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 "../internal.h"
34 
35 
36 typedef uint8_t u8;
37 typedef uint64_t u64;
38 typedef int64_t s64;
39 
40 /* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
41  * using 64-bit coefficients called 'limbs', and sometimes (for multiplication
42  * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 +
43  * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-limb
44  * representation is an 'felem'; a 7-widelimb representation is a 'widefelem'.
45  * Even within felems, bits of adjacent limbs overlap, and we don't always
46  * reduce the representations: we ensure that inputs to each felem
47  * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, and
48  * fit into a 128-bit word without overflow. The coefficients are then again
49  * partially reduced to obtain an felem satisfying a_i < 2^57. We only reduce
50  * to the unique minimal representation at the end of the computation. */
51 
52 typedef uint64_t limb;
53 typedef uint128_t widelimb;
54 
55 typedef limb felem[4];
56 typedef widelimb widefelem[7];
57 
58 /* Field element represented as a byte arrary. 28*8 = 224 bits is also the
59  * group order size for the elliptic curve, and we also use this type for
60  * scalars for point multiplication. */
61 typedef u8 felem_bytearray[28];
62 
63 /* Precomputed multiples of the standard generator
64  * Points are given in coordinates (X, Y, Z) where Z normally is 1
65  * (0 for the point at infinity).
66  * For each field element, slice a_0 is word 0, etc.
67  *
68  * The table has 2 * 16 elements, starting with the following:
69  * index | bits    | point
70  * ------+---------+------------------------------
71  *     0 | 0 0 0 0 | 0G
72  *     1 | 0 0 0 1 | 1G
73  *     2 | 0 0 1 0 | 2^56G
74  *     3 | 0 0 1 1 | (2^56 + 1)G
75  *     4 | 0 1 0 0 | 2^112G
76  *     5 | 0 1 0 1 | (2^112 + 1)G
77  *     6 | 0 1 1 0 | (2^112 + 2^56)G
78  *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
79  *     8 | 1 0 0 0 | 2^168G
80  *     9 | 1 0 0 1 | (2^168 + 1)G
81  *    10 | 1 0 1 0 | (2^168 + 2^56)G
82  *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
83  *    12 | 1 1 0 0 | (2^168 + 2^112)G
84  *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
85  *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
86  *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
87  * followed by a copy of this with each element multiplied by 2^28.
88  *
89  * The reason for this is so that we can clock bits into four different
90  * locations when doing simple scalar multiplies against the base point,
91  * and then another four locations using the second 16 elements. */
92 static const felem g_pre_comp[2][16][3] = {
93     {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
94      {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
95       {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
96       {1, 0, 0, 0}},
97      {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
98       {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
99       {1, 0, 0, 0}},
100      {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
101       {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
102       {1, 0, 0, 0}},
103      {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
104       {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
105       {1, 0, 0, 0}},
106      {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
107       {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
108       {1, 0, 0, 0}},
109      {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
110       {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
111       {1, 0, 0, 0}},
112      {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
113       {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
114       {1, 0, 0, 0}},
115      {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
116       {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
117       {1, 0, 0, 0}},
118      {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
119       {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
120       {1, 0, 0, 0}},
121      {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
122       {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
123       {1, 0, 0, 0}},
124      {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
125       {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
126       {1, 0, 0, 0}},
127      {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
128       {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
129       {1, 0, 0, 0}},
130      {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
131       {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
132       {1, 0, 0, 0}},
133      {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
134       {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
135       {1, 0, 0, 0}},
136      {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
137       {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
138       {1, 0, 0, 0}}},
139     {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
140      {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
141       {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
142       {1, 0, 0, 0}},
143      {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
144       {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
145       {1, 0, 0, 0}},
146      {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
147       {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
148       {1, 0, 0, 0}},
149      {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
150       {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
151       {1, 0, 0, 0}},
152      {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
153       {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
154       {1, 0, 0, 0}},
155      {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
156       {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
157       {1, 0, 0, 0}},
158      {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
159       {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
160       {1, 0, 0, 0}},
161      {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
162       {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
163       {1, 0, 0, 0}},
164      {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
165       {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
166       {1, 0, 0, 0}},
167      {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
168       {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
169       {1, 0, 0, 0}},
170      {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
171       {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
172       {1, 0, 0, 0}},
173      {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
174       {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
175       {1, 0, 0, 0}},
176      {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
177       {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
178       {1, 0, 0, 0}},
179      {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
180       {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
181       {1, 0, 0, 0}},
182      {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
183       {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
184       {1, 0, 0, 0}}}};
185 
186 /* Helper functions to convert field elements to/from internal representation */
bin28_to_felem(felem out,const u8 in[28])187 static void bin28_to_felem(felem out, const u8 in[28]) {
188   out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
189   out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
190   out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
191   out[3] = (*((const uint64_t *)(in + 20))) >> 8;
192 }
193 
felem_to_bin28(u8 out[28],const felem in)194 static void felem_to_bin28(u8 out[28], const felem in) {
195   for (size_t i = 0; i < 7; ++i) {
196     out[i] = in[0] >> (8 * i);
197     out[i + 7] = in[1] >> (8 * i);
198     out[i + 14] = in[2] >> (8 * i);
199     out[i + 21] = in[3] >> (8 * i);
200   }
201 }
202 
203 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
flip_endian(u8 * out,const u8 * in,size_t len)204 static void flip_endian(u8 *out, const u8 *in, size_t len) {
205   for (size_t i = 0; i < len; ++i) {
206     out[i] = in[len - 1 - i];
207   }
208 }
209 
210 /* From OpenSSL BIGNUM to internal representation */
BN_to_felem(felem out,const BIGNUM * bn)211 static int BN_to_felem(felem out, const BIGNUM *bn) {
212   /* BN_bn2bin eats leading zeroes */
213   felem_bytearray b_out;
214   OPENSSL_memset(b_out, 0, sizeof(b_out));
215   size_t num_bytes = BN_num_bytes(bn);
216   if (num_bytes > sizeof(b_out) ||
217       BN_is_negative(bn)) {
218     OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE);
219     return 0;
220   }
221 
222   felem_bytearray b_in;
223   num_bytes = BN_bn2bin(bn, b_in);
224   flip_endian(b_out, b_in, num_bytes);
225   bin28_to_felem(out, b_out);
226   return 1;
227 }
228 
229 /* From internal representation to OpenSSL BIGNUM */
felem_to_BN(BIGNUM * out,const felem in)230 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) {
231   felem_bytearray b_in, b_out;
232   felem_to_bin28(b_in, in);
233   flip_endian(b_out, b_in, sizeof(b_out));
234   return BN_bin2bn(b_out, sizeof(b_out), out);
235 }
236 
237 /* Field operations, using the internal representation of field elements.
