2 * Copyright 2011-2020 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (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 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication
29 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
30 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
31 * work which got its smarts from Daniel J. Bernstein's work on the same.
34 #include <openssl/e_os2.h>
35 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
36 NON_EMPTY_TRANSLATION_UNIT
40 # include <openssl/err.h>
41 # include "ec_local.h"
43 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
44 /* even with gcc, the typedef won't work for 32-bit platforms */
45 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
48 # error "Your compiler doesn't appear to support 128-bit integer types"
55 * The underlying field. P521 operates over GF(2^521-1). We can serialise an
56 * element of this field into 66 bytes where the most significant byte
57 * contains only a single bit. We call this an felem_bytearray.
60 typedef u8 felem_bytearray[66];
63 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
64 * These values are big-endian.
66 static const felem_bytearray nistp521_curve_params[5] = {
67 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
68 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
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,
76 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
77 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
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,
85 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
86 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
87 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
88 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
89 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
90 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
91 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
92 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
94 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
95 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
96 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
97 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
98 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
99 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
100 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
101 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
103 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
104 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
105 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
106 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
107 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
108 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
109 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
110 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
115 * The representation of field elements.
116 * ------------------------------------
118 * We represent field elements with nine values. These values are either 64 or
119 * 128 bits and the field element represented is:
120 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
121 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
122 * 58 bits apart, but are greater than 58 bits in length, the most significant
123 * bits of each limb overlap with the least significant bits of the next.
125 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
130 typedef uint64_t limb;
131 typedef limb limb_aX __attribute((__aligned__(1)));
132 typedef limb felem[NLIMBS];
133 typedef uint128_t largefelem[NLIMBS];
135 static const limb bottom57bits = 0x1ffffffffffffff;
136 static const limb bottom58bits = 0x3ffffffffffffff;
139 * bin66_to_felem takes a little-endian byte array and converts it into felem
140 * form. This assumes that the CPU is little-endian.
142 static void bin66_to_felem(felem out, const u8 in[66])
144 out[0] = (*((limb *) & in[0])) & bottom58bits;
145 out[1] = (*((limb_aX *) & in[7]) >> 2) & bottom58bits;
146 out[2] = (*((limb_aX *) & in[14]) >> 4) & bottom58bits;
147 out[3] = (*((limb_aX *) & in[21]) >> 6) & bottom58bits;
148 out[4] = (*((limb_aX *) & in[29])) & bottom58bits;
149 out[5] = (*((limb_aX *) & in[36]) >> 2) & bottom58bits;
150 out[6] = (*((limb_aX *) & in[43]) >> 4) & bottom58bits;
151 out[7] = (*((limb_aX *) & in[50]) >> 6) & bottom58bits;
152 out[8] = (*((limb_aX *) & in[58])) & bottom57bits;
156 * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
157 * array. This assumes that the CPU is little-endian.
159 static void felem_to_bin66(u8 out[66], const felem in)
162 (*((limb *) & out[0])) = in[0];
163 (*((limb_aX *) & out[7])) |= in[1] << 2;
164 (*((limb_aX *) & out[14])) |= in[2] << 4;
165 (*((limb_aX *) & out[21])) |= in[3] << 6;
166 (*((limb_aX *) & out[29])) = in[4];
167 (*((limb_aX *) & out[36])) |= in[5] << 2;
168 (*((limb_aX *) & out[43])) |= in[6] << 4;
169 (*((limb_aX *) & out[50])) |= in[7] << 6;
170 (*((limb_aX *) & out[58])) = in[8];
173 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
174 static int BN_to_felem(felem out, const BIGNUM *bn)
176 felem_bytearray b_out;
179 if (BN_is_negative(bn)) {
180 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
183 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
185 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
188 bin66_to_felem(out, b_out);
192 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
193 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
195 felem_bytearray b_out;
196 felem_to_bin66(b_out, in);
197 return BN_lebin2bn(b_out, sizeof(b_out), out);
205 static void felem_one(felem out)
218 static void felem_assign(felem out, const felem in)
231 /* felem_sum64 sets out = out + in. */
232 static void felem_sum64(felem out, const felem in)
245 /* felem_scalar sets out = in * scalar */
246 static void felem_scalar(felem out, const felem in, limb scalar)
248 out[0] = in[0] * scalar;
249 out[1] = in[1] * scalar;
250 out[2] = in[2] * scalar;
251 out[3] = in[3] * scalar;
252 out[4] = in[4] * scalar;
253 out[5] = in[5] * scalar;
254 out[6] = in[6] * scalar;
255 out[7] = in[7] * scalar;
256 out[8] = in[8] * scalar;
259 /* felem_scalar64 sets out = out * scalar */
260 static void felem_scalar64(felem out, limb scalar)
273 /* felem_scalar128 sets out = out * scalar */
274 static void felem_scalar128(largefelem out, limb scalar)
288 * felem_neg sets |out| to |-in|
290 * in[i] < 2^59 + 2^14
294 static void felem_neg(felem out, const felem in)
296 /* In order to prevent underflow, we subtract from 0 mod p. */
297 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
298 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
300 out[0] = two62m3 - in[0];
301 out[1] = two62m2 - in[1];
302 out[2] = two62m2 - in[2];
303 out[3] = two62m2 - in[3];
304 out[4] = two62m2 - in[4];
305 out[5] = two62m2 - in[5];
306 out[6] = two62m2 - in[6];
307 out[7] = two62m2 - in[7];
308 out[8] = two62m2 - in[8];
312 * felem_diff64 subtracts |in| from |out|
314 * in[i] < 2^59 + 2^14
316 * out[i] < out[i] + 2^62
318 static void felem_diff64(felem out, const felem in)
321 * In order to prevent underflow, we add 0 mod p before subtracting.
323 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
324 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
326 out[0] += two62m3 - in[0];
327 out[1] += two62m2 - in[1];
328 out[2] += two62m2 - in[2];
329 out[3] += two62m2 - in[3];
330 out[4] += two62m2 - in[4];
331 out[5] += two62m2 - in[5];
332 out[6] += two62m2 - in[6];
333 out[7] += two62m2 - in[7];
334 out[8] += two62m2 - in[8];
338 * felem_diff_128_64 subtracts |in| from |out|
340 * in[i] < 2^62 + 2^17
342 * out[i] < out[i] + 2^63
344 static void felem_diff_128_64(largefelem out, const felem in)
347 * In order to prevent underflow, we add 64p mod p (which is equivalent
348 * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521
349 * digit number with all bits set to 1. See "The representation of field
350 * elements" comment above for a description of how limbs are used to
351 * represent a number. 64p is represented with 8 limbs containing a number
352 * with 58 bits set and one limb with a number with 57 bits set.
354 static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6);
355 static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5);
357 out[0] += two63m6 - in[0];
358 out[1] += two63m5 - in[1];
359 out[2] += two63m5 - in[2];
360 out[3] += two63m5 - in[3];
361 out[4] += two63m5 - in[4];
362 out[5] += two63m5 - in[5];
363 out[6] += two63m5 - in[6];
364 out[7] += two63m5 - in[7];
365 out[8] += two63m5 - in[8];
369 * felem_diff_128_64 subtracts |in| from |out|
373 * out[i] < out[i] + 2^127 - 2^69
375 static void felem_diff128(largefelem out, const largefelem in)
378 * In order to prevent underflow, we add 0 mod p before subtracting.
