1 /* Copyright (c) 2020, 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 // An implementation of the NIST P-256 elliptic curve point multiplication.
16 // 256-bit Montgomery form for 64 and 32-bit. Field operations are generated by
17 // Fiat, which lives in //third_party/fiat.
18
19 #include <openssl/base.h>
20
21 #include <openssl/bn.h>
22 #include <openssl/ec.h>
23 #include <openssl/err.h>
24 #include <openssl/mem.h>
25 #include <openssl/type_check.h>
26
27 #include <assert.h>
28 #include <string.h>
29
30 #include "../../internal.h"
31 #include "../delocate.h"
32 #include "./internal.h"
33
34
35 // MSVC does not implement uint128_t, and crashes with intrinsics
36 #if defined(BORINGSSL_HAS_UINT128)
37 #define BORINGSSL_NISTP256_64BIT 1
38 #include "../../../third_party/fiat/p256_64.h"
39 #else
40 #include "../../../third_party/fiat/p256_32.h"
41 #endif
42
43
44 // utility functions, handwritten
45
46 #if defined(BORINGSSL_NISTP256_64BIT)
47
48 #define FIAT_P256_NLIMBS 4
49 typedef uint64_t fiat_p256_limb_t;
50 typedef uint64_t fiat_p256_felem[FIAT_P256_NLIMBS];
51 #else // 64BIT; else 32BIT
52
53 #define FIAT_P256_NLIMBS 8
54 typedef uint32_t fiat_p256_limb_t;
55 typedef uint32_t fiat_p256_felem[FIAT_P256_NLIMBS];
56
57 #endif // 64BIT
58
59
fiat_p256_nz(const fiat_p256_limb_t in1[FIAT_P256_NLIMBS])60 static fiat_p256_limb_t fiat_p256_nz(
61 const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) {
62 fiat_p256_limb_t ret;
63 fiat_p256_nonzero(&ret, in1);
64 return ret;
65 }
66
fiat_p256_copy(fiat_p256_limb_t out[FIAT_P256_NLIMBS],const fiat_p256_limb_t in1[FIAT_P256_NLIMBS])67 static void fiat_p256_copy(fiat_p256_limb_t out[FIAT_P256_NLIMBS],
68 const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) {
69 for (int i = 0; i < FIAT_P256_NLIMBS; i++) {
70 out[i] = in1[i];
71 }
72 }
73
fiat_p256_cmovznz(fiat_p256_limb_t out[FIAT_P256_NLIMBS],fiat_p256_limb_t t,const fiat_p256_limb_t z[FIAT_P256_NLIMBS],const fiat_p256_limb_t nz[FIAT_P256_NLIMBS])74 static void fiat_p256_cmovznz(fiat_p256_limb_t out[FIAT_P256_NLIMBS],
75 fiat_p256_limb_t t,
76 const fiat_p256_limb_t z[FIAT_P256_NLIMBS],
77 const fiat_p256_limb_t nz[FIAT_P256_NLIMBS]) {
78 fiat_p256_selectznz(out, !!t, z, nz);
79 }
80
fiat_p256_from_generic(fiat_p256_felem out,const EC_FELEM * in)81 static void fiat_p256_from_generic(fiat_p256_felem out, const EC_FELEM *in) {
82 fiat_p256_from_bytes(out, in->bytes);
83 }
84
fiat_p256_to_generic(EC_FELEM * out,const fiat_p256_felem in)85 static void fiat_p256_to_generic(EC_FELEM *out, const fiat_p256_felem in) {
86 // This works because 256 is a multiple of 64, so there are no excess bytes to
87 // zero when rounding up to |BN_ULONG|s.
88 OPENSSL_STATIC_ASSERT(
89 256 / 8 == sizeof(BN_ULONG) * ((256 + BN_BITS2 - 1) / BN_BITS2),
90 "fiat_p256_to_bytes leaves bytes uninitialized");
91 fiat_p256_to_bytes(out->bytes, in);
92 }
93
94 // fiat_p256_inv_square calculates |out| = |in|^{-2}
95 //
96 // Based on Fermat's Little Theorem:
97 // a^p = a (mod p)
98 // a^{p-1} = 1 (mod p)
99 // a^{p-3} = a^{-2} (mod p)
fiat_p256_inv_square(fiat_p256_felem out,const fiat_p256_felem in)100 static void fiat_p256_inv_square(fiat_p256_felem out,
101 const fiat_p256_felem in) {
102 // This implements the addition chain described in
103 // https://briansmith.org/ecc-inversion-addition-chains-01#p256_field_inversion
104 fiat_p256_felem x2, x3, x6, x12, x15, x30, x32;
105 fiat_p256_square(x2, in); // 2^2 - 2^1
106 fiat_p256_mul(x2, x2, in); // 2^2 - 2^0
107
108 fiat_p256_square(x3, x2); // 2^3 - 2^1
109 fiat_p256_mul(x3, x3, in); // 2^3 - 2^0
110
111 fiat_p256_square(x6, x3);
112 for (int i = 1; i < 3; i++) {
113 fiat_p256_square(x6, x6);
114 } // 2^6 - 2^3
115 fiat_p256_mul(x6, x6, x3); // 2^6 - 2^0
116
117 fiat_p256_square(x12, x6);
118 for (int i = 1; i < 6; i++) {
119 fiat_p256_square(x12, x12);
120 } // 2^12 - 2^6
121 fiat_p256_mul(x12, x12, x6); // 2^12 - 2^0
122
123 fiat_p256_square(x15, x12);
124 for (int i = 1; i < 3; i++) {
125 fiat_p256_square(x15, x15);
126 } // 2^15 - 2^3
127 fiat_p256_mul(x15, x15, x3); // 2^15 - 2^0
128
129 fiat_p256_square(x30, x15);
130 for (int i = 1; i < 15; i++) {
131 fiat_p256_square(x30, x30);
132 } // 2^30 - 2^15
133 fiat_p256_mul(x30, x30, x15); // 2^30 - 2^0
134
135 fiat_p256_square(x32, x30);
136 fiat_p256_square(x32, x32); // 2^32 - 2^2
137 fiat_p256_mul(x32, x32, x2); // 2^32 - 2^0
138
139 fiat_p256_felem ret;
140 fiat_p256_square(ret, x32);
141 for (int i = 1; i < 31 + 1; i++) {
142 fiat_p256_square(ret, ret);
143 } // 2^64 - 2^32
144 fiat_p256_mul(ret, ret, in); // 2^64 - 2^32 + 2^0
145
146 for (int i = 0; i < 96 + 32; i++) {
147 fiat_p256_square(ret, ret);
148 } // 2^192 - 2^160 + 2^128
149 fiat_p256_mul(ret, ret, x32); // 2^192 - 2^160 + 2^128 + 2^32 - 2^0
150
151 for (int i = 0; i < 32; i++) {
152 fiat_p256_square(ret, ret);
153 } // 2^224 - 2^192 + 2^160 + 2^64 - 2^32
154 fiat_p256_mul(ret, ret, x32); // 2^224 - 2^192 + 2^160 + 2^64 - 2^0
155
156 for (int i = 0; i < 30; i++) {
157 fiat_p256_square(ret, ret);
158 } // 2^254 - 2^222 + 2^190 + 2^94 - 2^30
159 fiat_p256_mul(ret, ret, x30); // 2^254 - 2^222 + 2^190 + 2^94 - 2^0
160
161 fiat_p256_square(ret, ret);
162 fiat_p256_square(out, ret); // 2^256 - 2^224 + 2^192 + 2^96 - 2^2
163 }
164
165 // Group operations
166 // ----------------
167 //
168 // Building on top of the field operations we have the operations on the
169 // elliptic curve group itself. Points on the curve are represented in Jacobian
170 // coordinates.
