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 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
173 static int BN_to_felem(felem out, const BIGNUM *bn)
175 felem_bytearray b_out;
178 if (BN_is_negative(bn)) {
179 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
182 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
184 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
187 bin66_to_felem(out, b_out);
191 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
192 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
194 felem_bytearray b_out;
195 felem_to_bin66(b_out, in);
196 return BN_lebin2bn(b_out, sizeof(b_out), out);
204 static void felem_one(felem out)
217 static void felem_assign(felem out, const felem in)
230 /* felem_sum64 sets out = out + in. */
231 static void felem_sum64(felem out, const felem in)
244 /* felem_scalar sets out = in * scalar */
245 static void felem_scalar(felem out, const felem in, limb scalar)
247 out[0] = in[0] * scalar;
248 out[1] = in[1] * scalar;
249 out[2] = in[2] * scalar;
250 out[3] = in[3] * scalar;
251 out[4] = in[4] * scalar;
252 out[5] = in[5] * scalar;
253 out[6] = in[6] * scalar;
254 out[7] = in[7] * scalar;
255 out[8] = in[8] * scalar;
258 /* felem_scalar64 sets out = out * scalar */
259 static void felem_scalar64(felem out, limb scalar)
272 /* felem_scalar128 sets out = out * scalar */
273 static void felem_scalar128(largefelem out, limb scalar)
287 * felem_neg sets |out| to |-in|
289 * in[i] < 2^59 + 2^14
293 static void felem_neg(felem out, const felem in)
295 /* In order to prevent underflow, we subtract from 0 mod p. */
296 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
297 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
299 out[0] = two62m3 - in[0];
300 out[1] = two62m2 - in[1];
301 out[2] = two62m2 - in[2];
302 out[3] = two62m2 - in[3];
303 out[4] = two62m2 - in[4];
304 out[5] = two62m2 - in[5];
305 out[6] = two62m2 - in[6];
306 out[7] = two62m2 - in[7];
307 out[8] = two62m2 - in[8];
311 * felem_diff64 subtracts |in| from |out|
313 * in[i] < 2^59 + 2^14
315 * out[i] < out[i] + 2^62
317 static void felem_diff64(felem out, const felem in)
320 * In order to prevent underflow, we add 0 mod p before subtracting.
322 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
323 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
325 out[0] += two62m3 - in[0];
326 out[1] += two62m2 - in[1];
327 out[2] += two62m2 - in[2];
328 out[3] += two62m2 - in[3];
329 out[4] += two62m2 - in[4];
330 out[5] += two62m2 - in[5];
331 out[6] += two62m2 - in[6];
332 out[7] += two62m2 - in[7];
333 out[8] += two62m2 - in[8];
337 * felem_diff_128_64 subtracts |in| from |out|
339 * in[i] < 2^62 + 2^17
341 * out[i] < out[i] + 2^63
343 static void felem_diff_128_64(largefelem out, const felem in)
346 * In order to prevent underflow, we add 64p mod p (which is equivalent
347 * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521
348 * digit number with all bits set to 1. See "The representation of field
349 * elements" comment above for a description of how limbs are used to
350 * represent a number. 64p is represented with 8 limbs containing a number
351 * with 58 bits set and one limb with a number with 57 bits set.
353 static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6);
354 static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5);
356 out[0] += two63m6 - in[0];
357 out[1] += two63m5 - in[1];
358 out[2] += two63m5 - in[2];
359 out[3] += two63m5 - in[3];
360 out[4] += two63m5 - in[4];
361 out[5] += two63m5 - in[5];
362 out[6] += two63m5 - in[6];
363 out[7] += two63m5 - in[7];
364 out[8] += two63m5 - in[8];
368 * felem_diff_128_64 subtracts |in| from |out|
372 * out[i] < out[i] + 2^127 - 2^69
374 static void felem_diff128(largefelem out, const largefelem in)
377 * In order to prevent underflow, we add 0 mod p before subtracting.
379 static const uint128_t two127m70 =
380 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
381 static const uint128_t two127m69 =
382 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
384 out[0] += (two127m70 - in[0]);
385 out[1] += (two127m69 - in[1]);
386 out[2] += (two127m69 - in[2]);
387 out[3] += (two127m69 - in[3]);
388 out[4] += (two127m69 - in[4]);
389 out[5] += (two127m69 - in[5]);
390 out[6] += (two127m69 - in[6]);
391 out[7] += (two127m69 - in[7]);
392 out[8] += (two127m69 - in[8]);
396 * felem_square sets |out| = |in|^2
400 * out[i] < 17 * max(in[i]) * max(in[i])
402 static void felem_square(largefelem out, const felem in)
405 felem_scalar(inx2, in, 2);
406 felem_scalar(inx4, in, 4);
409 * We have many cases were we want to do
412 * This is obviously just
414 * However, rather than do the doubling on the 128 bit result, we
415 * double one of the inputs to the multiplication by reading from
419 out[0] = ((uint128_t) in[0]) * in[0];
420 out[1] = ((uint128_t) in[0]) * inx2[1];
421 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
422 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
423 out[4] = ((uint128_t) in[0]) * inx2[4] +
424 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
425 out[5] = ((uint128_t) in[0]) * inx2[5] +
426 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
427 out[6] = ((uint128_t) in[0]) * inx2[6] +
428 ((uint128_t) in[1]) * inx2[5] +
429 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
430 out[7] = ((uint128_t) in[0]) * inx2[7] +
431 ((uint128_t) in[1]) * inx2[6] +
432 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
433 out[8] = ((uint128_t) in[0]) * inx2[8] +
434 ((uint128_t) in[1]) * inx2[7] +
435 ((uint128_t) in[2]) * inx2[6] +
436 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
439 * The remaining limbs fall above 2^521, with the first falling at 2^522.
