2 * Copyright 2011-2023 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * ECDSA low level APIs are deprecated for public use, but still ok for
30 #include "internal/deprecated.h"
33 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication
35 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
36 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
37 * work which got its smarts from Daniel J. Bernstein's work on the same.
40 #include <openssl/e_os2.h>
43 #include <openssl/err.h>
46 #include "internal/numbers.h"
49 # error "Your compiler doesn't appear to support 128-bit integer types"
56 * The underlying field. P521 operates over GF(2^521-1). We can serialize an
57 * element of this field into 66 bytes where the most significant byte
58 * contains only a single bit. We call this an felem_bytearray.
61 typedef u8 felem_bytearray[66];
64 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
65 * These values are big-endian.
67 static const felem_bytearray nistp521_curve_params[5] = {
68 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
69 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
70 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
71 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
72 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
73 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
74 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
75 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
77 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
78 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
79 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
80 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
81 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
82 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
83 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
84 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
86 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
87 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
88 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
89 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
90 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
91 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
92 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
93 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
95 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
96 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
97 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
98 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
99 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
100 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
101 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
102 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
104 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
105 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
106 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
107 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
108 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
109 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
110 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
111 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
116 * The representation of field elements.
117 * ------------------------------------
119 * We represent field elements with nine values. These values are either 64 or
120 * 128 bits and the field element represented is:
121 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
122 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
123 * 58 bits apart, but are greater than 58 bits in length, the most significant
124 * bits of each limb overlap with the least significant bits of the next.
126 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
131 typedef uint64_t limb;
132 typedef limb limb_aX __attribute((__aligned__(1)));
133 typedef limb felem[NLIMBS];
134 typedef uint128_t largefelem[NLIMBS];
136 static const limb bottom57bits = 0x1ffffffffffffff;
137 static const limb bottom58bits = 0x3ffffffffffffff;
140 * bin66_to_felem takes a little-endian byte array and converts it into felem
141 * form. This assumes that the CPU is little-endian.
143 static void bin66_to_felem(felem out, const u8 in[66])
145 out[0] = (*((limb *) & in[0])) & bottom58bits;
146 out[1] = (*((limb_aX *) & in[7]) >> 2) & bottom58bits;
147 out[2] = (*((limb_aX *) & in[14]) >> 4) & bottom58bits;
148 out[3] = (*((limb_aX *) & in[21]) >> 6) & bottom58bits;
149 out[4] = (*((limb_aX *) & in[29])) & bottom58bits;
150 out[5] = (*((limb_aX *) & in[36]) >> 2) & bottom58bits;
151 out[6] = (*((limb_aX *) & in[43]) >> 4) & bottom58bits;
152 out[7] = (*((limb_aX *) & in[50]) >> 6) & bottom58bits;
153 out[8] = (*((limb_aX *) & in[58])) & bottom57bits;
157 * felem_to_bin66 takes an felem and serializes into a little endian, 66 byte
158 * array. This assumes that the CPU is little-endian.
160 static void felem_to_bin66(u8 out[66], const felem in)
163 (*((limb *) & out[0])) = in[0];
164 (*((limb_aX *) & out[7])) |= in[1] << 2;
165 (*((limb_aX *) & out[14])) |= in[2] << 4;
166 (*((limb_aX *) & out[21])) |= in[3] << 6;
167 (*((limb_aX *) & out[29])) = in[4];
168 (*((limb_aX *) & out[36])) |= in[5] << 2;
169 (*((limb_aX *) & out[43])) |= in[6] << 4;
170 (*((limb_aX *) & out[50])) |= in[7] << 6;
171 (*((limb_aX *) & out[58])) = in[8];
174 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
175 static int BN_to_felem(felem out, const BIGNUM *bn)
177 felem_bytearray b_out;
180 if (BN_is_negative(bn)) {
181 ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
184 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
186 ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
189 bin66_to_felem(out, b_out);
193 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
194 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
196 felem_bytearray b_out;
197 felem_to_bin66(b_out, in);
198 return BN_lebin2bn(b_out, sizeof(b_out), out);
206 static void felem_one(felem out)
219 static void felem_assign(felem out, const felem in)
232 /* felem_sum64 sets out = out + in. */
233 static void felem_sum64(felem out, const felem in)
246 /* felem_scalar sets out = in * scalar */
247 static void felem_scalar(felem out, const felem in, limb scalar)
249 out[0] = in[0] * scalar;
250 out[1] = in[1] * scalar;
251 out[2] = in[2] * scalar;
252 out[3] = in[3] * scalar;
253 out[4] = in[4] * scalar;
254 out[5] = in[5] * scalar;
255 out[6] = in[6] * scalar;
256 out[7] = in[7] * scalar;
257 out[8] = in[8] * scalar;
260 /* felem_scalar64 sets out = out * scalar */
261 static void felem_scalar64(felem out, limb scalar)
274 /* felem_scalar128 sets out = out * scalar */
275 static void felem_scalar128(largefelem out, limb scalar)
289 * felem_neg sets |out| to |-in|
291 * in[i] < 2^59 + 2^14
295 static void felem_neg(felem out, const felem in)
297 /* In order to prevent underflow, we subtract from 0 mod p. */
298 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
299 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
301 out[0] = two62m3 - in[0];
302 out[1] = two62m2 - in[1];
303 out[2] = two62m2 - in[2];
304 out[3] = two62m2 - in[3];
305 out[4] = two62m2 - in[4];
306 out[5] = two62m2 - in[5];
307 out[6] = two62m2 - in[6];
308 out[7] = two62m2 - in[7];
309 out[8] = two62m2 - in[8];
313 * felem_diff64 subtracts |in| from |out|
315 * in[i] < 2^59 + 2^14
317 * out[i] < out[i] + 2^62
319 static void felem_diff64(felem out, const felem in)
322 * In order to prevent underflow, we add 0 mod p before subtracting.
324 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
325 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
327 out[0] += two62m3 - in[0];
328 out[1] += two62m2 - in[1];
329 out[2] += two62m2 - in[2];
330 out[3] += two62m2 - in[3];
331 out[4] += two62m2 - in[4];
332 out[5] += two62m2 - in[5];
333 out[6] += two62m2 - in[6];
334 out[7] += two62m2 - in[7];
335 out[8] += two62m2 - in[8];
339 * felem_diff_128_64 subtracts |in| from |out|
341 * in[i] < 2^62 + 2^17
343 * out[i] < out[i] + 2^63
345 static void felem_diff_128_64(largefelem out, const felem in)
348 * In order to prevent underflow, we add 64p mod p (which is equivalent
349 * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521
350 * digit number with all bits set to 1. See "The representation of field
351 * elements" comment above for a description of how limbs are used to
352 * represent a number. 64p is represented with 8 limbs containing a number
353 * with 58 bits set and one limb with a number with 57 bits set.
355 static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6);
356 static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5);
358 out[0] += two63m6 - in[0];
359 out[1] += two63m5 - in[1];
360 out[2] += two63m5 - in[2];
361 out[3] += two63m5 - in[3];
362 out[4] += two63m5 - in[4];
363 out[5] += two63m5 - in[5];
364 out[6] += two63m5 - in[6];
365 out[7] += two63m5 - in[7];
366 out[8] += two63m5 - in[8];
370 * felem_diff_128_64 subtracts |in| from |out|
374 * out[i] < out[i] + 2^127 - 2^69
376 static void felem_diff128(largefelem out, const largefelem in)
379 * In order to prevent underflow, we add 0 mod p before subtracting.
