, ftmp);
558 p224_felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
559 for (size_t i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
560 p224_felem_square(tmp, ftmp2);
561 p224_felem_reduce(ftmp2, tmp);
562 }
563 p224_felem_mul(tmp, ftmp2, ftmp);
564 p224_felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
565 p224_felem_square(tmp, ftmp2);
566 p224_felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
567 for (size_t i = 0; i < 11; ++i) {/* 2^24 - 2^12 */
568 p224_felem_square(tmp, ftmp3);
569 p224_felem_reduce(ftmp3, tmp);
570 }
571 p224_felem_mul(tmp, ftmp3, ftmp2);
572 p224_felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
573 p224_felem_square(tmp, ftmp2);
574 p224_felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
575 for (size_t i = 0; i < 23; ++i) {/* 2^48 - 2^24 */
576 p224_felem_square(tmp, ftmp3);
577 p224_felem_reduce(ftmp3, tmp);
578 }
579 p224_felem_mul(tmp, ftmp3, ftmp2);
580 p224_felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
581 p224_felem_square(tmp, ftmp3);
582 p224_felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
583 for (size_t i = 0; i < 47; ++i) {/* 2^96 - 2^48 */
584 p224_felem_square(tmp, ftmp4);
585 p224_felem_reduce(ftmp4, tmp);
586 }
587 p224_felem_mul(tmp, ftmp3, ftmp4);
588 p224_felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
589 p224_felem_square(tmp, ftmp3);
590 p224_felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
591 for (size_t i = 0; i < 23; ++i) {/* 2^120 - 2^24 */
592 p224_felem_square(tmp, ftmp4);
593 p224_felem_reduce(ftmp4, tmp);
594 }
595 p224_felem_mul(tmp, ftmp2, ftmp4);
596 p224_felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
597 for (size_t i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
598 p224_felem_square(tmp, ftmp2);
599 p224_felem_reduce(ftmp2, tmp);
600 }
601 p224_felem_mul(tmp, ftmp2, ftmp);
602 p224_felem_reduce(ftmp, tmp); /* 2^126 - 1 */
603 p224_felem_square(tmp, ftmp);
604 p224_felem_reduce(ftmp, tmp); /* 2^127 - 2 */
605 p224_felem_mul(tmp, ftmp, in);
606 p224_felem_reduce(ftmp, tmp); /* 2^127 - 1 */
607 for (size_t i = 0; i < 97; ++i) {/* 2^224 - 2^97 */
608 p224_felem_square(tmp, ftmp);
609 p224_felem_reduce(ftmp, tmp);
610 }
611 p224_felem_mul(tmp, ftmp, ftmp3);
612 p224_felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
613 }
614
615 /* Copy in constant time:
616 * if icopy == 1, copy in to out,
617 * if icopy == 0, copy out to itself. */
p224_copy_conditional(p224_felem out,const p224_felem in,p224_limb icopy)618 static void p224_copy_conditional(p224_felem out, const p224_felem in,
619 p224_limb icopy) {
620 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
621 const p224_limb copy = -icopy;
622 for (size_t i = 0; i < 4; ++i) {
623 const p224_limb tmp = copy & (in[i] ^ out[i]);
624 out[i] ^= tmp;
625 }
626 }
627
628 /* ELLIPTIC CURVE POINT OPERATIONS
629 *
630 * Points are represented in Jacobian projective coordinates:
631 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
632 * or to the point at infinity if Z == 0. */
633
634 /* Double an elliptic curve point:
635 * (X', Y', Z') = 2 * (X, Y, Z), where
636 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
637 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
638 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
639 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
640 * while x_out == y_in is not (maybe this works, but it's not tested). */
p224_point_double(p224_felem x_out,p224_felem y_out,p224_felem z_out,const p224_felem x_in,const p224_felem y_in,const p224_felem z_in)641 static void p224_point_double(p224_felem x_out, p224_felem y_out,
642 p224_felem z_out, const p224_felem x_in,
643 const p224_felem y_in, const p224_felem z_in) {
644 p224_widefelem tmp, tmp2;