440 * They correspond to locations one bit up from the limbs produced above
441 * so we would have to multiply by two to align them. Again, rather than
442 * operate on the 128-bit result, we double one of the inputs to the
443 * multiplication. If we want to double for both this reason, and the
444 * reason above, then we end up multiplying by four.
448 out[0] += ((uint128_t) in[1]) * inx4[8] +
449 ((uint128_t) in[2]) * inx4[7] +
450 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
453 out[1] += ((uint128_t) in[2]) * inx4[8] +
454 ((uint128_t) in[3]) * inx4[7] +
455 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
458 out[2] += ((uint128_t) in[3]) * inx4[8] +
459 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
462 out[3] += ((uint128_t) in[4]) * inx4[8] +
463 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
466 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
469 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
472 out[6] += ((uint128_t) in[7]) * inx4[8];
475 out[7] += ((uint128_t) in[8]) * inx2[8];
479 * felem_mul sets |out| = |in1| * |in2|
484 * out[i] < 17 * max(in1[i]) * max(in2[i])
486 static void felem_mul(largefelem out, const felem in1, const felem in2)
489 felem_scalar(in2x2, in2, 2);
491 out[0] = ((uint128_t) in1[0]) * in2[0];
493 out[1] = ((uint128_t) in1[0]) * in2[1] +
494 ((uint128_t) in1[1]) * in2[0];
496 out[2] = ((uint128_t) in1[0]) * in2[2] +
497 ((uint128_t) in1[1]) * in2[1] +
498 ((uint128_t) in1[2]) * in2[0];
500 out[3] = ((uint128_t) in1[0]) * in2[3] +
501 ((uint128_t) in1[1]) * in2[2] +
502 ((uint128_t) in1[2]) * in2[1] +
503 ((uint128_t) in1[3]) * in2[0];
505 out[4] = ((uint128_t) in1[0]) * in2[4] +
506 ((uint128_t) in1[1]) * in2[3] +
507 ((uint128_t) in1[2]) * in2[2] +
508 ((uint128_t) in1[3]) * in2[1] +
509 ((uint128_t) in1[4]) * in2[0];
511 out[5] = ((uint128_t) in1[0]) * in2[5] +
512 ((uint128_t) in1[1]) * in2[4] +
513 ((uint128_t) in1[2]) * in2[3] +
514 ((uint128_t) in1[3]) * in2[2] +
515 ((uint128_t) in1[4]) * in2[1] +
516 ((uint128_t) in1[5]) * in2[0];
518 out[6] = ((uint128_t) in1[0]) * in2[6] +
519 ((uint128_t) in1[1]) * in2[5] +
520 ((uint128_t) in1[2]) * in2[4] +
521 ((uint128_t) in1[3]) * in2[3] +
522 ((uint128_t) in1[4]) * in2[2] +
523 ((uint128_t) in1[5]) * in2[1] +
524 ((uint128_t) in1[6]) * in2[0];
526 out[7] = ((uint128_t) in1[0]) * in2[7] +
527 ((uint128_t) in1[1]) * in2[6] +
528 ((uint128_t) in1[2]) * in2[5] +
529 ((uint128_t) in1[3]) * in2[4] +
530 ((uint128_t) in1[4]) * in2[3] +
531 ((uint128_t) in1[5]) * in2[2] +
532 ((uint128_t) in1[6]) * in2[1] +
533 ((uint128_t) in1[7]) * in2[0];
535 out[8] = ((uint128_t) in1[0]) * in2[8] +
536 ((uint128_t) in1[1]) * in2[7] +
537 ((uint128_t) in1[2]) * in2[6] +
538 ((uint128_t) in1[3]) * in2[5] +
539 ((uint128_t) in1[4]) * in2[4] +
540 ((uint128_t) in1[5]) * in2[3] +
541 ((uint128_t) in1[6]) * in2[2] +
542 ((uint128_t) in1[7]) * in2[1] +
543 ((uint128_t) in1[8]) * in2[0];
545 /* See comment in felem_square about the use of in2x2 here */
547 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
548 ((uint128_t) in1[2]) * in2x2[7] +
549 ((uint128_t) in1[3]) * in2x2[6] +
550 ((uint128_t) in1[4]) * in2x2[5] +
551 ((uint128_t) in1[5]) * in2x2[4] +
552 ((uint128_t) in1[6]) * in2x2[3] +
553 ((uint128_t) in1[7]) * in2x2[2] +
554 ((uint128_t) in1[8]) * in2x2[1];
556 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
557 ((uint128_t) in1[3]) * in2x2[7] +
558 ((uint128_t) in1[4]) * in2x2[6] +
559 ((uint128_t) in1[5]) * in2x2[5] +
560 ((uint128_t) in1[6]) * in2x2[4] +
561 ((uint128_t) in1[7]) * in2x2[3] +
562 ((uint128_t) in1[8]) * in2x2[2];
564 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
565 ((uint128_t) in1[4]) * in2x2[7] +
566 ((uint128_t) in1[5]) * in2x2[6] +
567 ((uint128_t) in1[6]) * in2x2[5] +
568 ((uint128_t) in1[7]) * in2x2[4] +
569 ((uint128_t) in1[8]) * in2x2[3];
571 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
572 ((uint128_t) in1[5]) * in2x2[7] +
573 ((uint128_t) in1[6]) * in2x2[6] +
574 ((uint128_t) in1[7]) * in2x2[5] +
575 ((uint128_t) in1[8]) * in2x2[4];