381 static const uint128_t two127m70 =
382 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
383 static const uint128_t two127m69 =
384 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
386 out[0] += (two127m70 - in[0]);
387 out[1] += (two127m69 - in[1]);
388 out[2] += (two127m69 - in[2]);
389 out[3] += (two127m69 - in[3]);
390 out[4] += (two127m69 - in[4]);
391 out[5] += (two127m69 - in[5]);
392 out[6] += (two127m69 - in[6]);
393 out[7] += (two127m69 - in[7]);
394 out[8] += (two127m69 - in[8]);
398 * felem_square sets |out| = |in|^2
402 * out[i] < 17 * max(in[i]) * max(in[i])
404 static void felem_square_ref(largefelem out, const felem in)
407 felem_scalar(inx2, in, 2);
408 felem_scalar(inx4, in, 4);
411 * We have many cases were we want to do
414 * This is obviously just
416 * However, rather than do the doubling on the 128 bit result, we
417 * double one of the inputs to the multiplication by reading from
421 out[0] = ((uint128_t) in[0]) * in[0];
422 out[1] = ((uint128_t) in[0]) * inx2[1];
423 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
424 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
425 out[4] = ((uint128_t) in[0]) * inx2[4] +
426 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
427 out[5] = ((uint128_t) in[0]) * inx2[5] +
428 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
429 out[6] = ((uint128_t) in[0]) * inx2[6] +
430 ((uint128_t) in[1]) * inx2[5] +
431 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
432 out[7] = ((uint128_t) in[0]) * inx2[7] +
433 ((uint128_t) in[1]) * inx2[6] +
434 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
435 out[8] = ((uint128_t) in[0]) * inx2[8] +
436 ((uint128_t) in[1]) * inx2[7] +
437 ((uint128_t) in[2]) * inx2[6] +
438 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
441 * The remaining limbs fall above 2^521, with the first falling at 2^522.
442 * They correspond to locations one bit up from the limbs produced above
443 * so we would have to multiply by two to align them. Again, rather than
444 * operate on the 128-bit result, we double one of the inputs to the
445 * multiplication. If we want to double for both this reason, and the
446 * reason above, then we end up multiplying by four.
450 out[0] += ((uint128_t) in[1]) * inx4[8] +
451 ((uint128_t) in[2]) * inx4[7] +
452 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
455 out[1] += ((uint128_t) in[2]) * inx4[8] +
456 ((uint128_t) in[3]) * inx4[7] +
457 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
460 out[2] += ((uint128_t) in[3]) * inx4[8] +
461 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
464 out[3] += ((uint128_t) in[4]) * inx4[8] +
465 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
468 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
471 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
474 out[6] += ((uint128_t) in[7]) * inx4[8];
477 out[7] += ((uint128_t) in[8]) * inx2[8];
481 * felem_mul sets |out| = |in1| * |in2|
486 * out[i] < 17 * max(in1[i]) * max(in2[i])
488 static void felem_mul_ref(largefelem out, const felem in1, const felem in2)
491 felem_scalar(in2x2, in2, 2);
493 out[0] = ((uint128_t) in1[0]) * in2[0];
495 out[1] = ((uint128_t) in1[0]) * in2[1] +
496 ((uint128_t) in1[1]) * in2[0];
498 out[2] = ((uint128_t) in1[0]) * in2[2] +
499 ((uint128_t) in1[1]) * in2[1] +
500 ((uint128_t) in1[2]) * in2[0];
502 out[3] = ((uint128_t) in1[0]) * in2[3] +
503 ((uint128_t) in1[1]) * in2[2] +
504 ((uint128_t) in1[2]) * in2[1] +
505 ((uint128_t) in1[3]) * in2[0];
507 out[4] = ((uint128_t) in1[0]) * in2[4] +
508 ((uint128_t) in1[1]) * in2[3] +
509 ((uint128_t) in1[2]) * in2[2] +
510 ((uint128_t) in1[3]) * in2[1] +
511 ((uint128_t) in1[4]) * in2[0];
513 out[5] = ((uint128_t) in1[0]) * in2[5] +
514 ((uint128_t) in1[1]) * in2[4] +
515 ((uint128_t) in1[2]) * in2[3] +
516 ((uint128_t) in1[3]) * in2[2] +
517 ((uint128_t) in1[4]) * in2[1] +
518 ((uint128_t) in1[5]) * in2[0];
520 out[6] = ((uint128_t) in1[0]) * in2[6] +
521 ((uint128_t) in1[1]) * in2[5] +
522 ((uint128_t) in1[2]) * in2[4] +
523 ((uint128_t) in1[3]) * in2[3] +
524 ((uint128_t) in1[4]) * in2[2] +
525 ((uint128_t) in1[5]) * in2[1] +
526 ((uint128_t) in1[6]) * in2[0];
528 out[7] = ((uint128_t) in1[0]) * in2[7] +
529 ((uint128_t) in1[1]) * in2[6] +
530 ((uint128_t) in1[2]) * in2[5] +
531 ((uint128_t) in1[3]) * in2[4] +
532 ((uint128_t) in1[4]) * in2[3] +
533 ((uint128_t) in1[5]) * in2[2] +
534 ((uint128_t) in1[6]) * in2[1] +
535 ((uint128_t) in1[7]) * in2[0];
537 out[8] = ((uint128_t) in1[0]) * in2[8] +
538 ((uint128_t) in1[1]) * in2[7] +
539 ((uint128_t) in1[2]) * in2[6] +
540 ((uint128_t) in1[3]) * in2[5] +
541 ((uint128_t) in1[4]) * in2[4] +
542 ((uint128_t) in1[5]) * in2[3] +
543 ((uint128_t) in1[6]) * in2[2] +
544 ((uint128_t) in1[7]) * in2[1] +
545 ((uint128_t) in1[8]) * in2[0];
547 /* See comment in felem_square about the use of in2x2 here */
549 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
550 ((uint128_t) in1[2]) * in2x2[7] +
551 ((uint128_t) in1[3]) * in2x2[6] +
552 ((uint128_t) in1[4]) * in2x2[5] +
553 ((uint128_t) in1[5]) * in2x2[4] +
554 ((uint128_t) in1[6]) * in2x2[3] +
555 ((uint128_t) in1[7]) * in2x2[2] +
556 ((uint128_t) in1[8]) * in2x2[1];
558 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
559 ((uint128_t) in1[3]) * in2x2[7] +
560 ((uint128_t) in1[4]) * in2x2[6] +
561 ((uint128_t) in1[5]) * in2x2[5] +
562 ((uint128_t) in1[6]) * in2x2[4] +
563 ((uint128_t) in1[7]) * in2x2[3] +
564 ((uint128_t) in1[8]) * in2x2[2];
566 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
567 ((uint128_t) in1[4]) * in2x2[7] +
568 ((uint128_t) in1[5]) * in2x2[6] +
569 ((uint128_t) in1[6]) * in2x2[5] +
570 ((uint128_t) in1[7]) * in2x2[4] +
571 ((uint128_t) in1[8]) * in2x2[3];
573 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
574 ((uint128_t) in1[5]) * in2x2[7] +
575 ((uint128_t) in1[6]) * in2x2[6] +
576 ((uint128_t) in1[7]) * in2x2[5] +
577 ((uint128_t) in1[8]) * in2x2[4];
579 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
580 ((uint128_t) in1[6]) * in2x2[7] +
581 ((uint128_t) in1[7]) * in2x2[6] +
582 ((uint128_t) in1[8]) * in2x2[5];
584 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
585 ((uint128_t) in1[7]) * in2x2[7] +
586 ((uint128_t) in1[8]) * in2x2[6];
588 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
589 ((uint128_t) in1[8]) * in2x2[7];
591 out[7] += ((uint128_t) in1[8]) * in2x2[8];
594 static const limb bottom52bits = 0xfffffffffffff;
597 * felem_reduce converts a largefelem to an felem.