645 p224_felem delta, gamma, beta, alpha, ftmp, ftmp2;
646
647 p224_felem_assign(ftmp, x_in);
648 p224_felem_assign(ftmp2, x_in);
649
650 /* delta = z^2 */
651 p224_felem_square(tmp, z_in);
652 p224_felem_reduce(delta, tmp);
653
654 /* gamma = y^2 */
655 p224_felem_square(tmp, y_in);
656 p224_felem_reduce(gamma, tmp);
657
658 /* beta = x*gamma */
659 p224_felem_mul(tmp, x_in, gamma);
660 p224_felem_reduce(beta, tmp);
661
662 /* alpha = 3*(x-delta)*(x+delta) */
663 p224_felem_diff(ftmp, delta);
664 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
665 p224_felem_sum(ftmp2, delta);
666 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
667 p224_felem_scalar(ftmp2, 3);
668 /* ftmp2[i] < 3 * 2^58 < 2^60 */
669 p224_felem_mul(tmp, ftmp, ftmp2);
670 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
671 p224_felem_reduce(alpha, tmp);
672
673 /* x' = alpha^2 - 8*beta */
674 p224_felem_square(tmp, alpha);
675 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
676 p224_felem_assign(ftmp, beta);
677 p224_felem_scalar(ftmp, 8);
678 /* ftmp[i] < 8 * 2^57 = 2^60 */
679 p224_felem_diff_128_64(tmp, ftmp);
680 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
681 p224_felem_reduce(x_out, tmp);
682
683 /* z' = (y + z)^2 - gamma - delta */
684 p224_felem_sum(delta, gamma);
685 /* delta[i] < 2^57 + 2^57 = 2^58 */
686 p224_felem_assign(ftmp, y_in);
687 p224_felem_sum(ftmp, z_in);
688 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
689 p224_felem_square(tmp, ftmp);
690 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
691 p224_felem_diff_128_64(tmp, delta);
692 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
693 p224_felem_reduce(z_out, tmp);
694
695 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
696 p224_felem_scalar(beta, 4);
697 /* beta[i] < 4 * 2^57 = 2^59 */
698 p224_felem_diff(beta, x_out);
699 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
700 p224_felem_mul(tmp, alpha, beta);
701 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
702 p224_felem_square(tmp2, gamma);
703 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
704 p224_widefelem_scalar(tmp2, 8);
705 /* tmp2[i] < 8 * 2^116 = 2^119 */
706 p224_widefelem_diff(tmp, tmp2);
707 /* tmp[i] < 2^119 + 2^120 < 2^121 */
708 p224_felem_reduce(y_out, tmp);
709 }
710
711 /* Add two elliptic curve points:
712 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
713 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
714 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
715 * 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 *
716 * X_1)^2 - X_3) -
717 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
718 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
719 *
720 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */
721
722 /* This function is not entirely constant-time: it includes a branch for
723 * checking whether the two input points are equal, (while not equal to the
724 * point at infinity). This case never happens during single point
725 * multiplication, so there is no timing leak for ECDH or ECDSA signing. */
p224_point_add(p224_felem x3,p224_felem y3,p224_felem z3,const p224_felem x1,const p224_felem y1,const p224_felem z1,const int mixed,const p224_felem x2,const p224_felem y2,const p224_felem z2)726 static void p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3,
727 const p224_felem x1, const p224_felem y1,
728 const p224_felem z1, const int mixed,
729 const p224_felem x2, const p224_felem y2,
730 const p224_felem z2) {
731 p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
732 p224_widefelem tmp, tmp2;
733 p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal;
734
735 if (!mixed) {
736 /* ftmp2 = z2^2 */
737 p224_felem_square(tmp, z2);
738 p224_felem_reduce(ftmp2, tmp);
