2 * Copyright 2011-2016 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>
43 # if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
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 "Need GCC 3.1 or later to define type uint128_t"
56 * The underlying field. P521 operates over GF(2^521-1). We can serialise 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 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 *) & in[7]) >> 2) & bottom58bits;
146 out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits;
147 out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits;
148 out[4] = (*((limb *) & in[29])) & bottom58bits;
149 out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits;
150 out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits;
151 out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits;
152 out[8] = (*((limb *) & 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 *) & out[7])) |= in[1] << 2;
164 (*((limb *) & out[14])) |= in[2] << 4;
165 (*((limb *) & out[21])) |= in[3] << 6;
166 (*((limb *) & out[29])) = in[4];
167 (*((limb *) & out[36])) |= in[5] << 2;
168 (*((limb *) & out[43])) |= in[6] << 4;
169 (*((limb *) & out[50])) |= in[7] << 6;
170 (*((limb *) & out[58])) = in[8];
173 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
174 static void flip_endian(u8 *out, const u8 *in, unsigned len)
177 for (i = 0; i < len; ++i)
178 out[i] = in[len - 1 - i];
181 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
182 static int BN_to_felem(felem out, const BIGNUM *bn)
184 felem_bytearray b_in;
185 felem_bytearray b_out;
188 /* BN_bn2bin eats leading zeroes */
189 memset(b_out, 0, sizeof(b_out));
190 num_bytes = BN_num_bytes(bn);
191 if (num_bytes > sizeof b_out) {
192 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
195 if (BN_is_negative(bn)) {
196 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
199 num_bytes = BN_bn2bin(bn, b_in);
200 flip_endian(b_out, b_in, num_bytes);
201 bin66_to_felem(out, b_out);
205 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
206 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
208 felem_bytearray b_in, b_out;
209 felem_to_bin66(b_in, in);
210 flip_endian(b_out, b_in, sizeof b_out);
211 return BN_bin2bn(b_out, sizeof b_out, out);
219 static void felem_one(felem out)
232 static void felem_assign(felem out, const felem in)
245 /* felem_sum64 sets out = out + in. */
246 static void felem_sum64(felem out, const felem in)
259 /* felem_scalar sets out = in * scalar */
260 static void felem_scalar(felem out, const felem in, limb scalar)
262 out[0] = in[0] * scalar;
263 out[1] = in[1] * scalar;
264 out[2] = in[2] * scalar;
265 out[3] = in[3] * scalar;
266 out[4] = in[4] * scalar;
267 out[5] = in[5] * scalar;
268 out[6] = in[6] * scalar;
269 out[7] = in[7] * scalar;
270 out[8] = in[8] * scalar;
273 /* felem_scalar64 sets out = out * scalar */
274 static void felem_scalar64(felem out, limb scalar)
287 /* felem_scalar128 sets out = out * scalar */
288 static void felem_scalar128(largefelem out, limb scalar)
302 * felem_neg sets |out| to |-in|
304 * in[i] < 2^59 + 2^14
308 static void felem_neg(felem out, const felem in)
310 /* In order to prevent underflow, we subtract from 0 mod p. */
311 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
312 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
314 out[0] = two62m3 - in[0];
315 out[1] = two62m2 - in[1];
316 out[2] = two62m2 - in[2];
317 out[3] = two62m2 - in[3];
318 out[4] = two62m2 - in[4];
319 out[5] = two62m2 - in[5];
320 out[6] = two62m2 - in[6];
321 out[7] = two62m2 - in[7];
322 out[8] = two62m2 - in[8];
326 * felem_diff64 subtracts |in| from |out|
328 * in[i] < 2^59 + 2^14
330 * out[i] < out[i] + 2^62
332 static void felem_diff64(felem out, const felem in)
335 * In order to prevent underflow, we add 0 mod p before subtracting.
337 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
338 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
340 out[0] += two62m3 - in[0];
341 out[1] += two62m2 - in[1];
342 out[2] += two62m2 - in[2];
343 out[3] += two62m2 - in[3];
344 out[4] += two62m2 - in[4];
345 out[5] += two62m2 - in[5];
346 out[6] += two62m2 - in[6];
347 out[7] += two62m2 - in[7];
348 out[8] += two62m2 - in[8];
352 * felem_diff_128_64 subtracts |in| from |out|
354 * in[i] < 2^62 + 2^17
356 * out[i] < out[i] + 2^63
358 static void felem_diff_128_64(largefelem out, const felem in)
361 * In order to prevent underflow, we add 0 mod p before subtracting.
363 static const limb two63m6 = (((limb) 1) << 62) - (((limb) 1) << 5);
364 static const limb two63m5 = (((limb) 1) << 62) - (((limb) 1) << 4);
366 out[0] += two63m6 - in[0];
367 out[1] += two63m5 - in[1];
368 out[2] += two63m5 - in[2];
369 out[3] += two63m5 - in[3];
370 out[4] += two63m5 - in[4];
371 out[5] += two63m5 - in[5];
372 out[6] += two63m5 - in[6];
373 out[7] += two63m5 - in[7];
374 out[8] += two63m5 - in[8];
378 * felem_diff_128_64 subtracts |in| from |out|
382 * out[i] < out[i] + 2^127 - 2^69
384 static void felem_diff128(largefelem out, const largefelem in)
387 * In order to prevent underflow, we add 0 mod p before subtracting.
389 static const uint128_t two127m70 =
390 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
391 static const uint128_t two127m69 =
392 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
394 out[0] += (two127m70 - in[0]);
395 out[1] += (two127m69 - in[1]);
396 out[2] += (two127m69 - in[2]);
397 out[3] += (two127m69 - in[3]);
398 out[4] += (two127m69 - in[4]);
399 out[5] += (two127m69 - in[5]);
400 out[6] += (two127m69 - in[6]);
401 out[7] += (two127m69 - in[7]);
402 out[8] += (two127m69 - in[8]);
406 * felem_square sets |out| = |in|^2
410 * out[i] < 17 * max(in[i]) * max(in[i])
412 static void felem_square(largefelem out, const felem in)
415 felem_scalar(inx2, in, 2);
416 felem_scalar(inx4, in, 4);
419 * We have many cases were we want to do
422 * This is obviously just
424 * However, rather than do the doubling on the 128 bit result, we
425 * double one of the inputs to the multiplication by reading from
429 out[0] = ((uint128_t) in[0]) * in[0];
430 out[1] = ((uint128_t) in[0]) * inx2[1];
431 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
432 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
433 out[4] = ((uint128_t) in[0]) * inx2[4] +
434 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
435 out[5] = ((uint128_t) in[0]) * inx2[5] +
436 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
437 out[6] = ((uint128_t) in[0]) * inx2[6] +
438 ((uint128_t) in[1]) * inx2[5] +
439 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
440 out[7] = ((uint128_t) in[0]) * inx2[7] +
441 ((uint128_t) in[1]) * inx2[6] +
442 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
443 out[8] = ((uint128_t) in[0]) * inx2[8] +
444 ((uint128_t) in[1]) * inx2[7] +
445 ((uint128_t) in[2]) * inx2[6] +
446 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
449 * The remaining limbs fall above 2^521, with the first falling at 2^522.