601 * out[i] < 2^59 + 2^14
603 static void felem_reduce(felem out, const largefelem in)
605 u64 overflow1, overflow2;
607 out[0] = ((limb) in[0]) & bottom58bits;
608 out[1] = ((limb) in[1]) & bottom58bits;
609 out[2] = ((limb) in[2]) & bottom58bits;
610 out[3] = ((limb) in[3]) & bottom58bits;
611 out[4] = ((limb) in[4]) & bottom58bits;
612 out[5] = ((limb) in[5]) & bottom58bits;
613 out[6] = ((limb) in[6]) & bottom58bits;
614 out[7] = ((limb) in[7]) & bottom58bits;
615 out[8] = ((limb) in[8]) & bottom58bits;
619 out[1] += ((limb) in[0]) >> 58;
620 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
622 * out[1] < 2^58 + 2^6 + 2^58
625 out[2] += ((limb) (in[0] >> 64)) >> 52;
627 out[2] += ((limb) in[1]) >> 58;
628 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
629 out[3] += ((limb) (in[1] >> 64)) >> 52;
631 out[3] += ((limb) in[2]) >> 58;
632 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
633 out[4] += ((limb) (in[2] >> 64)) >> 52;
635 out[4] += ((limb) in[3]) >> 58;
636 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
637 out[5] += ((limb) (in[3] >> 64)) >> 52;
639 out[5] += ((limb) in[4]) >> 58;
640 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
641 out[6] += ((limb) (in[4] >> 64)) >> 52;
643 out[6] += ((limb) in[5]) >> 58;
644 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
645 out[7] += ((limb) (in[5] >> 64)) >> 52;
647 out[7] += ((limb) in[6]) >> 58;
648 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
649 out[8] += ((limb) (in[6] >> 64)) >> 52;
651 out[8] += ((limb) in[7]) >> 58;
652 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
654 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
657 overflow1 = ((limb) (in[7] >> 64)) >> 52;
659 overflow1 += ((limb) in[8]) >> 58;
660 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
661 overflow2 = ((limb) (in[8] >> 64)) >> 52;
663 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
664 overflow2 <<= 1; /* overflow2 < 2^13 */
666 out[0] += overflow1; /* out[0] < 2^60 */
667 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
669 out[1] += out[0] >> 58;
670 out[0] &= bottom58bits;
673 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
678 #if defined(ECP_NISTP521_ASM)
679 static void felem_square_wrapper(largefelem out, const felem in);
680 static void felem_mul_wrapper(largefelem out, const felem in1, const felem in2);
682 static void (*felem_square_p)(largefelem out, const felem in) =
683 felem_square_wrapper;
684 static void (*felem_mul_p)(largefelem out, const felem in1, const felem in2) =
687 void p521_felem_square(largefelem out, const felem in);
688 void p521_felem_mul(largefelem out, const felem in1, const felem in2);
690 # if defined(_ARCH_PPC64)
691 # include "crypto/ppc_arch.h"
694 static void felem_select(void)
696 # if defined(_ARCH_PPC64)
697 if ((OPENSSL_ppccap_P & PPC_MADD300) && (OPENSSL_ppccap_P & PPC_ALTIVEC)) {
698 felem_square_p = p521_felem_square;
699 felem_mul_p = p521_felem_mul;
706 felem_square_p = felem_square_ref;
707 felem_mul_p = felem_mul_ref;
710 static void felem_square_wrapper(largefelem out, const felem in)
713 felem_square_p(out, in);
716 static void felem_mul_wrapper(largefelem out, const felem in1, const felem in2)
719 felem_mul_p(out, in1, in2);
722 # define felem_square felem_square_p
723 # define felem_mul felem_mul_p
725 # define felem_square felem_square_ref
726 # define felem_mul felem_mul_ref
729 static void felem_square_reduce(felem out, const felem in)
732 felem_square(tmp, in);
733 felem_reduce(out, tmp);
736 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
739 felem_mul(tmp, in1, in2);
740 felem_reduce(out, tmp);
744 * felem_inv calculates |out| = |in|^{-1}
746 * Based on Fermat's Little Theorem:
748 * a^{p-1} = 1 (mod p)
749 * a^{p-2} = a^{-1} (mod p)
751 static void felem_inv(felem out, const felem in)
753 felem ftmp, ftmp2, ftmp3, ftmp4;
757 felem_square(tmp, in);
758 felem_reduce(ftmp, tmp); /* 2^1 */
759 felem_mul(tmp, in, ftmp);
760 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
761 felem_assign(ftmp2, ftmp);
762 felem_square(tmp, ftmp);
763 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
764 felem_mul(tmp, in, ftmp);
765 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
766 felem_square(tmp, ftmp);
767 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
769 felem_square(tmp, ftmp2);
770 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
771 felem_square(tmp, ftmp3);
772 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
773 felem_mul(tmp, ftmp3, ftmp2);
774 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
776 felem_assign(ftmp2, ftmp3);
777 felem_square(tmp, ftmp3);
778 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
779 felem_square(tmp, ftmp3);
780 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
781 felem_square(tmp, ftmp3);
782 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
783 felem_square(tmp, ftmp3);
784 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
785 felem_mul(tmp, ftmp3, ftmp);
786 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
787 felem_square(tmp, ftmp4);
788 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
789 felem_mul(tmp, ftmp3, ftmp2);
790 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
791 felem_assign(ftmp2, ftmp3);
793 for (i = 0; i < 8; i++) {
794 felem_square(tmp, ftmp3);
795 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
797 felem_mul(tmp, ftmp3, ftmp2);
798 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
799 felem_assign(ftmp2, ftmp3);
801 for (i = 0; i < 16; i++) {
802 felem_square(tmp, ftmp3);
803 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
805 felem_mul(tmp, ftmp3, ftmp2);
806 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
807 felem_assign(ftmp2, ftmp3);
809 for (i = 0; i < 32; i++) {
810 felem_square(tmp, ftmp3);
811 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
813 felem_mul(tmp, ftmp3, ftmp2);
814 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
815 felem_assign(ftmp2, ftmp3);
817 for (i = 0; i < 64; i++) {
818 felem_square(tmp, ftmp3);
819 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
821 felem_mul(tmp, ftmp3, ftmp2);
822 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
823 felem_assign(ftmp2, ftmp3);
825 for (i = 0; i < 128; i++) {
826 felem_square(tmp, ftmp3);
827 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
829 felem_mul(tmp, ftmp3, ftmp2);
830 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
831 felem_assign(ftmp2, ftmp3);
833 for (i = 0; i < 256; i++) {
834 felem_square(tmp, ftmp3);
835 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
837 felem_mul(tmp, ftmp3, ftmp2);
838 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
840 for (i = 0; i < 9; i++) {
841 felem_square(tmp, ftmp3);
842 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
844 felem_mul(tmp, ftmp3, ftmp4);
845 felem_reduce(ftmp3, tmp); /* 2^521 - 2^2 */
846 felem_mul(tmp, ftmp3, in);
847 felem_reduce(out, tmp); /* 2^521 - 3 */
850 /* This is 2^521-1, expressed as an felem */
851 static const felem kPrime = {
852 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
853 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
854 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
858 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
861 * in[i] < 2^59 + 2^14
863 static limb felem_is_zero(const felem in)
867 felem_assign(ftmp, in);
869 ftmp[0] += ftmp[8] >> 57;
870 ftmp[8] &= bottom57bits;
872 ftmp[1] += ftmp[0] >> 58;
873 ftmp[0] &= bottom58bits;
874 ftmp[2] += ftmp[1] >> 58;
875 ftmp[1] &= bottom58bits;
876 ftmp[3] += ftmp[2] >> 58;
877 ftmp[2] &= bottom58bits;
878 ftmp[4] += ftmp[3] >> 58;
879 ftmp[3] &= bottom58bits;
880 ftmp[5] += ftmp[4] >> 58;
881 ftmp[4] &= bottom58bits;
882 ftmp[6] += ftmp[5] >> 58;
883 ftmp[5] &= bottom58bits;
884 ftmp[7] += ftmp[6] >> 58;
885 ftmp[6] &= bottom58bits;
886 ftmp[8] += ftmp[7] >> 58;
887 ftmp[7] &= bottom58bits;
888 /* ftmp[8] < 2^57 + 4 */
891 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
892 * than our bound for ftmp[8]. Therefore we only have to check if the
893 * zero is zero or 2^521-1.