380 static const uint128_t two127m70 =
381 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
382 static const uint128_t two127m69 =
383 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
385 out[0] += (two127m70 - in[0]);
386 out[1] += (two127m69 - in[1]);
387 out[2] += (two127m69 - in[2]);
388 out[3] += (two127m69 - in[3]);
389 out[4] += (two127m69 - in[4]);
390 out[5] += (two127m69 - in[5]);
391 out[6] += (two127m69 - in[6]);
392 out[7] += (two127m69 - in[7]);
393 out[8] += (two127m69 - in[8]);
397 * felem_square sets |out| = |in|^2
401 * out[i] < 17 * max(in[i]) * max(in[i])
403 static void felem_square(largefelem out, const felem in)
406 felem_scalar(inx2, in, 2);
407 felem_scalar(inx4, in, 4);
410 * We have many cases were we want to do
413 * This is obviously just
415 * However, rather than do the doubling on the 128 bit result, we
416 * double one of the inputs to the multiplication by reading from
420 out[0] = ((uint128_t) in[0]) * in[0];
421 out[1] = ((uint128_t) in[0]) * inx2[1];
422 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
423 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
424 out[4] = ((uint128_t) in[0]) * inx2[4] +
425 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
426 out[5] = ((uint128_t) in[0]) * inx2[5] +
427 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
428 out[6] = ((uint128_t) in[0]) * inx2[6] +
429 ((uint128_t) in[1]) * inx2[5] +
430 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
431 out[7] = ((uint128_t) in[0]) * inx2[7] +
432 ((uint128_t) in[1]) * inx2[6] +
433 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
434 out[8] = ((uint128_t) in[0]) * inx2[8] +
435 ((uint128_t) in[1]) * inx2[7] +
436 ((uint128_t) in[2]) * inx2[6] +
437 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
440 * The remaining limbs fall above 2^521, with the first falling at 2^522.
441 * They correspond to locations one bit up from the limbs produced above
442 * so we would have to multiply by two to align them. Again, rather than
443 * operate on the 128-bit result, we double one of the inputs to the
444 * multiplication. If we want to double for both this reason, and the
445 * reason above, then we end up multiplying by four.
449 out[0] += ((uint128_t) in[1]) * inx4[8] +
450 ((uint128_t) in[2]) * inx4[7] +
451 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
454 out[1] += ((uint128_t) in[2]) * inx4[8] +
455 ((uint128_t) in[3]) * inx4[7] +
456 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
459 out[2] += ((uint128_t) in[3]) * inx4[8] +
460 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
463 out[3] += ((uint128_t) in[4]) * inx4[8] +
464 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
467 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
470 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
473 out[6] += ((uint128_t) in[7]) * inx4[8];
476 out[7] += ((uint128_t) in[8]) * inx2[8];
480 * felem_mul sets |out| = |in1| * |in2|
485 * out[i] < 17 * max(in1[i]) * max(in2[i])
487 static void felem_mul(largefelem out, const felem in1, const felem in2)
490 felem_scalar(in2x2, in2, 2);
492 out[0] = ((uint128_t) in1[0]) * in2[0];
494 out[1] = ((uint128_t) in1[0]) * in2[1] +
495 ((uint128_t) in1[1]) * in2[0];
497 out[2] = ((uint128_t) in1[0]) * in2[2] +
498 ((uint128_t) in1[1]) * in2[1] +
499 ((uint128_t) in1[2]) * in2[0];
501 out[3] = ((uint128_t) in1[0]) * in2[3] +
502 ((uint128_t) in1[1]) * in2[2] +
503 ((uint128_t) in1[2]) * in2[1] +
504 ((uint128_t) in1[3]) * in2[0];
506 out[4] = ((uint128_t) in1[0]) * in2[4] +
507 ((uint128_t) in1[1]) * in2[3] +
508 ((uint128_t) in1[2]) * in2[2] +
509 ((uint128_t) in1[3]) * in2[1] +
510 ((uint128_t) in1[4]) * in2[0];
512 out[5] = ((uint128_t) in1[0]) * in2[5] +
513 ((uint128_t) in1[1]) * in2[4] +
514 ((uint128_t) in1[2]) * in2[3] +
515 ((uint128_t) in1[3]) * in2[2] +
516 ((uint128_t) in1[4]) * in2[1] +
517 ((uint128_t) in1[5]) * in2[0];
519 out[6] = ((uint128_t) in1[0]) * in2[6] +
520 ((uint128_t) in1[1]) * in2[5] +
521 ((uint128_t) in1[2]) * in2[4] +
522 ((uint128_t) in1[3]) * in2[3] +
523 ((uint128_t) in1[4]) * in2[2] +
524 ((uint128_t) in1[5]) * in2[1] +
525 ((uint128_t) in1[6]) * in2[0];
527 out[7] = ((uint128_t) in1[0]) * in2[7] +
528 ((uint128_t) in1[1]) * in2[6] +
529 ((uint128_t) in1[2]) * in2[5] +
530 ((uint128_t) in1[3]) * in2[4] +
531 ((uint128_t) in1[4]) * in2[3] +
532 ((uint128_t) in1[5]) * in2[2] +
533 ((uint128_t) in1[6]) * in2[1] +
534 ((uint128_t) in1[7]) * in2[0];
536 out[8] = ((uint128_t) in1[0]) * in2[8] +
537 ((uint128_t) in1[1]) * in2[7] +
538 ((uint128_t) in1[2]) * in2[6] +
539 ((uint128_t) in1[3]) * in2[5] +
540 ((uint128_t) in1[4]) * in2[4] +
541 ((uint128_t) in1[5]) * in2[3] +
542 ((uint128_t) in1[6]) * in2[2] +
543 ((uint128_t) in1[7]) * in2[1] +
544 ((uint128_t) in1[8]) * in2[0];
546 /* See comment in felem_square about the use of in2x2 here */
548 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
549 ((uint128_t) in1[2]) * in2x2[7] +
550 ((uint128_t) in1[3]) * in2x2[6] +
551 ((uint128_t) in1[4]) * in2x2[5] +
552 ((uint128_t) in1[5]) * in2x2[4] +
553 ((uint128_t) in1[6]) * in2x2[3] +
554 ((uint128_t) in1[7]) * in2x2[2] +
555 ((uint128_t) in1[8]) * in2x2[1];
557 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
558 ((uint128_t) in1[3]) * in2x2[7] +
559 ((uint128_t) in1[4]) * in2x2[6] +
560 ((uint128_t) in1[5]) * in2x2[5] +
561 ((uint128_t) in1[6]) * in2x2[4] +
562 ((uint128_t) in1[7]) * in2x2[3] +
563 ((uint128_t) in1[8]) * in2x2[2];
565 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
566 ((uint128_t) in1[4]) * in2x2[7] +
567 ((uint128_t) in1[5]) * in2x2[6] +
568 ((uint128_t) in1[6]) * in2x2[5] +
569 ((uint128_t) in1[7]) * in2x2[4] +
570 ((uint128_t) in1[8]) * in2x2[3];
572 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
573 ((uint128_t) in1[5]) * in2x2[7] +
574 ((uint128_t) in1[6]) * in2x2[6] +
575 ((uint128_t) in1[7]) * in2x2[5] +
576 ((uint128_t) in1[8]) * in2x2[4];
578 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
579 ((uint128_t) in1[6]) * in2x2[7] +
580 ((uint128_t) in1[7]) * in2x2[6] +
581 ((uint128_t) in1[8]) * in2x2[5];
583 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
584 ((uint128_t) in1[7]) * in2x2[7] +
585 ((uint128_t) in1[8]) * in2x2[6];
587 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
588 ((uint128_t) in1[8]) * in2x2[7];
590 out[7] += ((uint128_t) in1[8]) * in2x2[8];
593 static const limb bottom52bits = 0xfffffffffffff;
596 * felem_reduce converts a largefelem to an felem.
