2 * Copyright 2011-2018 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * 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(__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 felem[NLIMBS];
132 typedef uint128_t largefelem[NLIMBS];
134 static const limb bottom57bits = 0x1ffffffffffffff;
135 static const limb bottom58bits = 0x3ffffffffffffff;
138 * bin66_to_felem takes a little-endian byte array and converts it into felem
139 * form. This assumes that the CPU is little-endian.
141 static void bin66_to_felem(felem out, const u8 in[66])
143 out[0] = (*((limb *) & in[0])) & bottom58bits;
144 out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits;
145 out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits;
146 out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits;
147 out[4] = (*((limb *) & in[29])) & bottom58bits;
148 out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits;
149 out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits;
150 out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits;
151 out[8] = (*((limb *) & in[58])) & bottom57bits;
155 * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
156 * array. This assumes that the CPU is little-endian.
158 static void felem_to_bin66(u8 out[66], const felem in)
161 (*((limb *) & out[0])) = in[0];
162 (*((limb *) & out[7])) |= in[1] << 2;
163 (*((limb *) & out[14])) |= in[2] << 4;
164 (*((limb *) & out[21])) |= in[3] << 6;
165 (*((limb *) & out[29])) = in[4];
166 (*((limb *) & out[36])) |= in[5] << 2;
167 (*((limb *) & out[43])) |= in[6] << 4;
168 (*((limb *) & out[50])) |= in[7] << 6;
169 (*((limb *) & out[58])) = in[8];
172 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
173 static void flip_endian(u8 *out, const u8 *in, unsigned len)
176 for (i = 0; i < len; ++i)
177 out[i] = in[len - 1 - i];
180 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
181 static int BN_to_felem(felem out, const BIGNUM *bn)
183 felem_bytearray b_in;
184 felem_bytearray b_out;
187 /* BN_bn2bin eats leading zeroes */
188 memset(b_out, 0, sizeof(b_out));
189 num_bytes = BN_num_bytes(bn);
190 if (num_bytes > sizeof(b_out)) {
191 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
194 if (BN_is_negative(bn)) {
195 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
198 num_bytes = BN_bn2bin(bn, b_in);
199 flip_endian(b_out, b_in, num_bytes);
200 bin66_to_felem(out, b_out);
204 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
205 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
207 felem_bytearray b_in, b_out;
208 felem_to_bin66(b_in, in);
209 flip_endian(b_out, b_in, sizeof(b_out));
210 return BN_bin2bn(b_out, sizeof(b_out), out);
218 static void felem_one(felem out)
231 static void felem_assign(felem out, const felem in)
244 /* felem_sum64 sets out = out + in. */
245 static void felem_sum64(felem out, const felem in)
258 /* felem_scalar sets out = in * scalar */
259 static void felem_scalar(felem out, const felem in, limb scalar)
261 out[0] = in[0] * scalar;
262 out[1] = in[1] * scalar;
263 out[2] = in[2] * scalar;
264 out[3] = in[3] * scalar;
265 out[4] = in[4] * scalar;
266 out[5] = in[5] * scalar;
267 out[6] = in[6] * scalar;
268 out[7] = in[7] * scalar;
269 out[8] = in[8] * scalar;
272 /* felem_scalar64 sets out = out * scalar */
273 static void felem_scalar64(felem out, limb scalar)
286 /* felem_scalar128 sets out = out * scalar */
287 static void felem_scalar128(largefelem out, limb scalar)
301 * felem_neg sets |out| to |-in|
303 * in[i] < 2^59 + 2^14
307 static void felem_neg(felem out, const felem in)
309 /* In order to prevent underflow, we subtract from 0 mod p. */
310 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
311 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
313 out[0] = two62m3 - in[0];
314 out[1] = two62m2 - in[1];
315 out[2] = two62m2 - in[2];
316 out[3] = two62m2 - in[3];
317 out[4] = two62m2 - in[4];
318 out[5] = two62m2 - in[5];
319 out[6] = two62m2 - in[6];
320 out[7] = two62m2 - in[7];
321 out[8] = two62m2 - in[8];
325 * felem_diff64 subtracts |in| from |out|
327 * in[i] < 2^59 + 2^14
329 * out[i] < out[i] + 2^62
331 static void felem_diff64(felem out, const felem in)
334 * In order to prevent underflow, we add 0 mod p before subtracting.
336 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
337 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
339 out[0] += two62m3 - in[0];
340 out[1] += two62m2 - in[1];
341 out[2] += two62m2 - in[2];
342 out[3] += two62m2 - in[3];
343 out[4] += two62m2 - in[4];
344 out[5] += two62m2 - in[5];
345 out[6] += two62m2 - in[6];
346 out[7] += two62m2 - in[7];
347 out[8] += two62m2 - in[8];
351 * felem_diff_128_64 subtracts |in| from |out|
353 * in[i] < 2^62 + 2^17
355 * out[i] < out[i] + 2^63
357 static void felem_diff_128_64(largefelem out, const felem in)
360 * In order to prevent underflow, we add 0 mod p before subtracting.
362 static const limb two63m6 = (((limb) 1) << 62) - (((limb) 1) << 5);
363 static const limb two63m5 = (((limb) 1) << 62) - (((limb) 1) << 4);
365 out[0] += two63m6 - in[0];
366 out[1] += two63m5 - in[1];
367 out[2] += two63m5 - in[2];
368 out[3] += two63m5 - in[3];
369 out[4] += two63m5 - in[4];
370 out[5] += two63m5 - in[5];
371 out[6] += two63m5 - in[6];
372 out[7] += two63m5 - in[7];
373 out[8] += two63m5 - in[8];
377 * felem_diff_128_64 subtracts |in| from |out|
381 * out[i] < out[i] + 2^127 - 2^69
383 static void felem_diff128(largefelem out, const largefelem in)
386 * In order to prevent underflow, we add 0 mod p before subtracting.
388 static const uint128_t two127m70 =
389 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
390 static const uint128_t two127m69 =
391 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
393 out[0] += (two127m70 - in[0]);
394 out[1] += (two127m69 - in[1]);
395 out[2] += (two127m69 - in[2]);
396 out[3] += (two127m69 - in[3]);
397 out[4] += (two127m69 - in[4]);
398 out[5] += (two127m69 - in[5]);
399 out[6] += (two127m69 - in[6]);
400 out[7] += (two127m69 - in[7]);
401 out[8] += (two127m69 - in[8]);
405 * felem_square sets |out| = |in|^2
409 * out[i] < 17 * max(in[i]) * max(in[i])
411 static void felem_square(largefelem out, const felem in)
414 felem_scalar(inx2, in, 2);
415 felem_scalar(inx4, in, 4);
418 * We have many cases were we want to do
421 * This is obviously just
423 * However, rather than do the doubling on the 128 bit result, we
424 * double one of the inputs to the multiplication by reading from
428 out[0] = ((uint128_t) in[0]) * in[0];
429 out[1] = ((uint128_t) in[0]) * inx2[1];
430 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
431 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
432 out[4] = ((uint128_t) in[0]) * inx2[4] +
433 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
434 out[5] = ((uint128_t) in[0]) * inx2[5] +
435 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
436 out[6] = ((uint128_t) in[0]) * inx2[6] +
437 ((uint128_t) in[1]) * inx2[5] +
438 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
439 out[7] = ((uint128_t) in[0]) * inx2[7] +
440 ((uint128_t) in[1]) * inx2[6] +
441 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
442 out[8] = ((uint128_t) in[0]) * inx2[8] +
443 ((uint128_t) in[1]) * inx2[7] +
444 ((uint128_t) in[2]) * inx2[6] +
445 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
448 * The remaining limbs fall above 2^521, with the first falling at 2^522.