450 * They correspond to locations one bit up from the limbs produced above
451 * so we would have to multiply by two to align them. Again, rather than
452 * operate on the 128-bit result, we double one of the inputs to the
453 * multiplication. If we want to double for both this reason, and the
454 * reason above, then we end up multiplying by four.
458 out[0] += ((uint128_t) in[1]) * inx4[8] +
459 ((uint128_t) in[2]) * inx4[7] +
460 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
463 out[1] += ((uint128_t) in[2]) * inx4[8] +
464 ((uint128_t) in[3]) * inx4[7] +
465 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
468 out[2] += ((uint128_t) in[3]) * inx4[8] +
469 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
472 out[3] += ((uint128_t) in[4]) * inx4[8] +
473 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
476 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
479 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
482 out[6] += ((uint128_t) in[7]) * inx4[8];
485 out[7] += ((uint128_t) in[8]) * inx2[8];
489 * felem_mul sets |out| = |in1| * |in2|
494 * out[i] < 17 * max(in1[i]) * max(in2[i])
496 static void felem_mul(largefelem out, const felem in1, const felem in2)
499 felem_scalar(in2x2, in2, 2);
501 out[0] = ((uint128_t) in1[0]) * in2[0];
503 out[1] = ((uint128_t) in1[0]) * in2[1] +
504 ((uint128_t) in1[1]) * in2[0];
506 out[2] = ((uint128_t) in1[0]) * in2[2] +
507 ((uint128_t) in1[1]) * in2[1] +
508 ((uint128_t) in1[2]) * in2[0];
510 out[3] = ((uint128_t) in1[0]) * in2[3] +
511 ((uint128_t) in1[1]) * in2[2] +
512 ((uint128_t) in1[2]) * in2[1] +
513 ((uint128_t) in1[3]) * in2[0];
515 out[4] = ((uint128_t) in1[0]) * in2[4] +
516 ((uint128_t) in1[1]) * in2[3] +
517 ((uint128_t) in1[2]) * in2[2] +
518 ((uint128_t) in1[3]) * in2[1] +
519 ((uint128_t) in1[4]) * in2[0];
521 out[5] = ((uint128_t) in1[0]) * in2[5] +
522 ((uint128_t) in1[1]) * in2[4] +
523 ((uint128_t) in1[2]) * in2[3] +
524 ((uint128_t) in1[3]) * in2[2] +
525 ((uint128_t) in1[4]) * in2[1] +
526 ((uint128_t) in1[5]) * in2[0];
528 out[6] = ((uint128_t) in1[0]) * in2[6] +
529 ((uint128_t) in1[1]) * in2[5] +
530 ((uint128_t) in1[2]) * in2[4] +
531 ((uint128_t) in1[3]) * in2[3] +
532 ((uint128_t) in1[4]) * in2[2] +
533 ((uint128_t) in1[5]) * in2[1] +
534 ((uint128_t) in1[6]) * in2[0];
536 out[7] = ((uint128_t) in1[0]) * in2[7] +
537 ((uint128_t) in1[1]) * in2[6] +
538 ((uint128_t) in1[2]) * in2[5] +
539 ((uint128_t) in1[3]) * in2[4] +
540 ((uint128_t) in1[4]) * in2[3] +
541 ((uint128_t) in1[5]) * in2[2] +
542 ((uint128_t) in1[6]) * in2[1] +
543 ((uint128_t) in1[7]) * in2[0];
545 out[8] = ((uint128_t) in1[0]) * in2[8] +
546 ((uint128_t) in1[1]) * in2[7] +
547 ((uint128_t) in1[2]) * in2[6] +
548 ((uint128_t) in1[3]) * in2[5] +
549 ((uint128_t) in1[4]) * in2[4] +
550 ((uint128_t) in1[5]) * in2[3] +
551 ((uint128_t) in1[6]) * in2[2] +
552 ((uint128_t) in1[7]) * in2[1] +
553 ((uint128_t) in1[8]) * in2[0];
555 /* See comment in felem_square about the use of in2x2 here */
557 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
558 ((uint128_t) in1[2]) * in2x2[7] +
559 ((uint128_t) in1[3]) * in2x2[6] +
560 ((uint128_t) in1[4]) * in2x2[5] +
561 ((uint128_t) in1[5]) * in2x2[4] +
562 ((uint128_t) in1[6]) * in2x2[3] +
563 ((uint128_t) in1[7]) * in2x2[2] +
564 ((uint128_t) in1[8]) * in2x2[1];
566 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
567 ((uint128_t) in1[3]) * in2x2[7] +
568 ((uint128_t) in1[4]) * in2x2[6] +
569 ((uint128_t) in1[5]) * in2x2[5] +
570 ((uint128_t) in1[6]) * in2x2[4] +
571 ((uint128_t) in1[7]) * in2x2[3] +
572 ((uint128_t) in1[8]) * in2x2[2];
574 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
575 ((uint128_t) in1[4]) * in2x2[7] +
576 ((uint128_t) in1[5]) * in2x2[6] +
577 ((uint128_t) in1[6]) * in2x2[5] +
578 ((uint128_t) in1[7]) * in2x2[4] +
579 ((uint128_t) in1[8]) * in2x2[3];
581 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
582 ((uint128_t) in1[5]) * in2x2[7] +
583 ((uint128_t) in1[6]) * in2x2[6] +
584 ((uint128_t) in1[7]) * in2x2[5] +
585 ((uint128_t) in1[8]) * in2x2[4];
587 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
588 ((uint128_t) in1[6]) * in2x2[7] +
589 ((uint128_t) in1[7]) * in2x2[6] +
590 ((uint128_t) in1[8]) * in2x2[5];
592 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
593 ((uint128_t) in1[7]) * in2x2[7] +
594 ((uint128_t) in1[8]) * in2x2[6];
596 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
597 ((uint128_t) in1[8]) * in2x2[7];
599 out[7] += ((uint128_t) in1[8]) * in2x2[8];
602 static const limb bottom52bits = 0xfffffffffffff;
605 * felem_reduce converts a largefelem to an felem.
