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 #include <openssl/obj.h>
30 
31 #include <string.h>
32 
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 static const felem_bytearray nistp224_curve_params[5] = {
64     {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
65      0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
66      0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
67     {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
68      0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
69      0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
70     {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
71      0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA, 0x27, 0x0B,
72      0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
73     {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
74      0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22, 0x34, 0x32,
75      0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
76     {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
77      0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64, 0x44, 0xd5,
78      0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}};
79 
80 /* Precomputed multiples of the standard generator
81  * Points are given in coordinates (X, Y, Z) where Z normally is 1
82  * (0 for the point at infinity).
83  * For each field element, slice a_0 is word 0, etc.
84  *
85  * The table has 2 * 16 elements, starting with the following:
86  * index | bits    | point
87  * ------+---------+------------------------------
88  *     0 | 0 0 0 0 | 0G
89  *     1 | 0 0 0 1 | 1G
90  *     2 | 0 0 1 0 | 2^56G
91  *     3 | 0 0 1 1 | (2^56 + 1)G
92  *     4 | 0 1 0 0 | 2^112G
93  *     5 | 0 1 0 1 | (2^112 + 1)G
94  *     6 | 0 1 1 0 | (2^112 + 2^56)G
95  *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
96  *     8 | 1 0 0 0 | 2^168G
97  *     9 | 1 0 0 1 | (2^168 + 1)G
98  *    10 | 1 0 1 0 | (2^168 + 2^56)G
99  *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
100  *    12 | 1 1 0 0 | (2^168 + 2^112)G
101  *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
102  *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
103  *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
104  * followed by a copy of this with each element multiplied by 2^28.
105  *
106  * The reason for this is so that we can clock bits into four different
107  * locations when doing simple scalar multiplies against the base point,
108  * and then another four locations using the second 16 elements. */
109 static const felem g_pre_comp[2][16][3] = {
110     {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
111      {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
112       {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
113       {1, 0, 0, 0}},
114      {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
115       {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
116       {1, 0, 0, 0}},
117      {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
118       {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
119       {1, 0, 0, 0}},
120      {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
121       {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
122       {1, 0, 0, 0}},
123      {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
124       {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
125       {1, 0, 0, 0}},
126      {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
127       {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
128       {1, 0, 0, 0}},
129      {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
130       {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
131       {1, 0, 0, 0}},
132      {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
133       {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
134       {1, 0, 0, 0}},
135      {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
136       {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
137       {1, 0, 0, 0}},
138      {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
139       {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
140       {1, 0, 0, 0}},
141      {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
142       {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
143       {1, 0, 0, 0}},
144      {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
145       {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
146       {1, 0, 0, 0}},
147      {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
148       {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
149       {1, 0, 0, 0}},
150      {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
151       {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
152       {1, 0, 0, 0}},
153      {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
154       {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
155       {1, 0, 0, 0}}},
156     {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
157      {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
158       {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
159       {1, 0, 0, 0}},
160      {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
161       {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
162       {1, 0, 0, 0}},
163      {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
164       {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
165       {1, 0, 0, 0}},
166      {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
167       {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
168       {1, 0, 0, 0}},
169      {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
170       {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
171       {1, 0, 0, 0}},
172      {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
173       {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
174       {1, 0, 0, 0}},
175      {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
176       {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
177       {1, 0, 0, 0}},
178      {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
179       {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
180       {1, 0, 0, 0}},
181      {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
182       {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
183       {1, 0, 0, 0}},
184      {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
185       {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
186       {1, 0, 0, 0}},
187      {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
188       {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
189       {1, 0, 0, 0}},
190      {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
191       {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
192       {1, 0, 0, 0}},
193      {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
194       {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
195       {1, 0, 0, 0}},
196      {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
197       {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
198       {1, 0, 0, 0}},
199      {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
200       {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
201       {1, 0, 0, 0}}}};
202 
203 /* Helper functions to convert field elements to/from internal representation */
bin28_to_felem(felem out,const u8 in[28])204 static void bin28_to_felem(felem out, const u8 in[28]) {
205   out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
206   out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
207   out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
208   out[3] = (*((const uint64_t *)(in + 20))) >> 8;
209 }
210 
felem_to_bin28(u8 out[28],const felem in)211 static void felem_to_bin28(u8 out[28], const felem in) {
212   unsigned i;
213   for (i = 0; i < 7; ++i) {
214     out[i] = in[0] >> (8 * i);
215     out[i + 7] = in[1] >> (8 * i);
216     out[i + 14] = in[2] >> (8 * i);
217     out[i + 21] = in[3] >> (8 * i);
218   }
219 }
220 
221 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
flip_endian(u8 * out,const u8 * in,unsigned len)222 static void flip_endian(u8 *out, const u8 *in, unsigned len) {
223   unsigned i;
224   for (i = 0; i < len; ++i) {
225     out[i] = in[len - 1 - i];
226   }
227 }
228 
229 /* From OpenSSL BIGNUM to internal representation */
BN_to_felem(felem out,const BIGNUM * bn)230 static int BN_to_felem(felem out, const BIGNUM *bn) {
231   /* BN_bn2bin eats leading zeroes */
232   felem_bytearray b_out;
233   memset(b_out, 0, sizeof(b_out));
234   unsigned num_bytes = BN_num_bytes(bn);
235   if (num_bytes > sizeof(b_out) ||
236       BN_is_negative(bn)) {
237     OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE);
238     return 0;
239   }
240 
241   felem_bytearray b_in;
242   num_bytes = BN_bn2bin(bn, b_in);
243   flip_endian(b_out, b_in, num_bytes);
244   bin28_to_felem(out, b_out);
245   return 1;
246 }
247 
248 /* From internal representation to OpenSSL BIGNUM */
felem_to_BN(BIGNUM * out,const felem in)249 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) {
250   felem_bytearray b_in, b_out;
251   felem_to_bin28(b_in, in);
252   flip_endian(b_out, b_in, sizeof(b_out));
253   return BN_bin2bn(b_out, sizeof(b_out), out);
254 }
255 
256 /* Field operations, using the internal representation of field elements.