238  * NB! These operations are specific to our point multiplication and cannot be
239  * expected to be correct in general - e.g., multiplication with a large scalar
240  * will cause an overflow. */
241 
felem_assign(felem out,const felem in)242 static void felem_assign(felem out, const felem in) {
243   out[0] = in[0];
244   out[1] = in[1];
245   out[2] = in[2];
246   out[3] = in[3];
247 }
248 
249 /* Sum two field elements: out += in */
felem_sum(felem out,const felem in)250 static void felem_sum(felem out, const felem in) {
251   out[0] += in[0];
252   out[1] += in[1];
253   out[2] += in[2];
254   out[3] += in[3];
255 }
256 
257 /* Get negative value: out = -in */
258 /* Assumes in[i] < 2^57 */
felem_neg(felem out,const felem in)259 static void felem_neg(felem out, const felem in) {
260   static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
261   static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
262   static const limb two58m42m2 =
263       (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
264 
265   /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
266   out[0] = two58p2 - in[0];
267   out[1] = two58m42m2 - in[1];
268   out[2] = two58m2 - in[2];
269   out[3] = two58m2 - in[3];
270 }
271 
272 /* Subtract field elements: out -= in */
273 /* Assumes in[i] < 2^57 */
felem_diff(felem out,const felem in)274 static void felem_diff(felem out, const felem in) {
275   static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
276   static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
277   static const limb two58m42m2 =
278       (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
279 
280   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
281   out[0] += two58p2;
282   out[1] += two58m42m2;
283   out[2] += two58m2;
284   out[3] += two58m2;
285 
286   out[0] -= in[0];
287   out[1] -= in[1];
288   out[2] -= in[2];
289   out[3] -= in[3];
290 }
291 
292 /* Subtract in unreduced 128-bit mode: out -= in */
293 /* Assumes in[i] < 2^119 */
widefelem_diff(widefelem out,const widefelem in)294 static void widefelem_diff(widefelem out, const widefelem in) {
295   static const widelimb two120 = ((widelimb)1) << 120;
296   static const widelimb two120m64 =
297       (((widelimb)1) << 120) - (((widelimb)1) << 64);
298   static const widelimb two120m104m64 =
299       (((widelimb)1) << 120) - (((widelimb)1) << 104) - (((widelimb)1) << 64);
300 
301   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
302   out[0] += two120;
303   out[1] += two120m64;
304   out[2] += two120m64;
305   out[3] += two120;
306   out[4] += two120m104m64;
307   out[5] += two120m64;
308   out[6] += two120m64;
309 
310   out[0] -= in[0];
311   out[1] -= in[1];
312   out[2] -= in[2];
313   out[3] -= in[3];
314   out[4] -= in[4];
315   out[5] -= in[5];
316   out[6] -= in[6];
317 }
318 
319 /* Subtract in mixed mode: out128 -= in64 */
320 /* in[i] < 2^63 */
felem_diff_128_64(widefelem out,const felem in)321 static void felem_diff_128_64(widefelem out, const felem in) {
322   static const widelimb two64p8 = (((widelimb)1) << 64) + (((widelimb)1) << 8);
323   static const widelimb two64m8 = (((widelimb)1) << 64) - (((widelimb)1) << 8);
324   static const widelimb two64m48m8 =
325       (((widelimb)1) << 64) - (((widelimb)1) << 48) - (((widelimb)1) << 8);
326 
327   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
328   out[0] += two64p8;
329   out[1] += two64m48m8;
330   out[2] += two64m8;
331   out[3] += two64m8;
332 
333   out[0] -= in[0];
334   out[1] -= in[1];
335   out[2] -= in[2];
336   out[3] -= in[3];
337 }
338 
339 /* Multiply a field element by a scalar: out = out * scalar
340  * The scalars we actually use are small, so results fit without overflow */
felem_scalar(felem out,const limb scalar)341 static void felem_scalar(felem out, const limb scalar) {
342   out[0] *= scalar;
343   out[1] *= scalar;
344   out[2] *= scalar;
345   out[3] *= scalar;
346 }
347 
348 /* Multiply an unreduced field element by a scalar: out = out * scalar
349  * The scalars we actually use are small, so results fit without overflow */