609 * out[i] < 2^59 + 2^14
611 static void felem_reduce(felem out, const largefelem in)
613 u64 overflow1, overflow2;
615 out[0] = ((limb) in[0]) & bottom58bits;
616 out[1] = ((limb) in[1]) & bottom58bits;
617 out[2] = ((limb) in[2]) & bottom58bits;
618 out[3] = ((limb) in[3]) & bottom58bits;
619 out[4] = ((limb) in[4]) & bottom58bits;
620 out[5] = ((limb) in[5]) & bottom58bits;
621 out[6] = ((limb) in[6]) & bottom58bits;
622 out[7] = ((limb) in[7]) & bottom58bits;
623 out[8] = ((limb) in[8]) & bottom58bits;
627 out[1] += ((limb) in[0]) >> 58;
628 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
630 * out[1] < 2^58 + 2^6 + 2^58
633 out[2] += ((limb) (in[0] >> 64)) >> 52;
635 out[2] += ((limb) in[1]) >> 58;
636 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
637 out[3] += ((limb) (in[1] >> 64)) >> 52;
639 out[3] += ((limb) in[2]) >> 58;
640 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
641 out[4] += ((limb) (in[2] >> 64)) >> 52;
643 out[4] += ((limb) in[3]) >> 58;
644 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
645 out[5] += ((limb) (in[3] >> 64)) >> 52;
647 out[5] += ((limb) in[4]) >> 58;
648 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
649 out[6] += ((limb) (in[4] >> 64)) >> 52;
651 out[6] += ((limb) in[5]) >> 58;
652 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
653 out[7] += ((limb) (in[5] >> 64)) >> 52;
655 out[7] += ((limb) in[6]) >> 58;
656 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
657 out[8] += ((limb) (in[6] >> 64)) >> 52;
659 out[8] += ((limb) in[7]) >> 58;
660 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
662 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
665 overflow1 = ((limb) (in[7] >> 64)) >> 52;
667 overflow1 += ((limb) in[8]) >> 58;
668 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
669 overflow2 = ((limb) (in[8] >> 64)) >> 52;
671 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
672 overflow2 <<= 1; /* overflow2 < 2^13 */
674 out[0] += overflow1; /* out[0] < 2^60 */
675 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
677 out[1] += out[0] >> 58;
678 out[0] &= bottom58bits;
681 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
686 static void felem_square_reduce(felem out, const felem in)
689 felem_square(tmp, in);
690 felem_reduce(out, tmp);
693 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
696 felem_mul(tmp, in1, in2);
697 felem_reduce(out, tmp);
701 * felem_inv calculates |out| = |in|^{-1}
703 * Based on Fermat's Little Theorem:
705 * a^{p-1} = 1 (mod p)
706 * a^{p-2} = a^{-1} (mod p)
708 static void felem_inv(felem out, const felem in)
710 felem ftmp, ftmp2, ftmp3, ftmp4;
714 felem_square(tmp, in);
715 felem_reduce(ftmp, tmp); /* 2^1 */
716 felem_mul(tmp, in, ftmp);
717 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
718 felem_assign(ftmp2, ftmp);
719 felem_square(tmp, ftmp);
720 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
721 felem_mul(tmp, in, ftmp);
722 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
723 felem_square(tmp, ftmp);
724 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
726 felem_square(tmp, ftmp2);
727 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
728 felem_square(tmp, ftmp3);
729 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
730 felem_mul(tmp, ftmp3, ftmp2);
731 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
733 felem_assign(ftmp2, ftmp3);
734 felem_square(tmp, ftmp3);
735 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
736 felem_square(tmp, ftmp3);
737 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
738 felem_square(tmp, ftmp3);
739 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
740 felem_square(tmp, ftmp3);
741 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
742 felem_assign(ftmp4, ftmp3);
743 felem_mul(tmp, ftmp3, ftmp);
744 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
745 felem_square(tmp, ftmp4);
746 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
747 felem_mul(tmp, ftmp3, ftmp2);
748 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
749 felem_assign(ftmp2, ftmp3);
751 for (i = 0; i < 8; i++) {
752 felem_square(tmp, ftmp3);
753 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
755 felem_mul(tmp, ftmp3, ftmp2);
756 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
757 felem_assign(ftmp2, ftmp3);
759 for (i = 0; i < 16; i++) {
760 felem_square(tmp, ftmp3);
761 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
763 felem_mul(tmp, ftmp3, ftmp2);
764 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
765 felem_assign(ftmp2, ftmp3);
767 for (i = 0; i < 32; i++) {
768 felem_square(tmp, ftmp3);
769 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
771 felem_mul(tmp, ftmp3, ftmp2);
772 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
773 felem_assign(ftmp2, ftmp3);
775 for (i = 0; i < 64; i++) {
776 felem_square(tmp, ftmp3);
777 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
779 felem_mul(tmp, ftmp3, ftmp2);
780 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
781 felem_assign(ftmp2, ftmp3);
783 for (i = 0; i < 128; i++) {
784 felem_square(tmp, ftmp3);
785 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
787 felem_mul(tmp, ftmp3, ftmp2);
788 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
789 felem_assign(ftmp2, ftmp3);
791 for (i = 0; i < 256; i++) {
792 felem_square(tmp, ftmp3);
793 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
795 felem_mul(tmp, ftmp3, ftmp2);
796 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
798 for (i = 0; i < 9; i++) {
799 felem_square(tmp, ftmp3);
800 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
802 felem_mul(tmp, ftmp3, ftmp4);
803 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
804 felem_mul(tmp, ftmp3, in);
805 felem_reduce(out, tmp); /* 2^512 - 3 */
808 /* This is 2^521-1, expressed as an felem */
809 static const felem kPrime = {
810 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
811 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
812 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
816 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
819 * in[i] < 2^59 + 2^14
821 static limb felem_is_zero(const felem in)
825 felem_assign(ftmp, in);
827 ftmp[0] += ftmp[8] >> 57;
828 ftmp[8] &= bottom57bits;
830 ftmp[1] += ftmp[0] >> 58;
831 ftmp[0] &= bottom58bits;
832 ftmp[2] += ftmp[1] >> 58;
833 ftmp[1] &= bottom58bits;
834 ftmp[3] += ftmp[2] >> 58;
835 ftmp[2] &= bottom58bits;
836 ftmp[4] += ftmp[3] >> 58;
837 ftmp[3] &= bottom58bits;
838 ftmp[5] += ftmp[4] >> 58;
839 ftmp[4] &= bottom58bits;
840 ftmp[6] += ftmp[5] >> 58;
841 ftmp[5] &= bottom58bits;
842 ftmp[7] += ftmp[6] >> 58;
843 ftmp[6] &= bottom58bits;
844 ftmp[8] += ftmp[7] >> 58;
845 ftmp[7] &= bottom58bits;
846 /* ftmp[8] < 2^57 + 4 */
849 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
850 * than our bound for ftmp[8]. Therefore we only have to check if the
851 * zero is zero or 2^521-1.