600 * out[i] < 2^59 + 2^14
602 static void felem_reduce(felem out, const largefelem in)
604 u64 overflow1, overflow2;
606 out[0] = ((limb) in[0]) & bottom58bits;
607 out[1] = ((limb) in[1]) & bottom58bits;
608 out[2] = ((limb) in[2]) & bottom58bits;
609 out[3] = ((limb) in[3]) & bottom58bits;
610 out[4] = ((limb) in[4]) & bottom58bits;
611 out[5] = ((limb) in[5]) & bottom58bits;
612 out[6] = ((limb) in[6]) & bottom58bits;
613 out[7] = ((limb) in[7]) & bottom58bits;
614 out[8] = ((limb) in[8]) & bottom58bits;
618 out[1] += ((limb) in[0]) >> 58;
619 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
621 * out[1] < 2^58 + 2^6 + 2^58
624 out[2] += ((limb) (in[0] >> 64)) >> 52;
626 out[2] += ((limb) in[1]) >> 58;
627 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
628 out[3] += ((limb) (in[1] >> 64)) >> 52;
630 out[3] += ((limb) in[2]) >> 58;
631 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
632 out[4] += ((limb) (in[2] >> 64)) >> 52;
634 out[4] += ((limb) in[3]) >> 58;
635 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
636 out[5] += ((limb) (in[3] >> 64)) >> 52;
638 out[5] += ((limb) in[4]) >> 58;
639 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
640 out[6] += ((limb) (in[4] >> 64)) >> 52;
642 out[6] += ((limb) in[5]) >> 58;
643 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
644 out[7] += ((limb) (in[5] >> 64)) >> 52;
646 out[7] += ((limb) in[6]) >> 58;
647 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
648 out[8] += ((limb) (in[6] >> 64)) >> 52;
650 out[8] += ((limb) in[7]) >> 58;
651 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
653 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
656 overflow1 = ((limb) (in[7] >> 64)) >> 52;
658 overflow1 += ((limb) in[8]) >> 58;
659 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
660 overflow2 = ((limb) (in[8] >> 64)) >> 52;
662 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
663 overflow2 <<= 1; /* overflow2 < 2^13 */
665 out[0] += overflow1; /* out[0] < 2^60 */
666 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
668 out[1] += out[0] >> 58;
669 out[0] &= bottom58bits;
672 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
677 static void felem_square_reduce(felem out, const felem in)
680 felem_square(tmp, in);
681 felem_reduce(out, tmp);
684 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
687 felem_mul(tmp, in1, in2);
688 felem_reduce(out, tmp);
692 * felem_inv calculates |out| = |in|^{-1}
694 * Based on Fermat's Little Theorem:
696 * a^{p-1} = 1 (mod p)
697 * a^{p-2} = a^{-1} (mod p)
699 static void felem_inv(felem out, const felem in)
701 felem ftmp, ftmp2, ftmp3, ftmp4;
705 felem_square(tmp, in);
706 felem_reduce(ftmp, tmp); /* 2^1 */
707 felem_mul(tmp, in, ftmp);
708 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
709 felem_assign(ftmp2, ftmp);
710 felem_square(tmp, ftmp);
711 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
712 felem_mul(tmp, in, ftmp);
713 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
714 felem_square(tmp, ftmp);
715 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
717 felem_square(tmp, ftmp2);
718 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
719 felem_square(tmp, ftmp3);
720 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
721 felem_mul(tmp, ftmp3, ftmp2);
722 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
724 felem_assign(ftmp2, ftmp3);
725 felem_square(tmp, ftmp3);
726 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
727 felem_square(tmp, ftmp3);
728 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
729 felem_square(tmp, ftmp3);
730 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
731 felem_square(tmp, ftmp3);
732 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
733 felem_assign(ftmp4, ftmp3);
734 felem_mul(tmp, ftmp3, ftmp);
735 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
736 felem_square(tmp, ftmp4);
737 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
738 felem_mul(tmp, ftmp3, ftmp2);
739 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
740 felem_assign(ftmp2, ftmp3);
742 for (i = 0; i < 8; i++) {
743 felem_square(tmp, ftmp3);
744 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
746 felem_mul(tmp, ftmp3, ftmp2);
747 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
748 felem_assign(ftmp2, ftmp3);
750 for (i = 0; i < 16; i++) {
751 felem_square(tmp, ftmp3);
752 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
754 felem_mul(tmp, ftmp3, ftmp2);
755 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
756 felem_assign(ftmp2, ftmp3);
758 for (i = 0; i < 32; i++) {
759 felem_square(tmp, ftmp3);
760 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
762 felem_mul(tmp, ftmp3, ftmp2);
763 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
764 felem_assign(ftmp2, ftmp3);
766 for (i = 0; i < 64; i++) {
767 felem_square(tmp, ftmp3);
768 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
770 felem_mul(tmp, ftmp3, ftmp2);
771 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
772 felem_assign(ftmp2, ftmp3);
774 for (i = 0; i < 128; i++) {
775 felem_square(tmp, ftmp3);
776 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
778 felem_mul(tmp, ftmp3, ftmp2);
779 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
780 felem_assign(ftmp2, ftmp3);
782 for (i = 0; i < 256; i++) {
783 felem_square(tmp, ftmp3);
784 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
786 felem_mul(tmp, ftmp3, ftmp2);
787 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
789 for (i = 0; i < 9; i++) {
790 felem_square(tmp, ftmp3);
791 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
793 felem_mul(tmp, ftmp3, ftmp4);
794 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
795 felem_mul(tmp, ftmp3, in);
796 felem_reduce(out, tmp); /* 2^512 - 3 */
799 /* This is 2^521-1, expressed as an felem */
800 static const felem kPrime = {
801 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
802 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
803 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
807 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
810 * in[i] < 2^59 + 2^14
812 static limb felem_is_zero(const felem in)
816 felem_assign(ftmp, in);
818 ftmp[0] += ftmp[8] >> 57;
819 ftmp[8] &= bottom57bits;
821 ftmp[1] += ftmp[0] >> 58;
822 ftmp[0] &= bottom58bits;
823 ftmp[2] += ftmp[1] >> 58;
824 ftmp[1] &= bottom58bits;
825 ftmp[3] += ftmp[2] >> 58;
826 ftmp[2] &= bottom58bits;
827 ftmp[4] += ftmp[3] >> 58;
828 ftmp[3] &= bottom58bits;
829 ftmp[5] += ftmp[4] >> 58;
830 ftmp[4] &= bottom58bits;
831 ftmp[6] += ftmp[5] >> 58;
832 ftmp[5] &= bottom58bits;
833 ftmp[7] += ftmp[6] >> 58;
834 ftmp[6] &= bottom58bits;
835 ftmp[8] += ftmp[7] >> 58;
836 ftmp[7] &= bottom58bits;
837 /* ftmp[8] < 2^57 + 4 */
840 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
841 * than our bound for ftmp[8]. Therefore we only have to check if the
842 * zero is zero or 2^521-1.