577 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
578 ((uint128_t) in1[6]) * in2x2[7] +
579 ((uint128_t) in1[7]) * in2x2[6] +
580 ((uint128_t) in1[8]) * in2x2[5];
582 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
583 ((uint128_t) in1[7]) * in2x2[7] +
584 ((uint128_t) in1[8]) * in2x2[6];
586 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
587 ((uint128_t) in1[8]) * in2x2[7];
589 out[7] += ((uint128_t) in1[8]) * in2x2[8];
592 static const limb bottom52bits = 0xfffffffffffff;
595 * felem_reduce converts a largefelem to an felem.
599 * out[i] < 2^59 + 2^14
601 static void felem_reduce(felem out, const largefelem in)
603 u64 overflow1, overflow2;
605 out[0] = ((limb) in[0]) & bottom58bits;
606 out[1] = ((limb) in[1]) & bottom58bits;
607 out[2] = ((limb) in[2]) & bottom58bits;
608 out[3] = ((limb) in[3]) & bottom58bits;
609 out[4] = ((limb) in[4]) & bottom58bits;
610 out[5] = ((limb) in[5]) & bottom58bits;
611 out[6] = ((limb) in[6]) & bottom58bits;
612 out[7] = ((limb) in[7]) & bottom58bits;
613 out[8] = ((limb) in[8]) & bottom58bits;
617 out[1] += ((limb) in[0]) >> 58;
618 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
620 * out[1] < 2^58 + 2^6 + 2^58
623 out[2] += ((limb) (in[0] >> 64)) >> 52;
625 out[2] += ((limb) in[1]) >> 58;
626 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
627 out[3] += ((limb) (in[1] >> 64)) >> 52;
629 out[3] += ((limb) in[2]) >> 58;
630 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
631 out[4] += ((limb) (in[2] >> 64)) >> 52;
633 out[4] += ((limb) in[3]) >> 58;
634 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
635 out[5] += ((limb) (in[3] >> 64)) >> 52;
637 out[5] += ((limb) in[4]) >> 58;
638 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
639 out[6] += ((limb) (in[4] >> 64)) >> 52;
641 out[6] += ((limb) in[5]) >> 58;
642 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
643 out[7] += ((limb) (in[5] >> 64)) >> 52;
645 out[7] += ((limb) in[6]) >> 58;
646 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
647 out[8] += ((limb) (in[6] >> 64)) >> 52;
649 out[8] += ((limb) in[7]) >> 58;
650 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
652 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
655 overflow1 = ((limb) (in[7] >> 64)) >> 52;
657 overflow1 += ((limb) in[8]) >> 58;
658 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
659 overflow2 = ((limb) (in[8] >> 64)) >> 52;
661 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
662 overflow2 <<= 1; /* overflow2 < 2^13 */
664 out[0] += overflow1; /* out[0] < 2^60 */
665 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
667 out[1] += out[0] >> 58;
668 out[0] &= bottom58bits;
671 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
676 static void felem_square_reduce(felem out, const felem in)
679 felem_square(tmp, in);
680 felem_reduce(out, tmp);
683 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
686 felem_mul(tmp, in1, in2);
687 felem_reduce(out, tmp);
691 * felem_inv calculates |out| = |in|^{-1}
693 * Based on Fermat's Little Theorem:
695 * a^{p-1} = 1 (mod p)
696 * a^{p-2} = a^{-1} (mod p)
698 static void felem_inv(felem out, const felem in)
700 felem ftmp, ftmp2, ftmp3, ftmp4;
704 felem_square(tmp, in);
705 felem_reduce(ftmp, tmp); /* 2^1 */
706 felem_mul(tmp, in, ftmp);
707 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
708 felem_assign(ftmp2, ftmp);
709 felem_square(tmp, ftmp);
710 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
711 felem_mul(tmp, in, ftmp);
712 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
713 felem_square(tmp, ftmp);
714 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
716 felem_square(tmp, ftmp2);
717 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
718 felem_square(tmp, ftmp3);
719 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
720 felem_mul(tmp, ftmp3, ftmp2);
721 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
723 felem_assign(ftmp2, ftmp3);
724 felem_square(tmp, ftmp3);
725 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
726 felem_square(tmp, ftmp3);