739
740 /* ftmp4 = z2^3 */
741 p224_felem_mul(tmp, ftmp2, z2);
742 p224_felem_reduce(ftmp4, tmp);
743
744 /* ftmp4 = z2^3*y1 */
745 p224_felem_mul(tmp2, ftmp4, y1);
746 p224_felem_reduce(ftmp4, tmp2);
747
748 /* ftmp2 = z2^2*x1 */
749 p224_felem_mul(tmp2, ftmp2, x1);
750 p224_felem_reduce(ftmp2, tmp2);
751 } else {
752 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
753
754 /* ftmp4 = z2^3*y1 */
755 p224_felem_assign(ftmp4, y1);
756
757 /* ftmp2 = z2^2*x1 */
758 p224_felem_assign(ftmp2, x1);
759 }
760
761 /* ftmp = z1^2 */
762 p224_felem_square(tmp, z1);
763 p224_felem_reduce(ftmp, tmp);
764
765 /* ftmp3 = z1^3 */
766 p224_felem_mul(tmp, ftmp, z1);
767 p224_felem_reduce(ftmp3, tmp);
768
769 /* tmp = z1^3*y2 */
770 p224_felem_mul(tmp, ftmp3, y2);
771 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
772
773 /* ftmp3 = z1^3*y2 - z2^3*y1 */
774 p224_felem_diff_128_64(tmp, ftmp4);
775 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
776 p224_felem_reduce(ftmp3, tmp);
777
778 /* tmp = z1^2*x2 */
779 p224_felem_mul(tmp, ftmp, x2);
780 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
781
782 /* ftmp = z1^2*x2 - z2^2*x1 */
783 p224_felem_diff_128_64(tmp, ftmp2);
784 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
785 p224_felem_reduce(ftmp, tmp);
786
787 /* the formulae are incorrect if the points are equal
788 * so we check for this and do doubling if this happens */
789 x_equal = p224_felem_is_zero(ftmp);
790 y_equal = p224_felem_is_zero(ftmp3);
791 z1_is_zero = p224_felem_is_zero(z1);
792 z2_is_zero = p224_felem_is_zero(z2);
793 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
794 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
795 p224_point_double(x3, y3, z3, x1, y1, z1);
796 return;
797 }
798
799 /* ftmp5 = z1*z2 */
800 if (!mixed) {
801 p224_felem_mul(tmp, z1, z2);
802 p224_felem_reduce(ftmp5, tmp);
803 } else {
804 /* special case z2 = 0 is handled later */
805 p224_felem_assign(ftmp5, z1);
806 }
807
808 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
809 p224_felem_mul(tmp, ftmp, ftmp5);
810 p224_felem_reduce(z_out, tmp);
811
812 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
813 p224_felem_assign(ftmp5, ftmp);
814 p224_felem_square(tmp, ftmp);
815 p224_felem_reduce(ftmp, tmp);
816
817 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
818 p224_felem_mul(tmp, ftmp, ftmp5);
819 p224_felem_reduce(ftmp5, tmp);
820
821 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
822 p224_felem_mul(tmp, ftmp2, ftmp);
823 p224_felem_reduce(ftmp2, tmp);
824
825 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
826 p224_felem_mul(tmp, ftmp4, ftmp5);
827 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
828
829 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
830 p224_felem_square(tmp2, ftmp3);
831 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
832
833 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
834 p224_felem_diff_128_64(tmp2, ftmp5);
835 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
836
837 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
838 p224_felem_assign(ftmp5, ftmp2);
839 p224_felem_scalar(ftmp5, 2);
840 /* ftmp5[i] < 2 * 2^57 = 2^58 */
841
842 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
843 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
844 p224_felem_diff_128_64(tmp2, ftmp5);
845 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
846 p224_felem_reduce(x_out, tmp2);
847
848 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
849 p224_felem_diff(ftmp2, x_out);
850 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
851
852 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
853 p224_felem_mul(tmp2, ftmp3, ftmp2);
854 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