171 //
172 // Both operations were transcribed to Coq and proven to correspond to naive
173 // implementations using Affine coordinates, for all suitable fields. In the
174 // Coq proofs, issues of constant-time execution and memory layout (aliasing)
175 // conventions were not considered. Specification of affine coordinates:
176 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Spec/WeierstrassCurve.v#L28>
177 // As a sanity check, a proof that these points form a commutative group:
178 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/AffineProofs.v#L33>
179
180 // fiat_p256_point_double calculates 2*(x_in, y_in, z_in)
181 //
182 // The method is taken from:
183 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
184 //
185 // Coq transcription and correctness proof:
186 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93>
187 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201>
188 //
189 // Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
190 // while x_out == y_in is not (maybe this works, but it's not tested).
fiat_p256_point_double(fiat_p256_felem x_out,fiat_p256_felem y_out,fiat_p256_felem z_out,const fiat_p256_felem x_in,const fiat_p256_felem y_in,const fiat_p256_felem z_in)191 static void fiat_p256_point_double(fiat_p256_felem x_out, fiat_p256_felem y_out,
192 fiat_p256_felem z_out,
193 const fiat_p256_felem x_in,
194 const fiat_p256_felem y_in,
195 const fiat_p256_felem z_in) {
196 fiat_p256_felem delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta;
197 // delta = z^2
198 fiat_p256_square(delta, z_in);
199 // gamma = y^2
200 fiat_p256_square(gamma, y_in);
201 // beta = x*gamma
202 fiat_p256_mul(beta, x_in, gamma);
203
204 // alpha = 3*(x-delta)*(x+delta)
205 fiat_p256_sub(ftmp, x_in, delta);
206 fiat_p256_add(ftmp2, x_in, delta);
207
208 fiat_p256_add(tmptmp, ftmp2, ftmp2);
209 fiat_p256_add(ftmp2, ftmp2, tmptmp);
210 fiat_p256_mul(alpha, ftmp, ftmp2);
211
212 // x' = alpha^2 - 8*beta
213 fiat_p256_square(x_out, alpha);
214 fiat_p256_add(fourbeta, beta, beta);
215 fiat_p256_add(fourbeta, fourbeta, fourbeta);
216 fiat_p256_add(tmptmp, fourbeta, fourbeta);
217 fiat_p256_sub(x_out, x_out, tmptmp);
218
219 // z' = (y + z)^2 - gamma - delta
220 fiat_p256_add(delta, gamma, delta);
221 fiat_p256_add(ftmp, y_in, z_in);
222 fiat_p256_square(z_out, ftmp);
223 fiat_p256_sub(z_out, z_out, delta);
224
225 // y' = alpha*(4*beta - x') - 8*gamma^2
226 fiat_p256_sub(y_out, fourbeta, x_out);
227 fiat_p256_add(gamma, gamma, gamma);
228 fiat_p256_square(gamma, gamma);
229 fiat_p256_mul(y_out, alpha, y_out);
230 fiat_p256_add(gamma, gamma, gamma);
231 fiat_p256_sub(y_out, y_out, gamma);
232 }
233
234 // fiat_p256_point_add calculates (x1, y1, z1) + (x2, y2, z2)
235 //
236 // The method is taken from:
237 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
238 // adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
239 //
240 // Coq transcription and correctness proof:
241 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L135>
242 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L205>
243 //
244 // This function includes a branch for checking whether the two input points
245 // are equal, (while not equal to the point at infinity). This case never
246 // happens during single point multiplication, so there is no timing leak for
247 // ECDH or ECDSA signing.
fiat_p256_point_add(fiat_p256_felem x3,fiat_p256_felem y3,fiat_p256_felem z3,const fiat_p256_felem x1,const fiat_p256_felem y1,const fiat_p256_felem z1,const int mixed,const fiat_p256_felem x2,const fiat_p256_felem y2,const fiat_p256_felem z2)248 static void fiat_p256_point_add(fiat_p256_felem x3, fiat_p256_felem y3,
249 fiat_p256_felem z3, const fiat_p256_felem x1,
250 const fiat_p256_felem y1,
251 const fiat_p256_felem z1, const int mixed,
252 const fiat_p256_felem x2,
253 const fiat_p256_felem y2,
254 const fiat_p256_felem z2) {
255 fiat_p256_felem x_out, y_out, z_out;
256 fiat_p256_limb_t z1nz = fiat_p256_nz(z1);
257 fiat_p256_limb_t z2nz = fiat_p256_nz(z2);
258
259 // z1z1 = z1z1 = z1**2
260 fiat_p256_felem z1z1;
261 fiat_p256_square(z1z1, z1);
262
263 fiat_p256_felem u1, s1, two_z1z2;
264 if (!mixed) {
265 // z2z2 = z2**2
266 fiat_p256_felem z2z2;
267 fiat_p256_square(z2z2, z2);
268
269 // u1 = x1*z2z2
270 fiat_p256_mul(u1, x1, z2z2);
271
272 // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2
273 fiat_p256_add(two_z1z2, z1, z2);
274 fiat_p256_square(two_z1z2, two_z1z2);
275 fiat_p256_sub(two_z1z2, two_z1z2, z1z1);
276 fiat_p256_sub(two_z1z2, two_z1z2, z2z2);
277
278 // s1 = y1 * z2**3
279 fiat_p256_mul(s1, z2, z2z2);
280 fiat_p256_mul(s1, s1, y1);
281 } else {
282 // We'll assume z2 = 1 (special case z2 = 0 is handled later).