727 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
728 felem_square(tmp, ftmp3);
729 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
730 felem_square(tmp, ftmp3);
731 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
732 felem_assign(ftmp4, ftmp3);
733 felem_mul(tmp, ftmp3, ftmp);
734 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
735 felem_square(tmp, ftmp4);
736 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
737 felem_mul(tmp, ftmp3, ftmp2);
738 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
739 felem_assign(ftmp2, ftmp3);
741 for (i = 0; i < 8; i++) {
742 felem_square(tmp, ftmp3);
743 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
745 felem_mul(tmp, ftmp3, ftmp2);
746 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
747 felem_assign(ftmp2, ftmp3);
749 for (i = 0; i < 16; i++) {
750 felem_square(tmp, ftmp3);
751 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
753 felem_mul(tmp, ftmp3, ftmp2);
754 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
755 felem_assign(ftmp2, ftmp3);
757 for (i = 0; i < 32; i++) {
758 felem_square(tmp, ftmp3);
759 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
761 felem_mul(tmp, ftmp3, ftmp2);
762 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
763 felem_assign(ftmp2, ftmp3);
765 for (i = 0; i < 64; i++) {
766 felem_square(tmp, ftmp3);
767 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
769 felem_mul(tmp, ftmp3, ftmp2);
770 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
771 felem_assign(ftmp2, ftmp3);
773 for (i = 0; i < 128; i++) {
774 felem_square(tmp, ftmp3);
775 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
777 felem_mul(tmp, ftmp3, ftmp2);
778 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
779 felem_assign(ftmp2, ftmp3);
781 for (i = 0; i < 256; i++) {
782 felem_square(tmp, ftmp3);
783 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
785 felem_mul(tmp, ftmp3, ftmp2);
786 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
788 for (i = 0; i < 9; i++) {
789 felem_square(tmp, ftmp3);
790 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
792 felem_mul(tmp, ftmp3, ftmp4);
793 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
794 felem_mul(tmp, ftmp3, in);
795 felem_reduce(out, tmp); /* 2^512 - 3 */
798 /* This is 2^521-1, expressed as an felem */
799 static const felem kPrime = {
800 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
801 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
802 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
806 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
809 * in[i] < 2^59 + 2^14
811 static limb felem_is_zero(const felem in)
815 felem_assign(ftmp, in);
817 ftmp[0] += ftmp[8] >> 57;
818 ftmp[8] &= bottom57bits;
820 ftmp[1] += ftmp[0] >> 58;
821 ftmp[0] &= bottom58bits;
822 ftmp[2] += ftmp[1] >> 58;
823 ftmp[1] &= bottom58bits;
824 ftmp[3] += ftmp[2] >> 58;
825 ftmp[2] &= bottom58bits;
826 ftmp[4] += ftmp[3] >> 58;
827 ftmp[3] &= bottom58bits;
828 ftmp[5] += ftmp[4] >> 58;
829 ftmp[4] &= bottom58bits;
830 ftmp[6] += ftmp[5] >> 58;
831 ftmp[5] &= bottom58bits;
832 ftmp[7] += ftmp[6] >> 58;
833 ftmp[6] &= bottom58bits;
834 ftmp[8] += ftmp[7] >> 58;
835 ftmp[7] &= bottom58bits;
836 /* ftmp[8] < 2^57 + 4 */
839 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
840 * than our bound for ftmp[8]. Therefore we only have to check if the
841 * zero is zero or 2^521-1.
857 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
858 * can be set is if is_zero was 0 before the decrement.
860 is_zero = 0 - (is_zero >> 63);
862 is_p = ftmp[0] ^ kPrime[0];
863 is_p |= ftmp[1] ^ kPrime[1];
864 is_p |= ftmp[2] ^ kPrime[2];
865 is_p |= ftmp[3] ^ kPrime[3];
866 is_p |= ftmp[4] ^ kPrime[4];
867 is_p |= ftmp[5] ^ kPrime[5];
868 is_p |= ftmp[6] ^ kPrime[6];
869 is_p |= ftmp[7] ^ kPrime[7];
870 is_p |= ftmp[8] ^ kPrime[8];
873 is_p = 0 - (is_p >> 63);
879 static int felem_is_zero_int(const void *in)
881 return (int)(felem_is_zero(in) & ((limb) 1));
885 * felem_contract converts |in| to its unique, minimal representation.