909 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
910 * can be set is if is_zero was 0 before the decrement.
912 is_zero = 0 - (is_zero >> 63);
914 is_p = ftmp[0] ^ kPrime[0];
915 is_p |= ftmp[1] ^ kPrime[1];
916 is_p |= ftmp[2] ^ kPrime[2];
917 is_p |= ftmp[3] ^ kPrime[3];
918 is_p |= ftmp[4] ^ kPrime[4];
919 is_p |= ftmp[5] ^ kPrime[5];
920 is_p |= ftmp[6] ^ kPrime[6];
921 is_p |= ftmp[7] ^ kPrime[7];
922 is_p |= ftmp[8] ^ kPrime[8];
925 is_p = 0 - (is_p >> 63);
931 static int felem_is_zero_int(const void *in)
933 return (int)(felem_is_zero(in) & ((limb) 1));
937 * felem_contract converts |in| to its unique, minimal representation.
939 * in[i] < 2^59 + 2^14
941 static void felem_contract(felem out, const felem in)
943 limb is_p, is_greater, sign;
944 static const limb two58 = ((limb) 1) << 58;
946 felem_assign(out, in);
948 out[0] += out[8] >> 57;
949 out[8] &= bottom57bits;
951 out[1] += out[0] >> 58;
952 out[0] &= bottom58bits;
953 out[2] += out[1] >> 58;
954 out[1] &= bottom58bits;
955 out[3] += out[2] >> 58;
956 out[2] &= bottom58bits;
957 out[4] += out[3] >> 58;
958 out[3] &= bottom58bits;
959 out[5] += out[4] >> 58;
960 out[4] &= bottom58bits;
961 out[6] += out[5] >> 58;
962 out[5] &= bottom58bits;
963 out[7] += out[6] >> 58;
964 out[6] &= bottom58bits;
965 out[8] += out[7] >> 58;
966 out[7] &= bottom58bits;
967 /* out[8] < 2^57 + 4 */
970 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
971 * out. See the comments in felem_is_zero regarding why we don't test for
972 * other multiples of the prime.
976 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
979 is_p = out[0] ^ kPrime[0];
980 is_p |= out[1] ^ kPrime[1];
981 is_p |= out[2] ^ kPrime[2];
982 is_p |= out[3] ^ kPrime[3];
983 is_p |= out[4] ^ kPrime[4];
984 is_p |= out[5] ^ kPrime[5];
985 is_p |= out[6] ^ kPrime[6];
986 is_p |= out[7] ^ kPrime[7];
987 is_p |= out[8] ^ kPrime[8];
996 is_p = 0 - (is_p >> 63);
999 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
1012 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
1013 * 57 is greater than zero as (2^521-1) + x >= 2^522
1015 is_greater = out[8] >> 57;
1016 is_greater |= is_greater << 32;
1017 is_greater |= is_greater << 16;
1018 is_greater |= is_greater << 8;
1019 is_greater |= is_greater << 4;
1020 is_greater |= is_greater << 2;
1021 is_greater |= is_greater << 1;
1022 is_greater = 0 - (is_greater >> 63);
1024 out[0] -= kPrime[0] & is_greater;
1025 out[1] -= kPrime[1] & is_greater;
1026 out[2] -= kPrime[2] & is_greater;
1027 out[3] -= kPrime[3] & is_greater;
1028 out[4] -= kPrime[4] & is_greater;
1029 out[5] -= kPrime[5] & is_greater;
1030 out[6] -= kPrime[6] & is_greater;
1031 out[7] -= kPrime[7] & is_greater;
1032 out[8] -= kPrime[8] & is_greater;
1034 /* Eliminate negative coefficients */
1035 sign = -(out[0] >> 63);
1036 out[0] += (two58 & sign);
1037 out[1] -= (1 & sign);
1038 sign = -(out[1] >> 63);
1039 out[1] += (two58 & sign);
1040 out[2] -= (1 & sign);
1041 sign = -(out[2] >> 63);
1042 out[2] += (two58 & sign);
1043 out[3] -= (1 & sign);
1044 sign = -(out[3] >> 63);
1045 out[3] += (two58 & sign);
1046 out[4] -= (1 & sign);
1047 sign = -(out[4] >> 63);
1048 out[4] += (two58 & sign);
1049 out[5] -= (1 & sign);
1050 sign = -(out[0] >> 63);
1051 out[5] += (two58 & sign);
1052 out[6] -= (1 & sign);
1053 sign = -(out[6] >> 63);
1054 out[6] += (two58 & sign);
1055 out[7] -= (1 & sign);
1056 sign = -(out[7] >> 63);
1057 out[7] += (two58 & sign);
1058 out[8] -= (1 & sign);
1059 sign = -(out[5] >> 63);
1060 out[5] += (two58 & sign);
1061 out[6] -= (1 & sign);
1062 sign = -(out[6] >> 63);
1063 out[6] += (two58 & sign);
1064 out[7] -= (1 & sign);
1065 sign = -(out[7] >> 63);
1066 out[7] += (two58 & sign);
1067 out[8] -= (1 & sign);
1074 * Building on top of the field operations we have the operations on the
1075 * elliptic curve group itself. Points on the curve are represented in Jacobian
1079 * point_double calculates 2*(x_in, y_in, z_in)
1081 * The method is taken from:
1082 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1084 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1085 * while x_out == y_in is not (maybe this works, but it's not tested). */
1087 point_double(felem x_out, felem y_out, felem z_out,
1088 const felem x_in, const felem y_in, const felem z_in)
1090 largefelem tmp, tmp2;
1091 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1093 felem_assign(ftmp, x_in);
1094 felem_assign(ftmp2, x_in);
1097 felem_square(tmp, z_in);
1098 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1101 felem_square(tmp, y_in);
1102 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1104 /* beta = x*gamma */
1105 felem_mul(tmp, x_in, gamma);
1106 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1108 /* alpha = 3*(x-delta)*(x+delta) */
1109 felem_diff64(ftmp, delta);
1110 /* ftmp[i] < 2^61 */
1111 felem_sum64(ftmp2, delta);
1112 /* ftmp2[i] < 2^60 + 2^15 */
1113 felem_scalar64(ftmp2, 3);
1114 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1115 felem_mul(tmp, ftmp, ftmp2);
1117 * tmp[i] < 17(3*2^121 + 3*2^76)
1118 * = 61*2^121 + 61*2^76
1119 * < 64*2^121 + 64*2^76
1123 felem_reduce(alpha, tmp);
1125 /* x' = alpha^2 - 8*beta */
1126 felem_square(tmp, alpha);
1128 * tmp[i] < 17*2^120 < 2^125