283
284 // u1 = x1*z2z2
285 fiat_p256_copy(u1, x1);
286 // two_z1z2 = 2z1z2
287 fiat_p256_add(two_z1z2, z1, z1);
288 // s1 = y1 * z2**3
289 fiat_p256_copy(s1, y1);
290 }
291
292 // u2 = x2*z1z1
293 fiat_p256_felem u2;
294 fiat_p256_mul(u2, x2, z1z1);
295
296 // h = u2 - u1
297 fiat_p256_felem h;
298 fiat_p256_sub(h, u2, u1);
299
300 fiat_p256_limb_t xneq = fiat_p256_nz(h);
301
302 // z_out = two_z1z2 * h
303 fiat_p256_mul(z_out, h, two_z1z2);
304
305 // z1z1z1 = z1 * z1z1
306 fiat_p256_felem z1z1z1;
307 fiat_p256_mul(z1z1z1, z1, z1z1);
308
309 // s2 = y2 * z1**3
310 fiat_p256_felem s2;
311 fiat_p256_mul(s2, y2, z1z1z1);
312
313 // r = (s2 - s1)*2
314 fiat_p256_felem r;
315 fiat_p256_sub(r, s2, s1);
316 fiat_p256_add(r, r, r);
317
318 fiat_p256_limb_t yneq = fiat_p256_nz(r);
319
320 fiat_p256_limb_t is_nontrivial_double = constant_time_is_zero_w(xneq | yneq) &
321 ~constant_time_is_zero_w(z1nz) &
322 ~constant_time_is_zero_w(z2nz);
323 if (is_nontrivial_double) {
324 fiat_p256_point_double(x3, y3, z3, x1, y1, z1);
325 return;
326 }
327
328 // I = (2h)**2
329 fiat_p256_felem i;
330 fiat_p256_add(i, h, h);
331 fiat_p256_square(i, i);
332
333 // J = h * I
334 fiat_p256_felem j;
335 fiat_p256_mul(j, h, i);
336
337 // V = U1 * I
338 fiat_p256_felem v;
339 fiat_p256_mul(v, u1, i);
340
341 // x_out = r**2 - J - 2V
342 fiat_p256_square(x_out, r);
343 fiat_p256_sub(x_out, x_out, j);
344 fiat_p256_sub(x_out, x_out, v);
345 fiat_p256_sub(x_out, x_out, v);
346
347 // y_out = r(V-x_out) - 2 * s1 * J
348 fiat_p256_sub(y_out, v, x_out);
349 fiat_p256_mul(y_out, y_out, r);
350 fiat_p256_felem s1j;
351 fiat_p256_mul(s1j, s1, j);
352 fiat_p256_sub(y_out, y_out, s1j);
353 fiat_p256_sub(y_out, y_out, s1j);
354
355 fiat_p256_cmovznz(x_out, z1nz, x2, x_out);
356 fiat_p256_cmovznz(x3, z2nz, x1, x_out);
357 fiat_p256_cmovznz(y_out, z1nz, y2, y_out);
358 fiat_p256_cmovznz(y3, z2nz, y1, y_out);
359 fiat_p256_cmovznz(z_out, z1nz, z2, z_out);
360 fiat_p256_cmovznz(z3, z2nz, z1, z_out);
361 }
362
363 // Base point pre computation
364 // --------------------------
365 //
366 // Two different sorts of precomputed tables are used in the following code.
367 // Each contain various points on the curve, where each point is three field
368 // elements (x, y, z).
369 //
370 // For the base point table, z is usually 1 (0 for the point at infinity).
371 // This table has 2 * 16 elements, starting with the following:
372 // index | bits | point
373 // ------+---------+------------------------------
374 // 0 | 0 0 0 0 | 0G
375 // 1 | 0 0 0 1 | 1G
376 // 2 | 0 0 1 0 | 2^64G
377 // 3 | 0 0 1 1 | (2^64 + 1)G
378 // 4 | 0 1 0 0 | 2^128G
379 // 5 | 0 1 0 1 | (2^128 + 1)G
380 // 6 | 0 1 1 0 | (2^128 + 2^64)G
381 // 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
382 // 8 | 1 0 0 0 | 2^192G
383 // 9 | 1 0 0 1 | (2^192 + 1)G
384 // 10 | 1 0 1 0 | (2^192 + 2^64)G
385 // 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
386 // 12 | 1 1 0 0 | (2^192 + 2^128)G
387 // 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
388 // 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
389 // 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
390 // followed by a copy of this with each element multiplied by 2^32.
391 //
392 // The reason for this is so that we can clock bits into four different
393 // locations when doing simple scalar multiplies against the base point,
394 // and then another four locations using the second 16 elements.
395 //
396 // Tables for other points have table[i] = iG for i in 0 .. 16.
397
398 // fiat_p256_g_pre_comp is the table of precomputed base points
399 #if defined(BORINGSSL_NISTP256_64BIT)
400 static const fiat_p256_felem fiat_p256_g_pre_comp[2][16][3] = {
401 {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}},
402 {{0x79e730d418a9143c, 0x75ba95fc5fedb601, 0x79fb732b77622510,
403 0x18905f76a53755c6},
404 {0xddf25357ce95560a, 0x8b4ab8e4ba19e45c, 0xd2e88688dd21f325,
405 0x8571ff1825885d85},
406 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
407 {{0x4f922fc516a0d2bb, 0xd5cc16c1a623499, 0x9241cf3a57c62c8b,
408 0x2f5e6961fd1b667f},
409 {0x5c15c70bf5a01797, 0x3d20b44d60956192, 0x4911b37071fdb52,
410 0xf648f9168d6f0f7b},
411 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
412 {{0x9e566847e137bbbc, 0xe434469e8a6a0bec, 0xb1c4276179d73463,
413 0x5abe0285133d0015},
414 {0x92aa837cc04c7dab, 0x573d9f4c43260c07, 0xc93156278e6cc37,
415 0x94bb725b6b6f7383},
416 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
417 {{0x62a8c244bfe20925, 0x91c19ac38fdce867, 0x5a96a5d5dd387063,
418 0x61d587d421d324f6},
419 {0xe87673a2a37173ea, 0x2384800853778b65, 0x10f8441e05bab43e,
420 0xfa11fe124621efbe},
421 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
422 {{0x1c891f2b2cb19ffd, 0x1ba8d5bb1923c23, 0xb6d03d678ac5ca8e,
423 0x586eb04c1f13bedc},
424 {0xc35c6e527e8ed09, 0x1e81a33c1819ede2, 0x278fd6c056c652fa,
425 0x19d5ac0870864f11},
426 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
427 {{0x62577734d2b533d5, 0x673b8af6a1bdddc0, 0x577e7c9aa79ec293,
428 0xbb6de651c3b266b1},
429 {0xe7e9303ab65259b3, 0xd6a0afd3d03a7480, 0xc5ac83d19b3cfc27,
430 0x60b4619a5d18b99b},
431 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
432 {{0xbd6a38e11ae5aa1c, 0xb8b7652b49e73658, 0xb130014ee5f87ed,
433 0x9d0f27b2aeebffcd},
434 {0xca9246317a730a55, 0x9c955b2fddbbc83a, 0x7c1dfe0ac019a71,
435 0x244a566d356ec48d},
436 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
437 {{0x56f8410ef4f8b16a, 0x97241afec47b266a, 0xa406b8e6d9c87c1,
438 0x803f3e02cd42ab1b},
439 {0x7f0309a804dbec69, 0xa83b85f73bbad05f, 0xc6097273ad8e197f,
440 0xc097440e5067adc1},
441 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
442 {{0x846a56f2c379ab34, 0xa8ee068b841df8d1, 0x20314459176c68ef,
443 0xf1af32d5915f1f30},
444 {0x99c375315d75bd50, 0x837cffbaf72f67bc, 0x613a41848d7723f,