887 * in[i] < 2^59 + 2^14
889 static void felem_contract(felem out, const felem in)
891 limb is_p, is_greater, sign;
892 static const limb two58 = ((limb) 1) << 58;
894 felem_assign(out, in);
896 out[0] += out[8] >> 57;
897 out[8] &= bottom57bits;
899 out[1] += out[0] >> 58;
900 out[0] &= bottom58bits;
901 out[2] += out[1] >> 58;
902 out[1] &= bottom58bits;
903 out[3] += out[2] >> 58;
904 out[2] &= bottom58bits;
905 out[4] += out[3] >> 58;
906 out[3] &= bottom58bits;
907 out[5] += out[4] >> 58;
908 out[4] &= bottom58bits;
909 out[6] += out[5] >> 58;
910 out[5] &= bottom58bits;
911 out[7] += out[6] >> 58;
912 out[6] &= bottom58bits;
913 out[8] += out[7] >> 58;
914 out[7] &= bottom58bits;
915 /* out[8] < 2^57 + 4 */
918 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
919 * out. See the comments in felem_is_zero regarding why we don't test for
920 * other multiples of the prime.
924 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
927 is_p = out[0] ^ kPrime[0];
928 is_p |= out[1] ^ kPrime[1];
929 is_p |= out[2] ^ kPrime[2];
930 is_p |= out[3] ^ kPrime[3];
931 is_p |= out[4] ^ kPrime[4];
932 is_p |= out[5] ^ kPrime[5];
933 is_p |= out[6] ^ kPrime[6];
934 is_p |= out[7] ^ kPrime[7];
935 is_p |= out[8] ^ kPrime[8];
944 is_p = 0 - (is_p >> 63);
947 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
960 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
961 * 57 is greater than zero as (2^521-1) + x >= 2^522
963 is_greater = out[8] >> 57;
964 is_greater |= is_greater << 32;
965 is_greater |= is_greater << 16;
966 is_greater |= is_greater << 8;
967 is_greater |= is_greater << 4;
968 is_greater |= is_greater << 2;
969 is_greater |= is_greater << 1;
970 is_greater = 0 - (is_greater >> 63);
972 out[0] -= kPrime[0] & is_greater;
973 out[1] -= kPrime[1] & is_greater;
974 out[2] -= kPrime[2] & is_greater;
975 out[3] -= kPrime[3] & is_greater;
976 out[4] -= kPrime[4] & is_greater;
977 out[5] -= kPrime[5] & is_greater;
978 out[6] -= kPrime[6] & is_greater;
979 out[7] -= kPrime[7] & is_greater;
980 out[8] -= kPrime[8] & is_greater;
982 /* Eliminate negative coefficients */
983 sign = -(out[0] >> 63);
984 out[0] += (two58 & sign);
985 out[1] -= (1 & sign);
986 sign = -(out[1] >> 63);
987 out[1] += (two58 & sign);
988 out[2] -= (1 & sign);
989 sign = -(out[2] >> 63);
990 out[2] += (two58 & sign);
991 out[3] -= (1 & sign);
992 sign = -(out[3] >> 63);
993 out[3] += (two58 & sign);
994 out[4] -= (1 & sign);
995 sign = -(out[4] >> 63);
996 out[4] += (two58 & sign);
997 out[5] -= (1 & sign);
998 sign = -(out[0] >> 63);
999 out[5] += (two58 & sign);
1000 out[6] -= (1 & sign);
1001 sign = -(out[6] >> 63);
1002 out[6] += (two58 & sign);
1003 out[7] -= (1 & sign);
1004 sign = -(out[7] >> 63);
1005 out[7] += (two58 & sign);
1006 out[8] -= (1 & sign);
1007 sign = -(out[5] >> 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);
1022 * Building on top of the field operations we have the operations on the
1023 * elliptic curve group itself. Points on the curve are represented in Jacobian
1027 * point_double calculates 2*(x_in, y_in, z_in)
1029 * The method is taken from:
1030 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1032 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1033 * while x_out == y_in is not (maybe this works, but it's not tested). */
1035 point_double(felem x_out, felem y_out, felem z_out,
1036 const felem x_in, const felem y_in, const felem z_in)
1038 largefelem tmp, tmp2;
1039 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1041 felem_assign(ftmp, x_in);
1042 felem_assign(ftmp2, x_in);
1045 felem_square(tmp, z_in);
1046 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1049 felem_square(tmp, y_in);
1050 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1052 /* beta = x*gamma */
1053 felem_mul(tmp, x_in, gamma);
1054 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1056 /* alpha = 3*(x-delta)*(x+delta) */
1057 felem_diff64(ftmp, delta);
1058 /* ftmp[i] < 2^61 */
1059 felem_sum64(ftmp2, delta);
1060 /* ftmp2[i] < 2^60 + 2^15 */
1061 felem_scalar64(ftmp2, 3);
1062 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1063 felem_mul(tmp, ftmp, ftmp2);
1065 * tmp[i] < 17(3*2^121 + 3*2^76)
1066 * = 61*2^121 + 61*2^76
1067 * < 64*2^121 + 64*2^76
1071 felem_reduce(alpha, tmp);
1073 /* x' = alpha^2 - 8*beta */
1074 felem_square(tmp, alpha);
1076 * tmp[i] < 17*2^120 < 2^125
1078 felem_assign(ftmp, beta);
1079 felem_scalar64(ftmp, 8);
1080 /* ftmp[i] < 2^62 + 2^17 */