257  * NB! These operations are specific to our point multiplication and cannot be
258  * expected to be correct in general - e.g., multiplication with a large scalar
259  * will cause an overflow. */
260 
felem_one(felem out)261 static void felem_one(felem out) {
262   out[0] = 1;
263   out[1] = 0;
264   out[2] = 0;
265   out[3] = 0;
266 }
267 
felem_assign(felem out,const felem in)268 static void felem_assign(felem out, const felem in) {
269   out[0] = in[0];
270   out[1] = in[1];
271   out[2] = in[2];
272   out[3] = in[3];
273 }
274 
275 /* Sum two field elements: out += in */
felem_sum(felem out,const felem in)276 static void felem_sum(felem out, const felem in) {
277   out[0] += in[0];
278   out[1] += in[1];
279   out[2] += in[2];
280   out[3] += in[3];
281 }
282 
283 /* Get negative value: out = -in */
284 /* Assumes in[i] < 2^57 */
felem_neg(felem out,const felem in)285 static void felem_neg(felem out, const felem in) {
286   static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
287   static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
288   static const limb two58m42m2 =
289       (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
290 
291   /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
292   out[0] = two58p2 - in[0];
293   out[1] = two58m42m2 - in[1];
294   out[2] = two58m2 - in[2];
295   out[3] = two58m2 - in[3];
296 }
297 
298 /* Subtract field elements: out -= in */
299 /* Assumes in[i] < 2^57 */
felem_diff(felem out,const felem in)300 static void felem_diff(felem out, const felem in) {
301   static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
302   static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
303   static const limb two58m42m2 =
304       (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
305 
306   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
307   out[0] += two58p2;
308   out[1] += two58m42m2;
309   out[2] += two58m2;
310   out[3] += two58m2;
311 
312   out[0] -= in[0];
313   out[1] -= in[1];
314   out[2] -= in[2];
315   out[3] -= in[3];
316 }
317 
318 /* Subtract in unreduced 128-bit mode: out -= in */
319 /* Assumes in[i] < 2^119 */
widefelem_diff(widefelem out,const widefelem in)320 static void widefelem_diff(widefelem out, const widefelem in) {
321   static const widelimb two120 = ((widelimb)1) << 120;
322   static const widelimb two120m64 =
323       (((widelimb)1) << 120) - (((widelimb)1) << 64);
324   static const widelimb two120m104m64 =
325       (((widelimb)1) << 120) - (((widelimb)1) << 104) - (((widelimb)1) << 64);
326 
327   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
328   out[0] += two120;
329   out[1] += two120m64;
330   out[2] += two120m64;
331   out[3] += two120;
332   out[4] += two120m104m64;
333   out[5] += two120m64;
334   out[6] += two120m64;
335 
336   out[0] -= in[0];
337   out[1] -= in[1];
338   out[2] -= in[2];
339   out[3] -= in[3];
340   out[4] -= in[4];
341   out[5] -= in[5];
342   out[6] -= in[6];
343 }
344 
345 /* Subtract in mixed mode: out128 -= in64 */
346 /* in[i] < 2^63 */
felem_diff_128_64(widefelem out,const felem in)347 static void felem_diff_128_64(widefelem out, const felem in) {
348   static const widelimb two64p8 = (((widelimb)1) << 64) + (((widelimb)1) << 8);
349   static const widelimb two64m8 = (((widelimb)1) << 64) - (((widelimb)1) << 8);
350   static const widelimb two64m48m8 =
351       (((widelimb)1) << 64) - (((widelimb)1) << 48) - (((widelimb)1) << 8);
352 
353   /* Add 0 mod 2^224-2^96+1 to ensure out > in */
354   out[0] += two64p8;
355   out[1] += two64m48m8;
356   out[2] += two64m8;
357   out[3] += two64m8;
358 
359   out[0] -= in[0];
360   out[1] -= in[1];
361   out[2] -= in[2];
362   out[3] -= in[3];
363 }
364 
365 /* Multiply a field element by a scalar: out = out * scalar
366  * The scalars we actually use are small, so results fit without overflow */
felem_scalar(felem out,const limb scalar)367 static void felem_scalar(felem out, const limb scalar) {
368   out[0] *= scalar;
369   out[1] *= scalar;
370   out[2] *= scalar;
371   out[3] *= scalar;
372 }
373 
374 /* Multiply an unreduced field element by a scalar: out = out * scalar
375  * The scalars we actually use are small, so results fit without overflow */
widefelem_scalar(widefelem out,const widelimb scalar)376 static void widefelem_scalar(widefelem out, const widelimb scalar) {
377   out[0] *= scalar;
378   out[1] *= scalar;
379   out[2] *= scalar;
380   out[3] *= scalar;
381   out[4] *= scalar;
382   out[5] *= scalar;
383   out[6] *= scalar;
384 }
385 
386 /* Square a field element: out = in^2 */
felem_square(widefelem out,const felem in)387 static void felem_square(widefelem out, const felem in) {
388   limb tmp0, tmp1, tmp2;
389   tmp0 = 2 * in[0];
390   tmp1 = 2 * in[1];
391   tmp2 = 2 * in[2];
392   out[0] = ((widelimb)in[0]) * in[0];
393   out[1] = ((widelimb)in[0]) * tmp1;
394   out[2] = ((widelimb)in[0]) * tmp2 + ((widelimb)in[1]) * in[1];
395   out[3] = ((widelimb)in[3]) * tmp0 + ((widelimb)in[1]) * tmp2;
396   out[4] = ((widelimb)in[3]) * tmp1 + ((widelimb)in[2]) * in[2];
397   out[5] = ((widelimb)in[3]) * tmp2;
398   out[6] = ((widelimb)in[3]) * in[3];
399 }
400 
401 /* Multiply two field elements: out = in1 * in2 */
felem_mul(widefelem out,const felem in1,const felem in2)402 static void felem_mul(widefelem out, const felem in1, const felem in2) {
403   out[0] = ((widelimb)in1[0]) * in2[0];
404   out[1] = ((widelimb)in1[0]) * in2[1] + ((widelimb)in1[1]) * in2[0];
405   out[2] = ((widelimb)in1[0]) * in2[2] + ((widelimb)in1[1]) * in2[1] +
406            ((widelimb)in1[2]) * in2[0];
407   out[3] = ((widelimb)in1[0]) * in2[3] + ((widelimb)in1[1]) * in2[2] +
408            ((widelimb)in1[2]) * in2[1] + ((widelimb)in1[3]) * in2[0];
409   out[4] = ((widelimb)in1[1]) * in2[3] + ((widelimb)in1[2]) * in2[2] +
410            ((widelimb)in1[3]) * in2[1];
411   out[5] = ((widelimb)in1[2]) * in2[3] + ((widelimb)in1[3]) * in2[2];
412   out[6] = ((widelimb)in1[3]) * in2[3];
413 }
414 
415 /* Reduce seven 128-bit coefficients to four 64-bit coefficients.