445 0x23d0f130e2d41c8b},
446 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
447 {{0xed93e225d5be5a2b, 0x6fe799835934f3c6, 0x4314092622626ffc,
448 0x50bbb4d97990216a},
449 {0x378191c6e57ec63e, 0x65422c40181dcdb2, 0x41a8099b0236e0f6,
450 0x2b10011801fe49c3},
451 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
452 {{0xfc68b5c59b391593, 0xc385f5a2598270fc, 0x7144f3aad19adcbb,
453 0xdd55899983fbae0c},
454 {0x93b88b8e74b82ff4, 0xd2e03c4071e734c9, 0x9a7a9eaf43c0322a,
455 0xe6e4c551149d6041},
456 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
457 {{0x5fe14bfe80ec21fe, 0xf6ce116ac255be82, 0x98bc5a072f4a5d67,
458 0xfad27148db7e63af},
459 {0x90c0b6ac29ab05b3, 0x37a9a83c4e251ae6, 0xa7dc875c2aade7d,
460 0x77387de39f0e1a84},
461 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
462 {{0x1e9ecc49a56c0dd7, 0xa5cffcd846086c74, 0x8f7a1408f505aece,
463 0xb37b85c0bef0c47e},
464 {0x3596b6e4cc0e6a8f, 0xfd6d4bbf6b388f23, 0xaba453fac39cef4e,
465 0x9c135ac8f9f628d5},
466 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
467 {{0xa1c729495c8f8be, 0x2961c4803bf362bf, 0x9e418403df63d4ac,
468 0xc109f9cb91ece900},
469 {0xc2d095d058945705, 0xb9083d96ddeb85c0, 0x84692b8d7a40449b,
470 0x9bc3344f2eee1ee1},
471 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
472 {{0xd5ae35642913074, 0x55491b2748a542b1, 0x469ca665b310732a,
473 0x29591d525f1a4cc1},
474 {0xe76f5b6bb84f983f, 0xbe7eef419f5f84e1, 0x1200d49680baa189,
475 0x6376551f18ef332c},
476 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}},
477 {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}},
478 {{0x202886024147519a, 0xd0981eac26b372f0, 0xa9d4a7caa785ebc8,
479 0xd953c50ddbdf58e9},
480 {0x9d6361ccfd590f8f, 0x72e9626b44e6c917, 0x7fd9611022eb64cf,
481 0x863ebb7e9eb288f3},
482 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
483 {{0x4fe7ee31b0e63d34, 0xf4600572a9e54fab, 0xc0493334d5e7b5a4,
484 0x8589fb9206d54831},
485 {0xaa70f5cc6583553a, 0x879094ae25649e5, 0xcc90450710044652,
486 0xebb0696d02541c4f},
487 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
488 {{0xabbaa0c03b89da99, 0xa6f2d79eb8284022, 0x27847862b81c05e8,
489 0x337a4b5905e54d63},
490 {0x3c67500d21f7794a, 0x207005b77d6d7f61, 0xa5a378104cfd6e8,
491 0xd65e0d5f4c2fbd6},
492 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
493 {{0xd433e50f6d3549cf, 0x6f33696ffacd665e, 0x695bfdacce11fcb4,
494 0x810ee252af7c9860},
495 {0x65450fe17159bb2c, 0xf7dfbebe758b357b, 0x2b057e74d69fea72,
496 0xd485717a92731745},
497 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
498 {{0xce1f69bbe83f7669, 0x9f8ae8272877d6b, 0x9548ae543244278d,
499 0x207755dee3c2c19c},
500 {0x87bd61d96fef1945, 0x18813cefb12d28c3, 0x9fbcd1d672df64aa,
501 0x48dc5ee57154b00d},
502 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
503 {{0xef0f469ef49a3154, 0x3e85a5956e2b2e9a, 0x45aaec1eaa924a9c,
504 0xaa12dfc8a09e4719},
505 {0x26f272274df69f1d, 0xe0e4c82ca2ff5e73, 0xb9d8ce73b7a9dd44,
506 0x6c036e73e48ca901},
507 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
508 {{0xe1e421e1a47153f0, 0xb86c3b79920418c9, 0x93bdce87705d7672,
509 0xf25ae793cab79a77},
510 {0x1f3194a36d869d0c, 0x9d55c8824986c264, 0x49fb5ea3096e945e,
511 0x39b8e65313db0a3e},
512 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
513 {{0xe3417bc035d0b34a, 0x440b386b8327c0a7, 0x8fb7262dac0362d1,
514 0x2c41114ce0cdf943},
515 {0x2ba5cef1ad95a0b1, 0xc09b37a867d54362, 0x26d6cdd201e486c9,
516 0x20477abf42ff9297},
517 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
518 {{0xf121b41bc0a67d2, 0x62d4760a444d248a, 0xe044f1d659b4737,
519 0x8fde365250bb4a8},
520 {0xaceec3da848bf287, 0xc2a62182d3369d6e, 0x3582dfdc92449482,
521 0x2f7e2fd2565d6cd7},
522 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
523 {{0xa0122b5178a876b, 0x51ff96ff085104b4, 0x50b31ab14f29f76,
524 0x84abb28b5f87d4e6},
525 {0xd5ed439f8270790a, 0x2d6cb59d85e3f46b, 0x75f55c1b6c1e2212,
526 0xe5436f6717655640},
527 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
528 {{0xc2965ecc9aeb596d, 0x1ea03e7023c92b4, 0x4704b4b62e013961,
529 0xca8fd3f905ea367},
530 {0x92523a42551b2b61, 0x1eb7a89c390fcd06, 0xe7f1d2be0392a63e,
531 0x96dca2644ddb0c33},
532 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
533 {{0x231c210e15339848, 0xe87a28e870778c8d, 0x9d1de6616956e170,
534 0x4ac3c9382bb09c0b},
535 {0x19be05516998987d, 0x8b2376c4ae09f4d6, 0x1de0b7651a3f933d,
536 0x380d94c7e39705f4},
537 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
538 {{0x3685954b8c31c31d, 0x68533d005bf21a0c, 0xbd7626e75c79ec9,
539 0xca17754742c69d54},
540 {0xcc6edafff6d2dbb2, 0xfd0d8cbd174a9d18, 0x875e8793aa4578e8,
541 0xa976a7139cab2ce6},
542 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
543 {{0xce37ab11b43ea1db, 0xa7ff1a95259d292, 0x851b02218f84f186,
544 0xa7222beadefaad13},
545 {0xa2ac78ec2b0a9144, 0x5a024051f2fa59c5, 0x91d1eca56147ce38,
546 0xbe94d523bc2ac690},
547 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}},
548 {{0x2d8daefd79ec1a0f, 0x3bbcd6fdceb39c97, 0xf5575ffc58f61a95,
549 0xdbd986c4adf7b420},
550 {0x81aa881415f39eb7, 0x6ee2fcf5b98d976c, 0x5465475dcf2f717d,
551 0x8e24d3c46860bbd0},
552 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}}};
553 #else
554 static const fiat_p256_felem fiat_p256_g_pre_comp[2][16][3] = {
555 {{{0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0},
556 {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0},
557 {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}},
558 {{0x18a9143c, 0x79e730d4, 0x5fedb601, 0x75ba95fc, 0x77622510, 0x79fb732b,
559 0xa53755c6, 0x18905f76},
560 {0xce95560a, 0xddf25357, 0xba19e45c, 0x8b4ab8e4, 0xdd21f325, 0xd2e88688,
561 0x25885d85, 0x8571ff18},
562 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
563 {{0x16a0d2bb, 0x4f922fc5, 0x1a623499, 0xd5cc16c, 0x57c62c8b, 0x9241cf3a,
564 0xfd1b667f, 0x2f5e6961},