widefelem_scalar(widefelem out,const widelimb scalar)350 static void widefelem_scalar(widefelem out, const widelimb scalar) {
351   out[0] *= scalar;
352   out[1] *= scalar;
353   out[2] *= scalar;
354   out[3] *= scalar;
355   out[4] *= scalar;
356   out[5] *= scalar;
357   out[6] *= scalar;
358 }
359 
360 /* Square a field element: out = in^2 */
felem_square(widefelem out,const felem in)361 static void felem_square(widefelem out, const felem in) {
362   limb tmp0, tmp1, tmp2;
363   tmp0 = 2 * in[0];
364   tmp1 = 2 * in[1];
365   tmp2 = 2 * in[2];
366   out[0] = ((widelimb)in[0]) * in[0];
367   out[1] = ((widelimb)in[0]) * tmp1;
368   out[2] = ((widelimb)in[0]) * tmp2 + ((widelimb)in[1]) * in[1];
369   out[3] = ((widelimb)in[3]) * tmp0 + ((widelimb)in[1]) * tmp2;
370   out[4] = ((widelimb)in[3]) * tmp1 + ((widelimb)in[2]) * in[2];
371   out[5] = ((widelimb)in[3]) * tmp2;
372   out[6] = ((widelimb)in[3]) * in[3];
373 }
374 
375 /* Multiply two field elements: out = in1 * in2 */
felem_mul(widefelem out,const felem in1,const felem in2)376 static void felem_mul(widefelem out, const felem in1, const felem in2) {
377   out[0] = ((widelimb)in1[0]) * in2[0];
378   out[1] = ((widelimb)in1[0]) * in2[1] + ((widelimb)in1[1]) * in2[0];
379   out[2] = ((widelimb)in1[0]) * in2[2] + ((widelimb)in1[1]) * in2[1] +
380            ((widelimb)in1[2]) * in2[0];
381   out[3] = ((widelimb)in1[0]) * in2[3] + ((widelimb)in1[1]) * in2[2] +
382            ((widelimb)in1[2]) * in2[1] + ((widelimb)in1[3]) * in2[0];
383   out[4] = ((widelimb)in1[1]) * in2[3] + ((widelimb)in1[2]) * in2[2] +
384            ((widelimb)in1[3]) * in2[1];
385   out[5] = ((widelimb)in1[2]) * in2[3] + ((widelimb)in1[3]) * in2[2];
386   out[6] = ((widelimb)in1[3]) * in2[3];
387 }
388 
389 /* Reduce seven 128-bit coefficients to four 64-bit coefficients.
390  * Requires in[i] < 2^126,
391  * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
felem_reduce(felem out,const widefelem in)392 static void felem_reduce(felem out, const widefelem in) {
393   static const widelimb two127p15 =
394       (((widelimb)1) << 127) + (((widelimb)1) << 15);
395   static const widelimb two127m71 =
396       (((widelimb)1) << 127) - (((widelimb)1) << 71);
397   static const widelimb two127m71m55 =
398       (((widelimb)1) << 127) - (((widelimb)1) << 71) - (((widelimb)1) << 55);
399   widelimb output[5];
400 
401   /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
402   output[0] = in[0] + two127p15;
403   output[1] = in[1] + two127m71m55;
404   output[2] = in[2] + two127m71;
405   output[3] = in[3];
406   output[4] = in[4];
407 
408   /* Eliminate in[4], in[5], in[6] */
409   output[4] += in[6] >> 16;
410   output[3] += (in[6] & 0xffff) << 40;
411   output[2] -= in[6];
412 
413   output[3] += in[5] >> 16;
414   output[2] += (in[5] & 0xffff) << 40;
415   output[1] -= in[5];
416 
417   output[2] += output[4] >> 16;
418   output[1] += (output[4] & 0xffff) << 40;
419   output[0] -= output[4];
420 
421   /* Carry 2 -> 3 -> 4 */
422   output[3] += output[2] >> 56;
423   output[2] &= 0x00ffffffffffffff;
424 
425   output[4] = output[3] >> 56;
426   output[3] &= 0x00ffffffffffffff;
427 
428   /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
429 
430   /* Eliminate output[4] */
431   output[2] += output[4] >> 16;
432   /* output[2] < 2^56 + 2^56 = 2^57 */
433   output[1] += (output[4] & 0xffff) << 40;
434   output[0] -= output[4];
435 
436   /* Carry 0 -> 1 -> 2 -> 3 */
437   output[1] += output[0] >> 56;
438   out[0] = output[0] & 0x00ffffffffffffff;
439 
440   output[2] += output[1] >> 56;
441   /* output[2] < 2^57 + 2^72 */
442   out[1] = output[1] & 0x00ffffffffffffff;
443   output[3] += output[2] >> 56;
444   /* output[3] <= 2^56 + 2^16 */
445   out[2] = output[2] & 0x00ffffffffffffff;
446 
447   /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
448    * out[3] <= 2^56 + 2^16 (due to final carry),
449    * so out < 2*p */
450   out[3] = output[3];
451 }
452 
453 /* Reduce to unique minimal representation.