867 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
868 * can be set is if is_zero was 0 before the decrement.
870 is_zero = ((s64) is_zero) >> 63;
872 is_p = ftmp[0] ^ kPrime[0];
873 is_p |= ftmp[1] ^ kPrime[1];
874 is_p |= ftmp[2] ^ kPrime[2];
875 is_p |= ftmp[3] ^ kPrime[3];
876 is_p |= ftmp[4] ^ kPrime[4];
877 is_p |= ftmp[5] ^ kPrime[5];
878 is_p |= ftmp[6] ^ kPrime[6];
879 is_p |= ftmp[7] ^ kPrime[7];
880 is_p |= ftmp[8] ^ kPrime[8];
883 is_p = ((s64) is_p) >> 63;
889 static int felem_is_zero_int(const felem in)
891 return (int)(felem_is_zero(in) & ((limb) 1));
895 * felem_contract converts |in| to its unique, minimal representation.
897 * in[i] < 2^59 + 2^14
899 static void felem_contract(felem out, const felem in)
901 limb is_p, is_greater, sign;
902 static const limb two58 = ((limb) 1) << 58;
904 felem_assign(out, in);
906 out[0] += out[8] >> 57;
907 out[8] &= bottom57bits;
909 out[1] += out[0] >> 58;
910 out[0] &= bottom58bits;
911 out[2] += out[1] >> 58;
912 out[1] &= bottom58bits;
913 out[3] += out[2] >> 58;
914 out[2] &= bottom58bits;
915 out[4] += out[3] >> 58;
916 out[3] &= bottom58bits;
917 out[5] += out[4] >> 58;
918 out[4] &= bottom58bits;
919 out[6] += out[5] >> 58;
920 out[5] &= bottom58bits;
921 out[7] += out[6] >> 58;
922 out[6] &= bottom58bits;
923 out[8] += out[7] >> 58;
924 out[7] &= bottom58bits;
925 /* out[8] < 2^57 + 4 */
928 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
929 * out. See the comments in felem_is_zero regarding why we don't test for
930 * other multiples of the prime.
934 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
937 is_p = out[0] ^ kPrime[0];
938 is_p |= out[1] ^ kPrime[1];
939 is_p |= out[2] ^ kPrime[2];
940 is_p |= out[3] ^ kPrime[3];
941 is_p |= out[4] ^ kPrime[4];
942 is_p |= out[5] ^ kPrime[5];
943 is_p |= out[6] ^ kPrime[6];
944 is_p |= out[7] ^ kPrime[7];
945 is_p |= out[8] ^ kPrime[8];
954 is_p = ((s64) is_p) >> 63;
957 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
970 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
971 * 57 is greater than zero as (2^521-1) + x >= 2^522
973 is_greater = out[8] >> 57;
974 is_greater |= is_greater << 32;
975 is_greater |= is_greater << 16;
976 is_greater |= is_greater << 8;
977 is_greater |= is_greater << 4;
978 is_greater |= is_greater << 2;
979 is_greater |= is_greater << 1;
980 is_greater = ((s64) is_greater) >> 63;
982 out[0] -= kPrime[0] & is_greater;
983 out[1] -= kPrime[1] & is_greater;
984 out[2] -= kPrime[2] & is_greater;
985 out[3] -= kPrime[3] & is_greater;
986 out[4] -= kPrime[4] & is_greater;
987 out[5] -= kPrime[5] & is_greater;
988 out[6] -= kPrime[6] & is_greater;
989 out[7] -= kPrime[7] & is_greater;
990 out[8] -= kPrime[8] & is_greater;
992 /* Eliminate negative coefficients */
993 sign = -(out[0] >> 63);
994 out[0] += (two58 & sign);
995 out[1] -= (1 & sign);
996 sign = -(out[1] >> 63);
997 out[1] += (two58 & sign);
998 out[2] -= (1 & sign);
999 sign = -(out[2] >> 63);
1000 out[2] += (two58 & sign);
1001 out[3] -= (1 & sign);
1002 sign = -(out[3] >> 63);
1003 out[3] += (two58 & sign);
1004 out[4] -= (1 & sign);
1005 sign = -(out[4] >> 63);
1006 out[4] += (two58 & sign);
1007 out[5] -= (1 & sign);
1008 sign = -(out[0] >> 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);
1017 sign = -(out[5] >> 63);
1018 out[5] += (two58 & sign);
1019 out[6] -= (1 & sign);
1020 sign = -(out[6] >> 63);
1021 out[6] += (two58 & sign);
1022 out[7] -= (1 & sign);
1023 sign = -(out[7] >> 63);
1024 out[7] += (two58 & sign);
1025 out[8] -= (1 & sign);
1032 * Building on top of the field operations we have the operations on the
1033 * elliptic curve group itself. Points on the curve are represented in Jacobian
1037 * point_double calculates 2*(x_in, y_in, z_in)
1039 * The method is taken from:
1040 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1042 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1043 * while x_out == y_in is not (maybe this works, but it's not tested). */
1045 point_double(felem x_out, felem y_out, felem z_out,
1046 const felem x_in, const felem y_in, const felem z_in)
1048 largefelem tmp, tmp2;
1049 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1051 felem_assign(ftmp, x_in);
1052 felem_assign(ftmp2, x_in);
1055 felem_square(tmp, z_in);
1056 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1059 felem_square(tmp, y_in);
1060 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1062 /* beta = x*gamma */
1063 felem_mul(tmp, x_in, gamma);
1064 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1066 /* alpha = 3*(x-delta)*(x+delta) */
1067 felem_diff64(ftmp, delta);
1068 /* ftmp[i] < 2^61 */
1069 felem_sum64(ftmp2, delta);
1070 /* ftmp2[i] < 2^60 + 2^15 */
1071 felem_scalar64(ftmp2, 3);
1072 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1073 felem_mul(tmp, ftmp, ftmp2);
1075 * tmp[i] < 17(3*2^121 + 3*2^76)
1076 * = 61*2^121 + 61*2^76
1077 * < 64*2^121 + 64*2^76
1081 felem_reduce(alpha, tmp);
1083 /* x' = alpha^2 - 8*beta */
1084 felem_square(tmp, alpha);
1086 * tmp[i] < 17*2^120 < 2^125
1088 felem_assign(ftmp, beta);
1089 felem_scalar64(ftmp, 8);
1090 /* ftmp[i] < 2^62 + 2^17 */
1091 felem_diff_128_64(tmp, ftmp);
1092 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1093 felem_reduce(x_out, tmp);
1095 /* z' = (y + z)^2 - gamma - delta */
1096 felem_sum64(delta, gamma);
1097 /* delta[i] < 2^60 + 2^15 */
1098 felem_assign(ftmp, y_in);
1099 felem_sum64(ftmp, z_in);
1100 /* ftmp[i] < 2^60 + 2^15 */
1101 felem_square(tmp, ftmp);
1103 * tmp[i] < 17(2^122) < 2^127
1105 felem_diff_128_64(tmp, delta);
1106 /* tmp[i] < 2^127 + 2^63 */
1107 felem_reduce(z_out, tmp);