565 {0xf5a01797, 0x5c15c70b, 0x60956192, 0x3d20b44d, 0x71fdb52, 0x4911b37,
566 0x8d6f0f7b, 0xf648f916},
567 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
568 {{0xe137bbbc, 0x9e566847, 0x8a6a0bec, 0xe434469e, 0x79d73463, 0xb1c42761,
569 0x133d0015, 0x5abe0285},
570 {0xc04c7dab, 0x92aa837c, 0x43260c07, 0x573d9f4c, 0x78e6cc37, 0xc931562,
571 0x6b6f7383, 0x94bb725b},
572 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
573 {{0xbfe20925, 0x62a8c244, 0x8fdce867, 0x91c19ac3, 0xdd387063, 0x5a96a5d5,
574 0x21d324f6, 0x61d587d4},
575 {0xa37173ea, 0xe87673a2, 0x53778b65, 0x23848008, 0x5bab43e, 0x10f8441e,
576 0x4621efbe, 0xfa11fe12},
577 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
578 {{0x2cb19ffd, 0x1c891f2b, 0xb1923c23, 0x1ba8d5b, 0x8ac5ca8e, 0xb6d03d67,
579 0x1f13bedc, 0x586eb04c},
580 {0x27e8ed09, 0xc35c6e5, 0x1819ede2, 0x1e81a33c, 0x56c652fa, 0x278fd6c0,
581 0x70864f11, 0x19d5ac08},
582 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
583 {{0xd2b533d5, 0x62577734, 0xa1bdddc0, 0x673b8af6, 0xa79ec293, 0x577e7c9a,
584 0xc3b266b1, 0xbb6de651},
585 {0xb65259b3, 0xe7e9303a, 0xd03a7480, 0xd6a0afd3, 0x9b3cfc27, 0xc5ac83d1,
586 0x5d18b99b, 0x60b4619a},
587 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
588 {{0x1ae5aa1c, 0xbd6a38e1, 0x49e73658, 0xb8b7652b, 0xee5f87ed, 0xb130014,
589 0xaeebffcd, 0x9d0f27b2},
590 {0x7a730a55, 0xca924631, 0xddbbc83a, 0x9c955b2f, 0xac019a71, 0x7c1dfe0,
591 0x356ec48d, 0x244a566d},
592 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
593 {{0xf4f8b16a, 0x56f8410e, 0xc47b266a, 0x97241afe, 0x6d9c87c1, 0xa406b8e,
594 0xcd42ab1b, 0x803f3e02},
595 {0x4dbec69, 0x7f0309a8, 0x3bbad05f, 0xa83b85f7, 0xad8e197f, 0xc6097273,
596 0x5067adc1, 0xc097440e},
597 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
598 {{0xc379ab34, 0x846a56f2, 0x841df8d1, 0xa8ee068b, 0x176c68ef, 0x20314459,
599 0x915f1f30, 0xf1af32d5},
600 {0x5d75bd50, 0x99c37531, 0xf72f67bc, 0x837cffba, 0x48d7723f, 0x613a418,
601 0xe2d41c8b, 0x23d0f130},
602 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
603 {{0xd5be5a2b, 0xed93e225, 0x5934f3c6, 0x6fe79983, 0x22626ffc, 0x43140926,
604 0x7990216a, 0x50bbb4d9},
605 {0xe57ec63e, 0x378191c6, 0x181dcdb2, 0x65422c40, 0x236e0f6, 0x41a8099b,
606 0x1fe49c3, 0x2b100118},
607 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
608 {{0x9b391593, 0xfc68b5c5, 0x598270fc, 0xc385f5a2, 0xd19adcbb, 0x7144f3aa,
609 0x83fbae0c, 0xdd558999},
610 {0x74b82ff4, 0x93b88b8e, 0x71e734c9, 0xd2e03c40, 0x43c0322a, 0x9a7a9eaf,
611 0x149d6041, 0xe6e4c551},
612 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
613 {{0x80ec21fe, 0x5fe14bfe, 0xc255be82, 0xf6ce116a, 0x2f4a5d67, 0x98bc5a07,
614 0xdb7e63af, 0xfad27148},
615 {0x29ab05b3, 0x90c0b6ac, 0x4e251ae6, 0x37a9a83c, 0xc2aade7d, 0xa7dc875,
616 0x9f0e1a84, 0x77387de3},
617 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
618 {{0xa56c0dd7, 0x1e9ecc49, 0x46086c74, 0xa5cffcd8, 0xf505aece, 0x8f7a1408,
619 0xbef0c47e, 0xb37b85c0},
620 {0xcc0e6a8f, 0x3596b6e4, 0x6b388f23, 0xfd6d4bbf, 0xc39cef4e, 0xaba453fa,
621 0xf9f628d5, 0x9c135ac8},
622 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
623 {{0x95c8f8be, 0xa1c7294, 0x3bf362bf, 0x2961c480, 0xdf63d4ac, 0x9e418403,
624 0x91ece900, 0xc109f9cb},
625 {0x58945705, 0xc2d095d0, 0xddeb85c0, 0xb9083d96, 0x7a40449b, 0x84692b8d,
626 0x2eee1ee1, 0x9bc3344f},
627 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
628 {{0x42913074, 0xd5ae356, 0x48a542b1, 0x55491b27, 0xb310732a, 0x469ca665,
629 0x5f1a4cc1, 0x29591d52},
630 {0xb84f983f, 0xe76f5b6b, 0x9f5f84e1, 0xbe7eef41, 0x80baa189, 0x1200d496,
631 0x18ef332c, 0x6376551f},
632 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}},
633 {{{0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0},
634 {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0},
635 {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}},
636 {{0x4147519a, 0x20288602, 0x26b372f0, 0xd0981eac, 0xa785ebc8, 0xa9d4a7ca,
637 0xdbdf58e9, 0xd953c50d},
638 {0xfd590f8f, 0x9d6361cc, 0x44e6c917, 0x72e9626b, 0x22eb64cf, 0x7fd96110,
639 0x9eb288f3, 0x863ebb7e},
640 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
641 {{0xb0e63d34, 0x4fe7ee31, 0xa9e54fab, 0xf4600572, 0xd5e7b5a4, 0xc0493334,
642 0x6d54831, 0x8589fb92},
643 {0x6583553a, 0xaa70f5cc, 0xe25649e5, 0x879094a, 0x10044652, 0xcc904507,
644 0x2541c4f, 0xebb0696d},
645 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
646 {{0x3b89da99, 0xabbaa0c0, 0xb8284022, 0xa6f2d79e, 0xb81c05e8, 0x27847862,
647 0x5e54d63, 0x337a4b59},
648 {0x21f7794a, 0x3c67500d, 0x7d6d7f61, 0x207005b7, 0x4cfd6e8, 0xa5a3781,
649 0xf4c2fbd6, 0xd65e0d5},
650 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
651 {{0x6d3549cf, 0xd433e50f, 0xfacd665e, 0x6f33696f, 0xce11fcb4, 0x695bfdac,
652 0xaf7c9860, 0x810ee252},
653 {0x7159bb2c, 0x65450fe1, 0x758b357b, 0xf7dfbebe, 0xd69fea72, 0x2b057e74,
654 0x92731745, 0xd485717a},
655 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
656 {{0xe83f7669, 0xce1f69bb, 0x72877d6b, 0x9f8ae82, 0x3244278d, 0x9548ae54,
657 0xe3c2c19c, 0x207755de},
658 {0x6fef1945, 0x87bd61d9, 0xb12d28c3, 0x18813cef, 0x72df64aa, 0x9fbcd1d6,
659 0x7154b00d, 0x48dc5ee5},
660 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
661 {{0xf49a3154, 0xef0f469e, 0x6e2b2e9a, 0x3e85a595, 0xaa924a9c, 0x45aaec1e,
662 0xa09e4719, 0xaa12dfc8},
663 {0x4df69f1d, 0x26f27227, 0xa2ff5e73, 0xe0e4c82c, 0xb7a9dd44, 0xb9d8ce73,
664 0xe48ca901, 0x6c036e73},
665 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
666 {{0xa47153f0, 0xe1e421e1, 0x920418c9, 0xb86c3b79, 0x705d7672, 0x93bdce87,
667 0xcab79a77, 0xf25ae793},
668 {0x6d869d0c, 0x1f3194a3, 0x4986c264, 0x9d55c882, 0x96e945e, 0x49fb5ea3,
669 0x13db0a3e, 0x39b8e653},
670 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
671 {{0x35d0b34a, 0xe3417bc0, 0x8327c0a7, 0x440b386b, 0xac0362d1, 0x8fb7262d,
672 0xe0cdf943, 0x2c41114c},