1081 felem_diff_128_64(tmp, ftmp);
1082 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1083 felem_reduce(x_out, tmp);
1085 /* z' = (y + z)^2 - gamma - delta */
1086 felem_sum64(delta, gamma);
1087 /* delta[i] < 2^60 + 2^15 */
1088 felem_assign(ftmp, y_in);
1089 felem_sum64(ftmp, z_in);
1090 /* ftmp[i] < 2^60 + 2^15 */
1091 felem_square(tmp, ftmp);
1093 * tmp[i] < 17(2^122) < 2^127
1095 felem_diff_128_64(tmp, delta);
1096 /* tmp[i] < 2^127 + 2^63 */
1097 felem_reduce(z_out, tmp);
1099 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1100 felem_scalar64(beta, 4);
1101 /* beta[i] < 2^61 + 2^16 */
1102 felem_diff64(beta, x_out);
1103 /* beta[i] < 2^61 + 2^60 + 2^16 */
1104 felem_mul(tmp, alpha, beta);
1106 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1107 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1108 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1111 felem_square(tmp2, gamma);
1113 * tmp2[i] < 17*(2^59 + 2^14)^2
1114 * = 17*(2^118 + 2^74 + 2^28)
1116 felem_scalar128(tmp2, 8);
1118 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1119 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1122 felem_diff128(tmp, tmp2);
1124 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1125 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1126 * 2^74 + 2^69 + 2^34 + 2^30
1129 felem_reduce(y_out, tmp);
1132 /* copy_conditional copies in to out iff mask is all ones. */
1133 static void copy_conditional(felem out, const felem in, limb mask)
1136 for (i = 0; i < NLIMBS; ++i) {
1137 const limb tmp = mask & (in[i] ^ out[i]);
1143 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1145 * The method is taken from
1146 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1147 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1149 * This function includes a branch for checking whether the two input points
1150 * are equal (while not equal to the point at infinity). See comment below
1153 static void point_add(felem x3, felem y3, felem z3,
1154 const felem x1, const felem y1, const felem z1,
1155 const int mixed, const felem x2, const felem y2,
1158 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1159 largefelem tmp, tmp2;
1160 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1162 z1_is_zero = felem_is_zero(z1);
1163 z2_is_zero = felem_is_zero(z2);
1165 /* ftmp = z1z1 = z1**2 */
1166 felem_square(tmp, z1);
1167 felem_reduce(ftmp, tmp);
1170 /* ftmp2 = z2z2 = z2**2 */
1171 felem_square(tmp, z2);
1172 felem_reduce(ftmp2, tmp);
1174 /* u1 = ftmp3 = x1*z2z2 */
1175 felem_mul(tmp, x1, ftmp2);
1176 felem_reduce(ftmp3, tmp);
1178 /* ftmp5 = z1 + z2 */
1179 felem_assign(ftmp5, z1);
1180 felem_sum64(ftmp5, z2);
1181 /* ftmp5[i] < 2^61 */
1183 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1184 felem_square(tmp, ftmp5);
1185 /* tmp[i] < 17*2^122 */
1186 felem_diff_128_64(tmp, ftmp);
1187 /* tmp[i] < 17*2^122 + 2^63 */
1188 felem_diff_128_64(tmp, ftmp2);
1189 /* tmp[i] < 17*2^122 + 2^64 */
1190 felem_reduce(ftmp5, tmp);
1192 /* ftmp2 = z2 * z2z2 */
1193 felem_mul(tmp, ftmp2, z2);
1194 felem_reduce(ftmp2, tmp);
1196 /* s1 = ftmp6 = y1 * z2**3 */
1197 felem_mul(tmp, y1, ftmp2);
1198 felem_reduce(ftmp6, tmp);
1201 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1204 /* u1 = ftmp3 = x1*z2z2 */
1205 felem_assign(ftmp3, x1);
1207 /* ftmp5 = 2*z1z2 */
1208 felem_scalar(ftmp5, z1, 2);
1210 /* s1 = ftmp6 = y1 * z2**3 */
1211 felem_assign(ftmp6, y1);
1215 felem_mul(tmp, x2, ftmp);
1216 /* tmp[i] < 17*2^120 */
1218 /* h = ftmp4 = u2 - u1 */
1219 felem_diff_128_64(tmp, ftmp3);
1220 /* tmp[i] < 17*2^120 + 2^63 */
1221 felem_reduce(ftmp4, tmp);
1223 x_equal = felem_is_zero(ftmp4);
1225 /* z_out = ftmp5 * h */
1226 felem_mul(tmp, ftmp5, ftmp4);
1227 felem_reduce(z_out, tmp);
1229 /* ftmp = z1 * z1z1 */
1230 felem_mul(tmp, ftmp, z1);
1231 felem_reduce(ftmp, tmp);
1233 /* s2 = tmp = y2 * z1**3 */
1234 felem_mul(tmp, y2, ftmp);
1235 /* tmp[i] < 17*2^120 */
1237 /* r = ftmp5 = (s2 - s1)*2 */
1238 felem_diff_128_64(tmp, ftmp6);
1239 /* tmp[i] < 17*2^120 + 2^63 */
1240 felem_reduce(ftmp5, tmp);
1241 y_equal = felem_is_zero(ftmp5);
1242 felem_scalar64(ftmp5, 2);
1243 /* ftmp5[i] < 2^61 */
1245 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1247 * This is obviously not constant-time but it will almost-never happen
1248 * for ECDH / ECDSA. The case where it can happen is during scalar-mult
1249 * where the intermediate value gets very close to the group order.