454  * Requires 0 <= in < 2*p (always call felem_reduce first) */
felem_contract(felem out,const felem in)455 static void felem_contract(felem out, const felem in) {
456   static const int64_t two56 = ((limb)1) << 56;
457   /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
458   /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
459   int64_t tmp[4], a;
460   tmp[0] = in[0];
461   tmp[1] = in[1];
462   tmp[2] = in[2];
463   tmp[3] = in[3];
464   /* Case 1: a = 1 iff in >= 2^224 */
465   a = (in[3] >> 56);
466   tmp[0] -= a;
467   tmp[1] += a << 40;
468   tmp[3] &= 0x00ffffffffffffff;
469   /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and
470    * the lower part is non-zero */
471   a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
472       (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
473   a &= 0x00ffffffffffffff;
474   /* turn a into an all-one mask (if a = 0) or an all-zero mask */
475   a = (a - 1) >> 63;
476   /* subtract 2^224 - 2^96 + 1 if a is all-one */
477   tmp[3] &= a ^ 0xffffffffffffffff;
478   tmp[2] &= a ^ 0xffffffffffffffff;
479   tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
480   tmp[0] -= 1 & a;
481 
482   /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
483    * be non-zero, so we only need one step */
484   a = tmp[0] >> 63;
485   tmp[0] += two56 & a;
486   tmp[1] -= 1 & a;
487 
488   /* carry 1 -> 2 -> 3 */
489   tmp[2] += tmp[1] >> 56;
490   tmp[1] &= 0x00ffffffffffffff;
491 
492   tmp[3] += tmp[2] >> 56;
493   tmp[2] &= 0x00ffffffffffffff;
494 
495   /* Now 0 <= out < p */
496   out[0] = tmp[0];
497   out[1] = tmp[1];
498   out[2] = tmp[2];
499   out[3] = tmp[3];
500 }
501 
502 /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
503  * elements are reduced to in < 2^225, so we only need to check three cases: 0,
504  * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */
felem_is_zero(const felem in)505 static limb felem_is_zero(const felem in) {
506   limb zero = in[0] | in[1] | in[2] | in[3];
507   zero = (((int64_t)(zero)-1) >> 63) & 1;
508 
509   limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) |
510                      (in[2] ^ 0x00ffffffffffffff) |
511                      (in[3] ^ 0x00ffffffffffffff);
512   two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1;
513   limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) |
514                      (in[2] ^ 0x00ffffffffffffff) |
515                      (in[3] ^ 0x01ffffffffffffff);
516   two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1;
517   return (zero | two224m96p1 | two225m97p2);
518 }
519 
520 /* Invert a field element */
521 /* Computation chain copied from djb's code */
felem_inv(felem out,const felem in)522 static void felem_inv(felem out, const felem in) {
523   felem ftmp, ftmp2, ftmp3, ftmp4;
524   widefelem tmp;
525 
526   felem_square(tmp, in);
527   felem_reduce(ftmp, tmp); /* 2 */
528   felem_mul(tmp, in, ftmp);
529   felem_reduce(ftmp, tmp); /* 2^2 - 1 */
530   felem_square(tmp, ftmp);
531   felem_reduce(ftmp, tmp); /* 2^3 - 2 */
532   felem_mul(tmp, in, ftmp);
533   felem_reduce(ftmp, tmp); /* 2^3 - 1 */
534   felem_square(tmp, ftmp);
535   felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
536   felem_square(tmp, ftmp2);
537   felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
538   felem_square(tmp, ftmp2);
539   felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
540   felem_mul(tmp, ftmp2, ftmp);
541   felem_reduce(ftmp, tmp); /* 2^6 - 1 */
542   felem_square(tmp, ftmp);
543   felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
544   for (size_t i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
545     felem_square(tmp, ftmp2);
546     felem_reduce(ftmp2, tmp);
547   }
548   felem_mul(tmp, ftmp2, ftmp);
549   felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
550   felem_square(tmp, ftmp2);
551   felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
552   for (size_t i = 0; i < 11; ++i) {/* 2^24 - 2^12 */
553     felem_square(tmp, ftmp3);
554     felem_reduce(ftmp3, tmp);
555   }
556   felem_mul(tmp, ftmp3, ftmp2);
557   felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
558   felem_square(tmp, ftmp2);
559   felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
560   for (size_t i = 0; i < 23; ++i) {/* 2^48 - 2^24 */
561     felem_square(tmp, ftmp3);
562     felem_reduce(ftmp3, tmp);
563   }
564   felem_mul(tmp, ftmp3, ftmp2);
565   felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
566   felem_square(tmp, ftmp3);
567   felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
568   for (size_t i = 0; i < 47; ++i) {/* 2^96 - 2^48 */
569     felem_square(tmp, ftmp4);
570     felem_reduce(ftmp4, tmp);
571   }
572   felem_mul(tmp, ftmp3, ftmp4);
573   felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
574   felem_square(tmp, ftmp3);
575   felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
576   for (size_t i = 0; i < 23; ++i) {/* 2^120 - 2^24 */
577     felem_square(tmp, ftmp4);
578     felem_reduce(ftmp4, tmp);
579   }
580   felem_mul(tmp, ftmp2, ftmp4);
581   felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
582   for (size_t i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
583     felem_square(tmp, ftmp2);
584     felem_reduce(ftmp2, tmp);
585   }
586   felem_mul(tmp, ftmp2, ftmp);
587   felem_reduce(ftmp, tmp); /* 2^126 - 1 */
588   felem_square(tmp, ftmp);
589   felem_reduce(ftmp, tmp); /* 2^127 - 2 */
590   felem_mul(tmp, ftmp, in);
591   felem_reduce(ftmp, tmp); /* 2^127 - 1 */
592   for (size_t i = 0; i < 97; ++i) {/* 2^224 - 2^97 */