416  * Requires in[i] < 2^126,
417  * 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)418 static void felem_reduce(felem out, const widefelem in) {
419   static const widelimb two127p15 =
420       (((widelimb)1) << 127) + (((widelimb)1) << 15);
421   static const widelimb two127m71 =
422       (((widelimb)1) << 127) - (((widelimb)1) << 71);
423   static const widelimb two127m71m55 =
424       (((widelimb)1) << 127) - (((widelimb)1) << 71) - (((widelimb)1) << 55);
425   widelimb output[5];
426 
427   /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
428   output[0] = in[0] + two127p15;
429   output[1] = in[1] + two127m71m55;
430   output[2] = in[2] + two127m71;
431   output[3] = in[3];
432   output[4] = in[4];
433 
434   /* Eliminate in[4], in[5], in[6] */
435   output[4] += in[6] >> 16;
436   output[3] += (in[6] & 0xffff) << 40;
437   output[2] -= in[6];
438 
439   output[3] += in[5] >> 16;
440   output[2] += (in[5] & 0xffff) << 40;
441   output[1] -= in[5];
442 
443   output[2] += output[4] >> 16;
444   output[1] += (output[4] & 0xffff) << 40;
445   output[0] -= output[4];
446 
447   /* Carry 2 -> 3 -> 4 */
448   output[3] += output[2] >> 56;
449   output[2] &= 0x00ffffffffffffff;
450 
451   output[4] = output[3] >> 56;
452   output[3] &= 0x00ffffffffffffff;
453 
454   /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
455 
456   /* Eliminate output[4] */
457   output[2] += output[4] >> 16;
458   /* output[2] < 2^56 + 2^56 = 2^57 */
459   output[1] += (output[4] & 0xffff) << 40;
460   output[0] -= output[4];
461 
462   /* Carry 0 -> 1 -> 2 -> 3 */
463   output[1] += output[0] >> 56;
464   out[0] = output[0] & 0x00ffffffffffffff;
465 
466   output[2] += output[1] >> 56;
467   /* output[2] < 2^57 + 2^72 */
468   out[1] = output[1] & 0x00ffffffffffffff;
469   output[3] += output[2] >> 56;
470   /* output[3] <= 2^56 + 2^16 */
471   out[2] = output[2] & 0x00ffffffffffffff;
472 
473   /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
474    * out[3] <= 2^56 + 2^16 (due to final carry),
475    * so out < 2*p */
476   out[3] = output[3];
477 }
478 
felem_square_reduce(felem out,const felem in)479 static void felem_square_reduce(felem out, const felem in) {
480   widefelem tmp;
481   felem_square(tmp, in);
482   felem_reduce(out, tmp);
483 }
484 
felem_mul_reduce(felem out,const felem in1,const felem in2)485 static void felem_mul_reduce(felem out, const felem in1, const felem in2) {
486   widefelem tmp;
487   felem_mul(tmp, in1, in2);
488   felem_reduce(out, tmp);
489 }
490 
491 /* Reduce to unique minimal representation.