1109 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1110 felem_scalar64(beta, 4);
1111 /* beta[i] < 2^61 + 2^16 */
1112 felem_diff64(beta, x_out);
1113 /* beta[i] < 2^61 + 2^60 + 2^16 */
1114 felem_mul(tmp, alpha, beta);
1116 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1117 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1118 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1121 felem_square(tmp2, gamma);
1123 * tmp2[i] < 17*(2^59 + 2^14)^2
1124 * = 17*(2^118 + 2^74 + 2^28)
1126 felem_scalar128(tmp2, 8);
1128 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1129 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1132 felem_diff128(tmp, tmp2);
1134 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1135 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1136 * 2^74 + 2^69 + 2^34 + 2^30
1139 felem_reduce(y_out, tmp);
1142 /* copy_conditional copies in to out iff mask is all ones. */
1143 static void copy_conditional(felem out, const felem in, limb mask)
1146 for (i = 0; i < NLIMBS; ++i) {
1147 const limb tmp = mask & (in[i] ^ out[i]);
1153 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1155 * The method is taken from
1156 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1157 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1159 * This function includes a branch for checking whether the two input points
1160 * are equal (while not equal to the point at infinity). This case never
1161 * happens during single point multiplication, so there is no timing leak for
1162 * ECDH or ECDSA signing. */
1163 static void point_add(felem x3, felem y3, felem z3,
1164 const felem x1, const felem y1, const felem z1,
1165 const int mixed, const felem x2, const felem y2,
1168 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1169 largefelem tmp, tmp2;
1170 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1172 z1_is_zero = felem_is_zero(z1);
1173 z2_is_zero = felem_is_zero(z2);
1175 /* ftmp = z1z1 = z1**2 */
1176 felem_square(tmp, z1);
1177 felem_reduce(ftmp, tmp);
1180 /* ftmp2 = z2z2 = z2**2 */
1181 felem_square(tmp, z2);
1182 felem_reduce(ftmp2, tmp);
1184 /* u1 = ftmp3 = x1*z2z2 */
1185 felem_mul(tmp, x1, ftmp2);
1186 felem_reduce(ftmp3, tmp);
1188 /* ftmp5 = z1 + z2 */
1189 felem_assign(ftmp5, z1);
1190 felem_sum64(ftmp5, z2);
1191 /* ftmp5[i] < 2^61 */
1193 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1194 felem_square(tmp, ftmp5);
1195 /* tmp[i] < 17*2^122 */
1196 felem_diff_128_64(tmp, ftmp);
1197 /* tmp[i] < 17*2^122 + 2^63 */
1198 felem_diff_128_64(tmp, ftmp2);
1199 /* tmp[i] < 17*2^122 + 2^64 */
1200 felem_reduce(ftmp5, tmp);
1202 /* ftmp2 = z2 * z2z2 */
1203 felem_mul(tmp, ftmp2, z2);
1204 felem_reduce(ftmp2, tmp);
1206 /* s1 = ftmp6 = y1 * z2**3 */
1207 felem_mul(tmp, y1, ftmp2);
1208 felem_reduce(ftmp6, tmp);
1211 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1214 /* u1 = ftmp3 = x1*z2z2 */
1215 felem_assign(ftmp3, x1);
1217 /* ftmp5 = 2*z1z2 */
1218 felem_scalar(ftmp5, z1, 2);
1220 /* s1 = ftmp6 = y1 * z2**3 */
1221 felem_assign(ftmp6, y1);
1225 felem_mul(tmp, x2, ftmp);
1226 /* tmp[i] < 17*2^120 */
1228 /* h = ftmp4 = u2 - u1 */
1229 felem_diff_128_64(tmp, ftmp3);
1230 /* tmp[i] < 17*2^120 + 2^63 */
1231 felem_reduce(ftmp4, tmp);
1233 x_equal = felem_is_zero(ftmp4);
1235 /* z_out = ftmp5 * h */
1236 felem_mul(tmp, ftmp5, ftmp4);
1237 felem_reduce(z_out, tmp);
1239 /* ftmp = z1 * z1z1 */
1240 felem_mul(tmp, ftmp, z1);
1241 felem_reduce(ftmp, tmp);
1243 /* s2 = tmp = y2 * z1**3 */
1244 felem_mul(tmp, y2, ftmp);
1245 /* tmp[i] < 17*2^120 */
1247 /* r = ftmp5 = (s2 - s1)*2 */
1248 felem_diff_128_64(tmp, ftmp6);
1249 /* tmp[i] < 17*2^120 + 2^63 */
1250 felem_reduce(ftmp5, tmp);
1251 y_equal = felem_is_zero(ftmp5);
1252 felem_scalar64(ftmp5, 2);
1253 /* ftmp5[i] < 2^61 */
1255 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1256 point_double(x3, y3, z3, x1, y1, z1);
1260 /* I = ftmp = (2h)**2 */
1261 felem_assign(ftmp, ftmp4);
1262 felem_scalar64(ftmp, 2);
1263 /* ftmp[i] < 2^61 */
1264 felem_square(tmp, ftmp);
1265 /* tmp[i] < 17*2^122 */
1266 felem_reduce(ftmp, tmp);
1268 /* J = ftmp2 = h * I */
1269 felem_mul(tmp, ftmp4, ftmp);
1270 felem_reduce(ftmp2, tmp);
1272 /* V = ftmp4 = U1 * I */
1273 felem_mul(tmp, ftmp3, ftmp);
1274 felem_reduce(ftmp4, tmp);
1276 /* x_out = r**2 - J - 2V */
1277 felem_square(tmp, ftmp5);
1278 /* tmp[i] < 17*2^122 */
1279 felem_diff_128_64(tmp, ftmp2);
1280 /* tmp[i] < 17*2^122 + 2^63 */
1281 felem_assign(ftmp3, ftmp4);
1282 felem_scalar64(ftmp4, 2);
1283 /* ftmp4[i] < 2^61 */
1284 felem_diff_128_64(tmp, ftmp4);
1285 /* tmp[i] < 17*2^122 + 2^64 */
1286 felem_reduce(x_out, tmp);
1288 /* y_out = r(V-x_out) - 2 * s1 * J */
1289 felem_diff64(ftmp3, x_out);
1291 * ftmp3[i] < 2^60 + 2^60 = 2^61
1293 felem_mul(tmp, ftmp5, ftmp3);
1294 /* tmp[i] < 17*2^122 */
1295 felem_mul(tmp2, ftmp6, ftmp2);
1296 /* tmp2[i] < 17*2^120 */
1297 felem_scalar128(tmp2, 2);
1298 /* tmp2[i] < 17*2^121 */
1299 felem_diff128(tmp, tmp2);
1301 * tmp[i] < 2^127 - 2^69 + 17*2^122
1302 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1305 felem_reduce(y_out, tmp);
1307 copy_conditional(x_out, x2, z1_is_zero);
1308 copy_conditional(x_out, x1, z2_is_zero);
1309 copy_conditional(y_out, y2, z1_is_zero);
1310 copy_conditional(y_out, y1, z2_is_zero);
1311 copy_conditional(z_out, z2, z1_is_zero);
1312 copy_conditional(z_out, z1, z2_is_zero);
1313 felem_assign(x3, x_out);
1314 felem_assign(y3, y_out);
1315 felem_assign(z3, z_out);
1319 * Base point pre computation
1320 * --------------------------
1322 * Two different sorts of precomputed tables are used in the following code.