1130 felem_assign(ftmp, beta);
1131 felem_scalar64(ftmp, 8);
1132 /* ftmp[i] < 2^62 + 2^17 */
1133 felem_diff_128_64(tmp, ftmp);
1134 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1135 felem_reduce(x_out, tmp);
1137 /* z' = (y + z)^2 - gamma - delta */
1138 felem_sum64(delta, gamma);
1139 /* delta[i] < 2^60 + 2^15 */
1140 felem_assign(ftmp, y_in);
1141 felem_sum64(ftmp, z_in);
1142 /* ftmp[i] < 2^60 + 2^15 */
1143 felem_square(tmp, ftmp);
1145 * tmp[i] < 17(2^122) < 2^127
1147 felem_diff_128_64(tmp, delta);
1148 /* tmp[i] < 2^127 + 2^63 */
1149 felem_reduce(z_out, tmp);
1151 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1152 felem_scalar64(beta, 4);
1153 /* beta[i] < 2^61 + 2^16 */
1154 felem_diff64(beta, x_out);
1155 /* beta[i] < 2^61 + 2^60 + 2^16 */
1156 felem_mul(tmp, alpha, beta);
1158 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1159 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1160 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1163 felem_square(tmp2, gamma);
1165 * tmp2[i] < 17*(2^59 + 2^14)^2
1166 * = 17*(2^118 + 2^74 + 2^28)
1168 felem_scalar128(tmp2, 8);
1170 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1171 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1174 felem_diff128(tmp, tmp2);
1176 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1177 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1178 * 2^74 + 2^69 + 2^34 + 2^30
1181 felem_reduce(y_out, tmp);
1184 /* copy_conditional copies in to out iff mask is all ones. */
1185 static void copy_conditional(felem out, const felem in, limb mask)
1188 for (i = 0; i < NLIMBS; ++i) {
1189 const limb tmp = mask & (in[i] ^ out[i]);
1195 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1197 * The method is taken from
1198 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1199 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1201 * This function includes a branch for checking whether the two input points
1202 * are equal (while not equal to the point at infinity). See comment below
1205 static void point_add(felem x3, felem y3, felem z3,
1206 const felem x1, const felem y1, const felem z1,
1207 const int mixed, const felem x2, const felem y2,
1210 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1211 largefelem tmp, tmp2;
1212 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1215 z1_is_zero = felem_is_zero(z1);
1216 z2_is_zero = felem_is_zero(z2);
1218 /* ftmp = z1z1 = z1**2 */
1219 felem_square(tmp, z1);
1220 felem_reduce(ftmp, tmp);
1223 /* ftmp2 = z2z2 = z2**2 */
1224 felem_square(tmp, z2);
1225 felem_reduce(ftmp2, tmp);
1227 /* u1 = ftmp3 = x1*z2z2 */
1228 felem_mul(tmp, x1, ftmp2);
1229 felem_reduce(ftmp3, tmp);
1231 /* ftmp5 = z1 + z2 */
1232 felem_assign(ftmp5, z1);
1233 felem_sum64(ftmp5, z2);
1234 /* ftmp5[i] < 2^61 */
1236 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1237 felem_square(tmp, ftmp5);
1238 /* tmp[i] < 17*2^122 */
1239 felem_diff_128_64(tmp, ftmp);
1240 /* tmp[i] < 17*2^122 + 2^63 */
1241 felem_diff_128_64(tmp, ftmp2);
1242 /* tmp[i] < 17*2^122 + 2^64 */
1243 felem_reduce(ftmp5, tmp);
1245 /* ftmp2 = z2 * z2z2 */
1246 felem_mul(tmp, ftmp2, z2);
1247 felem_reduce(ftmp2, tmp);
1249 /* s1 = ftmp6 = y1 * z2**3 */
1250 felem_mul(tmp, y1, ftmp2);
1251 felem_reduce(ftmp6, tmp);
1254 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1257 /* u1 = ftmp3 = x1*z2z2 */
1258 felem_assign(ftmp3, x1);
1260 /* ftmp5 = 2*z1z2 */
1261 felem_scalar(ftmp5, z1, 2);
1263 /* s1 = ftmp6 = y1 * z2**3 */
1264 felem_assign(ftmp6, y1);
1268 felem_mul(tmp, x2, ftmp);
1269 /* tmp[i] < 17*2^120 */
1271 /* h = ftmp4 = u2 - u1 */
1272 felem_diff_128_64(tmp, ftmp3);
1273 /* tmp[i] < 17*2^120 + 2^63 */
1274 felem_reduce(ftmp4, tmp);
1276 x_equal = felem_is_zero(ftmp4);
1278 /* z_out = ftmp5 * h */
1279 felem_mul(tmp, ftmp5, ftmp4);
1280 felem_reduce(z_out, tmp);
1282 /* ftmp = z1 * z1z1 */
1283 felem_mul(tmp, ftmp, z1);
1284 felem_reduce(ftmp, tmp);
1286 /* s2 = tmp = y2 * z1**3 */
1287 felem_mul(tmp, y2, ftmp);
1288 /* tmp[i] < 17*2^120 */
1290 /* r = ftmp5 = (s2 - s1)*2 */
1291 felem_diff_128_64(tmp, ftmp6);
1292 /* tmp[i] < 17*2^120 + 2^63 */
1293 felem_reduce(ftmp5, tmp);
1294 y_equal = felem_is_zero(ftmp5);
1295 felem_scalar64(ftmp5, 2);
1296 /* ftmp5[i] < 2^61 */
1299 * The formulae are incorrect if the points are equal, in affine coordinates
1300 * (X_1, Y_1) == (X_2, Y_2), so we check for this and do doubling if this
1303 * We use bitwise operations to avoid potential side-channels introduced by
1304 * the short-circuiting behaviour of boolean operators.
1306 * The special case of either point being the point at infinity (z1 and/or
1307 * z2 are zero), is handled separately later on in this function, so we
1308 * avoid jumping to point_double here in those special cases.
1310 * Notice the comment below on the implications of this branching for timing
1311 * leaks and why it is considered practically irrelevant.
1313 points_equal = (x_equal & y_equal & (~z1_is_zero) & (~z2_is_zero));
1317 * This is obviously not constant-time but it will almost-never happen
1318 * for ECDH / ECDSA. The case where it can happen is during scalar-mult
1319 * where the intermediate value gets very close to the group order.