858 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
859 * can be set is if is_zero was 0 before the decrement.
861 is_zero = 0 - (is_zero >> 63);
863 is_p = ftmp[0] ^ kPrime[0];
864 is_p |= ftmp[1] ^ kPrime[1];
865 is_p |= ftmp[2] ^ kPrime[2];
866 is_p |= ftmp[3] ^ kPrime[3];
867 is_p |= ftmp[4] ^ kPrime[4];
868 is_p |= ftmp[5] ^ kPrime[5];
869 is_p |= ftmp[6] ^ kPrime[6];
870 is_p |= ftmp[7] ^ kPrime[7];
871 is_p |= ftmp[8] ^ kPrime[8];
874 is_p = 0 - (is_p >> 63);
880 static int felem_is_zero_int(const void *in)
882 return (int)(felem_is_zero(in) & ((limb) 1));
886 * felem_contract converts |in| to its unique, minimal representation.
888 * in[i] < 2^59 + 2^14
890 static void felem_contract(felem out, const felem in)
892 limb is_p, is_greater, sign;
893 static const limb two58 = ((limb) 1) << 58;
895 felem_assign(out, in);
897 out[0] += out[8] >> 57;
898 out[8] &= bottom57bits;
900 out[1] += out[0] >> 58;
901 out[0] &= bottom58bits;
902 out[2] += out[1] >> 58;
903 out[1] &= bottom58bits;
904 out[3] += out[2] >> 58;
905 out[2] &= bottom58bits;
906 out[4] += out[3] >> 58;
907 out[3] &= bottom58bits;
908 out[5] += out[4] >> 58;
909 out[4] &= bottom58bits;
910 out[6] += out[5] >> 58;
911 out[5] &= bottom58bits;
912 out[7] += out[6] >> 58;
913 out[6] &= bottom58bits;
914 out[8] += out[7] >> 58;
915 out[7] &= bottom58bits;
916 /* out[8] < 2^57 + 4 */
919 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
920 * out. See the comments in felem_is_zero regarding why we don't test for
921 * other multiples of the prime.
925 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
928 is_p = out[0] ^ kPrime[0];
929 is_p |= out[1] ^ kPrime[1];
930 is_p |= out[2] ^ kPrime[2];
931 is_p |= out[3] ^ kPrime[3];
932 is_p |= out[4] ^ kPrime[4];
933 is_p |= out[5] ^ kPrime[5];
934 is_p |= out[6] ^ kPrime[6];
935 is_p |= out[7] ^ kPrime[7];
936 is_p |= out[8] ^ kPrime[8];
945 is_p = 0 - (is_p >> 63);
948 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
961 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
962 * 57 is greater than zero as (2^521-1) + x >= 2^522
964 is_greater = out[8] >> 57;
965 is_greater |= is_greater << 32;
966 is_greater |= is_greater << 16;
967 is_greater |= is_greater << 8;
968 is_greater |= is_greater << 4;
969 is_greater |= is_greater << 2;
970 is_greater |= is_greater << 1;
971 is_greater = 0 - (is_greater >> 63);
973 out[0] -= kPrime[0] & is_greater;
974 out[1] -= kPrime[1] & is_greater;
975 out[2] -= kPrime[2] & is_greater;
976 out[3] -= kPrime[3] & is_greater;
977 out[4] -= kPrime[4] & is_greater;
978 out[5] -= kPrime[5] & is_greater;
979 out[6] -= kPrime[6] & is_greater;
980 out[7] -= kPrime[7] & is_greater;
981 out[8] -= kPrime[8] & is_greater;
983 /* Eliminate negative coefficients */
984 sign = -(out[0] >> 63);
985 out[0] += (two58 & sign);
986 out[1] -= (1 & sign);
987 sign = -(out[1] >> 63);
988 out[1] += (two58 & sign);
989 out[2] -= (1 & sign);
990 sign = -(out[2] >> 63);
991 out[2] += (two58 & sign);
992 out[3] -= (1 & sign);
993 sign = -(out[3] >> 63);
994 out[3] += (two58 & sign);
995 out[4] -= (1 & sign);
996 sign = -(out[4] >> 63);
997 out[4] += (two58 & sign);
998 out[5] -= (1 & sign);
999 sign = -(out[0] >> 63);
1000 out[5] += (two58 & sign);
1001 out[6] -= (1 & sign);
1002 sign = -(out[6] >> 63);
1003 out[6] += (two58 & sign);
1004 out[7] -= (1 & sign);
1005 sign = -(out[7] >> 63);
1006 out[7] += (two58 & sign);
1007 out[8] -= (1 & sign);
1008 sign = -(out[5] >> 63);
1009 out[5] += (two58 & sign);
1010 out[6] -= (1 & sign);
1011 sign = -(out[6] >> 63);
1012 out[6] += (two58 & sign);
1013 out[7] -= (1 & sign);
1014 sign = -(out[7] >> 63);
1015 out[7] += (two58 & sign);
1016 out[8] -= (1 & sign);
1023 * Building on top of the field operations we have the operations on the
1024 * elliptic curve group itself. Points on the curve are represented in Jacobian
1028 * point_double calculates 2*(x_in, y_in, z_in)
1030 * The method is taken from:
1031 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1033 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1034 * while x_out == y_in is not (maybe this works, but it's not tested). */
1036 point_double(felem x_out, felem y_out, felem z_out,
1037 const felem x_in, const felem y_in, const felem z_in)
1039 largefelem tmp, tmp2;
1040 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1042 felem_assign(ftmp, x_in);
1043 felem_assign(ftmp2, x_in);
1046 felem_square(tmp, z_in);
1047 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1050 felem_square(tmp, y_in);
1051 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1053 /* beta = x*gamma */
1054 felem_mul(tmp, x_in, gamma);
1055 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1057 /* alpha = 3*(x-delta)*(x+delta) */
1058 felem_diff64(ftmp, delta);
1059 /* ftmp[i] < 2^61 */
1060 felem_sum64(ftmp2, delta);
1061 /* ftmp2[i] < 2^60 + 2^15 */
1062 felem_scalar64(ftmp2, 3);
1063 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1064 felem_mul(tmp, ftmp, ftmp2);
1066 * tmp[i] < 17(3*2^121 + 3*2^76)
1067 * = 61*2^121 + 61*2^76
1068 * < 64*2^121 + 64*2^76
1072 felem_reduce(alpha, tmp);
1074 /* x' = alpha^2 - 8*beta */
1075 felem_square(tmp, alpha);
1077 * tmp[i] < 17*2^120 < 2^125