449 * They correspond to locations one bit up from the limbs produced above
450 * so we would have to multiply by two to align them. Again, rather than
451 * operate on the 128-bit result, we double one of the inputs to the
452 * multiplication. If we want to double for both this reason, and the
453 * reason above, then we end up multiplying by four.
457 out[0] += ((uint128_t) in[1]) * inx4[8] +
458 ((uint128_t) in[2]) * inx4[7] +
459 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
462 out[1] += ((uint128_t) in[2]) * inx4[8] +
463 ((uint128_t) in[3]) * inx4[7] +
464 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
467 out[2] += ((uint128_t) in[3]) * inx4[8] +
468 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
471 out[3] += ((uint128_t) in[4]) * inx4[8] +
472 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
475 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
478 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
481 out[6] += ((uint128_t) in[7]) * inx4[8];
484 out[7] += ((uint128_t) in[8]) * inx2[8];
488 * felem_mul sets |out| = |in1| * |in2|
493 * out[i] < 17 * max(in1[i]) * max(in2[i])
495 static void felem_mul(largefelem out, const felem in1, const felem in2)
498 felem_scalar(in2x2, in2, 2);
500 out[0] = ((uint128_t) in1[0]) * in2[0];
502 out[1] = ((uint128_t) in1[0]) * in2[1] +
503 ((uint128_t) in1[1]) * in2[0];
505 out[2] = ((uint128_t) in1[0]) * in2[2] +
506 ((uint128_t) in1[1]) * in2[1] +
507 ((uint128_t) in1[2]) * in2[0];
509 out[3] = ((uint128_t) in1[0]) * in2[3] +
510 ((uint128_t) in1[1]) * in2[2] +
511 ((uint128_t) in1[2]) * in2[1] +
512 ((uint128_t) in1[3]) * in2[0];
514 out[4] = ((uint128_t) in1[0]) * in2[4] +
515 ((uint128_t) in1[1]) * in2[3] +
516 ((uint128_t) in1[2]) * in2[2] +
517 ((uint128_t) in1[3]) * in2[1] +
518 ((uint128_t) in1[4]) * in2[0];
520 out[5] = ((uint128_t) in1[0]) * in2[5] +
521 ((uint128_t) in1[1]) * in2[4] +
522 ((uint128_t) in1[2]) * in2[3] +
523 ((uint128_t) in1[3]) * in2[2] +
524 ((uint128_t) in1[4]) * in2[1] +
525 ((uint128_t) in1[5]) * in2[0];
527 out[6] = ((uint128_t) in1[0]) * in2[6] +
528 ((uint128_t) in1[1]) * in2[5] +
529 ((uint128_t) in1[2]) * in2[4] +
530 ((uint128_t) in1[3]) * in2[3] +
531 ((uint128_t) in1[4]) * in2[2] +
532 ((uint128_t) in1[5]) * in2[1] +
533 ((uint128_t) in1[6]) * in2[0];
535 out[7] = ((uint128_t) in1[0]) * in2[7] +
536 ((uint128_t) in1[1]) * in2[6] +
537 ((uint128_t) in1[2]) * in2[5] +
538 ((uint128_t) in1[3]) * in2[4] +
539 ((uint128_t) in1[4]) * in2[3] +
540 ((uint128_t) in1[5]) * in2[2] +
541 ((uint128_t) in1[6]) * in2[1] +
542 ((uint128_t) in1[7]) * in2[0];
544 out[8] = ((uint128_t) in1[0]) * in2[8] +
545 ((uint128_t) in1[1]) * in2[7] +
546 ((uint128_t) in1[2]) * in2[6] +
547 ((uint128_t) in1[3]) * in2[5] +
548 ((uint128_t) in1[4]) * in2[4] +
549 ((uint128_t) in1[5]) * in2[3] +
550 ((uint128_t) in1[6]) * in2[2] +
551 ((uint128_t) in1[7]) * in2[1] +
552 ((uint128_t) in1[8]) * in2[0];
554 /* See comment in felem_square about the use of in2x2 here */
556 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
557 ((uint128_t) in1[2]) * in2x2[7] +
558 ((uint128_t) in1[3]) * in2x2[6] +
559 ((uint128_t) in1[4]) * in2x2[5] +
560 ((uint128_t) in1[5]) * in2x2[4] +
561 ((uint128_t) in1[6]) * in2x2[3] +
562 ((uint128_t) in1[7]) * in2x2[2] +
563 ((uint128_t) in1[8]) * in2x2[1];
565 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
566 ((uint128_t) in1[3]) * in2x2[7] +
567 ((uint128_t) in1[4]) * in2x2[6] +
568 ((uint128_t) in1[5]) * in2x2[5] +
569 ((uint128_t) in1[6]) * in2x2[4] +
570 ((uint128_t) in1[7]) * in2x2[3] +
571 ((uint128_t) in1[8]) * in2x2[2];
573 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
574 ((uint128_t) in1[4]) * in2x2[7] +
575 ((uint128_t) in1[5]) * in2x2[6] +
576 ((uint128_t) in1[6]) * in2x2[5] +
577 ((uint128_t) in1[7]) * in2x2[4] +
578 ((uint128_t) in1[8]) * in2x2[3];
580 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
581 ((uint128_t) in1[5]) * in2x2[7] +
582 ((uint128_t) in1[6]) * in2x2[6] +
583 ((uint128_t) in1[7]) * in2x2[5] +
584 ((uint128_t) in1[8]) * in2x2[4];
586 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
587 ((uint128_t) in1[6]) * in2x2[7] +
588 ((uint128_t) in1[7]) * in2x2[6] +
589 ((uint128_t) in1[8]) * in2x2[5];
591 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
592 ((uint128_t) in1[7]) * in2x2[7] +
593 ((uint128_t) in1[8]) * in2x2[6];
595 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
596 ((uint128_t) in1[8]) * in2x2[7];
598 out[7] += ((uint128_t) in1[8]) * in2x2[8];
601 static const limb bottom52bits = 0xfffffffffffff;
604 * felem_reduce converts a largefelem to an felem.