593     felem_square(tmp, ftmp);
594     felem_reduce(ftmp, tmp);
595   }
596   felem_mul(tmp, ftmp, ftmp3);
597   felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
598 }
599 
600 /* Copy in constant time:
601  * if icopy == 1, copy in to out,
602  * if icopy == 0, copy out to itself. */
copy_conditional(felem out,const felem in,limb icopy)603 static void copy_conditional(felem out, const felem in, limb icopy) {
604   /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
605   const limb copy = -icopy;
606   for (size_t i = 0; i < 4; ++i) {
607     const limb tmp = copy & (in[i] ^ out[i]);
608     out[i] ^= tmp;
609   }
610 }
611 
612 /* ELLIPTIC CURVE POINT OPERATIONS
613  *
614  * Points are represented in Jacobian projective coordinates:
615  * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
616  * or to the point at infinity if Z == 0. */
617 
618 /* Double an elliptic curve point:
619  * (X', Y', Z') = 2 * (X, Y, Z), where
620  * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
621  * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
622  * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
623  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
624  * while x_out == y_in is not (maybe this works, but it's not tested). */
point_double(felem x_out,felem y_out,felem z_out,const felem x_in,const felem y_in,const felem z_in)625 static void point_double(felem x_out, felem y_out, felem z_out,
626                          const felem x_in, const felem y_in, const felem z_in) {
627   widefelem tmp, tmp2;
628   felem delta, gamma, beta, alpha, ftmp, ftmp2;
629 
630   felem_assign(ftmp, x_in);
631   felem_assign(ftmp2, x_in);
632 
633   /* delta = z^2 */
634   felem_square(tmp, z_in);
635   felem_reduce(delta, tmp);
636 
637   /* gamma = y^2 */
638   felem_square(tmp, y_in);
639   felem_reduce(gamma, tmp);
640 
641   /* beta = x*gamma */
642   felem_mul(tmp, x_in, gamma);
643   felem_reduce(beta, tmp);
644 
645   /* alpha = 3*(x-delta)*(x+delta) */
646   felem_diff(ftmp, delta);
647   /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
648   felem_sum(ftmp2, delta);
649   /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
650   felem_scalar(ftmp2, 3);
651   /* ftmp2[i] < 3 * 2^58 < 2^60 */
652   felem_mul(tmp, ftmp, ftmp2);
653   /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
654   felem_reduce(alpha, tmp);
655 
656   /* x' = alpha^2 - 8*beta */
657   felem_square(tmp, alpha);
658   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
659   felem_assign(ftmp, beta);
660   felem_scalar(ftmp, 8);
661   /* ftmp[i] < 8 * 2^57 = 2^60 */
662   felem_diff_128_64(tmp, ftmp);
663   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
664   felem_reduce(x_out, tmp);
665 
666   /* z' = (y + z)^2 - gamma - delta */
667   felem_sum(delta, gamma);
668   /* delta[i] < 2^57 + 2^57 = 2^58 */
669   felem_assign(ftmp, y_in);
670   felem_sum(ftmp, z_in);
671   /* ftmp[i] < 2^57 + 2^57 = 2^58 */
672   felem_square(tmp, ftmp);
673   /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
674   felem_diff_128_64(tmp, delta);
675   /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
676   felem_reduce(z_out, tmp);
677 
678   /* y' = alpha*(4*beta - x') - 8*gamma^2 */
679   felem_scalar(beta, 4);
680   /* beta[i] < 4 * 2^57 = 2^59 */
681   felem_diff(beta, x_out);
682   /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
683   felem_mul(tmp, alpha, beta);
684   /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
685   felem_square(tmp2, gamma);
686   /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
687   widefelem_scalar(tmp2, 8);
688   /* tmp2[i] < 8 * 2^116 = 2^119 */
689   widefelem_diff(tmp, tmp2);
690   /* tmp[i] < 2^119 + 2^120 < 2^121 */
691   felem_reduce(y_out, tmp);
692 }
693 
694 /* Add two elliptic curve points:
695  * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
696  * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
697  * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
698  * 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 *
699  * X_1)^2 - X_3) -
700  *        Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
701  * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
702  *
703  * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */
704 
705 /* This function is not entirely constant-time: it includes a branch for
706  * checking whether the two input points are equal, (while not equal to the
707  * point at infinity). This case never happens during single point
708  * multiplication, so there is no timing leak for ECDH or ECDSA signing. */
point_add(felem x3,felem y3,felem z3,const felem x1,const felem y1,const felem z1,const int mixed,const felem x2,const felem y2,const felem z2)709 static void point_add(felem x3, felem y3, felem z3, const felem x1,
710                       const felem y1, const felem z1, const int mixed,
711                       const felem x2, const felem y2, const felem z2) {
712   felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
713   widefelem tmp, tmp2;
714   limb z1_is_zero, z2_is_zero, x_equal, y_equal;
715 
716   if (!mixed) {
717     /* ftmp2 = z2^2 */
718     felem_square(tmp, z2);
719     felem_reduce(ftmp2, tmp);
720 
721     /* ftmp4 = z2^3 */
722     felem_mul(tmp, ftmp2, z2);
723     felem_reduce(ftmp4, tmp);
724 
725     /* ftmp4 = z2^3*y1 */
726     felem_mul(tmp2, ftmp4, y1);
727     felem_reduce(ftmp4, tmp2);
728 
729     /* ftmp2 = z2^2*x1 */
730     felem_mul(tmp2, ftmp2, x1);
731     felem_reduce(ftmp2, tmp2);
732   } else {
733     /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