1079 felem_assign(ftmp, beta);
1080 felem_scalar64(ftmp, 8);
1081 /* ftmp[i] < 2^62 + 2^17 */
1082 felem_diff_128_64(tmp, ftmp);
1083 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1084 felem_reduce(x_out, tmp);
1086 /* z' = (y + z)^2 - gamma - delta */
1087 felem_sum64(delta, gamma);
1088 /* delta[i] < 2^60 + 2^15 */
1089 felem_assign(ftmp, y_in);
1090 felem_sum64(ftmp, z_in);
1091 /* ftmp[i] < 2^60 + 2^15 */
1092 felem_square(tmp, ftmp);
1094 * tmp[i] < 17(2^122) < 2^127
1096 felem_diff_128_64(tmp, delta);
1097 /* tmp[i] < 2^127 + 2^63 */
1098 felem_reduce(z_out, tmp);
1100 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1101 felem_scalar64(beta, 4);
1102 /* beta[i] < 2^61 + 2^16 */
1103 felem_diff64(beta, x_out);
1104 /* beta[i] < 2^61 + 2^60 + 2^16 */
1105 felem_mul(tmp, alpha, beta);
1107 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1108 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1109 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1112 felem_square(tmp2, gamma);
1114 * tmp2[i] < 17*(2^59 + 2^14)^2
1115 * = 17*(2^118 + 2^74 + 2^28)
1117 felem_scalar128(tmp2, 8);
1119 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1120 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1123 felem_diff128(tmp, tmp2);
1125 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1126 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1127 * 2^74 + 2^69 + 2^34 + 2^30
1130 felem_reduce(y_out, tmp);
1133 /* copy_conditional copies in to out iff mask is all ones. */
1134 static void copy_conditional(felem out, const felem in, limb mask)
1137 for (i = 0; i < NLIMBS; ++i) {
1138 const limb tmp = mask & (in[i] ^ out[i]);
1144 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1146 * The method is taken from
1147 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1148 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1150 * This function includes a branch for checking whether the two input points
1151 * are equal (while not equal to the point at infinity). See comment below
1154 static void point_add(felem x3, felem y3, felem z3,
1155 const felem x1, const felem y1, const felem z1,
1156 const int mixed, const felem x2, const felem y2,
1159 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1160 largefelem tmp, tmp2;
1161 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1164 z1_is_zero = felem_is_zero(z1);
1165 z2_is_zero = felem_is_zero(z2);
1167 /* ftmp = z1z1 = z1**2 */
1168 felem_square(tmp, z1);
1169 felem_reduce(ftmp, tmp);
1172 /* ftmp2 = z2z2 = z2**2 */
1173 felem_square(tmp, z2);
1174 felem_reduce(ftmp2, tmp);
1176 /* u1 = ftmp3 = x1*z2z2 */
1177 felem_mul(tmp, x1, ftmp2);
1178 felem_reduce(ftmp3, tmp);
1180 /* ftmp5 = z1 + z2 */
1181 felem_assign(ftmp5, z1);
1182 felem_sum64(ftmp5, z2);
1183 /* ftmp5[i] < 2^61 */
1185 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1186 felem_square(tmp, ftmp5);
1187 /* tmp[i] < 17*2^122 */
1188 felem_diff_128_64(tmp, ftmp);
1189 /* tmp[i] < 17*2^122 + 2^63 */
1190 felem_diff_128_64(tmp, ftmp2);
1191 /* tmp[i] < 17*2^122 + 2^64 */
1192 felem_reduce(ftmp5, tmp);
1194 /* ftmp2 = z2 * z2z2 */
1195 felem_mul(tmp, ftmp2, z2);
1196 felem_reduce(ftmp2, tmp);
1198 /* s1 = ftmp6 = y1 * z2**3 */
1199 felem_mul(tmp, y1, ftmp2);
1200 felem_reduce(ftmp6, tmp);
1203 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1206 /* u1 = ftmp3 = x1*z2z2 */
1207 felem_assign(ftmp3, x1);
1209 /* ftmp5 = 2*z1z2 */
1210 felem_scalar(ftmp5, z1, 2);
1212 /* s1 = ftmp6 = y1 * z2**3 */
1213 felem_assign(ftmp6, y1);
1217 felem_mul(tmp, x2, ftmp);
1218 /* tmp[i] < 17*2^120 */
1220 /* h = ftmp4 = u2 - u1 */
1221 felem_diff_128_64(tmp, ftmp3);
1222 /* tmp[i] < 17*2^120 + 2^63 */
1223 felem_reduce(ftmp4, tmp);
1225 x_equal = felem_is_zero(ftmp4);
1227 /* z_out = ftmp5 * h */
1228 felem_mul(tmp, ftmp5, ftmp4);
1229 felem_reduce(z_out, tmp);
1231 /* ftmp = z1 * z1z1 */
1232 felem_mul(tmp, ftmp, z1);
1233 felem_reduce(ftmp, tmp);
1235 /* s2 = tmp = y2 * z1**3 */
1236 felem_mul(tmp, y2, ftmp);
1237 /* tmp[i] < 17*2^120 */
1239 /* r = ftmp5 = (s2 - s1)*2 */
1240 felem_diff_128_64(tmp, ftmp6);
1241 /* tmp[i] < 17*2^120 + 2^63 */
1242 felem_reduce(ftmp5, tmp);
1243 y_equal = felem_is_zero(ftmp5);
1244 felem_scalar64(ftmp5, 2);
1245 /* ftmp5[i] < 2^61 */
1248 * The formulae are incorrect if the points are equal, in affine coordinates
1249 * (X_1, Y_1) == (X_2, Y_2), so we check for this and do doubling if this
1252 * We use bitwise operations to avoid potential side-channels introduced by
1253 * the short-circuiting behaviour of boolean operators.
1255 * The special case of either point being the point at infinity (z1 and/or
1256 * z2 are zero), is handled separately later on in this function, so we
1257 * avoid jumping to point_double here in those special cases.
1259 * Notice the comment below on the implications of this branching for timing
1260 * leaks and why it is considered practically irrelevant.
1262 points_equal = (x_equal & y_equal & (~z1_is_zero) & (~z2_is_zero));
1266 * This is obviously not constant-time but it will almost-never happen
1267 * for ECDH / ECDSA. The case where it can happen is during scalar-mult
1268 * where the intermediate value gets very close to the group order.