608 * out[i] < 2^59 + 2^14
610 static void felem_reduce(felem out, const largefelem in)
612 u64 overflow1, overflow2;
614 out[0] = ((limb) in[0]) & bottom58bits;
615 out[1] = ((limb) in[1]) & bottom58bits;
616 out[2] = ((limb) in[2]) & bottom58bits;
617 out[3] = ((limb) in[3]) & bottom58bits;
618 out[4] = ((limb) in[4]) & bottom58bits;
619 out[5] = ((limb) in[5]) & bottom58bits;
620 out[6] = ((limb) in[6]) & bottom58bits;
621 out[7] = ((limb) in[7]) & bottom58bits;
622 out[8] = ((limb) in[8]) & bottom58bits;
626 out[1] += ((limb) in[0]) >> 58;
627 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
629 * out[1] < 2^58 + 2^6 + 2^58
632 out[2] += ((limb) (in[0] >> 64)) >> 52;
634 out[2] += ((limb) in[1]) >> 58;
635 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
636 out[3] += ((limb) (in[1] >> 64)) >> 52;
638 out[3] += ((limb) in[2]) >> 58;
639 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
640 out[4] += ((limb) (in[2] >> 64)) >> 52;
642 out[4] += ((limb) in[3]) >> 58;
643 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
644 out[5] += ((limb) (in[3] >> 64)) >> 52;
646 out[5] += ((limb) in[4]) >> 58;
647 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
648 out[6] += ((limb) (in[4] >> 64)) >> 52;
650 out[6] += ((limb) in[5]) >> 58;
651 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
652 out[7] += ((limb) (in[5] >> 64)) >> 52;
654 out[7] += ((limb) in[6]) >> 58;
655 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
656 out[8] += ((limb) (in[6] >> 64)) >> 52;
658 out[8] += ((limb) in[7]) >> 58;
659 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
661 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
664 overflow1 = ((limb) (in[7] >> 64)) >> 52;
666 overflow1 += ((limb) in[8]) >> 58;
667 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
668 overflow2 = ((limb) (in[8] >> 64)) >> 52;
670 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
671 overflow2 <<= 1; /* overflow2 < 2^13 */
673 out[0] += overflow1; /* out[0] < 2^60 */
674 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
676 out[1] += out[0] >> 58;
677 out[0] &= bottom58bits;
680 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
685 static void felem_square_reduce(felem out, const felem in)
688 felem_square(tmp, in);
689 felem_reduce(out, tmp);
692 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
695 felem_mul(tmp, in1, in2);
696 felem_reduce(out, tmp);
700 * felem_inv calculates |out| = |in|^{-1}
702 * Based on Fermat's Little Theorem:
704 * a^{p-1} = 1 (mod p)
705 * a^{p-2} = a^{-1} (mod p)
707 static void felem_inv(felem out, const felem in)
709 felem ftmp, ftmp2, ftmp3, ftmp4;
713 felem_square(tmp, in);
714 felem_reduce(ftmp, tmp); /* 2^1 */
715 felem_mul(tmp, in, ftmp);
716 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
717 felem_assign(ftmp2, ftmp);
718 felem_square(tmp, ftmp);
719 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
720 felem_mul(tmp, in, ftmp);
721 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
722 felem_square(tmp, ftmp);
723 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
725 felem_square(tmp, ftmp2);
726 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
727 felem_square(tmp, ftmp3);
728 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
729 felem_mul(tmp, ftmp3, ftmp2);
730 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
732 felem_assign(ftmp2, ftmp3);
733 felem_square(tmp, ftmp3);
734 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
735 felem_square(tmp, ftmp3);
736 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
737 felem_square(tmp, ftmp3);
738 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
739 felem_square(tmp, ftmp3);
740 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
741 felem_assign(ftmp4, ftmp3);
742 felem_mul(tmp, ftmp3, ftmp);
743 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
744 felem_square(tmp, ftmp4);
745 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
746 felem_mul(tmp, ftmp3, ftmp2);
747 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
748 felem_assign(ftmp2, ftmp3);
750 for (i = 0; i < 8; i++) {
751 felem_square(tmp, ftmp3);
752 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
754 felem_mul(tmp, ftmp3, ftmp2);
755 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
756 felem_assign(ftmp2, ftmp3);
758 for (i = 0; i < 16; i++) {
759 felem_square(tmp, ftmp3);
760 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
762 felem_mul(tmp, ftmp3, ftmp2);
763 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
764 felem_assign(ftmp2, ftmp3);
766 for (i = 0; i < 32; i++) {
767 felem_square(tmp, ftmp3);
768 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
770 felem_mul(tmp, ftmp3, ftmp2);
771 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
772 felem_assign(ftmp2, ftmp3);
774 for (i = 0; i < 64; i++) {
775 felem_square(tmp, ftmp3);
776 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
778 felem_mul(tmp, ftmp3, ftmp2);
779 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
780 felem_assign(ftmp2, ftmp3);
782 for (i = 0; i < 128; i++) {
783 felem_square(tmp, ftmp3);
784 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
786 felem_mul(tmp, ftmp3, ftmp2);
787 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
788 felem_assign(ftmp2, ftmp3);
790 for (i = 0; i < 256; i++) {
791 felem_square(tmp, ftmp3);
792 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
794 felem_mul(tmp, ftmp3, ftmp2);
795 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
797 for (i = 0; i < 9; i++) {
798 felem_square(tmp, ftmp3);
799 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
801 felem_mul(tmp, ftmp3, ftmp4);
802 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
803 felem_mul(tmp, ftmp3, in);
804 felem_reduce(out, tmp); /* 2^512 - 3 */
807 /* This is 2^521-1, expressed as an felem */
808 static const felem kPrime = {
809 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
810 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
811 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
815 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
818 * in[i] < 2^59 + 2^14
820 static limb felem_is_zero(const felem in)
824 felem_assign(ftmp, in);
826 ftmp[0] += ftmp[8] >> 57;
827 ftmp[8] &= bottom57bits;
829 ftmp[1] += ftmp[0] >> 58;
830 ftmp[0] &= bottom58bits;
831 ftmp[2] += ftmp[1] >> 58;
832 ftmp[1] &= bottom58bits;
833 ftmp[3] += ftmp[2] >> 58;
834 ftmp[2] &= bottom58bits;
835 ftmp[4] += ftmp[3] >> 58;
836 ftmp[3] &= bottom58bits;
837 ftmp[5] += ftmp[4] >> 58;
838 ftmp[4] &= bottom58bits;
839 ftmp[6] += ftmp[5] >> 58;
840 ftmp[5] &= bottom58bits;
841 ftmp[7] += ftmp[6] >> 58;
842 ftmp[6] &= bottom58bits;
843 ftmp[8] += ftmp[7] >> 58;
844 ftmp[7] &= bottom58bits;
845 /* ftmp[8] < 2^57 + 4 */
848 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
849 * than our bound for ftmp[8]. Therefore we only have to check if the
850 * zero is zero or 2^521-1.