1323 * Each contain various points on the curve, where each point is three field
1324 * elements (x, y, z).
1326 * For the base point table, z is usually 1 (0 for the point at infinity).
1327 * This table has 16 elements:
1328 * index | bits | point
1329 * ------+---------+------------------------------
1332 * 2 | 0 0 1 0 | 2^130G
1333 * 3 | 0 0 1 1 | (2^130 + 1)G
1334 * 4 | 0 1 0 0 | 2^260G
1335 * 5 | 0 1 0 1 | (2^260 + 1)G
1336 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1337 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1338 * 8 | 1 0 0 0 | 2^390G
1339 * 9 | 1 0 0 1 | (2^390 + 1)G
1340 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1341 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1342 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1343 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1344 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1345 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1347 * The reason for this is so that we can clock bits into four different
1348 * locations when doing simple scalar multiplies against the base point.
1350 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1352 /* gmul is the table of precomputed base points */
1353 static const felem gmul[16][3] = {
1354 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1355 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1356 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1357 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1358 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1359 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1360 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1361 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1362 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1363 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1364 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1365 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1366 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1367 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1368 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1369 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1370 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1371 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1372 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1373 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1374 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1375 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1376 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1377 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1378 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1379 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1380 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1381 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1382 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1383 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1384 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1385 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1386 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1387 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1388 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1389 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1390 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1391 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1392 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1393 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1394 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1395 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1396 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1397 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1398 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1399 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1400 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1401 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1402 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1403 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1404 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1405 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1406 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1407 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1408 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1409 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1410 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1411 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1412 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1413 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1414 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1415 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1416 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1417 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1418 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1419 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1420 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1421 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1422 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1423 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1424 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1425 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1426 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1427 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1428 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1429 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1430 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1431 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1432 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1433 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1434 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1435 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1436 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1437 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1438 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1439 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1440 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1441 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1442 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1443 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1444 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1445 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1446 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1447 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1448 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1449 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1450 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1451 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1452 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1453 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1454 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1455 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1456 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1457 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1458 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1459 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1460 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1461 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1465 * select_point selects the |idx|th point from a precomputation table and
1468 /* pre_comp below is of the size provided in |size| */
1469 static void select_point(const limb idx, unsigned int size,
1470 const felem pre_comp[][3], felem out[3])
1473 limb *outlimbs = &out[0][0];
1475 memset(out, 0, sizeof(*out) * 3);
1477 for (i = 0; i < size; i++) {
1478 const limb *inlimbs = &pre_comp[i][0][0];
1479 limb mask = i ^ idx;
1485 for (j = 0; j < NLIMBS * 3; j++)
1486 outlimbs[j] |= inlimbs[j] & mask;
1490 /* get_bit returns the |i|th bit in |in| */
1491 static char get_bit(const felem_bytearray in, int i)
1495 return (in[i >> 3] >> (i & 7)) & 1;
1499 * Interleaved point multiplication using precomputed point multiples: The
1500 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1501 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1502 * generator, using certain (large) precomputed multiples in g_pre_comp.
1503 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1505 static void batch_mul(felem x_out, felem y_out, felem z_out,
1506 const felem_bytearray scalars[],
1507 const unsigned num_points, const u8 *g_scalar,
1508 const int mixed, const felem pre_comp[][17][3],
1509 const felem g_pre_comp[16][3])
1512 unsigned num, gen_mul = (g_scalar != NULL);
1513 felem nq[3], tmp[4];
1517 /* set nq to the point at infinity */
1518 memset(nq, 0, sizeof(nq));
1521 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1522 * of the generator (last quarter of rounds) and additions of other
1523 * points multiples (every 5th round).