734 
735     /* ftmp4 = z2^3*y1 */
736     felem_assign(ftmp4, y1);
737 
738     /* ftmp2 = z2^2*x1 */
739     felem_assign(ftmp2, x1);
740   }
741 
742   /* ftmp = z1^2 */
743   felem_square(tmp, z1);
744   felem_reduce(ftmp, tmp);
745 
746   /* ftmp3 = z1^3 */
747   felem_mul(tmp, ftmp, z1);
748   felem_reduce(ftmp3, tmp);
749 
750   /* tmp = z1^3*y2 */
751   felem_mul(tmp, ftmp3, y2);
752   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
753 
754   /* ftmp3 = z1^3*y2 - z2^3*y1 */
755   felem_diff_128_64(tmp, ftmp4);
756   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
757   felem_reduce(ftmp3, tmp);
758 
759   /* tmp = z1^2*x2 */
760   felem_mul(tmp, ftmp, x2);
761   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
762 
763   /* ftmp = z1^2*x2 - z2^2*x1 */
764   felem_diff_128_64(tmp, ftmp2);
765   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
766   felem_reduce(ftmp, tmp);
767 
768   /* the formulae are incorrect if the points are equal
769    * so we check for this and do doubling if this happens */
770   x_equal = felem_is_zero(ftmp);
771   y_equal = felem_is_zero(ftmp3);
772   z1_is_zero = felem_is_zero(z1);
773   z2_is_zero = felem_is_zero(z2);
774   /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
775   if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
776     point_double(x3, y3, z3, x1, y1, z1);
777     return;
778   }
779 
780   /* ftmp5 = z1*z2 */
781   if (!mixed) {
782     felem_mul(tmp, z1, z2);
783     felem_reduce(ftmp5, tmp);
784   } else {
785     /* special case z2 = 0 is handled later */
786     felem_assign(ftmp5, z1);
787   }
788 
789   /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
790   felem_mul(tmp, ftmp, ftmp5);
791   felem_reduce(z_out, tmp);
792 
793   /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
794   felem_assign(ftmp5, ftmp);
795   felem_square(tmp, ftmp);
796   felem_reduce(ftmp, tmp);
797 
798   /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
799   felem_mul(tmp, ftmp, ftmp5);
800   felem_reduce(ftmp5, tmp);
801 
802   /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
803   felem_mul(tmp, ftmp2, ftmp);
804   felem_reduce(ftmp2, tmp);
805 
806   /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
807   felem_mul(tmp, ftmp4, ftmp5);
808   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
809 
810   /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
811   felem_square(tmp2, ftmp3);
812   /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
813 
814   /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
815   felem_diff_128_64(tmp2, ftmp5);
816   /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
817 
818   /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
819   felem_assign(ftmp5, ftmp2);
820   felem_scalar(ftmp5, 2);
821   /* ftmp5[i] < 2 * 2^57 = 2^58 */
822 
823   /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
824      2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
825   felem_diff_128_64(tmp2, ftmp5);
826   /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
827   felem_reduce(x_out, tmp2);
828 
829   /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
830   felem_diff(ftmp2, x_out);
831   /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
832 
833   /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
834   felem_mul(tmp2, ftmp3, ftmp2);
835   /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
836 
837   /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
838      z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
839   widefelem_diff(tmp2, tmp);
840   /* tmp2[i] < 2^118 + 2^120 < 2^121 */
841   felem_reduce(y_out, tmp2);
842 
843   /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
844    * the point at infinity, so we need to check for this separately */
845 
846   /* if point 1 is at infinity, copy point 2 to output, and vice versa */
847   copy_conditional(x_out, x2, z1_is_zero);
848   copy_conditional(x_out, x1, z2_is_zero);
849   copy_conditional(y_out, y2, z1_is_zero);
850   copy_conditional(y_out, y1, z2_is_zero);
851   copy_conditional(z_out, z2, z1_is_zero);
852   copy_conditional(z_out, z1, z2_is_zero);
853   felem_assign(x3, x_out);
854   felem_assign(y3, y_out);
855   felem_assign(z3, z_out);
856 }
857 
858 /* select_point selects the |idx|th point from a precomputation table and
859  * copies it to out. */
select_point(const u64 idx,size_t size,const felem pre_comp[][3],felem out[3])860 static void select_point(const u64 idx, size_t size,
861                          const felem pre_comp[/*size*/][3], felem out[3]) {
862   limb *outlimbs = &out[0][0];
863   OPENSSL_memset(outlimbs, 0, 3 * sizeof(felem));
864 
865   for (size_t i = 0; i < size; i++) {
866     const limb *inlimbs = &pre_comp[i][0][0];
867     u64 mask = i ^ idx;
868     mask |= mask >> 4;
869     mask |= mask >> 2;
870     mask |= mask >> 1;
871     mask &= 1;
872     mask--;
873     for (size_t j = 0; j < 4 * 3; j++) {
874       outlimbs[j] |= inlimbs[j] & mask;
875     }
876   }
877 }
878 
879 /* get_bit returns the |i|th bit in |in| */
get_bit(const felem_bytearray in,size_t i)880 static char get_bit(const felem_bytearray in, size_t i) {
881   if (i >= 224) {
882     return 0;
883   }
884   return (in[i >> 3] >> (i & 7)) & 1;
885 }
886 
887 /* Interleaved point multiplication using precomputed point multiples:
888  * The small point multiples 0*P, 1*P, ..., 16*P are in p_pre_comp, the scalars
889  * in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple
890  * of the generator, using certain (large) precomputed multiples in g_pre_comp.