866 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
867 * can be set is if is_zero was 0 before the decrement.
869 is_zero = 0 - (is_zero >> 63);
871 is_p = ftmp[0] ^ kPrime[0];
872 is_p |= ftmp[1] ^ kPrime[1];
873 is_p |= ftmp[2] ^ kPrime[2];
874 is_p |= ftmp[3] ^ kPrime[3];
875 is_p |= ftmp[4] ^ kPrime[4];
876 is_p |= ftmp[5] ^ kPrime[5];
877 is_p |= ftmp[6] ^ kPrime[6];
878 is_p |= ftmp[7] ^ kPrime[7];
879 is_p |= ftmp[8] ^ kPrime[8];
882 is_p = 0 - (is_p >> 63);
888 static int felem_is_zero_int(const void *in)
890 return (int)(felem_is_zero(in) & ((limb) 1));
894 * felem_contract converts |in| to its unique, minimal representation.
896 * in[i] < 2^59 + 2^14
898 static void felem_contract(felem out, const felem in)
900 limb is_p, is_greater, sign;
901 static const limb two58 = ((limb) 1) << 58;
903 felem_assign(out, in);
905 out[0] += out[8] >> 57;
906 out[8] &= bottom57bits;
908 out[1] += out[0] >> 58;
909 out[0] &= bottom58bits;
910 out[2] += out[1] >> 58;
911 out[1] &= bottom58bits;
912 out[3] += out[2] >> 58;
913 out[2] &= bottom58bits;
914 out[4] += out[3] >> 58;
915 out[3] &= bottom58bits;
916 out[5] += out[4] >> 58;
917 out[4] &= bottom58bits;
918 out[6] += out[5] >> 58;
919 out[5] &= bottom58bits;
920 out[7] += out[6] >> 58;
921 out[6] &= bottom58bits;
922 out[8] += out[7] >> 58;
923 out[7] &= bottom58bits;
924 /* out[8] < 2^57 + 4 */
927 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
928 * out. See the comments in felem_is_zero regarding why we don't test for
929 * other multiples of the prime.
933 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
936 is_p = out[0] ^ kPrime[0];
937 is_p |= out[1] ^ kPrime[1];
938 is_p |= out[2] ^ kPrime[2];
939 is_p |= out[3] ^ kPrime[3];
940 is_p |= out[4] ^ kPrime[4];
941 is_p |= out[5] ^ kPrime[5];
942 is_p |= out[6] ^ kPrime[6];
943 is_p |= out[7] ^ kPrime[7];
944 is_p |= out[8] ^ kPrime[8];
953 is_p = 0 - (is_p >> 63);
956 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
969 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
970 * 57 is greater than zero as (2^521-1) + x >= 2^522
972 is_greater = out[8] >> 57;
973 is_greater |= is_greater << 32;
974 is_greater |= is_greater << 16;
975 is_greater |= is_greater << 8;
976 is_greater |= is_greater << 4;
977 is_greater |= is_greater << 2;
978 is_greater |= is_greater << 1;
979 is_greater = 0 - (is_greater >> 63);
981 out[0] -= kPrime[0] & is_greater;
982 out[1] -= kPrime[1] & is_greater;
983 out[2] -= kPrime[2] & is_greater;
984 out[3] -= kPrime[3] & is_greater;
985 out[4] -= kPrime[4] & is_greater;
986 out[5] -= kPrime[5] & is_greater;
987 out[6] -= kPrime[6] & is_greater;
988 out[7] -= kPrime[7] & is_greater;
989 out[8] -= kPrime[8] & is_greater;
991 /* Eliminate negative coefficients */
992 sign = -(out[0] >> 63);
993 out[0] += (two58 & sign);
994 out[1] -= (1 & sign);
995 sign = -(out[1] >> 63);
996 out[1] += (two58 & sign);
997 out[2] -= (1 & sign);
998 sign = -(out[2] >> 63);
999 out[2] += (two58 & sign);
1000 out[3] -= (1 & sign);
1001 sign = -(out[3] >> 63);
1002 out[3] += (two58 & sign);
1003 out[4] -= (1 & sign);
1004 sign = -(out[4] >> 63);
1005 out[4] += (two58 & sign);
1006 out[5] -= (1 & sign);
1007 sign = -(out[0] >> 63);
1008 out[5] += (two58 & sign);
1009 out[6] -= (1 & sign);
1010 sign = -(out[6] >> 63);
1011 out[6] += (two58 & sign);
1012 out[7] -= (1 & sign);
1013 sign = -(out[7] >> 63);
1014 out[7] += (two58 & sign);
1015 out[8] -= (1 & sign);
1016 sign = -(out[5] >> 63);
1017 out[5] += (two58 & sign);
1018 out[6] -= (1 & sign);
1019 sign = -(out[6] >> 63);
1020 out[6] += (two58 & sign);
1021 out[7] -= (1 & sign);
1022 sign = -(out[7] >> 63);
1023 out[7] += (two58 & sign);
1024 out[8] -= (1 & sign);
1031 * Building on top of the field operations we have the operations on the
1032 * elliptic curve group itself. Points on the curve are represented in Jacobian
1036 * point_double calculates 2*(x_in, y_in, z_in)
1038 * The method is taken from:
1039 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1041 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1042 * while x_out == y_in is not (maybe this works, but it's not tested). */
1044 point_double(felem x_out, felem y_out, felem z_out,
1045 const felem x_in, const felem y_in, const felem z_in)
1047 largefelem tmp, tmp2;
1048 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1050 felem_assign(ftmp, x_in);
1051 felem_assign(ftmp2, x_in);
1054 felem_square(tmp, z_in);
1055 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1058 felem_square(tmp, y_in);
1059 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1061 /* beta = x*gamma */
1062 felem_mul(tmp, x_in, gamma);
1063 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1065 /* alpha = 3*(x-delta)*(x+delta) */
1066 felem_diff64(ftmp, delta);
1067 /* ftmp[i] < 2^61 */
1068 felem_sum64(ftmp2, delta);
1069 /* ftmp2[i] < 2^60 + 2^15 */