1250 * Since |ec_GFp_nistp_recode_scalar_bits| produces signed digits for
1251 * the scalar, it's possible for the intermediate value to be a small
1252 * negative multiple of the base point, and for the final signed digit
1253 * to be the same value. We believe that this only occurs for the scalar
1254 * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
1255 * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb
1256 * 71e913863f7, in that case the penultimate intermediate is -9G and
1257 * the final digit is also -9G. Since this only happens for a single
1258 * scalar, the timing leak is irrelevant. (Any attacker who wanted to
1259 * check whether a secret scalar was that exact value, can already do
1262 point_double(x3, y3, z3, x1, y1, z1);
1266 /* I = ftmp = (2h)**2 */
1267 felem_assign(ftmp, ftmp4);
1268 felem_scalar64(ftmp, 2);
1269 /* ftmp[i] < 2^61 */
1270 felem_square(tmp, ftmp);
1271 /* tmp[i] < 17*2^122 */
1272 felem_reduce(ftmp, tmp);
1274 /* J = ftmp2 = h * I */
1275 felem_mul(tmp, ftmp4, ftmp);
1276 felem_reduce(ftmp2, tmp);
1278 /* V = ftmp4 = U1 * I */
1279 felem_mul(tmp, ftmp3, ftmp);
1280 felem_reduce(ftmp4, tmp);
1282 /* x_out = r**2 - J - 2V */
1283 felem_square(tmp, ftmp5);
1284 /* tmp[i] < 17*2^122 */
1285 felem_diff_128_64(tmp, ftmp2);
1286 /* tmp[i] < 17*2^122 + 2^63 */
1287 felem_assign(ftmp3, ftmp4);
1288 felem_scalar64(ftmp4, 2);
1289 /* ftmp4[i] < 2^61 */
1290 felem_diff_128_64(tmp, ftmp4);
1291 /* tmp[i] < 17*2^122 + 2^64 */
1292 felem_reduce(x_out, tmp);
1294 /* y_out = r(V-x_out) - 2 * s1 * J */
1295 felem_diff64(ftmp3, x_out);
1297 * ftmp3[i] < 2^60 + 2^60 = 2^61
1299 felem_mul(tmp, ftmp5, ftmp3);
1300 /* tmp[i] < 17*2^122 */
1301 felem_mul(tmp2, ftmp6, ftmp2);
1302 /* tmp2[i] < 17*2^120 */
1303 felem_scalar128(tmp2, 2);
1304 /* tmp2[i] < 17*2^121 */
1305 felem_diff128(tmp, tmp2);
1307 * tmp[i] < 2^127 - 2^69 + 17*2^122
1308 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1311 felem_reduce(y_out, tmp);
1313 copy_conditional(x_out, x2, z1_is_zero);
1314 copy_conditional(x_out, x1, z2_is_zero);
1315 copy_conditional(y_out, y2, z1_is_zero);
1316 copy_conditional(y_out, y1, z2_is_zero);
1317 copy_conditional(z_out, z2, z1_is_zero);
1318 copy_conditional(z_out, z1, z2_is_zero);
1319 felem_assign(x3, x_out);
1320 felem_assign(y3, y_out);
1321 felem_assign(z3, z_out);
1325 * Base point pre computation
1326 * --------------------------
1328 * Two different sorts of precomputed tables are used in the following code.
1329 * Each contain various points on the curve, where each point is three field
1330 * elements (x, y, z).
1332 * For the base point table, z is usually 1 (0 for the point at infinity).