855
856 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
857 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
858 p224_widefelem_diff(tmp2, tmp);
859 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
860 p224_felem_reduce(y_out, tmp2);
861
862 /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
863 * the point at infinity, so we need to check for this separately */
864
865 /* if point 1 is at infinity, copy point 2 to output, and vice versa */
866 p224_copy_conditional(x_out, x2, z1_is_zero);
867 p224_copy_conditional(x_out, x1, z2_is_zero);
868 p224_copy_conditional(y_out, y2, z1_is_zero);
869 p224_copy_conditional(y_out, y1, z2_is_zero);
870 p224_copy_conditional(z_out, z2, z1_is_zero);
871 p224_copy_conditional(z_out, z1, z2_is_zero);
872 p224_felem_assign(x3, x_out);
873 p224_felem_assign(y3, y_out);
874 p224_felem_assign(z3, z_out);
875 }
876
877 /* p224_select_point selects the |idx|th point from a precomputation table and
878 * copies it to out. */
p224_select_point(const uint64_t idx,size_t size,const p224_felem pre_comp[][3],p224_felem out[3])879 static void p224_select_point(const uint64_t idx, size_t size,
880 const p224_felem pre_comp[/*size*/][3],
881 p224_felem out[3]) {
882 p224_limb *outlimbs = &out[0][0];
883 OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem));
884
885 for (size_t i = 0; i < size; i++) {
886 const p224_limb *inlimbs = &pre_comp[i][0][0];
887 uint64_t mask = i ^ idx;
888 mask |= mask >> 4;
889 mask |= mask >> 2;
890 mask |= mask >> 1;
891 mask &= 1;
892 mask--;
893 for (size_t j = 0; j < 4 * 3; j++) {
894 outlimbs[j] |= inlimbs[j] & mask;
895 }
896 }
897 }
898
899 /* p224_get_bit returns the |i|th bit in |in| */
p224_get_bit(const p224_felem_bytearray in,size_t i)900 static char p224_get_bit(const p224_felem_bytearray in, size_t i) {
901 if (i >= 224) {
902 return 0;
903 }
904 return (in[i >> 3] >> (i & 7)) & 1;
905 }
906
907 /* Interleaved point multiplication using precomputed point multiples:
908 * The small point multiples 0*P, 1*P, ..., 16*P are in p_pre_comp, the scalars
909 * in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple
910 * of the generator, using certain (large) precomputed multiples in
911 * g_p224_pre_comp. Output point (X, Y, Z) is stored in x_out, y_out, z_out */
p224_batch_mul(p224_felem x_out,p224_felem y_out,p224_felem z_out,const uint8_t * p_scalar,const uint8_t * g_scalar,const p224_felem p_pre_comp[17][3])912 static void p224_batch_mul(p224_felem x_out, p224_felem y_out, p224_felem z_out,
913 const uint8_t *p_scalar, const uint8_t *g_scalar,
914 const p224_felem p_pre_comp[17][3]) {
915 p224_felem nq[3], tmp[4];
916 uint64_t bits;
917 uint8_t sign, digit;
918
919 /* set nq to the point at infinity */
920 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));
921
922 /* Loop over both scalars msb-to-lsb, interleaving additions of multiples of
923 * the generator (two in each of the last 28 rounds) and additions of p (every
924 * 5th round). */
925 int skip = 1; /* save two point operations in the first round */
926 size_t i = p_scalar != NULL ? 220 : 27;
927 for (;;) {
928 /* double */
929 if (!skip) {
930 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
931 }
932
933 /* add multiples of the generator */
934 if (g_scalar != NULL && i <= 27) {
935 /* first, look 28 bits upwards */
936 bits = p224_get_bit(g_scalar, i + 196) << 3;
937 bits |= p224_get_bit(g_scalar, i + 140) << 2;
938 bits |= p224_get_bit(g_scalar, i + 84) << 1;
939 bits |= p224_get_bit(g_scalar, i + 28);
940 /* select the point to add, in constant time */
941 p224_select_point(bits, 16, g_p224_pre_comp[1], tmp);
942
943 if (!skip) {
944 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
945 tmp[0], tmp[1], tmp[2]);