492  * Requires 0 <= in < 2*p (always call felem_reduce first) */
felem_contract(felem out,const felem in)493 static void felem_contract(felem out, const felem in) {
494   static const int64_t two56 = ((limb)1) << 56;
495   /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
496   /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
497   int64_t tmp[4], a;
498   tmp[0] = in[0];
499   tmp[1] = in[1];
500   tmp[2] = in[2];
501   tmp[3] = in[3];
502   /* Case 1: a = 1 iff in >= 2^224 */
503   a = (in[3] >> 56);
504   tmp[0] -= a;
505   tmp[1] += a << 40;
506   tmp[3] &= 0x00ffffffffffffff;
507   /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and
508    * the lower part is non-zero */
509   a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
510       (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
511   a &= 0x00ffffffffffffff;
512   /* turn a into an all-one mask (if a = 0) or an all-zero mask */
513   a = (a - 1) >> 63;
514   /* subtract 2^224 - 2^96 + 1 if a is all-one */
515   tmp[3] &= a ^ 0xffffffffffffffff;
516   tmp[2] &= a ^ 0xffffffffffffffff;
517   tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
518   tmp[0] -= 1 & a;
519 
520   /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
521    * be non-zero, so we only need one step */
522   a = tmp[0] >> 63;
523   tmp[0] += two56 & a;
524   tmp[1] -= 1 & a;
525 
526   /* carry 1 -> 2 -> 3 */
527   tmp[2] += tmp[1] >> 56;
528   tmp[1] &= 0x00ffffffffffffff;
529 
530   tmp[3] += tmp[2] >> 56;
531   tmp[2] &= 0x00ffffffffffffff;
532 
533   /* Now 0 <= out < p */
534   out[0] = tmp[0];
535   out[1] = tmp[1];
536   out[2] = tmp[2];
537   out[3] = tmp[3];
538 }
539 
540 /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
541  * elements are reduced to in < 2^225, so we only need to check three cases: 0,
542  * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */
felem_is_zero(const felem in)543 static limb felem_is_zero(const felem in) {
544   limb zero = in[0] | in[1] | in[2] | in[3];
545   zero = (((int64_t)(zero)-1) >> 63) & 1;
546 
547   limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) |
548                      (in[2] ^ 0x00ffffffffffffff) |
549                      (in[3] ^ 0x00ffffffffffffff);
550   two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1;
551   limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) |
552                      (in[2] ^ 0x00ffffffffffffff) |
553                      (in[3] ^ 0x01ffffffffffffff);
554   two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1;
555   return (zero | two224m96p1 | two225m97p2);
556 }
557 
felem_is_zero_int(const felem in)558 static limb felem_is_zero_int(const felem in) {
559   return (int)(felem_is_zero(in) & ((limb)1));
560 }
561 
562 /* Invert a field element */
563 /* Computation chain copied from djb's code */
felem_inv(felem out,const felem in)564 static void felem_inv(felem out, const felem in) {
565   felem ftmp, ftmp2, ftmp3, ftmp4;
566   widefelem tmp;
567   unsigned i;
568 
569   felem_square(tmp, in);
570   felem_reduce(ftmp, tmp); /* 2 */
571   felem_mul(tmp, in, ftmp);
572   felem_reduce(ftmp, tmp); /* 2^2 - 1 */
573   felem_square(tmp, ftmp);
574   felem_reduce(ftmp, tmp); /* 2^3 - 2 */
575   felem_mul(tmp, in, ftmp);
576   felem_reduce(ftmp, tmp); /* 2^3 - 1 */
577   felem_square(tmp, ftmp);
578   felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
579   felem_square(tmp, ftmp2);
580   felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
581   felem_square(tmp, ftmp2);
582   felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
583   felem_mul(tmp, ftmp2, ftmp);
584   felem_reduce(ftmp, tmp); /* 2^6 - 1 */
585   felem_square(tmp, ftmp);
586   felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
587   for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
588     felem_square(tmp, ftmp2);
589     felem_reduce(ftmp2, tmp);
590   }
591   felem_mul(tmp, ftmp2, ftmp);
592   felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
593   felem_square(tmp, ftmp2);
594   felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
595   for (i = 0; i < 11; ++i) {/* 2^24 - 2^12 */
596     felem_square(tmp, ftmp3);
597     felem_reduce(ftmp3, tmp);
598   }
599   felem_mul(tmp, ftmp3, ftmp2);
600   felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
601   felem_square(tmp, ftmp2);
602   felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
603   for (i = 0; i < 23; ++i) {/* 2^48 - 2^24 */
604     felem_square(tmp, ftmp3);
605     felem_reduce(ftmp3, tmp);
606   }
607   felem_mul(tmp, ftmp3, ftmp2);
608   felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
609   felem_square(tmp, ftmp3);
610   felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
611   for (i = 0; i < 47; ++i) {/* 2^96 - 2^48 */
612     felem_square(tmp, ftmp4);
613     felem_reduce(ftmp4, tmp);
614   }
615   felem_mul(tmp, ftmp3, ftmp4);
616   felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
617   felem_square(tmp, ftmp3);
618   felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
619   for (i = 0; i < 23; ++i) {/* 2^120 - 2^24 */
620     felem_square(tmp, ftmp4);
621     felem_reduce(ftmp4, tmp);
622   }
623   felem_mul(tmp, ftmp2, ftmp4);
624   felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
625   for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
626     felem_square(tmp, ftmp2);
627     felem_reduce(ftmp2, tmp);
628   }
629   felem_mul(tmp, ftmp2, ftmp);
630   felem_reduce(ftmp, tmp); /* 2^126 - 1 */
631   felem_square(tmp, ftmp);
632   felem_reduce(ftmp, tmp); /* 2^127 - 2 */
633   felem_mul(tmp, ftmp, in);
634   felem_reduce(ftmp, tmp); /* 2^127 - 1 */
635   for (i = 0; i < 97; ++i) {/* 2^224 - 2^97 */
636     felem_square(tmp, ftmp);
637     felem_reduce(ftmp, tmp);
638   }
639   felem_mul(tmp, ftmp, ftmp3);
640   felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
641 }
642 
643 /* Copy in constant time:
644  * if icopy == 1, copy in to out,
645  * if icopy == 0, copy out to itself. */
copy_conditional(felem out,const felem in,limb icopy)646 static void copy_conditional(felem out, const felem in, limb icopy) {
647   unsigned i;
648   /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
649   const limb copy = -icopy;