891  * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
batch_mul(felem x_out,felem y_out,felem z_out,const u8 * p_scalar,const u8 * g_scalar,const felem p_pre_comp[17][3])892 static void batch_mul(felem x_out, felem y_out, felem z_out, const u8 *p_scalar,
893                       const u8 *g_scalar, const felem p_pre_comp[17][3]) {
894   felem nq[3], tmp[4];
895   u64 bits;
896   u8 sign, digit;
897 
898   /* set nq to the point at infinity */
899   OPENSSL_memset(nq, 0, 3 * sizeof(felem));
900 
901   /* Loop over both scalars msb-to-lsb, interleaving additions of multiples of
902    * the generator (two in each of the last 28 rounds) and additions of p (every
903    * 5th round). */
904   int skip = 1; /* save two point operations in the first round */
905   size_t i = p_scalar != NULL ? 220 : 27;
906   for (;;) {
907     /* double */
908     if (!skip) {
909       point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
910     }
911 
912     /* add multiples of the generator */
913     if (g_scalar != NULL && i <= 27) {
914       /* first, look 28 bits upwards */
915       bits = get_bit(g_scalar, i + 196) << 3;
916       bits |= get_bit(g_scalar, i + 140) << 2;
917       bits |= get_bit(g_scalar, i + 84) << 1;
918       bits |= get_bit(g_scalar, i + 28);
919       /* select the point to add, in constant time */
920       select_point(bits, 16, g_pre_comp[1], tmp);
921 
922       if (!skip) {
923         point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
924                   tmp[0], tmp[1], tmp[2]);
925       } else {
926         OPENSSL_memcpy(nq, tmp, 3 * sizeof(felem));
927         skip = 0;
928       }
929 
930       /* second, look at the current position */
931       bits = get_bit(g_scalar, i + 168) << 3;
932       bits |= get_bit(g_scalar, i + 112) << 2;
933       bits |= get_bit(g_scalar, i + 56) << 1;
934       bits |= get_bit(g_scalar, i);
935       /* select the point to add, in constant time */
936       select_point(bits, 16, g_pre_comp[0], tmp);
937       point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0],
938                 tmp[1], tmp[2]);
939     }
940 
941     /* do other additions every 5 doublings */
942     if (p_scalar != NULL && i % 5 == 0) {
943       bits = get_bit(p_scalar, i + 4) << 5;
944       bits |= get_bit(p_scalar, i + 3) << 4;
945       bits |= get_bit(p_scalar, i + 2) << 3;
946       bits |= get_bit(p_scalar, i + 1) << 2;
947       bits |= get_bit(p_scalar, i) << 1;
948       bits |= get_bit(p_scalar, i - 1);
949       ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
950 
951       /* select the point to add or subtract */
952       select_point(digit, 17, p_pre_comp, tmp);
953       felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
954       copy_conditional(tmp[1], tmp[3], sign);
955 
956       if (!skip) {
957         point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */,
958                   tmp[0], tmp[1], tmp[2]);
959       } else {
960         OPENSSL_memcpy(nq, tmp, 3 * sizeof(felem));
961         skip = 0;
962       }
963     }
964 
965     if (i == 0) {
966       break;
967     }
968     --i;
969   }
970   felem_assign(x_out, nq[0]);
971   felem_assign(y_out, nq[1]);
972   felem_assign(z_out, nq[2]);
973 }
974 
975 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
976  * (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)977 static int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
978                                                         const EC_POINT *point,
979                                                         BIGNUM *x, BIGNUM *y,
980                                                         BN_CTX *ctx) {
981   felem z1, z2, x_in, y_in, x_out, y_out;
982   widefelem tmp;
983 
984   if (EC_POINT_is_at_infinity(group, point)) {
985     OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
986     return 0;
987   }
988 
989   if (!BN_to_felem(x_in, &point->X) ||
990       !BN_to_felem(y_in, &point->Y) ||
991       !BN_to_felem(z1, &point->Z)) {
992     return 0;
993   }
994 
995   felem_inv(z2, z1);
996   felem_square(tmp, z2);
997   felem_reduce(z1, tmp);
998   felem_mul(tmp, x_in, z1);
999   felem_reduce(x_in, tmp);
1000   felem_contract(x_out, x_in);
1001   if (x != NULL && !felem_to_BN(x, x_out)) {
1002     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1003     return 0;
1004   }
1005 
1006   felem_mul(tmp, z1, z2);
1007   felem_reduce(z1, tmp);
1008   felem_mul(tmp, y_in, z1);
1009   felem_reduce(y_in, tmp);
1010   felem_contract(y_out, y_in);
1011   if (y != NULL && !felem_to_BN(y, y_out)) {
1012     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1013     return 0;
1014   }
1015 
1016   return 1;
1017 }
1018 