1070 felem_scalar64(ftmp2, 3);
1071 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1072 felem_mul(tmp, ftmp, ftmp2);
1074 * tmp[i] < 17(3*2^121 + 3*2^76)
1075 * = 61*2^121 + 61*2^76
1076 * < 64*2^121 + 64*2^76
1080 felem_reduce(alpha, tmp);
1082 /* x' = alpha^2 - 8*beta */
1083 felem_square(tmp, alpha);
1085 * tmp[i] < 17*2^120 < 2^125
1087 felem_assign(ftmp, beta);
1088 felem_scalar64(ftmp, 8);
1089 /* ftmp[i] < 2^62 + 2^17 */
1090 felem_diff_128_64(tmp, ftmp);
1091 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1092 felem_reduce(x_out, tmp);
1094 /* z' = (y + z)^2 - gamma - delta */
1095 felem_sum64(delta, gamma);
1096 /* delta[i] < 2^60 + 2^15 */
1097 felem_assign(ftmp, y_in);
1098 felem_sum64(ftmp, z_in);
1099 /* ftmp[i] < 2^60 + 2^15 */
1100 felem_square(tmp, ftmp);
1102 * tmp[i] < 17(2^122) < 2^127
1104 felem_diff_128_64(tmp, delta);
1105 /* tmp[i] < 2^127 + 2^63 */
1106 felem_reduce(z_out, tmp);
1108 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1109 felem_scalar64(beta, 4);
1110 /* beta[i] < 2^61 + 2^16 */
1111 felem_diff64(beta, x_out);
1112 /* beta[i] < 2^61 + 2^60 + 2^16 */
1113 felem_mul(tmp, alpha, beta);
1115 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1116 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1117 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1120 felem_square(tmp2, gamma);
1122 * tmp2[i] < 17*(2^59 + 2^14)^2
1123 * = 17*(2^118 + 2^74 + 2^28)
1125 felem_scalar128(tmp2, 8);
1127 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1128 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1131 felem_diff128(tmp, tmp2);
1133 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1134 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1135 * 2^74 + 2^69 + 2^34 + 2^30
1138 felem_reduce(y_out, tmp);
1141 /* copy_conditional copies in to out iff mask is all ones. */
1142 static void copy_conditional(felem out, const felem in, limb mask)
1145 for (i = 0; i < NLIMBS; ++i) {
1146 const limb tmp = mask & (in[i] ^ out[i]);
1152 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1154 * The method is taken from
1155 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1156 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1158 * This function includes a branch for checking whether the two input points
1159 * are equal (while not equal to the point at infinity). See comment below
1162 static void point_add(felem x3, felem y3, felem z3,
1163 const felem x1, const felem y1, const felem z1,
1164 const int mixed, const felem x2, const felem y2,
1167 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1168 largefelem tmp, tmp2;
1169 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1171 z1_is_zero = felem_is_zero(z1);
1172 z2_is_zero = felem_is_zero(z2);
1174 /* ftmp = z1z1 = z1**2 */
1175 felem_square(tmp, z1);
1176 felem_reduce(ftmp, tmp);
1179 /* ftmp2 = z2z2 = z2**2 */
1180 felem_square(tmp, z2);
1181 felem_reduce(ftmp2, tmp);
1183 /* u1 = ftmp3 = x1*z2z2 */
1184 felem_mul(tmp, x1, ftmp2);
1185 felem_reduce(ftmp3, tmp);
1187 /* ftmp5 = z1 + z2 */
1188 felem_assign(ftmp5, z1);
1189 felem_sum64(ftmp5, z2);
1190 /* ftmp5[i] < 2^61 */
1192 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1193 felem_square(tmp, ftmp5);
1194 /* tmp[i] < 17*2^122 */
1195 felem_diff_128_64(tmp, ftmp);
1196 /* tmp[i] < 17*2^122 + 2^63 */
1197 felem_diff_128_64(tmp, ftmp2);
1198 /* tmp[i] < 17*2^122 + 2^64 */
1199 felem_reduce(ftmp5, tmp);
1201 /* ftmp2 = z2 * z2z2 */
1202 felem_mul(tmp, ftmp2, z2);
1203 felem_reduce(ftmp2, tmp);
1205 /* s1 = ftmp6 = y1 * z2**3 */
1206 felem_mul(tmp, y1, ftmp2);
1207 felem_reduce(ftmp6, tmp);
1210 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1213 /* u1 = ftmp3 = x1*z2z2 */
1214 felem_assign(ftmp3, x1);
1216 /* ftmp5 = 2*z1z2 */
1217 felem_scalar(ftmp5, z1, 2);
1219 /* s1 = ftmp6 = y1 * z2**3 */
1220 felem_assign(ftmp6, y1);
1224 felem_mul(tmp, x2, ftmp);
1225 /* tmp[i] < 17*2^120 */
1227 /* h = ftmp4 = u2 - u1 */
1228 felem_diff_128_64(tmp, ftmp3);
1229 /* tmp[i] < 17*2^120 + 2^63 */
1230 felem_reduce(ftmp4, tmp);
1232 x_equal = felem_is_zero(ftmp4);
1234 /* z_out = ftmp5 * h */
1235 felem_mul(tmp, ftmp5, ftmp4);
1236 felem_reduce(z_out, tmp);
1238 /* ftmp = z1 * z1z1 */
1239 felem_mul(tmp, ftmp, z1);
1240 felem_reduce(ftmp, tmp);
1242 /* s2 = tmp = y2 * z1**3 */
1243 felem_mul(tmp, y2, ftmp);
1244 /* tmp[i] < 17*2^120 */
1246 /* r = ftmp5 = (s2 - s1)*2 */
1247 felem_diff_128_64(tmp, ftmp6);
1248 /* tmp[i] < 17*2^120 + 2^63 */
1249 felem_reduce(ftmp5, tmp);
1250 y_equal = felem_is_zero(ftmp5);
1251 felem_scalar64(ftmp5, 2);
1252 /* ftmp5[i] < 2^61 */
1254 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1256 * This is obviously not constant-time but it will almost-never happen
1257 * for ECDH / ECDSA. The case where it can happen is during scalar-mult
1258 * where the intermediate value gets very close to the group order.