650   for (i = 0; i < 4; ++i) {
651     const limb tmp = copy & (in[i] ^ out[i]);
652     out[i] ^= tmp;
653   }
654 }
655 
656 /* ELLIPTIC CURVE POINT OPERATIONS
657  *
658  * Points are represented in Jacobian projective coordinates:
659  * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
660  * or to the point at infinity if Z == 0. */
661 
662 /* Double an elliptic curve point:
663  * (X', Y', Z') = 2 * (X, Y, Z), where
664  * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
665  * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
666  * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
667  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
668  * 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)669 static void point_double(felem x_out, felem y_out, felem z_out,
670                          const felem x_in, const felem y_in, const felem z_in) {
671   widefelem tmp, tmp2;
672   felem delta, gamma, beta, alpha, ftmp, ftmp2;
673 
674   felem_assign(ftmp, x_in);
675   felem_assign(ftmp2, x_in);
676 
677   /* delta = z^2 */
678   felem_square(tmp, z_in);
679   felem_reduce(delta, tmp);
680 
681   /* gamma = y^2 */
682   felem_square(tmp, y_in);
683   felem_reduce(gamma, tmp);
684 
685   /* beta = x*gamma */
686   felem_mul(tmp, x_in, gamma);
687   felem_reduce(beta, tmp);
688 
689   /* alpha = 3*(x-delta)*(x+delta) */
690   felem_diff(ftmp, delta);
691   /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
692   felem_sum(ftmp2, delta);
693   /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
694   felem_scalar(ftmp2, 3);
695   /* ftmp2[i] < 3 * 2^58 < 2^60 */
696   felem_mul(tmp, ftmp, ftmp2);
697   /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
698   felem_reduce(alpha, tmp);
699 
700   /* x' = alpha^2 - 8*beta */
701   felem_square(tmp, alpha);
702   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
703   felem_assign(ftmp, beta);
704   felem_scalar(ftmp, 8);
705   /* ftmp[i] < 8 * 2^57 = 2^60 */
706   felem_diff_128_64(tmp, ftmp);
707   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
708   felem_reduce(x_out, tmp);
709 
710   /* z' = (y + z)^2 - gamma - delta */
711   felem_sum(delta, gamma);
712   /* delta[i] < 2^57 + 2^57 = 2^58 */
713   felem_assign(ftmp, y_in);
714   felem_sum(ftmp, z_in);
715   /* ftmp[i] < 2^57 + 2^57 = 2^58 */
716   felem_square(tmp, ftmp);
717   /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
718   felem_diff_128_64(tmp, delta);
719   /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
720   felem_reduce(z_out, tmp);
721 
722   /* y' = alpha*(4*beta - x') - 8*gamma^2 */
723   felem_scalar(beta, 4);
724   /* beta[i] < 4 * 2^57 = 2^59 */
725   felem_diff(beta, x_out);
726   /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
727   felem_mul(tmp, alpha, beta);
728   /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
729   felem_square(tmp2, gamma);
730   /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
731   widefelem_scalar(tmp2, 8);
732   /* tmp2[i] < 8 * 2^116 = 2^119 */
733   widefelem_diff(tmp, tmp2);
734   /* tmp[i] < 2^119 + 2^120 < 2^121 */
735   felem_reduce(y_out, tmp);
736 }
737 
738 /* Add two elliptic curve points:
739  * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
740  * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
741  * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
742  * 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 *
743  * X_1)^2 - X_3) -
744  *        Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
745  * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
746  *
747  * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */
748 
749 /* This function is not entirely constant-time: it includes a branch for
750  * checking whether the two input points are equal, (while not equal to the
751  * point at infinity). This case never happens during single point
752  * 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)753 static void point_add(felem x3, felem y3, felem z3, const felem x1,
754                       const felem y1, const felem z1, const int mixed,
755                       const felem x2, const felem y2, const felem z2) {
756   felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
757   widefelem tmp, tmp2;
758   limb z1_is_zero, z2_is_zero, x_equal, y_equal;
759 
760   if (!mixed) {
761     /* ftmp2 = z2^2 */
762     felem_square(tmp, z2);
763     felem_reduce(ftmp2, tmp);
764 
765     /* ftmp4 = z2^3 */
766     felem_mul(tmp, ftmp2, z2);
767     felem_reduce(ftmp4, tmp);
768 
769     /* ftmp4 = z2^3*y1 */
770     felem_mul(tmp2, ftmp4, y1);
771     felem_reduce(ftmp4, tmp2);
772 
773     /* ftmp2 = z2^2*x1 */
774     felem_mul(tmp2, ftmp2, x1);
775     felem_reduce(ftmp2, tmp2);
776   } else {
777     /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
778 
779     /* ftmp4 = z2^3*y1 */
780     felem_assign(ftmp4, y1);
781 
782     /* ftmp2 = z2^2*x1 */
783     felem_assign(ftmp2, x1);
784   }
785 
786   /* ftmp = z1^2 */
787   felem_square(tmp, z1);
788   felem_reduce(ftmp, tmp);
789 
790   /* ftmp3 = z1^3 */
791   felem_mul(tmp, ftmp, z1);
792   felem_reduce(ftmp3, tmp);
793 
794   /* tmp = z1^3*y2 */
795   felem_mul(tmp, ftmp3, y2);
796   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
797 
798   /* ftmp3 = z1^3*y2 - z2^3*y1 */
799   felem_diff_128_64(tmp, ftmp4);
800   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
801   felem_reduce(ftmp3, tmp);
802 
803   /* tmp = z1^2*x2 */
804   felem_mul(tmp, ftmp, x2);
805   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
806 
807   /* ftmp = z1^2*x2 - z2^2*x1 */
808   felem_diff_128_64(tmp, ftmp2);
809   /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
810   felem_reduce(ftmp, tmp);
811 
812   /* the formulae are incorrect if the points are equal
813    * so we check for this and do doubling if this happens */
814   x_equal = felem_is_zero(ftmp);
815   y_equal = felem_is_zero(ftmp3);
816   z1_is_zero = felem_is_zero(z1);
817   z2_is_zero = felem_is_zero(z2);
818   /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
819   if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
820     point_double(x3, y3, z3, x1, y1, z1);
821     return;
822   }
823 
824   /* ftmp5 = z1*z2 */
825   if (!mixed) {