1269 * Since |ec_GFp_nistp_recode_scalar_bits| produces signed digits for
1270 * the scalar, it's possible for the intermediate value to be a small
1271 * negative multiple of the base point, and for the final signed digit
1272 * to be the same value. We believe that this only occurs for the scalar
1273 * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
1274 * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb
1275 * 71e913863f7, in that case the penultimate intermediate is -9G and
1276 * the final digit is also -9G. Since this only happens for a single
1277 * scalar, the timing leak is irrelevant. (Any attacker who wanted to
1278 * check whether a secret scalar was that exact value, can already do
1281 point_double(x3, y3, z3, x1, y1, z1);
1285 /* I = ftmp = (2h)**2 */
1286 felem_assign(ftmp, ftmp4);
1287 felem_scalar64(ftmp, 2);
1288 /* ftmp[i] < 2^61 */
1289 felem_square(tmp, ftmp);
1290 /* tmp[i] < 17*2^122 */
1291 felem_reduce(ftmp, tmp);
1293 /* J = ftmp2 = h * I */
1294 felem_mul(tmp, ftmp4, ftmp);
1295 felem_reduce(ftmp2, tmp);
1297 /* V = ftmp4 = U1 * I */
1298 felem_mul(tmp, ftmp3, ftmp);
1299 felem_reduce(ftmp4, tmp);
1301 /* x_out = r**2 - J - 2V */
1302 felem_square(tmp, ftmp5);
1303 /* tmp[i] < 17*2^122 */
1304 felem_diff_128_64(tmp, ftmp2);
1305 /* tmp[i] < 17*2^122 + 2^63 */
1306 felem_assign(ftmp3, ftmp4);
1307 felem_scalar64(ftmp4, 2);
1308 /* ftmp4[i] < 2^61 */
1309 felem_diff_128_64(tmp, ftmp4);
1310 /* tmp[i] < 17*2^122 + 2^64 */
1311 felem_reduce(x_out, tmp);
1313 /* y_out = r(V-x_out) - 2 * s1 * J */
1314 felem_diff64(ftmp3, x_out);
1316 * ftmp3[i] < 2^60 + 2^60 = 2^61
1318 felem_mul(tmp, ftmp5, ftmp3);
1319 /* tmp[i] < 17*2^122 */
1320 felem_mul(tmp2, ftmp6, ftmp2);
1321 /* tmp2[i] < 17*2^120 */
1322 felem_scalar128(tmp2, 2);
1323 /* tmp2[i] < 17*2^121 */
1324 felem_diff128(tmp, tmp2);
1326 * tmp[i] < 2^127 - 2^69 + 17*2^122
1327 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1330 felem_reduce(y_out, tmp);
1332 copy_conditional(x_out, x2, z1_is_zero);
1333 copy_conditional(x_out, x1, z2_is_zero);
1334 copy_conditional(y_out, y2, z1_is_zero);
1335 copy_conditional(y_out, y1, z2_is_zero);
1336 copy_conditional(z_out, z2, z1_is_zero);
1337 copy_conditional(z_out, z1, z2_is_zero);
1338 felem_assign(x3, x_out);
1339 felem_assign(y3, y_out);
1340 felem_assign(z3, z_out);
1344 * Base point pre computation
1345 * --------------------------
1347 * Two different sorts of precomputed tables are used in the following code.
1348 * Each contain various points on the curve, where each point is three field
1349 * elements (x, y, z).
1351 * For the base point table, z is usually 1 (0 for the point at infinity).
1352 * This table has 16 elements:
1353 * index | bits | point
1354 * ------+---------+------------------------------
1357 * 2 | 0 0 1 0 | 2^130G
1358 * 3 | 0 0 1 1 | (2^130 + 1)G
1359 * 4 | 0 1 0 0 | 2^260G
1360 * 5 | 0 1 0 1 | (2^260 + 1)G
1361 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1362 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1363 * 8 | 1 0 0 0 | 2^390G
1364 * 9 | 1 0 0 1 | (2^390 + 1)G
1365 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1366 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1367 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1368 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1369 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1370 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1372 * The reason for this is so that we can clock bits into four different
1373 * locations when doing simple scalar multiplies against the base point.
1375 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1377 /* gmul is the table of precomputed base points */
1378 static const felem gmul[16][3] = {
1379 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1380 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1381 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1382 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1383 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1384 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1385 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1386 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1387 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1388 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1389 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1390 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1391 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1392 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1393 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1394 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1395 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1396 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1397 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1398 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1399 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1400 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1401 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1402 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1403 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1404 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1405 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1406 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1407 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1408 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1409 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1410 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1411 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1412 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1413 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1414 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1415 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1416 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1417 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1418 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1419 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1420 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1421 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1422 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1423 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1424 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1425 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1426 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1427 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1428 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1429 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1430 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1431 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1432 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1433 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1434 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1435 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1436 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1437 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1438 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1439 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1440 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1441 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1442 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1443 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1444 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1445 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1446 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1447 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1448 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1449 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1450 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1451 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1452 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1453 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1454 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1455 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1456 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1457 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1458 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1459 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1460 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1461 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1462 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1463 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1464 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1465 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1466 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1467 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1468 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1469 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1470 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1471 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1472 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1473 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1474 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1475 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1476 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1477 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1478 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1479 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1480 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1481 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1482 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1483 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1484 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1485 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1486 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1490 * select_point selects the |idx|th point from a precomputation table and
1493 /* pre_comp below is of the size provided in |size| */
1494 static void select_point(const limb idx, unsigned int size,
1495 const felem pre_comp[][3], felem out[3])
1498 limb *outlimbs = &out[0][0];
1500 memset(out, 0, sizeof(*out) * 3);
1502 for (i = 0; i < size; i++) {
1503 const limb *inlimbs = &pre_comp[i][0][0];
1504 limb mask = i ^ idx;
1510 for (j = 0; j < NLIMBS * 3; j++)
1511 outlimbs[j] |= inlimbs[j] & mask;
1515 /* get_bit returns the |i|th bit in |in| */
1516 static char get_bit(const felem_bytearray in, int i)
1520 return (in[i >> 3] >> (i & 7)) & 1;
1524 * Interleaved point multiplication using precomputed point multiples: The
1525 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1526 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1527 * generator, using certain (large) precomputed multiples in g_pre_comp.
1528 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1530 static void batch_mul(felem x_out, felem y_out, felem z_out,
1531 const felem_bytearray scalars[],
1532 const unsigned num_points, const u8 *g_scalar,
1533 const int mixed, const felem pre_comp[][17][3],
1534 const felem g_pre_comp[16][3])
1537 unsigned num, gen_mul = (g_scalar != NULL);
1538 felem nq[3], tmp[4];
1542 /* set nq to the point at infinity */
1543 memset(nq, 0, sizeof(nq));
1546 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1547 * of the generator (last quarter of rounds) and additions of other
1548 * points multiples (every 5th round).