673 {0xad95a0b1, 0x2ba5cef1, 0x67d54362, 0xc09b37a8, 0x1e486c9, 0x26d6cdd2,
674 0x42ff9297, 0x20477abf},
675 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
676 {{0xbc0a67d2, 0xf121b41, 0x444d248a, 0x62d4760a, 0x659b4737, 0xe044f1d,
677 0x250bb4a8, 0x8fde365},
678 {0x848bf287, 0xaceec3da, 0xd3369d6e, 0xc2a62182, 0x92449482, 0x3582dfdc,
679 0x565d6cd7, 0x2f7e2fd2},
680 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
681 {{0x178a876b, 0xa0122b5, 0x85104b4, 0x51ff96ff, 0x14f29f76, 0x50b31ab,
682 0x5f87d4e6, 0x84abb28b},
683 {0x8270790a, 0xd5ed439f, 0x85e3f46b, 0x2d6cb59d, 0x6c1e2212, 0x75f55c1b,
684 0x17655640, 0xe5436f67},
685 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
686 {{0x9aeb596d, 0xc2965ecc, 0x23c92b4, 0x1ea03e7, 0x2e013961, 0x4704b4b6,
687 0x905ea367, 0xca8fd3f},
688 {0x551b2b61, 0x92523a42, 0x390fcd06, 0x1eb7a89c, 0x392a63e, 0xe7f1d2be,
689 0x4ddb0c33, 0x96dca264},
690 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
691 {{0x15339848, 0x231c210e, 0x70778c8d, 0xe87a28e8, 0x6956e170, 0x9d1de661,
692 0x2bb09c0b, 0x4ac3c938},
693 {0x6998987d, 0x19be0551, 0xae09f4d6, 0x8b2376c4, 0x1a3f933d, 0x1de0b765,
694 0xe39705f4, 0x380d94c7},
695 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
696 {{0x8c31c31d, 0x3685954b, 0x5bf21a0c, 0x68533d00, 0x75c79ec9, 0xbd7626e,
697 0x42c69d54, 0xca177547},
698 {0xf6d2dbb2, 0xcc6edaff, 0x174a9d18, 0xfd0d8cbd, 0xaa4578e8, 0x875e8793,
699 0x9cab2ce6, 0xa976a713},
700 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
701 {{0xb43ea1db, 0xce37ab11, 0x5259d292, 0xa7ff1a9, 0x8f84f186, 0x851b0221,
702 0xdefaad13, 0xa7222bea},
703 {0x2b0a9144, 0xa2ac78ec, 0xf2fa59c5, 0x5a024051, 0x6147ce38, 0x91d1eca5,
704 0xbc2ac690, 0xbe94d523},
705 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}},
706 {{0x79ec1a0f, 0x2d8daefd, 0xceb39c97, 0x3bbcd6fd, 0x58f61a95, 0xf5575ffc,
707 0xadf7b420, 0xdbd986c4},
708 {0x15f39eb7, 0x81aa8814, 0xb98d976c, 0x6ee2fcf5, 0xcf2f717d, 0x5465475d,
709 0x6860bbd0, 0x8e24d3c4},
710 {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}}};
711 #endif
712
713 // fiat_p256_select_point selects the |idx|th point from a precomputation table
714 // and copies it to out.
fiat_p256_select_point(const fiat_p256_limb_t idx,size_t size,const fiat_p256_felem pre_comp[][3],fiat_p256_felem out[3])715 static void fiat_p256_select_point(const fiat_p256_limb_t idx, size_t size,
716 const fiat_p256_felem pre_comp[/*size*/][3],
717 fiat_p256_felem out[3]) {
718 OPENSSL_memset(out, 0, sizeof(fiat_p256_felem) * 3);
719 for (size_t i = 0; i < size; i++) {
720 fiat_p256_limb_t mismatch = i ^ idx;
721 fiat_p256_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]);
722 fiat_p256_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]);
723 fiat_p256_cmovznz(out[2], mismatch, pre_comp[i][2], out[2]);
724 }
725 }
726
727 // fiat_p256_get_bit returns the |i|th bit in |in|
fiat_p256_get_bit(const uint8_t * in,int i)728 static char fiat_p256_get_bit(const uint8_t *in, int i) {
729 if (i < 0 || i >= 256) {
730 return 0;
731 }
732 return (in[i >> 3] >> (i & 7)) & 1;
733 }
734
735 // OPENSSL EC_METHOD FUNCTIONS
736
737 // Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
738 // (X/Z^2, Y/Z^3).
ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP * group,const EC_RAW_POINT * point,EC_FELEM * x_out,EC_FELEM * y_out)739 static int ec_GFp_nistp256_point_get_affine_coordinates(
740 const EC_GROUP *group, const EC_RAW_POINT *point, EC_FELEM *x_out,
741 EC_FELEM *y_out) {
742 if (ec_GFp_simple_is_at_infinity(group, point)) {
743 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
744 return 0;
745 }
746
747 fiat_p256_felem z1, z2;
748 fiat_p256_from_generic(z1, &point->Z);
749 fiat_p256_inv_square(z2, z1);
750
751 if (x_out != NULL) {
752 fiat_p256_felem x;
753 fiat_p256_from_generic(x, &point->X);
754 fiat_p256_mul(x, x, z2);
755 fiat_p256_from_montgomery(x, x);
756 fiat_p256_to_generic(x_out, x);
757 }
758
759 if (y_out != NULL) {
760 fiat_p256_felem y;
761 fiat_p256_from_generic(y, &point->Y);
762 fiat_p256_square(z2, z2); // z^-4
763 fiat_p256_mul(y, y, z1); // y * z
764 fiat_p256_mul(y, y, z2); // y * z^-3
765 fiat_p256_from_montgomery(y, y);
766 fiat_p256_to_generic(y_out, y);
767 }
768
769 return 1;
770 }
771
ec_GFp_nistp256_add(const EC_GROUP * group,EC_RAW_POINT * r,const EC_RAW_POINT * a,const EC_RAW_POINT * b)772 static void ec_GFp_nistp256_add(const EC_GROUP *group, EC_RAW_POINT *r,
773 const EC_RAW_POINT *a, const EC_RAW_POINT *b) {
774 fiat_p256_felem x1, y1, z1, x2, y2, z2;
775 fiat_p256_from_generic(x1, &a->X);
776 fiat_p256_from_generic(y1, &a->Y);
777 fiat_p256_from_generic(z1, &a->Z);
778 fiat_p256_from_generic(x2, &b->X);
779 fiat_p256_from_generic(y2, &b->Y);
780 fiat_p256_from_generic(z2, &b->Z);
781 fiat_p256_point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2,
782 z2);
783 fiat_p256_to_generic(&r->X, x1);
784 fiat_p256_to_generic(&r->Y, y1);
785 fiat_p256_to_generic(&r->Z, z1);
786 }
787
ec_GFp_nistp256_dbl(const EC_GROUP * group,EC_RAW_POINT * r,const EC_RAW_POINT * a)788 static void ec_GFp_nistp256_dbl(const EC_GROUP *group, EC_RAW_POINT *r,
789 const EC_RAW_POINT *a) {
790 fiat_p256_felem x, y, z;
791 fiat_p256_from_generic(x, &a->X);
792 fiat_p256_from_generic(y, &a->Y);
793 fiat_p256_from_generic(z, &a->Z);
794 fiat_p256_point_double(x, y, z, x, y, z);
795 fiat_p256_to_generic(&r->X, x);
796 fiat_p256_to_generic(&r->Y, y);
797 fiat_p256_to_generic(&r->Z, z);
798 }
799
ec_GFp_nistp256_point_mul(const EC_GROUP * group,EC_RAW_POINT * r,const EC_RAW_POINT * p,const EC_SCALAR * scalar)800 static void ec_GFp_nistp256_point_mul(const EC_GROUP *group, EC_RAW_POINT *r,
801 const EC_RAW_POINT *p,
802 const EC_SCALAR *scalar) {
803 fiat_p256_felem p_pre_comp[17][3];
804 OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp));
805 // Precompute multiples.