946 } else {
947 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));
948 skip = 0;
949 }
950
951 /* second, look at the current position */
952 bits = p224_get_bit(g_scalar, i + 168) << 3;
953 bits |= p224_get_bit(g_scalar, i + 112) << 2;
954 bits |= p224_get_bit(g_scalar, i + 56) << 1;
955 bits |= p224_get_bit(g_scalar, i);
956 /* select the point to add, in constant time */
957 p224_select_point(bits, 16, g_p224_pre_comp[0], tmp);
958 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
959 tmp[0], tmp[1], tmp[2]);
960 }
961
962 /* do other additions every 5 doublings */
963 if (p_scalar != NULL && i % 5 == 0) {
964 bits = p224_get_bit(p_scalar, i + 4) << 5;
965 bits |= p224_get_bit(p_scalar, i + 3) << 4;
966 bits |= p224_get_bit(p_scalar, i + 2) << 3;
967 bits |= p224_get_bit(p_scalar, i + 1) << 2;
968 bits |= p224_get_bit(p_scalar, i) << 1;
969 bits |= p224_get_bit(p_scalar, i - 1);
970 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
971
972 /* select the point to add or subtract */
973 p224_select_point(digit, 17, p_pre_comp, tmp);
974 p224_felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
975 p224_copy_conditional(tmp[1], tmp[3], sign);
976
977 if (!skip) {
978 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */,
979 tmp[0], tmp[1], tmp[2]);
980 } else {
981 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));
982 skip = 0;
983 }
984 }
985
986 if (i == 0) {
987 break;
988 }
989 --i;
990 }
991 p224_felem_assign(x_out, nq[0]);
992 p224_felem_assign(y_out, nq[1]);
993 p224_felem_assign(z_out, nq[2]);
994 }
995
996 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
997 * (X', Y') = (X/Z^2, Y/Z^3) */
ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BN_CTX * ctx)998 static int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
999 const EC_POINT *point,
1000 BIGNUM *x, BIGNUM *y,
1001 BN_CTX *ctx) {
1002 p224_felem z1, z2, x_in, y_in, x_out, y_out;
1003 p224_widefelem tmp;
1004
1005 if (EC_POINT_is_at_infinity(group, point)) {
1006 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
1007 return 0;
1008 }
1009
1010 if (!p224_BN_to_felem(x_in, &point->X) ||
1011 !p224_BN_to_felem(y_in, &point->Y) ||
1012 !p224_BN_to_felem(z1, &point->Z)) {
1013 return 0;
1014 }
1015
1016 p224_felem_inv(z2, z1);
1017 p224_felem_square(tmp, z2);
1018 p224_felem_reduce(z1, tmp);
1019 p224_felem_mul(tmp, x_in, z1);
1020 p224_felem_reduce(x_in, tmp);
1021 p224_felem_contract(x_out, x_in);
1022 if (x != NULL && !p224_felem_to_BN(x, x_out)) {
1023 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1024 return 0;
1025 }
1026
1027 p224_felem_mul(tmp, z1, z2);
1028 p224_felem_reduce(z1, tmp);
1029 p224_felem_mul(tmp, y_in, z1);
1030 p224_felem_reduce(y_in, tmp);
1031 p224_felem_contract(y_out, y_in);
1032 if (y != NULL && !p224_felem_to_BN(y, y_out)) {
1033 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1034 return 0;
1035 }
1036
1037 return 1;
1038 }
1039
ec_GFp_nistp224_points_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * g_scalar,const EC_POINT * p,const BIGNUM * p_scalar,BN_CTX * ctx)1040 static int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1041 const BIGNUM *g_scalar, const EC_POINT *p,
1042 const BIGNUM *p_scalar, BN_CTX *ctx) {
1043 int ret = 0;
1044 BN_CTX *new_ctx = NULL;
1045 BIGNUM *x, *y, *z, *tmp_scalar;
1046 p224_felem_bytearray g_secret, p_secret;
1047 p224_felem p_pre_comp[17][3];
1048 p224_felem_bytearray tmp;
1049 p224_felem x_in, y_in, z_in, x_out, y_out, z_out;
1050
1051 if (ctx == NULL) {
1052 ctx = BN_CTX_new();
1053 new_ctx = ctx;
1054 if (ctx == NULL) {
1055 return 0;
1056 }
1057 }
1058
1059 BN_CTX_start(ctx);
1060 if ((x = BN_CTX_get(ctx)) == NULL ||
1061 (y = BN_CTX_get(ctx)) == NULL ||
1062 (z = BN_CTX_get(ctx)) == NULL ||