1259 * Since |ec_GFp_nistp_recode_scalar_bits| produces signed digits for
1260 * the scalar, it's possible for the intermediate value to be a small
1261 * negative multiple of the base point, and for the final signed digit
1262 * to be the same value. We believe that this only occurs for the scalar
1263 * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
1264 * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb
1265 * 71e913863f7, in that case the penultimate intermediate is -9G and
1266 * the final digit is also -9G. Since this only happens for a single
1267 * scalar, the timing leak is irrelevent. (Any attacker who wanted to
1268 * check whether a secret scalar was that exact value, can already do
1271 point_double(x3, y3, z3, x1, y1, z1);
1275 /* I = ftmp = (2h)**2 */
1276 felem_assign(ftmp, ftmp4);
1277 felem_scalar64(ftmp, 2);
1278 /* ftmp[i] < 2^61 */
1279 felem_square(tmp, ftmp);
1280 /* tmp[i] < 17*2^122 */
1281 felem_reduce(ftmp, tmp);
1283 /* J = ftmp2 = h * I */
1284 felem_mul(tmp, ftmp4, ftmp);
1285 felem_reduce(ftmp2, tmp);
1287 /* V = ftmp4 = U1 * I */
1288 felem_mul(tmp, ftmp3, ftmp);
1289 felem_reduce(ftmp4, tmp);
1291 /* x_out = r**2 - J - 2V */
1292 felem_square(tmp, ftmp5);
1293 /* tmp[i] < 17*2^122 */
1294 felem_diff_128_64(tmp, ftmp2);
1295 /* tmp[i] < 17*2^122 + 2^63 */
1296 felem_assign(ftmp3, ftmp4);
1297 felem_scalar64(ftmp4, 2);
1298 /* ftmp4[i] < 2^61 */
1299 felem_diff_128_64(tmp, ftmp4);
1300 /* tmp[i] < 17*2^122 + 2^64 */
1301 felem_reduce(x_out, tmp);
1303 /* y_out = r(V-x_out) - 2 * s1 * J */
1304 felem_diff64(ftmp3, x_out);
1306 * ftmp3[i] < 2^60 + 2^60 = 2^61
1308 felem_mul(tmp, ftmp5, ftmp3);
1309 /* tmp[i] < 17*2^122 */
1310 felem_mul(tmp2, ftmp6, ftmp2);
1311 /* tmp2[i] < 17*2^120 */
1312 felem_scalar128(tmp2, 2);
1313 /* tmp2[i] < 17*2^121 */
1314 felem_diff128(tmp, tmp2);
1316 * tmp[i] < 2^127 - 2^69 + 17*2^122
1317 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1320 felem_reduce(y_out, tmp);
1322 copy_conditional(x_out, x2, z1_is_zero);
1323 copy_conditional(x_out, x1, z2_is_zero);
1324 copy_conditional(y_out, y2, z1_is_zero);
1325 copy_conditional(y_out, y1, z2_is_zero);
1326 copy_conditional(z_out, z2, z1_is_zero);
1327 copy_conditional(z_out, z1, z2_is_zero);
1328 felem_assign(x3, x_out);
1329 felem_assign(y3, y_out);
1330 felem_assign(z3, z_out);
1334 * Base point pre computation
1335 * --------------------------
1337 * Two different sorts of precomputed tables are used in the following code.
1338 * Each contain various points on the curve, where each point is three field
1339 * elements (x, y, z).
1341 * For the base point table, z is usually 1 (0 for the point at infinity).
1342 * This table has 16 elements:
1343 * index | bits | point
1344 * ------+---------+------------------------------
1347 * 2 | 0 0 1 0 | 2^130G
1348 * 3 | 0 0 1 1 | (2^130 + 1)G
1349 * 4 | 0 1 0 0 | 2^260G
1350 * 5 | 0 1 0 1 | (2^260 + 1)G
1351 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1352 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1353 * 8 | 1 0 0 0 | 2^390G
1354 * 9 | 1 0 0 1 | (2^390 + 1)G
1355 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1356 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1357 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1358 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1359 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1360 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1362 * The reason for this is so that we can clock bits into four different
1363 * locations when doing simple scalar multiplies against the base point.
1365 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1367 /* gmul is the table of precomputed base points */
1368 static const felem gmul[16][3] = {
1369 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1370 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1371 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1372 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1373 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1374 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1375 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1376 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1377 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1378 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1379 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1380 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1381 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1382 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1383 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1384 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1385 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1386 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1387 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1388 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1389 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1390 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1391 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1392 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1393 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1394 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1395 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1396 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1397 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1398 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1399 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1400 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1401 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1402 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1403 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1404 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1405 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1406 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1407 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1408 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1409 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1410 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1411 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1412 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1413 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1414 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1415 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1416 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1417 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1418 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1419 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1420 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1421 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1422 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1423 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1424 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1425 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1426 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1427 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1428 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1429 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1430 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1431 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1432 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1433 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1434 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1435 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1436 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1437 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1438 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1439 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1440 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1441 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1442 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1443 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1444 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1445 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1446 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1447 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1448 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1449 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1450 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1451 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1452 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1453 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1454 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1455 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1456 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1457 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1458 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1459 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1460 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1461 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1462 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1463 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1464 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1465 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1466 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1467 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1468 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1469 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1470 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1471 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1472 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1473 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1474 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1475 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1476 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1480 * select_point selects the |idx|th point from a precomputation table and
1483 /* pre_comp below is of the size provided in |size| */
1484 static void select_point(const limb idx, unsigned int size,
1485 const felem pre_comp[][3], felem out[3])
1488 limb *outlimbs = &out[0][0];
1490 memset(out, 0, sizeof(*out) * 3);
1492 for (i = 0; i < size; i++) {
1493 const limb *inlimbs = &pre_comp[i][0][0];
1494 limb mask = i ^ idx;
1500 for (j = 0; j < NLIMBS * 3; j++)
1501 outlimbs[j] |= inlimbs[j] & mask;
1505 /* get_bit returns the |i|th bit in |in| */
1506 static char get_bit(const felem_bytearray in, int i)
1510 return (in[i >> 3] >> (i & 7)) & 1;
1514 * Interleaved point multiplication using precomputed point multiples: The
1515 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1516 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1517 * generator, using certain (large) precomputed multiples in g_pre_comp.
1518 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1520 static void batch_mul(felem x_out, felem y_out, felem z_out,
1521 const felem_bytearray scalars[],
1522 const unsigned num_points, const u8 *g_scalar,
1523 const int mixed, const felem pre_comp[][17][3],
1524 const felem g_pre_comp[16][3])
1527 unsigned num, gen_mul = (g_scalar != NULL);
1528 felem nq[3], tmp[4];
1532 /* set nq to the point at infinity */
1533 memset(nq, 0, sizeof(nq));
1536 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1537 * of the generator (last quarter of rounds) and additions of other
1538 * points multiples (every 5th round).