826     felem_mul(tmp, z1, z2);
827     felem_reduce(ftmp5, tmp);
828   } else {
829     /* special case z2 = 0 is handled later */
830     felem_assign(ftmp5, z1);
831   }
832 
833   /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
834   felem_mul(tmp, ftmp, ftmp5);
835   felem_reduce(z_out, tmp);
836 
837   /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
838   felem_assign(ftmp5, ftmp);
839   felem_square(tmp, ftmp);
840   felem_reduce(ftmp, tmp);
841 
842   /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
843   felem_mul(tmp, ftmp, ftmp5);
844   felem_reduce(ftmp5, tmp);
845 
846   /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
847   felem_mul(tmp, ftmp2, ftmp);
848   felem_reduce(ftmp2, tmp);
849 
850   /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
851   felem_mul(tmp, ftmp4, ftmp5);
852   /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
853 
854   /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
855   felem_square(tmp2, ftmp3);
856   /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
857 
858   /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
859   felem_diff_128_64(tmp2, ftmp5);
860   /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
861 
862   /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
863   felem_assign(ftmp5, ftmp2);
864   felem_scalar(ftmp5, 2);
865   /* ftmp5[i] < 2 * 2^57 = 2^58 */
866 
867   /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
868      2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
869   felem_diff_128_64(tmp2, ftmp5);
870   /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
871   felem_reduce(x_out, tmp2);
872 
873   /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
874   felem_diff(ftmp2, x_out);
875   /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
876 
877   /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
878   felem_mul(tmp2, ftmp3, ftmp2);
879   /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
880 
881   /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
882      z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
883   widefelem_diff(tmp2, tmp);
884   /* tmp2[i] < 2^118 + 2^120 < 2^121 */
885   felem_reduce(y_out, tmp2);
886 
887   /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
888    * the point at infinity, so we need to check for this separately */
889 
890   /* if point 1 is at infinity, copy point 2 to output, and vice versa */
891   copy_conditional(x_out, x2, z1_is_zero);
892   copy_conditional(x_out, x1, z2_is_zero);
893   copy_conditional(y_out, y2, z1_is_zero);
894   copy_conditional(y_out, y1, z2_is_zero);
895   copy_conditional(z_out, z2, z1_is_zero);
896   copy_conditional(z_out, z1, z2_is_zero);
897   felem_assign(x3, x_out);
898   felem_assign(y3, y_out);
899   felem_assign(z3, z_out);
900 }
901 
902 /* select_point selects the |idx|th point from a precomputation table and
903  * copies it to out. */
select_point(const u64 idx,unsigned int size,const felem pre_comp[][3],felem out[3])904 static void select_point(const u64 idx, unsigned int size,
905                          const felem pre_comp[/*size*/][3], felem out[3]) {
906   unsigned i, j;
907   limb *outlimbs = &out[0][0];
908   memset(outlimbs, 0, 3 * sizeof(felem));
909 
910   for (i = 0; i < size; i++) {
911     const limb *inlimbs = &pre_comp[i][0][0];
912     u64 mask = i ^ idx;
913     mask |= mask >> 4;
914     mask |= mask >> 2;
915     mask |= mask >> 1;
916     mask &= 1;
917     mask--;
918     for (j = 0; j < 4 * 3; j++) {
919       outlimbs[j] |= inlimbs[j] & mask;
920     }
921   }
922 }
923 
924 /* get_bit returns the |i|th bit in |in| */
get_bit(const felem_bytearray in,unsigned i)925 static char get_bit(const felem_bytearray in, unsigned i) {
926   if (i >= 224) {
927     return 0;
928   }
929   return (in[i >> 3] >> (i & 7)) & 1;
930 }
931 
932 /* Interleaved point multiplication using precomputed point multiples:
933  * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
934  * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
935  * of the generator, using certain (large) precomputed multiples in g_pre_comp.
936  * 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 felem_bytearray scalars[],const unsigned num_points,const u8 * g_scalar,const int mixed,const felem pre_comp[][17][3])937 static void batch_mul(felem x_out, felem y_out, felem z_out,
938                       const felem_bytearray scalars[],
939                       const unsigned num_points, const u8 *g_scalar,
940                       const int mixed, const felem pre_comp[][17][3]) {
941   int i, skip;
942   unsigned num;
943   unsigned gen_mul = (g_scalar != NULL);
944   felem nq[3], tmp[4];
945   u64 bits;
946   u8 sign, digit;
947 
948   /* set nq to the point at infinity */
949   memset(nq, 0, 3 * sizeof(felem));
950 
951   /* Loop over all scalars msb-to-lsb, interleaving additions
952    * of multiples of the generator (two in each of the last 28 rounds)
953    * and additions of other points multiples (every 5th round). */
954   skip = 1; /* save two point operations in the first round */
955   for (i = (num_points ? 220 : 27); i >= 0; --i) {
956     /* double */
957     if (!skip) {
958       point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
959     }
960 
961     /* add multiples of the generator */
962     if (gen_mul && (i <= 27)) {
963       /* first, look 28 bits upwards */
964       bits = get_bit(g_scalar, i + 196) << 3;
965       bits |= get_bit(g_scalar, i + 140) << 2;
966       bits |= get_bit(g_scalar, i + 84) << 1;
967       bits |= get_bit(g_scalar, i + 28);
968       /* select the point to add, in constant time */
969       select_point(bits, 16, g_pre_comp[1], tmp);
970 
971       if (!skip) {
972         point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
973                   tmp[0], tmp[1], tmp[2]);
974       } else {
975         memcpy(nq, tmp, 3 * sizeof(felem));
976         skip = 0;
977       }
978 
979       /* second, look at the current position */
980       bits = get_bit(g_scalar, i + 168) << 3;
981       bits |= get_bit(g_scalar, i + 112) << 2;
982       bits |= get_bit(g_scalar, i + 56) << 1;
983       bits |= get_bit(g_scalar, i);
984       /* select the point to add, in constant time */