1063 (tmp_scalar = BN_CTX_get(ctx)) == NULL) {
1064 goto err;
1065 }
1066
1067 if (p != NULL && p_scalar != NULL) {
1068 /* We treat NULL scalars as 0, and NULL points as points at infinity, i.e.,
1069 * they contribute nothing to the linear combination. */
1070 OPENSSL_memset(&p_secret, 0, sizeof(p_secret));
1071 OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp));
1072 size_t num_bytes;
1073 /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1074 if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) {
1075 /* this is an unusual input, and we don't guarantee
1076 * constant-timeness */
1077 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
1078 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1079 goto err;
1080 }
1081 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1082 } else {
1083 num_bytes = BN_bn2bin(p_scalar, tmp);
1084 }
1085
1086 p224_flip_endian(p_secret, tmp, num_bytes);
1087 /* precompute multiples */
1088 if (!p224_BN_to_felem(x_out, &p->X) ||
1089 !p224_BN_to_felem(y_out, &p->Y) ||
1090 !p224_BN_to_felem(z_out, &p->Z)) {
1091 goto err;
1092 }
1093
1094 p224_felem_assign(p_pre_comp[1][0], x_out);
1095 p224_felem_assign(p_pre_comp[1][1], y_out);
1096 p224_felem_assign(p_pre_comp[1][2], z_out);
1097
1098 for (size_t j = 2; j <= 16; ++j) {
1099 if (j & 1) {
1100 p224_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2],
1101 p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2],
1102 0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1],
1103 p_pre_comp[j - 1][2]);
1104 } else {
1105 p224_point_double(p_pre_comp[j][0], p_pre_comp[j][1],
1106 p_pre_comp[j][2], p_pre_comp[j / 2][0],
1107 p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]);
1108 }
1109 }
1110 }
1111
1112 if (g_scalar != NULL) {
1113 OPENSSL_memset(g_secret, 0, sizeof(g_secret));
1114 size_t num_bytes;
1115 /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1116 if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) {
1117 /* this is an unusual input, and we don't guarantee constant-timeness */
1118 if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) {
1119 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1120 goto err;
1121 }
1122 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1123 } else {
1124 num_bytes = BN_bn2bin(g_scalar, tmp);
1125 }
1126
1127 p224_flip_endian(g_secret, tmp, num_bytes);
1128 }
1129 p224_batch_mul(
1130 x_out, y_out, z_out, (p != NULL && p_scalar != NULL) ? p_secret : NULL,
1131 g_scalar != NULL ? g_secret : NULL, (const p224_felem(*)[3])p_pre_comp);
1132
1133 /* reduce the output to its unique minimal representation */
1134 p224_felem_contract(x_in, x_out);
1135 p224_felem_contract(y_in, y_out);
1136 p224_felem_contract(z_in, z_out);
1137 if (!p224_felem_to_BN(x, x_in) ||
1138 !p224_felem_to_BN(y, y_in) ||
1139 !p224_felem_to_BN(z, z_in)) {
1140 OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1141 goto err;
1142 }
1143 ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1144
1145 err:
1146 BN_CTX_end(ctx);
1147 BN_CTX_free(new_ctx);
1148 return ret;
1149 }
1150
DEFINE_METHOD_FUNCTION(EC_METHOD,EC_GFp_nistp224_method)1151 DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) {
1152 out->group_init = ec_GFp_simple_group_init;
1153 out->group_finish = ec_GFp_simple_group_finish;
1154 out->group_copy = ec_GFp_simple_group_copy;
1155 out->group_set_curve = ec_GFp_simple_group_set_curve;
1156 out->point_get_affine_coordinates =
1157 ec_GFp_nistp224_point_get_affine_coordinates;
1158 out->mul = ec_GFp_nistp224_points_mul;
1159 out->field_mul = ec_GFp_simple_field_mul;
1160 out->field_sqr = ec_GFp_simple_field_sqr;
1161 out->field_encode = NULL;
1162 out->field_decode = NULL;
1163 };
1164
1165 #endif /* 64_BIT && !WINDOWS && !SMALL */
1166