1540 skip = 1; /* save two point operations in the first
1542 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1545 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1547 /* add multiples of the generator */
1548 if (gen_mul && (i <= 130)) {
1549 bits = get_bit(g_scalar, i + 390) << 3;
1551 bits |= get_bit(g_scalar, i + 260) << 2;
1552 bits |= get_bit(g_scalar, i + 130) << 1;
1553 bits |= get_bit(g_scalar, i);
1555 /* select the point to add, in constant time */
1556 select_point(bits, 16, g_pre_comp, tmp);
1558 /* The 1 argument below is for "mixed" */
1559 point_add(nq[0], nq[1], nq[2],
1560 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1562 memcpy(nq, tmp, 3 * sizeof(felem));
1567 /* do other additions every 5 doublings */
1568 if (num_points && (i % 5 == 0)) {
1569 /* loop over all scalars */
1570 for (num = 0; num < num_points; ++num) {
1571 bits = get_bit(scalars[num], i + 4) << 5;
1572 bits |= get_bit(scalars[num], i + 3) << 4;
1573 bits |= get_bit(scalars[num], i + 2) << 3;
1574 bits |= get_bit(scalars[num], i + 1) << 2;
1575 bits |= get_bit(scalars[num], i) << 1;
1576 bits |= get_bit(scalars[num], i - 1);
1577 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1580 * select the point to add or subtract, in constant time
1582 select_point(digit, 17, pre_comp[num], tmp);
1583 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1585 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1588 point_add(nq[0], nq[1], nq[2],
1589 nq[0], nq[1], nq[2],
1590 mixed, tmp[0], tmp[1], tmp[2]);
1592 memcpy(nq, tmp, 3 * sizeof(felem));
1598 felem_assign(x_out, nq[0]);
1599 felem_assign(y_out, nq[1]);
1600 felem_assign(z_out, nq[2]);
1603 /* Precomputation for the group generator. */
1604 struct nistp521_pre_comp_st {
1605 felem g_pre_comp[16][3];
1606 CRYPTO_REF_COUNT references;
1607 CRYPTO_RWLOCK *lock;
1610 const EC_METHOD *EC_GFp_nistp521_method(void)
1612 static const EC_METHOD ret = {
1613 EC_FLAGS_DEFAULT_OCT,
1614 NID_X9_62_prime_field,
1615 ec_GFp_nistp521_group_init,
1616 ec_GFp_simple_group_finish,
1617 ec_GFp_simple_group_clear_finish,
1618 ec_GFp_nist_group_copy,
1619 ec_GFp_nistp521_group_set_curve,
1620 ec_GFp_simple_group_get_curve,
1621 ec_GFp_simple_group_get_degree,
1622 ec_group_simple_order_bits,
1623 ec_GFp_simple_group_check_discriminant,
1624 ec_GFp_simple_point_init,
1625 ec_GFp_simple_point_finish,
1626 ec_GFp_simple_point_clear_finish,
1627 ec_GFp_simple_point_copy,
1628 ec_GFp_simple_point_set_to_infinity,
1629 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1630 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1631 ec_GFp_simple_point_set_affine_coordinates,
1632 ec_GFp_nistp521_point_get_affine_coordinates,
1633 0 /* point_set_compressed_coordinates */ ,
1638 ec_GFp_simple_invert,
1639 ec_GFp_simple_is_at_infinity,
1640 ec_GFp_simple_is_on_curve,
1642 ec_GFp_simple_make_affine,
1643 ec_GFp_simple_points_make_affine,
1644 ec_GFp_nistp521_points_mul,
1645 ec_GFp_nistp521_precompute_mult,
1646 ec_GFp_nistp521_have_precompute_mult,
1647 ec_GFp_nist_field_mul,
1648 ec_GFp_nist_field_sqr,
1650 0 /* field_encode */ ,
1651 0 /* field_decode */ ,
1652 0, /* field_set_to_one */
1653 ec_key_simple_priv2oct,
1654 ec_key_simple_oct2priv,
1655 0, /* set private */
1656 ec_key_simple_generate_key,
1657 ec_key_simple_check_key,
1658 ec_key_simple_generate_public_key,
1661 ecdh_simple_compute_key,
1662 0, /* field_inverse_mod_ord */
1663 0, /* blind_coordinates */
1665 0, /* ladder_step */
1672 /******************************************************************************/
1674 * FUNCTIONS TO MANAGE PRECOMPUTATION
1677 static NISTP521_PRE_COMP *nistp521_pre_comp_new(void)
1679 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1682 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1686 ret->references = 1;
1688 ret->lock = CRYPTO_THREAD_lock_new();
1689 if (ret->lock == NULL) {
1690 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1697 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1701 CRYPTO_UP_REF(&p->references, &i, p->lock);
1705 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1712 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1713 REF_PRINT_COUNT("EC_nistp521", x);
1716 REF_ASSERT_ISNT(i < 0);
1718 CRYPTO_THREAD_lock_free(p->lock);
1722 /******************************************************************************/
1724 * OPENSSL EC_METHOD FUNCTIONS
1727 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1730 ret = ec_GFp_simple_group_init(group);
1731 group->a_is_minus3 = 1;
1735 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1736 const BIGNUM *a, const BIGNUM *b,
1740 BN_CTX *new_ctx = NULL;
1741 BIGNUM *curve_p, *curve_a, *curve_b;
1744 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1747 curve_p = BN_CTX_get(ctx);
1748 curve_a = BN_CTX_get(ctx);
1749 curve_b = BN_CTX_get(ctx);
1750 if (curve_b == NULL)
1752 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1753 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1754 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1755 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1756 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1757 EC_R_WRONG_CURVE_PARAMETERS);
1760 group->field_mod_func = BN_nist_mod_521;
1761 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1764 BN_CTX_free(new_ctx);
1769 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1772 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1773 const EC_POINT *point,
1774 BIGNUM *x, BIGNUM *y,
1777 felem z1, z2, x_in, y_in, x_out, y_out;
1780 if (EC_POINT_is_at_infinity(group, point)) {
1781 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1782 EC_R_POINT_AT_INFINITY);
1785 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1786 (!BN_to_felem(z1, point->Z)))
1789 felem_square(tmp, z2);
1790 felem_reduce(z1, tmp);
1791 felem_mul(tmp, x_in, z1);
1792 felem_reduce(x_in, tmp);
1793 felem_contract(x_out, x_in);
1795 if (!felem_to_BN(x, x_out)) {
1796 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1801 felem_mul(tmp, z1, z2);
1802 felem_reduce(z1, tmp);
1803 felem_mul(tmp, y_in, z1);
1804 felem_reduce(y_in, tmp);
1805 felem_contract(y_out, y_in);
1807 if (!felem_to_BN(y, y_out)) {
1808 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1816 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1817 static void make_points_affine(size_t num, felem points[][3],
1821 * Runs in constant time, unless an input is the point at infinity (which
1822 * normally shouldn't happen).
1824 ec_GFp_nistp_points_make_affine_internal(num,
1828 (void (*)(void *))felem_one,
1830 (void (*)(void *, const void *))
1832 (void (*)(void *, const void *))
1833 felem_square_reduce, (void (*)
1840 (void (*)(void *, const void *))
1842 (void (*)(void *, const void *))
1847 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1848 * values Result is stored in r (r can equal one of the inputs).