1550 skip = 1; /* save two point operations in the first
1552 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1555 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1557 /* add multiples of the generator */
1558 if (gen_mul && (i <= 130)) {
1559 bits = get_bit(g_scalar, i + 390) << 3;
1561 bits |= get_bit(g_scalar, i + 260) << 2;
1562 bits |= get_bit(g_scalar, i + 130) << 1;
1563 bits |= get_bit(g_scalar, i);
1565 /* select the point to add, in constant time */
1566 select_point(bits, 16, g_pre_comp, tmp);
1568 /* The 1 argument below is for "mixed" */
1569 point_add(nq[0], nq[1], nq[2],
1570 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1572 memcpy(nq, tmp, 3 * sizeof(felem));
1577 /* do other additions every 5 doublings */
1578 if (num_points && (i % 5 == 0)) {
1579 /* loop over all scalars */
1580 for (num = 0; num < num_points; ++num) {
1581 bits = get_bit(scalars[num], i + 4) << 5;
1582 bits |= get_bit(scalars[num], i + 3) << 4;
1583 bits |= get_bit(scalars[num], i + 2) << 3;
1584 bits |= get_bit(scalars[num], i + 1) << 2;
1585 bits |= get_bit(scalars[num], i) << 1;
1586 bits |= get_bit(scalars[num], i - 1);
1587 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1590 * select the point to add or subtract, in constant time
1592 select_point(digit, 17, pre_comp[num], tmp);
1593 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1595 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1598 point_add(nq[0], nq[1], nq[2],
1599 nq[0], nq[1], nq[2],
1600 mixed, tmp[0], tmp[1], tmp[2]);
1602 memcpy(nq, tmp, 3 * sizeof(felem));
1608 felem_assign(x_out, nq[0]);
1609 felem_assign(y_out, nq[1]);
1610 felem_assign(z_out, nq[2]);
1613 /* Precomputation for the group generator. */
1614 struct nistp521_pre_comp_st {
1615 felem g_pre_comp[16][3];
1616 CRYPTO_REF_COUNT references;
1617 CRYPTO_RWLOCK *lock;
1620 const EC_METHOD *EC_GFp_nistp521_method(void)
1622 static const EC_METHOD ret = {
1623 EC_FLAGS_DEFAULT_OCT,
1624 NID_X9_62_prime_field,
1625 ec_GFp_nistp521_group_init,
1626 ec_GFp_simple_group_finish,
1627 ec_GFp_simple_group_clear_finish,
1628 ec_GFp_nist_group_copy,
1629 ec_GFp_nistp521_group_set_curve,
1630 ec_GFp_simple_group_get_curve,
1631 ec_GFp_simple_group_get_degree,
1632 ec_group_simple_order_bits,
1633 ec_GFp_simple_group_check_discriminant,
1634 ec_GFp_simple_point_init,
1635 ec_GFp_simple_point_finish,
1636 ec_GFp_simple_point_clear_finish,
1637 ec_GFp_simple_point_copy,
1638 ec_GFp_simple_point_set_to_infinity,
1639 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1640 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1641 ec_GFp_simple_point_set_affine_coordinates,
1642 ec_GFp_nistp521_point_get_affine_coordinates,
1643 0 /* point_set_compressed_coordinates */ ,
1648 ec_GFp_simple_invert,
1649 ec_GFp_simple_is_at_infinity,
1650 ec_GFp_simple_is_on_curve,
1652 ec_GFp_simple_make_affine,
1653 ec_GFp_simple_points_make_affine,
1654 ec_GFp_nistp521_points_mul,
1655 ec_GFp_nistp521_precompute_mult,
1656 ec_GFp_nistp521_have_precompute_mult,
1657 ec_GFp_nist_field_mul,
1658 ec_GFp_nist_field_sqr,
1660 ec_GFp_simple_field_inv,
1661 0 /* field_encode */ ,
1662 0 /* field_decode */ ,
1663 0, /* field_set_to_one */
1664 ec_key_simple_priv2oct,
1665 ec_key_simple_oct2priv,
1666 0, /* set private */
1667 ec_key_simple_generate_key,
1668 ec_key_simple_check_key,
1669 ec_key_simple_generate_public_key,
1672 ecdh_simple_compute_key,
1673 0, /* field_inverse_mod_ord */
1674 0, /* blind_coordinates */
1676 0, /* ladder_step */
1683 /******************************************************************************/
1685 * FUNCTIONS TO MANAGE PRECOMPUTATION
1688 static NISTP521_PRE_COMP *nistp521_pre_comp_new(void)
1690 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1693 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1697 ret->references = 1;
1699 ret->lock = CRYPTO_THREAD_lock_new();
1700 if (ret->lock == NULL) {
1701 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1708 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1712 CRYPTO_UP_REF(&p->references, &i, p->lock);
1716 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1723 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1724 REF_PRINT_COUNT("EC_nistp521", x);
1727 REF_ASSERT_ISNT(i < 0);
1729 CRYPTO_THREAD_lock_free(p->lock);
1733 /******************************************************************************/
1735 * OPENSSL EC_METHOD FUNCTIONS
1738 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1741 ret = ec_GFp_simple_group_init(group);
1742 group->a_is_minus3 = 1;
1746 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1747 const BIGNUM *a, const BIGNUM *b,
1751 BN_CTX *new_ctx = NULL;
1752 BIGNUM *curve_p, *curve_a, *curve_b;
1755 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1758 curve_p = BN_CTX_get(ctx);
1759 curve_a = BN_CTX_get(ctx);
1760 curve_b = BN_CTX_get(ctx);
1761 if (curve_b == NULL)
1763 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1764 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1765 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1766 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1767 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1768 EC_R_WRONG_CURVE_PARAMETERS);
1771 group->field_mod_func = BN_nist_mod_521;
1772 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1775 BN_CTX_free(new_ctx);
1780 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1783 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1784 const EC_POINT *point,
1785 BIGNUM *x, BIGNUM *y,
1788 felem z1, z2, x_in, y_in, x_out, y_out;
1791 if (EC_POINT_is_at_infinity(group, point)) {
1792 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1793 EC_R_POINT_AT_INFINITY);
1796 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1797 (!BN_to_felem(z1, point->Z)))
1800 felem_square(tmp, z2);
1801 felem_reduce(z1, tmp);
1802 felem_mul(tmp, x_in, z1);
1803 felem_reduce(x_in, tmp);
1804 felem_contract(x_out, x_in);
1806 if (!felem_to_BN(x, x_out)) {
1807 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1812 felem_mul(tmp, z1, z2);
1813 felem_reduce(z1, tmp);
1814 felem_mul(tmp, y_in, z1);
1815 felem_reduce(y_in, tmp);
1816 felem_contract(y_out, y_in);
1818 if (!felem_to_BN(y, y_out)) {
1819 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1827 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1828 static void make_points_affine(size_t num, felem points[][3],
1832 * Runs in constant time, unless an input is the point at infinity (which
1833 * normally shouldn't happen).
1835 ec_GFp_nistp_points_make_affine_internal(num,
1839 (void (*)(void *))felem_one,
1841 (void (*)(void *, const void *))
1843 (void (*)(void *, const void *))
1844 felem_square_reduce, (void (*)
1851 (void (*)(void *, const void *))
1853 (void (*)(void *, const void *))
1858 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1859 * values Result is stored in r (r can equal one of the inputs).