985       select_point(bits, 16, g_pre_comp[0], tmp);
986       point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0],
987                 tmp[1], tmp[2]);
988     }
989 
990     /* do other additions every 5 doublings */
991     if (num_points && (i % 5 == 0)) {
992       /* loop over all scalars */
993       for (num = 0; num < num_points; ++num) {
994         bits = get_bit(scalars[num], i + 4) << 5;
995         bits |= get_bit(scalars[num], i + 3) << 4;
996         bits |= get_bit(scalars[num], i + 2) << 3;
997         bits |= get_bit(scalars[num], i + 1) << 2;
998         bits |= get_bit(scalars[num], i) << 1;
999         bits |= get_bit(scalars[num], i - 1);
1000         ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1001 
1002         /* select the point to add or subtract */
1003         select_point(digit, 17, pre_comp[num], tmp);
1004         felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
1005         copy_conditional(tmp[1], tmp[3], sign);
1006 
1007         if (!skip) {
1008           point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], mixed, tmp[0],
1009                     tmp[1], tmp[2]);
1010         } else {
1011           memcpy(nq, tmp, 3 * sizeof(felem));
1012           skip = 0;
1013         }
1014       }
1015     }
1016   }
1017   felem_assign(x_out, nq[0]);
1018   felem_assign(y_out, nq[1]);
1019   felem_assign(z_out, nq[2]);
1020 }
1021 
ec_GFp_nistp224_group_init(EC_GROUP * group)1022 int ec_GFp_nistp224_group_init(EC_GROUP *group) {
1023   int ret;
1024   ret = ec_GFp_simple_group_init(group);
1025   group->a_is_minus3 = 1;
1026   return ret;
1027 }
1028 
ec_GFp_nistp224_group_set_curve(EC_GROUP * group,const BIGNUM * p,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)1029 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1030                                     const BIGNUM *a, const BIGNUM *b,
1031                                     BN_CTX *ctx) {
1032   int ret = 0;
1033   BN_CTX *new_ctx = NULL;
1034   BIGNUM *curve_p, *curve_a, *curve_b;
1035 
1036   if (ctx == NULL) {
1037     ctx = BN_CTX_new();
1038     new_ctx = ctx;
1039     if (ctx == NULL) {
1040       return 0;
1041     }
1042   }
1043   BN_CTX_start(ctx);
1044   if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1045       ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1046       ((curve_b = BN_CTX_get(ctx)) == NULL)) {
1047     goto err;
1048   }
1049   BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1050   BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1051   BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1052   if (BN_cmp(curve_p, p) ||
1053       BN_cmp(curve_a, a) ||
1054       BN_cmp(curve_b, b)) {
1055     OPENSSL_PUT_ERROR(EC, EC_R_WRONG_CURVE_PARAMETERS);
1056     goto err;
1057   }
1058   ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1059 
1060 err:
1061   BN_CTX_end(ctx);
1062   BN_CTX_free(new_ctx);
1063   return ret;
1064 }
1065 
1066 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1067  * (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)1068 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1069                                                  const EC_POINT *point,
1070                                                  BIGNUM *x, BIGNUM *y,
1071                                                  BN_CTX *ctx) {
1072   felem z1, z2, x_in, y_in, x_out, y_out;
1073   widefelem tmp;
1074 
1075   if (EC_POINT_is_at_infinity(group, point)) {
1076     OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
1077     return 0;
1078   }
1079 
1080   if (!BN_to_felem(x_in, &point->X) ||
1081       !BN_to_felem(y_in, &point->Y) ||
1082       !BN_to_felem(z1, &point->Z)) {
1083     return 0;
1084   }
1085 
1086   felem_inv(z2, z1);
1087   felem_square(tmp, z2);
1088   felem_reduce(z1, tmp);
1089   felem_mul(tmp, x_in, z1);
1090   felem_reduce(x_in, tmp);
1091   felem_contract(x_out, x_in);
1092   if (x != NULL && !felem_to_BN(x, x_out)) {
1093     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1094     return 0;
1095   }
1096 
1097   felem_mul(tmp, z1, z2);
1098   felem_reduce(z1, tmp);
1099   felem_mul(tmp, y_in, z1);
1100   felem_reduce(y_in, tmp);
1101   felem_contract(y_out, y_in);
1102   if (y != NULL && !felem_to_BN(y, y_out)) {
1103     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1104     return 0;
1105   }
1106 
1107   return 1;
1108 }
1109 
make_points_affine(size_t num,felem points[][3],felem tmp_felems[])1110 static void make_points_affine(size_t num, felem points[/*num*/][3],
1111                                felem tmp_felems[/*num+1*/]) {
1112   /* Runs in constant time, unless an input is the point at infinity
1113    * (which normally shouldn't happen). */
1114   ec_GFp_nistp_points_make_affine_internal(
1115       num, points, sizeof(felem), tmp_felems, (void (*)(void *))felem_one,
1116       (int (*)(const void *))felem_is_zero_int,
1117       (void (*)(void *, const void *))felem_assign,
1118       (void (*)(void *, const void *))felem_square_reduce,
1119       (void (*)(void *, const void *, const void *))felem_mul_reduce,
1120       (void (*)(void *, const void *))felem_inv,
1121       (void (*)(void *, const void *))felem_contract);
1122 }
1123 
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)1124 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1125                                const BIGNUM *g_scalar, const EC_POINT *p_,
1126                                const BIGNUM *p_scalar_, BN_CTX *ctx) {
1127   /* TODO: This function used to take |points| and |scalars| as arrays of
1128    * |num| elements. The code below should be simplified to work in terms of
1129    * |p_| and |p_scalar_|. */
1130   size_t num = p_ != NULL ? 1 : 0;
1131   const EC_POINT **points = p_ != NULL ? &p_ : NULL;
1132   BIGNUM const *const *scalars = p_ != NULL ? &p_scalar_ : NULL;
1133 
1134   int ret = 0;
1135   int j;
1136   unsigned i;
1137   int mixed = 0;
1138   BN_CTX *new_ctx = NULL;
1139   BIGNUM *x, *y, *z, *tmp_scalar;
1140   felem_bytearray g_secret;
1141   felem_bytearray *secrets = NULL;
1142   felem(*pre_comp)[17][3] = NULL;
1143   felem *tmp_felems = NULL;
1144   felem_bytearray tmp;