1320 * Since |ossl_ec_GFp_nistp_recode_scalar_bits| produces signed digits
1321 * for the scalar, it's possible for the intermediate value to be a small
1322 * negative multiple of the base point, and for the final signed digit
1323 * to be the same value. We believe that this only occurs for the scalar
1324 * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
1325 * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb
1326 * 71e913863f7, in that case the penultimate intermediate is -9G and
1327 * the final digit is also -9G. Since this only happens for a single
1328 * scalar, the timing leak is irrelevant. (Any attacker who wanted to
1329 * check whether a secret scalar was that exact value, can already do
1332 point_double(x3, y3, z3, x1, y1, z1);
1336 /* I = ftmp = (2h)**2 */
1337 felem_assign(ftmp, ftmp4);
1338 felem_scalar64(ftmp, 2);
1339 /* ftmp[i] < 2^61 */
1340 felem_square(tmp, ftmp);
1341 /* tmp[i] < 17*2^122 */
1342 felem_reduce(ftmp, tmp);
1344 /* J = ftmp2 = h * I */
1345 felem_mul(tmp, ftmp4, ftmp);
1346 felem_reduce(ftmp2, tmp);
1348 /* V = ftmp4 = U1 * I */
1349 felem_mul(tmp, ftmp3, ftmp);
1350 felem_reduce(ftmp4, tmp);
1352 /* x_out = r**2 - J - 2V */
1353 felem_square(tmp, ftmp5);
1354 /* tmp[i] < 17*2^122 */
1355 felem_diff_128_64(tmp, ftmp2);
1356 /* tmp[i] < 17*2^122 + 2^63 */
1357 felem_assign(ftmp3, ftmp4);
1358 felem_scalar64(ftmp4, 2);
1359 /* ftmp4[i] < 2^61 */
1360 felem_diff_128_64(tmp, ftmp4);
1361 /* tmp[i] < 17*2^122 + 2^64 */
1362 felem_reduce(x_out, tmp);
1364 /* y_out = r(V-x_out) - 2 * s1 * J */
1365 felem_diff64(ftmp3, x_out);
1367 * ftmp3[i] < 2^60 + 2^60 = 2^61
1369 felem_mul(tmp, ftmp5, ftmp3);
1370 /* tmp[i] < 17*2^122 */
1371 felem_mul(tmp2, ftmp6, ftmp2);
1372 /* tmp2[i] < 17*2^120 */
1373 felem_scalar128(tmp2, 2);
1374 /* tmp2[i] < 17*2^121 */
1375 felem_diff128(tmp, tmp2);
1377 * tmp[i] < 2^127 - 2^69 + 17*2^122
1378 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1381 felem_reduce(y_out, tmp);
1383 copy_conditional(x_out, x2, z1_is_zero);
1384 copy_conditional(x_out, x1, z2_is_zero);
1385 copy_conditional(y_out, y2, z1_is_zero);
1386 copy_conditional(y_out, y1, z2_is_zero);
1387 copy_conditional(z_out, z2, z1_is_zero);
1388 copy_conditional(z_out, z1, z2_is_zero);
1389 felem_assign(x3, x_out);
1390 felem_assign(y3, y_out);
1391 felem_assign(z3, z_out);
1395 * Base point pre computation
1396 * --------------------------
1398 * Two different sorts of precomputed tables are used in the following code.
1399 * Each contain various points on the curve, where each point is three field
1400 * elements (x, y, z).
1402 * For the base point table, z is usually 1 (0 for the point at infinity).
1403 * This table has 16 elements:
1404 * index | bits | point
1405 * ------+---------+------------------------------
1408 * 2 | 0 0 1 0 | 2^130G
1409 * 3 | 0 0 1 1 | (2^130 + 1)G
1410 * 4 | 0 1 0 0 | 2^260G
1411 * 5 | 0 1 0 1 | (2^260 + 1)G
1412 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1413 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1414 * 8 | 1 0 0 0 | 2^390G
1415 * 9 | 1 0 0 1 | (2^390 + 1)G
1416 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1417 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1418 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1419 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1420 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1421 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1423 * The reason for this is so that we can clock bits into four different
1424 * locations when doing simple scalar multiplies against the base point.
1426 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1428 /* gmul is the table of precomputed base points */
1429 static const felem gmul[16][3] = {
1430 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1431 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1432 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1433 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1434 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1435 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1436 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1437 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1438 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1439 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1440 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1441 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1442 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1443 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1444 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1445 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1446 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1447 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1448 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1449 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1450 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1451 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1452 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1453 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1454 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1455 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1456 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1457 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1458 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1459 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1460 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1461 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1462 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1463 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1464 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1465 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1466 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1467 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1468 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1469 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1470 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1471 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1472 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1473 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1474 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1475 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1476 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1477 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1478 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1479 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1480 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1481 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1482 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1483 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1484 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1485 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1486 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1487 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1488 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1489 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1490 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1491 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1492 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1493 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1494 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1495 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1496 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1497 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1498 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1499 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1500 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1501 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1502 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1503 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1504 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1505 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1506 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1507 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1508 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1509 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1510 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1511 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1512 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1513 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1514 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1515 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1516 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1517 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1518 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1519 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1520 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1521 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1522 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1523 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1524 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1525 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1526 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1527 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1528 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1529 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1530 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1531 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1532 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1533 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1534 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1535 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1536 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1537 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1541 * select_point selects the |idx|th point from a precomputation table and
1544 /* pre_comp below is of the size provided in |size| */
1545 static void select_point(const limb idx, unsigned int size,
1546 const felem pre_comp[][3], felem out[3])
1549 limb *outlimbs = &out[0][0];
1551 memset(out, 0, sizeof(*out) * 3);
1553 for (i = 0; i < size; i++) {
1554 const limb *inlimbs = &pre_comp[i][0][0];
1555 limb mask = i ^ idx;
1561 for (j = 0; j < NLIMBS * 3; j++)
1562 outlimbs[j] |= inlimbs[j] & mask;
1566 /* get_bit returns the |i|th bit in |in| */
1567 static char get_bit(const felem_bytearray in, int i)
1571 return (in[i >> 3] >> (i & 7)) & 1;
1575 * Interleaved point multiplication using precomputed point multiples: The
1576 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1577 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1578 * generator, using certain (large) precomputed multiples in g_pre_comp.
1579 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1581 static void batch_mul(felem x_out, felem y_out, felem z_out,
1582 const felem_bytearray scalars[],
1583 const unsigned num_points, const u8 *g_scalar,
1584 const int mixed, const felem pre_comp[][17][3],
1585 const felem g_pre_comp[16][3])
1588 unsigned num, gen_mul = (g_scalar != NULL);
1589 felem nq[3], tmp[4];
1593 /* set nq to the point at infinity */
1594 memset(nq, 0, sizeof(nq));
1597 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1598 * of the generator (last quarter of rounds) and additions of other
1599 * points multiples (every 5th round).