806 fiat_p256_from_generic(p_pre_comp[1][0], &p->X);
807 fiat_p256_from_generic(p_pre_comp[1][1], &p->Y);
808 fiat_p256_from_generic(p_pre_comp[1][2], &p->Z);
809 for (size_t j = 2; j <= 16; ++j) {
810 if (j & 1) {
811 fiat_p256_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2],
812 p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2],
813 0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1],
814 p_pre_comp[j - 1][2]);
815 } else {
816 fiat_p256_point_double(p_pre_comp[j][0], p_pre_comp[j][1],
817 p_pre_comp[j][2], p_pre_comp[j / 2][0],
818 p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]);
819 }
820 }
821
822 // Set nq to the point at infinity.
823 fiat_p256_felem nq[3] = {{0}, {0}, {0}}, ftmp, tmp[3];
824
825 // Loop over |scalar| msb-to-lsb, incorporating |p_pre_comp| every 5th round.
826 int skip = 1; // Save two point operations in the first round.
827 for (size_t i = 255; i < 256; i--) {
828 // double
829 if (!skip) {
830 fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
831 }
832
833 // do other additions every 5 doublings
834 if (i % 5 == 0) {
835 uint64_t bits = fiat_p256_get_bit(scalar->bytes, i + 4) << 5;
836 bits |= fiat_p256_get_bit(scalar->bytes, i + 3) << 4;
837 bits |= fiat_p256_get_bit(scalar->bytes, i + 2) << 3;
838 bits |= fiat_p256_get_bit(scalar->bytes, i + 1) << 2;
839 bits |= fiat_p256_get_bit(scalar->bytes, i) << 1;
840 bits |= fiat_p256_get_bit(scalar->bytes, i - 1);
841 uint8_t sign, digit;
842 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
843
844 // select the point to add or subtract, in constant time.
845 fiat_p256_select_point(digit, 17, (const fiat_p256_felem(*)[3])p_pre_comp,
846 tmp);
847 fiat_p256_opp(ftmp, tmp[1]); // (X, -Y, Z) is the negative point.
848 fiat_p256_cmovznz(tmp[1], sign, tmp[1], ftmp);
849
850 if (!skip) {
851 fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2],
852 0 /* mixed */, tmp[0], tmp[1], tmp[2]);
853 } else {
854 fiat_p256_copy(nq[0], tmp[0]);
855 fiat_p256_copy(nq[1], tmp[1]);
856 fiat_p256_copy(nq[2], tmp[2]);
857 skip = 0;
858 }
859 }
860 }
861
862 fiat_p256_to_generic(&r->X, nq[0]);
863 fiat_p256_to_generic(&r->Y, nq[1]);
864 fiat_p256_to_generic(&r->Z, nq[2]);
865 }
866
ec_GFp_nistp256_point_mul_base(const EC_GROUP * group,EC_RAW_POINT * r,const EC_SCALAR * scalar)867 static void ec_GFp_nistp256_point_mul_base(const EC_GROUP *group,
868 EC_RAW_POINT *r,
869 const EC_SCALAR *scalar) {
870 // Set nq to the point at infinity.
871 fiat_p256_felem nq[3] = {{0}, {0}, {0}}, tmp[3];
872
873 int skip = 1; // Save two point operations in the first round.
874 for (size_t i = 31; i < 32; i--) {
875 if (!skip) {
876 fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
877 }
878
879 // First, look 32 bits upwards.
880 uint64_t bits = fiat_p256_get_bit(scalar->bytes, i + 224) << 3;
881 bits |= fiat_p256_get_bit(scalar->bytes, i + 160) << 2;
882 bits |= fiat_p256_get_bit(scalar->bytes, i + 96) << 1;
883 bits |= fiat_p256_get_bit(scalar->bytes, i + 32);
884 // Select the point to add, in constant time.
885 fiat_p256_select_point(bits, 16, fiat_p256_g_pre_comp[1], tmp);
886
887 if (!skip) {
888 fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2],
889 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
890 } else {
891 fiat_p256_copy(nq[0], tmp[0]);
892 fiat_p256_copy(nq[1], tmp[1]);
893 fiat_p256_copy(nq[2], tmp[2]);
894 skip = 0;
895 }
896
897 // Second, look at the current position.
898 bits = fiat_p256_get_bit(scalar->bytes, i + 192) << 3;
899 bits |= fiat_p256_get_bit(scalar->bytes, i + 128) << 2;
900 bits |= fiat_p256_get_bit(scalar->bytes, i + 64) << 1;
901 bits |= fiat_p256_get_bit(scalar->bytes, i);
902 // Select the point to add, in constant time.
903 fiat_p256_select_point(bits, 16, fiat_p256_g_pre_comp[0], tmp);
904 fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
905 tmp[0], tmp[1], tmp[2]);
906 }
907
908 fiat_p256_to_generic(&r->X, nq[0]);
909 fiat_p256_to_generic(&r->Y, nq[1]);
910 fiat_p256_to_generic(&r->Z, nq[2]);
911 }
912
ec_GFp_nistp256_point_mul_public(const EC_GROUP * group,EC_RAW_POINT * r,const EC_SCALAR * g_scalar,const EC_RAW_POINT * p,const EC_SCALAR * p_scalar)913 static void ec_GFp_nistp256_point_mul_public(const EC_GROUP *group,
914 EC_RAW_POINT *r,
915 const EC_SCALAR *g_scalar,
916 const EC_RAW_POINT *p,
917 const EC_SCALAR *p_scalar) {
918 #define P256_WSIZE_PUBLIC 4
919 // Precompute multiples of |p|. p_pre_comp[i] is (2*i+1) * |p|.