1145   unsigned num_bytes;
1146   size_t num_points = num;
1147   felem x_in, y_in, z_in, x_out, y_out, z_out;
1148   const EC_POINT *p = NULL;
1149   const BIGNUM *p_scalar = NULL;
1150 
1151   if (ctx == NULL) {
1152     ctx = BN_CTX_new();
1153     new_ctx = ctx;
1154     if (ctx == NULL) {
1155       return 0;
1156     }
1157   }
1158 
1159   BN_CTX_start(ctx);
1160   if ((x = BN_CTX_get(ctx)) == NULL ||
1161       (y = BN_CTX_get(ctx)) == NULL ||
1162       (z = BN_CTX_get(ctx)) == NULL ||
1163       (tmp_scalar = BN_CTX_get(ctx)) == NULL) {
1164     goto err;
1165   }
1166 
1167   if (num_points > 0) {
1168     if (num_points >= 3) {
1169       /* unless we precompute multiples for just one or two points,
1170        * converting those into affine form is time well spent  */
1171       mixed = 1;
1172     }
1173     secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1174     pre_comp = OPENSSL_malloc(num_points * sizeof(felem[17][3]));
1175     if (mixed) {
1176       tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1177     }
1178     if (secrets == NULL ||
1179         pre_comp == NULL ||
1180         (mixed && tmp_felems == NULL)) {
1181       OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
1182       goto err;
1183     }
1184 
1185     /* we treat NULL scalars as 0, and NULL points as points at infinity,
1186      * i.e., they contribute nothing to the linear combination */
1187     memset(secrets, 0, num_points * sizeof(felem_bytearray));
1188     memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1189     for (i = 0; i < num_points; ++i) {
1190       if (i == num) {
1191         /* the generator */
1192         p = EC_GROUP_get0_generator(group);
1193         p_scalar = g_scalar;
1194       } else {
1195         /* the i^th point */
1196         p = points[i];
1197         p_scalar = scalars[i];
1198       }
1199 
1200       if (p_scalar != NULL && p != NULL) {
1201         /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1202         if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) {
1203           /* this is an unusual input, and we don't guarantee
1204            * constant-timeness */
1205           if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
1206             OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1207             goto err;
1208           }
1209           num_bytes = BN_bn2bin(tmp_scalar, tmp);
1210         } else {
1211           num_bytes = BN_bn2bin(p_scalar, tmp);
1212         }
1213 
1214         flip_endian(secrets[i], tmp, num_bytes);
1215         /* precompute multiples */
1216         if (!BN_to_felem(x_out, &p->X) ||
1217             !BN_to_felem(y_out, &p->Y) ||
1218             !BN_to_felem(z_out, &p->Z)) {
1219           goto err;
1220         }
1221 
1222         felem_assign(pre_comp[i][1][0], x_out);
1223         felem_assign(pre_comp[i][1][1], y_out);
1224         felem_assign(pre_comp[i][1][2], z_out);
1225 
1226         for (j = 2; j <= 16; ++j) {
1227           if (j & 1) {
1228             point_add(pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1229                       pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
1230                       0, pre_comp[i][j - 1][0], pre_comp[i][j - 1][1],
1231                       pre_comp[i][j - 1][2]);
1232           } else {
1233             point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1234                          pre_comp[i][j][2], pre_comp[i][j / 2][0],
1235                          pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]);
1236           }
1237         }
1238       }
1239     }
1240 
1241     if (mixed) {
1242       make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1243     }
1244   }
1245 
1246   if (g_scalar != NULL) {
1247     memset(g_secret, 0, sizeof(g_secret));
1248     /* reduce g_scalar to 0 <= g_scalar < 2^224 */
1249     if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) {
1250       /* this is an unusual input, and we don't guarantee constant-timeness */
1251       if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) {
1252         OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1253         goto err;
1254       }
1255       num_bytes = BN_bn2bin(tmp_scalar, tmp);
1256     } else {
1257       num_bytes = BN_bn2bin(g_scalar, tmp);
1258     }
1259 
1260     flip_endian(g_secret, tmp, num_bytes);
1261   }
1262   batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets,
1263             num_points, g_scalar != NULL ? g_secret : NULL, mixed,
1264             (const felem(*)[17][3])pre_comp);
1265 
1266   /* reduce the output to its unique minimal representation */
1267   felem_contract(x_in, x_out);
1268   felem_contract(y_in, y_out);
1269   felem_contract(z_in, z_out);
1270   if (!felem_to_BN(x, x_in) ||
1271       !felem_to_BN(y, y_in) ||
1272       !felem_to_BN(z, z_in)) {
1273     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
1274     goto err;
1275   }
1276   ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1277 
1278 err:
1279   BN_CTX_end(ctx);
1280   BN_CTX_free(new_ctx);
1281   OPENSSL_free(secrets);
1282   OPENSSL_free(pre_comp);
1283   OPENSSL_free(tmp_felems);
1284   return ret;
1285 }
1286 
EC_GFp_nistp224_method(void)1287 const EC_METHOD *EC_GFp_nistp224_method(void) {
1288   static const EC_METHOD ret = {ec_GFp_nistp224_group_init,
1289                                 ec_GFp_simple_group_finish,
1290                                 ec_GFp_simple_group_clear_finish,
1291                                 ec_GFp_simple_group_copy,
1292                                 ec_GFp_nistp224_group_set_curve,
1293                                 ec_GFp_nistp224_point_get_affine_coordinates,
1294                                 ec_GFp_nistp224_points_mul,
1295                                 0 /* check_pub_key_order */,
1296                                 ec_GFp_simple_field_mul,
1297                                 ec_GFp_simple_field_sqr,
1298                                 0 /* field_encode */,
1299                                 0 /* field_decode */,
1300                                 0 /* field_set_to_one */};
1301 
1302   return &ret;
1303 }
1304 
1305 #endif  /* 64_BIT && !WINDOWS && !SMALL */
1306