1601 skip = 1; /* save two point operations in the first
1603 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1606 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1608 /* add multiples of the generator */
1609 if (gen_mul && (i <= 130)) {
1610 bits = get_bit(g_scalar, i + 390) << 3;
1612 bits |= get_bit(g_scalar, i + 260) << 2;
1613 bits |= get_bit(g_scalar, i + 130) << 1;
1614 bits |= get_bit(g_scalar, i);
1616 /* select the point to add, in constant time */
1617 select_point(bits, 16, g_pre_comp, tmp);
1619 /* The 1 argument below is for "mixed" */
1620 point_add(nq[0], nq[1], nq[2],
1621 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1623 memcpy(nq, tmp, 3 * sizeof(felem));
1628 /* do other additions every 5 doublings */
1629 if (num_points && (i % 5 == 0)) {
1630 /* loop over all scalars */
1631 for (num = 0; num < num_points; ++num) {
1632 bits = get_bit(scalars[num], i + 4) << 5;
1633 bits |= get_bit(scalars[num], i + 3) << 4;
1634 bits |= get_bit(scalars[num], i + 2) << 3;
1635 bits |= get_bit(scalars[num], i + 1) << 2;
1636 bits |= get_bit(scalars[num], i) << 1;
1637 bits |= get_bit(scalars[num], i - 1);
1638 ossl_ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1641 * select the point to add or subtract, in constant time
1643 select_point(digit, 17, pre_comp[num], tmp);
1644 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1646 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1649 point_add(nq[0], nq[1], nq[2],
1650 nq[0], nq[1], nq[2],
1651 mixed, tmp[0], tmp[1], tmp[2]);
1653 memcpy(nq, tmp, 3 * sizeof(felem));
1659 felem_assign(x_out, nq[0]);
1660 felem_assign(y_out, nq[1]);
1661 felem_assign(z_out, nq[2]);
1664 /* Precomputation for the group generator. */
1665 struct nistp521_pre_comp_st {
1666 felem g_pre_comp[16][3];
1667 CRYPTO_REF_COUNT references;
1670 const EC_METHOD *EC_GFp_nistp521_method(void)
1672 static const EC_METHOD ret = {
1673 EC_FLAGS_DEFAULT_OCT,
1674 NID_X9_62_prime_field,
1675 ossl_ec_GFp_nistp521_group_init,
1676 ossl_ec_GFp_simple_group_finish,
1677 ossl_ec_GFp_simple_group_clear_finish,
1678 ossl_ec_GFp_nist_group_copy,
1679 ossl_ec_GFp_nistp521_group_set_curve,
1680 ossl_ec_GFp_simple_group_get_curve,
1681 ossl_ec_GFp_simple_group_get_degree,
1682 ossl_ec_group_simple_order_bits,
1683 ossl_ec_GFp_simple_group_check_discriminant,
1684 ossl_ec_GFp_simple_point_init,
1685 ossl_ec_GFp_simple_point_finish,
1686 ossl_ec_GFp_simple_point_clear_finish,
1687 ossl_ec_GFp_simple_point_copy,
1688 ossl_ec_GFp_simple_point_set_to_infinity,
1689 ossl_ec_GFp_simple_point_set_affine_coordinates,
1690 ossl_ec_GFp_nistp521_point_get_affine_coordinates,
1691 0 /* point_set_compressed_coordinates */ ,
1694 ossl_ec_GFp_simple_add,
1695 ossl_ec_GFp_simple_dbl,
1696 ossl_ec_GFp_simple_invert,
1697 ossl_ec_GFp_simple_is_at_infinity,
1698 ossl_ec_GFp_simple_is_on_curve,
1699 ossl_ec_GFp_simple_cmp,
1700 ossl_ec_GFp_simple_make_affine,
1701 ossl_ec_GFp_simple_points_make_affine,
1702 ossl_ec_GFp_nistp521_points_mul,
1703 ossl_ec_GFp_nistp521_precompute_mult,
1704 ossl_ec_GFp_nistp521_have_precompute_mult,
1705 ossl_ec_GFp_nist_field_mul,
1706 ossl_ec_GFp_nist_field_sqr,
1708 ossl_ec_GFp_simple_field_inv,
1709 0 /* field_encode */ ,
1710 0 /* field_decode */ ,
1711 0, /* field_set_to_one */
1712 ossl_ec_key_simple_priv2oct,
1713 ossl_ec_key_simple_oct2priv,
1714 0, /* set private */
1715 ossl_ec_key_simple_generate_key,
1716 ossl_ec_key_simple_check_key,
1717 ossl_ec_key_simple_generate_public_key,
1720 ossl_ecdh_simple_compute_key,
1721 ossl_ecdsa_simple_sign_setup,
1722 ossl_ecdsa_simple_sign_sig,
1723 ossl_ecdsa_simple_verify_sig,
1724 0, /* field_inverse_mod_ord */
1725 0, /* blind_coordinates */
1727 0, /* ladder_step */
1734 /******************************************************************************/
1736 * FUNCTIONS TO MANAGE PRECOMPUTATION
1739 static NISTP521_PRE_COMP *nistp521_pre_comp_new(void)
1741 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1746 if (!CRYPTO_NEW_REF(&ret->references, 1)) {
1753 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1757 CRYPTO_UP_REF(&p->references, &i);
1761 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1768 CRYPTO_DOWN_REF(&p->references, &i);
1769 REF_PRINT_COUNT("EC_nistp521", p);
1772 REF_ASSERT_ISNT(i < 0);
1774 CRYPTO_FREE_REF(&p->references);
1778 /******************************************************************************/
1780 * OPENSSL EC_METHOD FUNCTIONS
1783 int ossl_ec_GFp_nistp521_group_init(EC_GROUP *group)
1786 ret = ossl_ec_GFp_simple_group_init(group);
1787 group->a_is_minus3 = 1;
1791 int ossl_ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1792 const BIGNUM *a, const BIGNUM *b,
1796 BIGNUM *curve_p, *curve_a, *curve_b;
1798 BN_CTX *new_ctx = NULL;
1801 ctx = new_ctx = BN_CTX_new();
1807 curve_p = BN_CTX_get(ctx);
1808 curve_a = BN_CTX_get(ctx);
1809 curve_b = BN_CTX_get(ctx);
1810 if (curve_b == NULL)
1812 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1813 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1814 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1815 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1816 ERR_raise(ERR_LIB_EC, EC_R_WRONG_CURVE_PARAMETERS);
1819 group->field_mod_func = BN_nist_mod_521;
1820 ret = ossl_ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1824 BN_CTX_free(new_ctx);
1830 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1833 int ossl_ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1834 const EC_POINT *point,
1835 BIGNUM *x, BIGNUM *y,
1838 felem z1, z2, x_in, y_in, x_out, y_out;
1841 if (EC_POINT_is_at_infinity(group, point)) {
1842 ERR_raise(ERR_LIB_EC, EC_R_POINT_AT_INFINITY);
1845 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1846 (!BN_to_felem(z1, point->Z)))
1849 felem_square(tmp, z2);
1850 felem_reduce(z1, tmp);
1851 felem_mul(tmp, x_in, z1);
1852 felem_reduce(x_in, tmp);
1853 felem_contract(x_out, x_in);
1855 if (!felem_to_BN(x, x_out)) {
1856 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1860 felem_mul(tmp, z1, z2);
1861 felem_reduce(z1, tmp);
1862 felem_mul(tmp, y_in, z1);
1863 felem_reduce(y_in, tmp);
1864 felem_contract(y_out, y_in);
1866 if (!felem_to_BN(y, y_out)) {
1867 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1874 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1875 static void make_points_affine(size_t num, felem points[][3],
1879 * Runs in constant time, unless an input is the point at infinity (which
1880 * normally shouldn't happen).
1882 ossl_ec_GFp_nistp_points_make_affine_internal(num,
1886 (void (*)(void *))felem_one,
1888 (void (*)(void *, const void *))
1890 (void (*)(void *, const void *))
1891 felem_square_reduce, (void (*)
1898 (void (*)(void *, const void *))
1900 (void (*)(void *, const void *))
1905 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1906 * values Result is stored in r (r can equal one of the inputs).