1525 skip = 1; /* save two point operations in the first
1527 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1530 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1532 /* add multiples of the generator */
1533 if (gen_mul && (i <= 130)) {
1534 bits = get_bit(g_scalar, i + 390) << 3;
1536 bits |= get_bit(g_scalar, i + 260) << 2;
1537 bits |= get_bit(g_scalar, i + 130) << 1;
1538 bits |= get_bit(g_scalar, i);
1540 /* select the point to add, in constant time */
1541 select_point(bits, 16, g_pre_comp, tmp);
1543 /* The 1 argument below is for "mixed" */
1544 point_add(nq[0], nq[1], nq[2],
1545 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1547 memcpy(nq, tmp, 3 * sizeof(felem));
1552 /* do other additions every 5 doublings */
1553 if (num_points && (i % 5 == 0)) {
1554 /* loop over all scalars */
1555 for (num = 0; num < num_points; ++num) {
1556 bits = get_bit(scalars[num], i + 4) << 5;
1557 bits |= get_bit(scalars[num], i + 3) << 4;
1558 bits |= get_bit(scalars[num], i + 2) << 3;
1559 bits |= get_bit(scalars[num], i + 1) << 2;
1560 bits |= get_bit(scalars[num], i) << 1;
1561 bits |= get_bit(scalars[num], i - 1);
1562 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1565 * select the point to add or subtract, in constant time
1567 select_point(digit, 17, pre_comp[num], tmp);
1568 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1570 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1573 point_add(nq[0], nq[1], nq[2],
1574 nq[0], nq[1], nq[2],
1575 mixed, tmp[0], tmp[1], tmp[2]);
1577 memcpy(nq, tmp, 3 * sizeof(felem));
1583 felem_assign(x_out, nq[0]);
1584 felem_assign(y_out, nq[1]);
1585 felem_assign(z_out, nq[2]);
1588 /* Precomputation for the group generator. */
1589 struct nistp521_pre_comp_st {
1590 felem g_pre_comp[16][3];
1591 CRYPTO_REF_COUNT references;
1592 CRYPTO_RWLOCK *lock;
1595 const EC_METHOD *EC_GFp_nistp521_method(void)
1597 static const EC_METHOD ret = {
1598 EC_FLAGS_DEFAULT_OCT,
1599 NID_X9_62_prime_field,
1600 ec_GFp_nistp521_group_init,
1601 ec_GFp_simple_group_finish,
1602 ec_GFp_simple_group_clear_finish,
1603 ec_GFp_nist_group_copy,
1604 ec_GFp_nistp521_group_set_curve,
1605 ec_GFp_simple_group_get_curve,
1606 ec_GFp_simple_group_get_degree,
1607 ec_group_simple_order_bits,
1608 ec_GFp_simple_group_check_discriminant,
1609 ec_GFp_simple_point_init,
1610 ec_GFp_simple_point_finish,
1611 ec_GFp_simple_point_clear_finish,
1612 ec_GFp_simple_point_copy,
1613 ec_GFp_simple_point_set_to_infinity,
1614 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1615 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1616 ec_GFp_simple_point_set_affine_coordinates,
1617 ec_GFp_nistp521_point_get_affine_coordinates,
1618 0 /* point_set_compressed_coordinates */ ,
1623 ec_GFp_simple_invert,
1624 ec_GFp_simple_is_at_infinity,
1625 ec_GFp_simple_is_on_curve,
1627 ec_GFp_simple_make_affine,
1628 ec_GFp_simple_points_make_affine,
1629 ec_GFp_nistp521_points_mul,
1630 ec_GFp_nistp521_precompute_mult,
1631 ec_GFp_nistp521_have_precompute_mult,
1632 ec_GFp_nist_field_mul,
1633 ec_GFp_nist_field_sqr,
1635 0 /* field_encode */ ,
1636 0 /* field_decode */ ,
1637 0, /* field_set_to_one */
1638 ec_key_simple_priv2oct,
1639 ec_key_simple_oct2priv,
1640 0, /* set private */
1641 ec_key_simple_generate_key,
1642 ec_key_simple_check_key,
1643 ec_key_simple_generate_public_key,
1646 ecdh_simple_compute_key
1652 /******************************************************************************/
1654 * FUNCTIONS TO MANAGE PRECOMPUTATION
1657 static NISTP521_PRE_COMP *nistp521_pre_comp_new()
1659 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1662 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1666 ret->references = 1;
1668 ret->lock = CRYPTO_THREAD_lock_new();
1669 if (ret->lock == NULL) {
1670 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1677 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1681 CRYPTO_UP_REF(&p->references, &i, p->lock);
1685 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1692 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1693 REF_PRINT_COUNT("EC_nistp521", x);
1696 REF_ASSERT_ISNT(i < 0);
1698 CRYPTO_THREAD_lock_free(p->lock);
1702 /******************************************************************************/
1704 * OPENSSL EC_METHOD FUNCTIONS
1707 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1710 ret = ec_GFp_simple_group_init(group);
1711 group->a_is_minus3 = 1;
1715 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1716 const BIGNUM *a, const BIGNUM *b,
1720 BN_CTX *new_ctx = NULL;
1721 BIGNUM *curve_p, *curve_a, *curve_b;
1724 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1727 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1728 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1729 ((curve_b = BN_CTX_get(ctx)) == NULL))
1731 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1732 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1733 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1734 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1735 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1736 EC_R_WRONG_CURVE_PARAMETERS);
1739 group->field_mod_func = BN_nist_mod_521;
1740 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1743 BN_CTX_free(new_ctx);
1748 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1751 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1752 const EC_POINT *point,
1753 BIGNUM *x, BIGNUM *y,
1756 felem z1, z2, x_in, y_in, x_out, y_out;
1759 if (EC_POINT_is_at_infinity(group, point)) {
1760 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1761 EC_R_POINT_AT_INFINITY);
1764 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1765 (!BN_to_felem(z1, point->Z)))
1768 felem_square(tmp, z2);
1769 felem_reduce(z1, tmp);
1770 felem_mul(tmp, x_in, z1);
1771 felem_reduce(x_in, tmp);
1772 felem_contract(x_out, x_in);
1774 if (!felem_to_BN(x, x_out)) {
1775 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1780 felem_mul(tmp, z1, z2);
1781 felem_reduce(z1, tmp);
1782 felem_mul(tmp, y_in, z1);
1783 felem_reduce(y_in, tmp);
1784 felem_contract(y_out, y_in);
1786 if (!felem_to_BN(y, y_out)) {
1787 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1795 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1796 static void make_points_affine(size_t num, felem points[][3],
1800 * Runs in constant time, unless an input is the point at infinity (which
1801 * normally shouldn't happen).
1803 ec_GFp_nistp_points_make_affine_internal(num,
1807 (void (*)(void *))felem_one,
1808 (int (*)(const void *))
1810 (void (*)(void *, const void *))
1812 (void (*)(void *, const void *))
1813 felem_square_reduce, (void (*)
1820 (void (*)(void *, const void *))
1822 (void (*)(void *, const void *))
1827 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1828 * values Result is stored in r (r can equal one of the inputs).