920 fiat_p256_felem p_pre_comp[1 << (P256_WSIZE_PUBLIC - 1)][3];
921 fiat_p256_from_generic(p_pre_comp[0][0], &p->X);
922 fiat_p256_from_generic(p_pre_comp[0][1], &p->Y);
923 fiat_p256_from_generic(p_pre_comp[0][2], &p->Z);
924 fiat_p256_felem p2[3];
925 fiat_p256_point_double(p2[0], p2[1], p2[2], p_pre_comp[0][0],
926 p_pre_comp[0][1], p_pre_comp[0][2]);
927 for (size_t i = 1; i < OPENSSL_ARRAY_SIZE(p_pre_comp); i++) {
928 fiat_p256_point_add(p_pre_comp[i][0], p_pre_comp[i][1], p_pre_comp[i][2],
929 p_pre_comp[i - 1][0], p_pre_comp[i - 1][1],
930 p_pre_comp[i - 1][2], 0 /* not mixed */, p2[0], p2[1],
931 p2[2]);
932 }
933
934 // Set up the coefficients for |p_scalar|.
935 int8_t p_wNAF[257];
936 ec_compute_wNAF(group, p_wNAF, p_scalar, 256, P256_WSIZE_PUBLIC);
937
938 // Set |ret| to the point at infinity.
939 int skip = 1; // Save some point operations.
940 fiat_p256_felem ret[3] = {{0}, {0}, {0}};
941 for (int i = 256; i >= 0; i--) {
942 if (!skip) {
943 fiat_p256_point_double(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2]);
944 }
945
946 // For the |g_scalar|, we use the precomputed table without the
947 // constant-time lookup.
948 if (i <= 31) {
949 // First, look 32 bits upwards.
950 uint64_t bits = fiat_p256_get_bit(g_scalar->bytes, i + 224) << 3;
951 bits |= fiat_p256_get_bit(g_scalar->bytes, i + 160) << 2;
952 bits |= fiat_p256_get_bit(g_scalar->bytes, i + 96) << 1;
953 bits |= fiat_p256_get_bit(g_scalar->bytes, i + 32);
954 fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2],
955 1 /* mixed */, fiat_p256_g_pre_comp[1][bits][0],
956 fiat_p256_g_pre_comp[1][bits][1],
957 fiat_p256_g_pre_comp[1][bits][2]);
958 skip = 0;
959
960 // Second, look at the current position.
961 bits = fiat_p256_get_bit(g_scalar->bytes, i + 192) << 3;
962 bits |= fiat_p256_get_bit(g_scalar->bytes, i + 128) << 2;
963 bits |= fiat_p256_get_bit(g_scalar->bytes, i + 64) << 1;
964 bits |= fiat_p256_get_bit(g_scalar->bytes, i);
965 fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2],
966 1 /* mixed */, fiat_p256_g_pre_comp[0][bits][0],
967 fiat_p256_g_pre_comp[0][bits][1],
968 fiat_p256_g_pre_comp[0][bits][2]);
969 }
970
971 int digit = p_wNAF[i];
972 if (digit != 0) {
973 assert(digit & 1);
974 int idx = digit < 0 ? (-digit) >> 1 : digit >> 1;
975 fiat_p256_felem *y = &p_pre_comp[idx][1], tmp;
976 if (digit < 0) {
977 fiat_p256_opp(tmp, p_pre_comp[idx][1]);
978 y = &tmp;
979 }
980 if (!skip) {
981 fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2],
982 0 /* not mixed */, p_pre_comp[idx][0], *y,
983 p_pre_comp[idx][2]);
984 } else {
985 fiat_p256_copy(ret[0], p_pre_comp[idx][0]);
986 fiat_p256_copy(ret[1], *y);
987 fiat_p256_copy(ret[2], p_pre_comp[idx][2]);
988 skip = 0;
989 }
990 }
991 }
992
993 fiat_p256_to_generic(&r->X, ret[0]);
994 fiat_p256_to_generic(&r->Y, ret[1]);
995 fiat_p256_to_generic(&r->Z, ret[2]);
996 }
997
ec_GFp_nistp256_cmp_x_coordinate(const EC_GROUP * group,const EC_RAW_POINT * p,const EC_SCALAR * r)998 static int ec_GFp_nistp256_cmp_x_coordinate(const EC_GROUP *group,
999 const EC_RAW_POINT *p,
1000 const EC_SCALAR *r) {
1001 if (ec_GFp_simple_is_at_infinity(group, p)) {
1002 return 0;
1003 }
1004
1005 // We wish to compare X/Z^2 with r. This is equivalent to comparing X with
1006 // r*Z^2. Note that X and Z are represented in Montgomery form, while r is
1007 // not.
1008 fiat_p256_felem Z2_mont;
1009 fiat_p256_from_generic(Z2_mont, &p->Z);
1010 fiat_p256_mul(Z2_mont, Z2_mont, Z2_mont);
1011
1012 fiat_p256_felem r_Z2;
1013 fiat_p256_from_bytes(r_Z2, r->bytes); // r < order < p, so this is valid.
1014 fiat_p256_mul(r_Z2, r_Z2, Z2_mont);
1015
1016 fiat_p256_felem X;
1017 fiat_p256_from_generic(X, &p->X);
1018 fiat_p256_from_montgomery(X, X);
1019
1020 if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) {
1021 return 1;
1022 }
1023
1024 // During signing the x coefficient is reduced modulo the group order.
1025 // Therefore there is a small possibility, less than 1/2^128, that group_order
1026 // < p.x < P. in that case we need not only to compare against |r| but also to
1027 // compare against r+group_order.
1028 assert(group->field.width == group->order.width);
1029 if (bn_less_than_words(r->words, group->field_minus_order.words,
1030 group->field.width)) {
1031 // We can ignore the carry because: r + group_order < p < 2^256.
1032 EC_FELEM tmp;
1033 bn_add_words(tmp.words, r->words, group->order.d, group->order.width);
1034 fiat_p256_from_generic(r_Z2, &tmp);
1035 fiat_p256_mul(r_Z2, r_Z2, Z2_mont);
1036 if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) {
1037 return 1;
1038 }
1039 }
1040
1041 return 0;
1042 }
1043
DEFINE_METHOD_FUNCTION(EC_METHOD,EC_GFp_nistp256_method)1044 DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp256_method) {
1045 out->group_init = ec_GFp_mont_group_init;
1046 out->group_finish = ec_GFp_mont_group_finish;
1047 out->group_set_curve = ec_GFp_mont_group_set_curve;
1048 out->point_get_affine_coordinates =
1049 ec_GFp_nistp256_point_get_affine_coordinates;
1050 out->add = ec_GFp_nistp256_add;
1051 out->dbl = ec_GFp_nistp256_dbl;
1052 out->mul = ec_GFp_nistp256_point_mul;
1053 out->mul_base = ec_GFp_nistp256_point_mul_base;
1054 out->mul_public = ec_GFp_nistp256_point_mul_public;
1055 out->felem_mul = ec_GFp_mont_felem_mul;
1056 out->felem_sqr = ec_GFp_mont_felem_sqr;
1057 out->felem_to_bytes = ec_GFp_mont_felem_to_bytes;
1058 out->felem_from_bytes = ec_GFp_mont_felem_from_bytes;
1059 out->scalar_inv0_montgomery = ec_simple_scalar_inv0_montgomery;
1060 out->scalar_to_montgomery_inv_vartime =
1061 ec_simple_scalar_to_montgomery_inv_vartime;
1062 out->cmp_x_coordinate = ec_GFp_nistp256_cmp_x_coordinate;
1063 }
1064
1065 #undef BORINGSSL_NISTP256_64BIT
1066