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)1019 static int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1020                                       const BIGNUM *g_scalar, const EC_POINT *p,
1021                                       const BIGNUM *p_scalar, BN_CTX *ctx) {
1022   int ret = 0;
1023   BN_CTX *new_ctx = NULL;
1024   BIGNUM *x, *y, *z, *tmp_scalar;
1025   felem_bytearray g_secret, p_secret;
1026   felem p_pre_comp[17][3];
1027   felem_bytearray tmp;
1028   felem x_in, y_in, z_in, x_out, y_out, z_out;
1029 
1030   if (ctx == NULL) {
1031     ctx = BN_CTX_new();
1032     new_ctx = ctx;
1033     if (ctx == NULL) {
1034       return 0;
1035     }
1036   }
1037 
1038   BN_CTX_start(ctx);
1039   if ((x = BN_CTX_get(ctx)) == NULL ||
1040       (y = BN_CTX_get(ctx)) == NULL ||
1041       (z = BN_CTX_get(ctx)) == NULL ||
1042       (tmp_scalar = BN_CTX_get(ctx)) == NULL) {
1043     goto err;
1044   }
1045 
1046   if (p != NULL && p_scalar != NULL) {
1047     /* We treat NULL scalars as 0, and NULL points as points at infinity, i.e.,
1048      * they contribute nothing to the linear combination. */
1049     OPENSSL_memset(&p_secret, 0, sizeof(p_secret));
1050     OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp));
1051     size_t num_bytes;
1052     /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1053     if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) {
1054       /* this is an unusual input, and we don't guarantee
1055        * constant-timeness */
1056       if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
1057         OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1058         goto err;
1059       }
1060       num_bytes = BN_bn2bin(tmp_scalar, tmp);
1061     } else {
1062       num_bytes = BN_bn2bin(p_scalar, tmp);
1063     }
1064 
1065     flip_endian(p_secret, tmp, num_bytes);
1066     /* precompute multiples */
1067     if (!BN_to_felem(x_out, &p->X) ||
1068         !BN_to_felem(y_out, &p->Y) ||
1069         !BN_to_felem(z_out, &p->Z)) {
1070       goto err;
1071     }
1072 
1073     felem_assign(p_pre_comp[1][0], x_out);
1074     felem_assign(p_pre_comp[1][1], y_out);
1075     felem_assign(p_pre_comp[1][2], z_out);
1076 
1077     for (size_t j = 2; j <= 16; ++j) {
1078       if (j & 1) {
1079         point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2],
1080                   p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2],
1081                   0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1],
1082                   p_pre_comp[j - 1][2]);
1083       } else {
1084         point_double(p_pre_comp[j][0], p_pre_comp[j][1],
1085                      p_pre_comp[j][2], p_pre_comp[j / 2][0],
1086                      p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]);
1087       }
1088     }
1089   }
1090 
1091   if (g_scalar != NULL) {
1092     OPENSSL_memset(g_secret, 0, sizeof(g_secret));
1093     size_t num_bytes;
1094     /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1095     if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) {
1096       /* this is an unusual input, and we don't guarantee constant-timeness */
1097       if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) {
1098         OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1099         goto err;
1100       }
1101       num_bytes = BN_bn2bin(tmp_scalar, tmp);
1102     } else {
1103       num_bytes = BN_bn2bin(g_scalar, tmp);
1104     }
1105 
1106     flip_endian(g_secret, tmp, num_bytes);
1107   }
1108   batch_mul(x_out, y_out, z_out,
1109             (p != NULL && p_scalar != NULL) ? p_secret : NULL,
1110             g_scalar != NULL ? g_secret : NULL, (const felem(*)[3])p_pre_comp);
1111 
1112   /* reduce the output to its unique minimal representation */
1113   felem_contract(x_in, x_out);
1114   felem_contract(y_in, y_out);
1115   felem_contract(z_in, z_out);
1116   if (!felem_to_BN(x, x_in) ||
1117       !felem_to_BN(y, y_in) ||
1118       !felem_to_BN(z, z_in)) {
1119     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1120     goto err;
1121   }
1122   ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1123 
1124 err:
1125   BN_CTX_end(ctx);
1126   BN_CTX_free(new_ctx);
1127   return ret;
1128 }
1129 
1130 const EC_METHOD EC_GFp_nistp224_method = {
1131     ec_GFp_simple_group_init,
1132     ec_GFp_simple_group_finish,
1133     ec_GFp_simple_group_copy,
1134     ec_GFp_simple_group_set_curve,
1135     ec_GFp_nistp224_point_get_affine_coordinates,
1136     ec_GFp_nistp224_points_mul,
1137     ec_GFp_simple_field_mul,
1138     ec_GFp_simple_field_sqr,
1139     NULL /* field_encode */,
1140     NULL /* field_decode */,
1141 };
1142 
1143 #endif  /* 64_BIT && !WINDOWS && !SMALL */
1144