1830 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1831 const BIGNUM *scalar, size_t num,
1832 const EC_POINT *points[],
1833 const BIGNUM *scalars[], BN_CTX *ctx)
1838 BN_CTX *new_ctx = NULL;
1839 BIGNUM *x, *y, *z, *tmp_scalar;
1840 felem_bytearray g_secret;
1841 felem_bytearray *secrets = NULL;
1842 felem (*pre_comp)[17][3] = NULL;
1843 felem *tmp_felems = NULL;
1844 felem_bytearray tmp;
1845 unsigned i, num_bytes;
1846 int have_pre_comp = 0;
1847 size_t num_points = num;
1848 felem x_in, y_in, z_in, x_out, y_out, z_out;
1849 NISTP521_PRE_COMP *pre = NULL;
1850 felem(*g_pre_comp)[3] = NULL;
1851 EC_POINT *generator = NULL;
1852 const EC_POINT *p = NULL;
1853 const BIGNUM *p_scalar = NULL;
1856 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1859 if (((x = BN_CTX_get(ctx)) == NULL) ||
1860 ((y = BN_CTX_get(ctx)) == NULL) ||
1861 ((z = BN_CTX_get(ctx)) == NULL) ||
1862 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1865 if (scalar != NULL) {
1866 pre = group->pre_comp.nistp521;
1868 /* we have precomputation, try to use it */
1869 g_pre_comp = &pre->g_pre_comp[0];
1871 /* try to use the standard precomputation */
1872 g_pre_comp = (felem(*)[3]) gmul;
1873 generator = EC_POINT_new(group);
1874 if (generator == NULL)
1876 /* get the generator from precomputation */
1877 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1878 !felem_to_BN(y, g_pre_comp[1][1]) ||
1879 !felem_to_BN(z, g_pre_comp[1][2])) {
1880 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1883 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1887 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1888 /* precomputation matches generator */
1892 * we don't have valid precomputation: treat the generator as a
1898 if (num_points > 0) {
1899 if (num_points >= 2) {
1901 * unless we precompute multiples for just one point, converting
1902 * those into affine form is time well spent
1906 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1907 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1910 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1911 if ((secrets == NULL) || (pre_comp == NULL)
1912 || (mixed && (tmp_felems == NULL))) {
1913 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1918 * we treat NULL scalars as 0, and NULL points as points at infinity,
1919 * i.e., they contribute nothing to the linear combination
1921 for (i = 0; i < num_points; ++i) {
1924 * we didn't have a valid precomputation, so we pick the
1928 p = EC_GROUP_get0_generator(group);
1931 /* the i^th point */
1934 p_scalar = scalars[i];
1936 if ((p_scalar != NULL) && (p != NULL)) {
1937 /* reduce scalar to 0 <= scalar < 2^521 */
1938 if ((BN_num_bits(p_scalar) > 521)
1939 || (BN_is_negative(p_scalar))) {
1941 * this is an unusual input, and we don't guarantee
1944 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1945 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1948 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1950 num_bytes = BN_bn2bin(p_scalar, tmp);
1951 flip_endian(secrets[i], tmp, num_bytes);
1952 /* precompute multiples */
1953 if ((!BN_to_felem(x_out, p->X)) ||
1954 (!BN_to_felem(y_out, p->Y)) ||
1955 (!BN_to_felem(z_out, p->Z)))
1957 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1958 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1959 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1960 for (j = 2; j <= 16; ++j) {
1962 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1963 pre_comp[i][j][2], pre_comp[i][1][0],
1964 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1965 pre_comp[i][j - 1][0],
1966 pre_comp[i][j - 1][1],
1967 pre_comp[i][j - 1][2]);
1969 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1970 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1971 pre_comp[i][j / 2][1],
1972 pre_comp[i][j / 2][2]);
1978 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1981 /* the scalar for the generator */
1982 if ((scalar != NULL) && (have_pre_comp)) {
1983 memset(g_secret, 0, sizeof(g_secret));
1984 /* reduce scalar to 0 <= scalar < 2^521 */
1985 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
1987 * this is an unusual input, and we don't guarantee
1990 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1991 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1994 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1996 num_bytes = BN_bn2bin(scalar, tmp);
1997 flip_endian(g_secret, tmp, num_bytes);
1998 /* do the multiplication with generator precomputation */
1999 batch_mul(x_out, y_out, z_out,
2000 (const felem_bytearray(*))secrets, num_points,
2002 mixed, (const felem(*)[17][3])pre_comp,
2003 (const felem(*)[3])g_pre_comp);
2005 /* do the multiplication without generator precomputation */
2006 batch_mul(x_out, y_out, z_out,
2007 (const felem_bytearray(*))secrets, num_points,
2008 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2009 /* reduce the output to its unique minimal representation */
2010 felem_contract(x_in, x_out);
2011 felem_contract(y_in, y_out);
2012 felem_contract(z_in, z_out);
2013 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2014 (!felem_to_BN(z, z_in))) {
2015 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2018 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2022 EC_POINT_free(generator);
2023 BN_CTX_free(new_ctx);
2024 OPENSSL_free(secrets);
2025 OPENSSL_free(pre_comp);
2026 OPENSSL_free(tmp_felems);
2030 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2033 NISTP521_PRE_COMP *pre = NULL;
2035 BN_CTX *new_ctx = NULL;
2037 EC_POINT *generator = NULL;
2038 felem tmp_felems[16];
2040 /* throw away old precomputation */
2041 EC_pre_comp_free(group);
2043 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2046 if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
2048 /* get the generator */
2049 if (group->generator == NULL)
2051 generator = EC_POINT_new(group);
2052 if (generator == NULL)
2054 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2055 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2056 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
2058 if ((pre = nistp521_pre_comp_new()) == NULL)
2061 * if the generator is the standard one, use built-in precomputation
2063 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2064 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2067 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2068 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2069 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2071 /* compute 2^130*G, 2^260*G, 2^390*G */
2072 for (i = 1; i <= 4; i <<= 1) {
2073 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2074 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2075 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2076 for (j = 0; j < 129; ++j) {
2077 point_double(pre->g_pre_comp[2 * i][0],
2078 pre->g_pre_comp[2 * i][1],
2079 pre->g_pre_comp[2 * i][2],
2080 pre->g_pre_comp[2 * i][0],
2081 pre->g_pre_comp[2 * i][1],
2082 pre->g_pre_comp[2 * i][2]);
2085 /* g_pre_comp[0] is the point at infinity */
2086 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2087 /* the remaining multiples */
2088 /* 2^130*G + 2^260*G */
2089 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2090 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2091 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2092 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2093 pre->g_pre_comp[2][2]);
2094 /* 2^130*G + 2^390*G */
2095 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2096 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2097 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2098 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2099 pre->g_pre_comp[2][2]);
2100 /* 2^260*G + 2^390*G */
2101 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2102 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2103 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2104 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2105 pre->g_pre_comp[4][2]);
2106 /* 2^130*G + 2^260*G + 2^390*G */
2107 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2108 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2109 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2110 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2111 pre->g_pre_comp[2][2]);
2112 for (i = 1; i < 8; ++i) {
2113 /* odd multiples: add G */
2114 point_add(pre->g_pre_comp[2 * i + 1][0],
2115 pre->g_pre_comp[2 * i + 1][1],
2116 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2117 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2118 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2119 pre->g_pre_comp[1][2]);
2121 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2124 SETPRECOMP(group, nistp521, pre);
2129 EC_POINT_free(generator);
2130 BN_CTX_free(new_ctx);
2131 EC_nistp521_pre_comp_free(pre);
2135 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2137 return HAVEPRECOMP(group, nistp521);