1850 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1851 const BIGNUM *scalar, size_t num,
1852 const EC_POINT *points[],
1853 const BIGNUM *scalars[], BN_CTX *ctx)
1858 BIGNUM *x, *y, *z, *tmp_scalar;
1859 felem_bytearray g_secret;
1860 felem_bytearray *secrets = NULL;
1861 felem (*pre_comp)[17][3] = NULL;
1862 felem *tmp_felems = NULL;
1863 felem_bytearray tmp;
1864 unsigned i, num_bytes;
1865 int have_pre_comp = 0;
1866 size_t num_points = num;
1867 felem x_in, y_in, z_in, x_out, y_out, z_out;
1868 NISTP521_PRE_COMP *pre = NULL;
1869 felem(*g_pre_comp)[3] = NULL;
1870 EC_POINT *generator = NULL;
1871 const EC_POINT *p = NULL;
1872 const BIGNUM *p_scalar = NULL;
1875 x = BN_CTX_get(ctx);
1876 y = BN_CTX_get(ctx);
1877 z = BN_CTX_get(ctx);
1878 tmp_scalar = BN_CTX_get(ctx);
1879 if (tmp_scalar == NULL)
1882 if (scalar != NULL) {
1883 pre = group->pre_comp.nistp521;
1885 /* we have precomputation, try to use it */
1886 g_pre_comp = &pre->g_pre_comp[0];
1888 /* try to use the standard precomputation */
1889 g_pre_comp = (felem(*)[3]) gmul;
1890 generator = EC_POINT_new(group);
1891 if (generator == NULL)
1893 /* get the generator from precomputation */
1894 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1895 !felem_to_BN(y, g_pre_comp[1][1]) ||
1896 !felem_to_BN(z, g_pre_comp[1][2])) {
1897 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1900 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1904 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1905 /* precomputation matches generator */
1909 * we don't have valid precomputation: treat the generator as a
1915 if (num_points > 0) {
1916 if (num_points >= 2) {
1918 * unless we precompute multiples for just one point, converting
1919 * those into affine form is time well spent
1923 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1924 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1927 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1928 if ((secrets == NULL) || (pre_comp == NULL)
1929 || (mixed && (tmp_felems == NULL))) {
1930 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1935 * we treat NULL scalars as 0, and NULL points as points at infinity,
1936 * i.e., they contribute nothing to the linear combination
1938 for (i = 0; i < num_points; ++i) {
1941 * we didn't have a valid precomputation, so we pick the
1945 p = EC_GROUP_get0_generator(group);
1948 /* the i^th point */
1951 p_scalar = scalars[i];
1953 if ((p_scalar != NULL) && (p != NULL)) {
1954 /* reduce scalar to 0 <= scalar < 2^521 */
1955 if ((BN_num_bits(p_scalar) > 521)
1956 || (BN_is_negative(p_scalar))) {
1958 * this is an unusual input, and we don't guarantee
1961 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1962 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1965 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1967 num_bytes = BN_bn2bin(p_scalar, tmp);
1968 flip_endian(secrets[i], tmp, num_bytes);
1969 /* precompute multiples */
1970 if ((!BN_to_felem(x_out, p->X)) ||
1971 (!BN_to_felem(y_out, p->Y)) ||
1972 (!BN_to_felem(z_out, p->Z)))
1974 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1975 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1976 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1977 for (j = 2; j <= 16; ++j) {
1979 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1980 pre_comp[i][j][2], pre_comp[i][1][0],
1981 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1982 pre_comp[i][j - 1][0],
1983 pre_comp[i][j - 1][1],
1984 pre_comp[i][j - 1][2]);
1986 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1987 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1988 pre_comp[i][j / 2][1],
1989 pre_comp[i][j / 2][2]);
1995 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1998 /* the scalar for the generator */
1999 if ((scalar != NULL) && (have_pre_comp)) {
2000 memset(g_secret, 0, sizeof(g_secret));
2001 /* reduce scalar to 0 <= scalar < 2^521 */
2002 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
2004 * this is an unusual input, and we don't guarantee
2007 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
2008 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2011 num_bytes = BN_bn2bin(tmp_scalar, tmp);
2013 num_bytes = BN_bn2bin(scalar, tmp);
2014 flip_endian(g_secret, tmp, num_bytes);
2015 /* do the multiplication with generator precomputation */
2016 batch_mul(x_out, y_out, z_out,
2017 (const felem_bytearray(*))secrets, num_points,
2019 mixed, (const felem(*)[17][3])pre_comp,
2020 (const felem(*)[3])g_pre_comp);
2022 /* do the multiplication without generator precomputation */
2023 batch_mul(x_out, y_out, z_out,
2024 (const felem_bytearray(*))secrets, num_points,
2025 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2026 /* reduce the output to its unique minimal representation */
2027 felem_contract(x_in, x_out);
2028 felem_contract(y_in, y_out);
2029 felem_contract(z_in, z_out);
2030 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2031 (!felem_to_BN(z, z_in))) {
2032 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2035 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2039 EC_POINT_free(generator);
2040 OPENSSL_free(secrets);
2041 OPENSSL_free(pre_comp);
2042 OPENSSL_free(tmp_felems);
2046 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2049 NISTP521_PRE_COMP *pre = NULL;
2051 BN_CTX *new_ctx = NULL;
2053 EC_POINT *generator = NULL;
2054 felem tmp_felems[16];
2056 /* throw away old precomputation */
2057 EC_pre_comp_free(group);
2059 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2062 x = BN_CTX_get(ctx);
2063 y = BN_CTX_get(ctx);
2066 /* get the generator */
2067 if (group->generator == NULL)
2069 generator = EC_POINT_new(group);
2070 if (generator == NULL)
2072 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2073 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2074 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
2076 if ((pre = nistp521_pre_comp_new()) == NULL)
2079 * if the generator is the standard one, use built-in precomputation
2081 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2082 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2085 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2086 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2087 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2089 /* compute 2^130*G, 2^260*G, 2^390*G */
2090 for (i = 1; i <= 4; i <<= 1) {
2091 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2092 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2093 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2094 for (j = 0; j < 129; ++j) {
2095 point_double(pre->g_pre_comp[2 * i][0],
2096 pre->g_pre_comp[2 * i][1],
2097 pre->g_pre_comp[2 * i][2],
2098 pre->g_pre_comp[2 * i][0],
2099 pre->g_pre_comp[2 * i][1],
2100 pre->g_pre_comp[2 * i][2]);
2103 /* g_pre_comp[0] is the point at infinity */
2104 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2105 /* the remaining multiples */
2106 /* 2^130*G + 2^260*G */
2107 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2108 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2109 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2110 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2111 pre->g_pre_comp[2][2]);
2112 /* 2^130*G + 2^390*G */
2113 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2114 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2115 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2116 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2117 pre->g_pre_comp[2][2]);
2118 /* 2^260*G + 2^390*G */
2119 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2120 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2121 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2122 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2123 pre->g_pre_comp[4][2]);
2124 /* 2^130*G + 2^260*G + 2^390*G */
2125 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2126 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2127 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2128 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2129 pre->g_pre_comp[2][2]);
2130 for (i = 1; i < 8; ++i) {
2131 /* odd multiples: add G */
2132 point_add(pre->g_pre_comp[2 * i + 1][0],
2133 pre->g_pre_comp[2 * i + 1][1],
2134 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2135 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2136 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2137 pre->g_pre_comp[1][2]);
2139 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2142 SETPRECOMP(group, nistp521, pre);
2147 EC_POINT_free(generator);
2148 BN_CTX_free(new_ctx);
2149 EC_nistp521_pre_comp_free(pre);
2153 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2155 return HAVEPRECOMP(group, nistp521);