1861 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1862 const BIGNUM *scalar, size_t num,
1863 const EC_POINT *points[],
1864 const BIGNUM *scalars[], BN_CTX *ctx)
1869 BIGNUM *x, *y, *z, *tmp_scalar;
1870 felem_bytearray g_secret;
1871 felem_bytearray *secrets = NULL;
1872 felem (*pre_comp)[17][3] = NULL;
1873 felem *tmp_felems = NULL;
1876 int have_pre_comp = 0;
1877 size_t num_points = num;
1878 felem x_in, y_in, z_in, x_out, y_out, z_out;
1879 NISTP521_PRE_COMP *pre = NULL;
1880 felem(*g_pre_comp)[3] = NULL;
1881 EC_POINT *generator = NULL;
1882 const EC_POINT *p = NULL;
1883 const BIGNUM *p_scalar = NULL;
1886 x = BN_CTX_get(ctx);
1887 y = BN_CTX_get(ctx);
1888 z = BN_CTX_get(ctx);
1889 tmp_scalar = BN_CTX_get(ctx);
1890 if (tmp_scalar == NULL)
1893 if (scalar != NULL) {
1894 pre = group->pre_comp.nistp521;
1896 /* we have precomputation, try to use it */
1897 g_pre_comp = &pre->g_pre_comp[0];
1899 /* try to use the standard precomputation */
1900 g_pre_comp = (felem(*)[3]) gmul;
1901 generator = EC_POINT_new(group);
1902 if (generator == NULL)
1904 /* get the generator from precomputation */
1905 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1906 !felem_to_BN(y, g_pre_comp[1][1]) ||
1907 !felem_to_BN(z, g_pre_comp[1][2])) {
1908 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1911 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1915 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1916 /* precomputation matches generator */
1920 * we don't have valid precomputation: treat the generator as a
1926 if (num_points > 0) {
1927 if (num_points >= 2) {
1929 * unless we precompute multiples for just one point, converting
1930 * those into affine form is time well spent
1934 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1935 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1938 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1939 if ((secrets == NULL) || (pre_comp == NULL)
1940 || (mixed && (tmp_felems == NULL))) {
1941 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1946 * we treat NULL scalars as 0, and NULL points as points at infinity,
1947 * i.e., they contribute nothing to the linear combination
1949 for (i = 0; i < num_points; ++i) {
1952 * we didn't have a valid precomputation, so we pick the
1955 p = EC_GROUP_get0_generator(group);
1958 /* the i^th point */
1960 p_scalar = scalars[i];
1962 if ((p_scalar != NULL) && (p != NULL)) {
1963 /* reduce scalar to 0 <= scalar < 2^521 */
1964 if ((BN_num_bits(p_scalar) > 521)
1965 || (BN_is_negative(p_scalar))) {
1967 * this is an unusual input, and we don't guarantee
1970 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1971 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1974 num_bytes = BN_bn2lebinpad(tmp_scalar,
1975 secrets[i], sizeof(secrets[i]));
1977 num_bytes = BN_bn2lebinpad(p_scalar,
1978 secrets[i], sizeof(secrets[i]));
1980 if (num_bytes < 0) {
1981 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1984 /* precompute multiples */
1985 if ((!BN_to_felem(x_out, p->X)) ||
1986 (!BN_to_felem(y_out, p->Y)) ||
1987 (!BN_to_felem(z_out, p->Z)))
1989 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1990 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1991 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1992 for (j = 2; j <= 16; ++j) {
1994 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1995 pre_comp[i][j][2], pre_comp[i][1][0],
1996 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1997 pre_comp[i][j - 1][0],
1998 pre_comp[i][j - 1][1],
1999 pre_comp[i][j - 1][2]);
2001 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
2002 pre_comp[i][j][2], pre_comp[i][j / 2][0],
2003 pre_comp[i][j / 2][1],
2004 pre_comp[i][j / 2][2]);
2010 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
2013 /* the scalar for the generator */
2014 if ((scalar != NULL) && (have_pre_comp)) {
2015 memset(g_secret, 0, sizeof(g_secret));
2016 /* reduce scalar to 0 <= scalar < 2^521 */
2017 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
2019 * this is an unusual input, and we don't guarantee
2022 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
2023 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2026 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
2028 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
2030 /* do the multiplication with generator precomputation */
2031 batch_mul(x_out, y_out, z_out,
2032 (const felem_bytearray(*))secrets, num_points,
2034 mixed, (const felem(*)[17][3])pre_comp,
2035 (const felem(*)[3])g_pre_comp);
2037 /* do the multiplication without generator precomputation */
2038 batch_mul(x_out, y_out, z_out,
2039 (const felem_bytearray(*))secrets, num_points,
2040 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2042 /* reduce the output to its unique minimal representation */
2043 felem_contract(x_in, x_out);
2044 felem_contract(y_in, y_out);
2045 felem_contract(z_in, z_out);
2046 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2047 (!felem_to_BN(z, z_in))) {
2048 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2051 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2055 EC_POINT_free(generator);
2056 OPENSSL_free(secrets);
2057 OPENSSL_free(pre_comp);
2058 OPENSSL_free(tmp_felems);
2062 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2065 NISTP521_PRE_COMP *pre = NULL;
2067 BN_CTX *new_ctx = NULL;
2069 EC_POINT *generator = NULL;
2070 felem tmp_felems[16];
2072 /* throw away old precomputation */
2073 EC_pre_comp_free(group);
2075 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2078 x = BN_CTX_get(ctx);
2079 y = BN_CTX_get(ctx);
2082 /* get the generator */
2083 if (group->generator == NULL)
2085 generator = EC_POINT_new(group);
2086 if (generator == NULL)
2088 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2089 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2090 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
2092 if ((pre = nistp521_pre_comp_new()) == NULL)
2095 * if the generator is the standard one, use built-in precomputation
2097 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2098 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2101 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2102 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2103 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2105 /* compute 2^130*G, 2^260*G, 2^390*G */
2106 for (i = 1; i <= 4; i <<= 1) {
2107 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2108 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2109 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2110 for (j = 0; j < 129; ++j) {
2111 point_double(pre->g_pre_comp[2 * i][0],
2112 pre->g_pre_comp[2 * i][1],
2113 pre->g_pre_comp[2 * i][2],
2114 pre->g_pre_comp[2 * i][0],
2115 pre->g_pre_comp[2 * i][1],
2116 pre->g_pre_comp[2 * i][2]);
2119 /* g_pre_comp[0] is the point at infinity */
2120 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2121 /* the remaining multiples */
2122 /* 2^130*G + 2^260*G */
2123 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2124 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2125 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2126 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2127 pre->g_pre_comp[2][2]);
2128 /* 2^130*G + 2^390*G */
2129 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2130 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2131 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2132 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2133 pre->g_pre_comp[2][2]);
2134 /* 2^260*G + 2^390*G */
2135 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2136 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2137 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2138 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2139 pre->g_pre_comp[4][2]);
2140 /* 2^130*G + 2^260*G + 2^390*G */
2141 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2142 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2143 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2144 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2145 pre->g_pre_comp[2][2]);
2146 for (i = 1; i < 8; ++i) {
2147 /* odd multiples: add G */
2148 point_add(pre->g_pre_comp[2 * i + 1][0],
2149 pre->g_pre_comp[2 * i + 1][1],
2150 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2151 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2152 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2153 pre->g_pre_comp[1][2]);
2155 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2158 SETPRECOMP(group, nistp521, pre);
2163 EC_POINT_free(generator);
2164 BN_CTX_free(new_ctx);
2165 EC_nistp521_pre_comp_free(pre);
2169 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2171 return HAVEPRECOMP(group, nistp521);