1908 int ossl_ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1909 const BIGNUM *scalar, size_t num,
1910 const EC_POINT *points[],
1911 const BIGNUM *scalars[], BN_CTX *ctx)
1916 BIGNUM *x, *y, *z, *tmp_scalar;
1917 felem_bytearray g_secret;
1918 felem_bytearray *secrets = NULL;
1919 felem (*pre_comp)[17][3] = NULL;
1920 felem *tmp_felems = NULL;
1923 int have_pre_comp = 0;
1924 size_t num_points = num;
1925 felem x_in, y_in, z_in, x_out, y_out, z_out;
1926 NISTP521_PRE_COMP *pre = NULL;
1927 felem(*g_pre_comp)[3] = NULL;
1928 EC_POINT *generator = NULL;
1929 const EC_POINT *p = NULL;
1930 const BIGNUM *p_scalar = NULL;
1933 x = BN_CTX_get(ctx);
1934 y = BN_CTX_get(ctx);
1935 z = BN_CTX_get(ctx);
1936 tmp_scalar = BN_CTX_get(ctx);
1937 if (tmp_scalar == NULL)
1940 if (scalar != NULL) {
1941 pre = group->pre_comp.nistp521;
1943 /* we have precomputation, try to use it */
1944 g_pre_comp = &pre->g_pre_comp[0];
1946 /* try to use the standard precomputation */
1947 g_pre_comp = (felem(*)[3]) gmul;
1948 generator = EC_POINT_new(group);
1949 if (generator == NULL)
1951 /* get the generator from precomputation */
1952 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1953 !felem_to_BN(y, g_pre_comp[1][1]) ||
1954 !felem_to_BN(z, g_pre_comp[1][2])) {
1955 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1958 if (!ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group,
1962 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1963 /* precomputation matches generator */
1967 * we don't have valid precomputation: treat the generator as a
1973 if (num_points > 0) {
1974 if (num_points >= 2) {
1976 * unless we precompute multiples for just one point, converting
1977 * those into affine form is time well spent
1981 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1982 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1985 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1986 if ((secrets == NULL) || (pre_comp == NULL)
1987 || (mixed && (tmp_felems == NULL)))
1991 * we treat NULL scalars as 0, and NULL points as points at infinity,
1992 * i.e., they contribute nothing to the linear combination
1994 for (i = 0; i < num_points; ++i) {
1997 * we didn't have a valid precomputation, so we pick the
2000 p = EC_GROUP_get0_generator(group);
2003 /* the i^th point */
2005 p_scalar = scalars[i];
2007 if ((p_scalar != NULL) && (p != NULL)) {
2008 /* reduce scalar to 0 <= scalar < 2^521 */
2009 if ((BN_num_bits(p_scalar) > 521)
2010 || (BN_is_negative(p_scalar))) {
2012 * this is an unusual input, and we don't guarantee
2015 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
2016 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
2019 num_bytes = BN_bn2lebinpad(tmp_scalar,
2020 secrets[i], sizeof(secrets[i]));
2022 num_bytes = BN_bn2lebinpad(p_scalar,
2023 secrets[i], sizeof(secrets[i]));
2025 if (num_bytes < 0) {
2026 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
2029 /* precompute multiples */
2030 if ((!BN_to_felem(x_out, p->X)) ||
2031 (!BN_to_felem(y_out, p->Y)) ||
2032 (!BN_to_felem(z_out, p->Z)))
2034 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
2035 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
2036 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
2037 for (j = 2; j <= 16; ++j) {
2039 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
2040 pre_comp[i][j][2], pre_comp[i][1][0],
2041 pre_comp[i][1][1], pre_comp[i][1][2], 0,
2042 pre_comp[i][j - 1][0],
2043 pre_comp[i][j - 1][1],
2044 pre_comp[i][j - 1][2]);
2046 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
2047 pre_comp[i][j][2], pre_comp[i][j / 2][0],
2048 pre_comp[i][j / 2][1],
2049 pre_comp[i][j / 2][2]);
2055 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
2058 /* the scalar for the generator */
2059 if ((scalar != NULL) && (have_pre_comp)) {
2060 memset(g_secret, 0, sizeof(g_secret));
2061 /* reduce scalar to 0 <= scalar < 2^521 */
2062 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
2064 * this is an unusual input, and we don't guarantee
2067 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
2068 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
2071 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
2073 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
2075 /* do the multiplication with generator precomputation */
2076 batch_mul(x_out, y_out, z_out,
2077 (const felem_bytearray(*))secrets, num_points,
2079 mixed, (const felem(*)[17][3])pre_comp,
2080 (const felem(*)[3])g_pre_comp);
2082 /* do the multiplication without generator precomputation */
2083 batch_mul(x_out, y_out, z_out,
2084 (const felem_bytearray(*))secrets, num_points,
2085 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2087 /* reduce the output to its unique minimal representation */
2088 felem_contract(x_in, x_out);
2089 felem_contract(y_in, y_out);
2090 felem_contract(z_in, z_out);
2091 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2092 (!felem_to_BN(z, z_in))) {
2093 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
2096 ret = ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group, r, x, y, z,
2101 EC_POINT_free(generator);
2102 OPENSSL_free(secrets);
2103 OPENSSL_free(pre_comp);
2104 OPENSSL_free(tmp_felems);
2108 int ossl_ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2111 NISTP521_PRE_COMP *pre = NULL;
2114 EC_POINT *generator = NULL;
2115 felem tmp_felems[16];
2117 BN_CTX *new_ctx = NULL;
2120 /* throw away old precomputation */
2121 EC_pre_comp_free(group);
2125 ctx = new_ctx = BN_CTX_new();
2131 x = BN_CTX_get(ctx);
2132 y = BN_CTX_get(ctx);
2135 /* get the generator */
2136 if (group->generator == NULL)
2138 generator = EC_POINT_new(group);
2139 if (generator == NULL)
2141 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2142 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2143 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
2145 if ((pre = nistp521_pre_comp_new()) == NULL)
2148 * if the generator is the standard one, use built-in precomputation
2150 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2151 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2154 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2155 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2156 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2158 /* compute 2^130*G, 2^260*G, 2^390*G */
2159 for (i = 1; i <= 4; i <<= 1) {
2160 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2161 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2162 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2163 for (j = 0; j < 129; ++j) {
2164 point_double(pre->g_pre_comp[2 * i][0],
2165 pre->g_pre_comp[2 * i][1],
2166 pre->g_pre_comp[2 * i][2],
2167 pre->g_pre_comp[2 * i][0],
2168 pre->g_pre_comp[2 * i][1],
2169 pre->g_pre_comp[2 * i][2]);
2172 /* g_pre_comp[0] is the point at infinity */
2173 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2174 /* the remaining multiples */
2175 /* 2^130*G + 2^260*G */
2176 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2177 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2178 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2179 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2180 pre->g_pre_comp[2][2]);
2181 /* 2^130*G + 2^390*G */
2182 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2183 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2184 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2185 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2186 pre->g_pre_comp[2][2]);
2187 /* 2^260*G + 2^390*G */
2188 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2189 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2190 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2191 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2192 pre->g_pre_comp[4][2]);
2193 /* 2^130*G + 2^260*G + 2^390*G */
2194 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2195 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2196 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2197 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2198 pre->g_pre_comp[2][2]);
2199 for (i = 1; i < 8; ++i) {
2200 /* odd multiples: add G */
2201 point_add(pre->g_pre_comp[2 * i + 1][0],
2202 pre->g_pre_comp[2 * i + 1][1],
2203 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2204 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2205 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2206 pre->g_pre_comp[1][2]);
2208 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2211 SETPRECOMP(group, nistp521, pre);
2216 EC_POINT_free(generator);
2218 BN_CTX_free(new_ctx);
2220 EC_nistp521_pre_comp_free(pre);
2224 int ossl_ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2226 return HAVEPRECOMP(group, nistp521);