1333 * This table has 16 elements:
1334 * index | bits | point
1335 * ------+---------+------------------------------
1338 * 2 | 0 0 1 0 | 2^130G
1339 * 3 | 0 0 1 1 | (2^130 + 1)G
1340 * 4 | 0 1 0 0 | 2^260G
1341 * 5 | 0 1 0 1 | (2^260 + 1)G
1342 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1343 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1344 * 8 | 1 0 0 0 | 2^390G
1345 * 9 | 1 0 0 1 | (2^390 + 1)G
1346 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1347 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1348 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1349 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1350 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1351 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1353 * The reason for this is so that we can clock bits into four different
1354 * locations when doing simple scalar multiplies against the base point.
1356 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1358 /* gmul is the table of precomputed base points */
1359 static const felem gmul[16][3] = {
1360 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1361 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1362 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1363 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1364 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1365 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1366 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1367 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1368 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1369 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1370 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1371 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1372 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1373 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1374 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1375 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1376 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1377 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1378 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1379 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1380 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1381 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1382 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1383 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1384 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1385 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1386 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1387 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1388 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1389 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1390 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1391 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1392 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1393 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1394 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1395 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1396 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1397 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1398 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1399 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1400 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1401 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1402 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1403 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1404 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1405 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1406 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1407 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1408 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1409 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1410 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1411 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1412 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1413 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1414 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1415 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1416 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1417 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1418 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1419 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1420 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1421 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1422 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1423 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1424 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1425 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1426 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1427 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1428 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1429 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1430 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1431 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1432 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1433 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1434 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1435 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1436 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1437 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1438 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1439 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1440 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1441 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1442 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1443 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1444 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1445 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1446 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1447 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1448 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1449 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1450 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1451 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1452 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1453 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1454 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1455 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1456 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1457 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1458 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1459 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1460 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1461 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1462 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1463 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1464 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1465 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1466 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1467 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1471 * select_point selects the |idx|th point from a precomputation table and
1474 /* pre_comp below is of the size provided in |size| */
1475 static void select_point(const limb idx, unsigned int size,
1476 const felem pre_comp[][3], felem out[3])
1479 limb *outlimbs = &out[0][0];
1481 memset(out, 0, sizeof(*out) * 3);
1483 for (i = 0; i < size; i++) {
1484 const limb *inlimbs = &pre_comp[i][0][0];
1485 limb mask = i ^ idx;
1491 for (j = 0; j < NLIMBS * 3; j++)
1492 outlimbs[j] |= inlimbs[j] & mask;
1496 /* get_bit returns the |i|th bit in |in| */
1497 static char get_bit(const felem_bytearray in, int i)
1501 return (in[i >> 3] >> (i & 7)) & 1;
1505 * Interleaved point multiplication using precomputed point multiples: The
1506 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1507 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1508 * generator, using certain (large) precomputed multiples in g_pre_comp.
1509 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1511 static void batch_mul(felem x_out, felem y_out, felem z_out,
1512 const felem_bytearray scalars[],
1513 const unsigned num_points, const u8 *g_scalar,
1514 const int mixed, const felem pre_comp[][17][3],
1515 const felem g_pre_comp[16][3])
1518 unsigned num, gen_mul = (g_scalar != NULL);
1519 felem nq[3], tmp[4];
1523 /* set nq to the point at infinity */
1524 memset(nq, 0, sizeof(nq));
1527 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1528 * of the generator (last quarter of rounds) and additions of other
1529 * points multiples (every 5th round).
1531 skip = 1; /* save two point operations in the first
1533 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1536 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1538 /* add multiples of the generator */
1539 if (gen_mul && (i <= 130)) {
1540 bits = get_bit(g_scalar, i + 390) << 3;
1542 bits |= get_bit(g_scalar, i + 260) << 2;
1543 bits |= get_bit(g_scalar, i + 130) << 1;
1544 bits |= get_bit(g_scalar, i);
1546 /* select the point to add, in constant time */
1547 select_point(bits, 16, g_pre_comp, tmp);
1549 /* The 1 argument below is for "mixed" */
1550 point_add(nq[0], nq[1], nq[2],
1551 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1553 memcpy(nq, tmp, 3 * sizeof(felem));
1558 /* do other additions every 5 doublings */
1559 if (num_points && (i % 5 == 0)) {
1560 /* loop over all scalars */
1561 for (num = 0; num < num_points; ++num) {
1562 bits = get_bit(scalars[num], i + 4) << 5;
1563 bits |= get_bit(scalars[num], i + 3) << 4;
1564 bits |= get_bit(scalars[num], i + 2) << 3;
1565 bits |= get_bit(scalars[num], i + 1) << 2;
1566 bits |= get_bit(scalars[num], i) << 1;
1567 bits |= get_bit(scalars[num], i - 1);
1568 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1571 * select the point to add or subtract, in constant time
1573 select_point(digit, 17, pre_comp[num], tmp);
1574 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1576 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1579 point_add(nq[0], nq[1], nq[2],
1580 nq[0], nq[1], nq[2],
1581 mixed, tmp[0], tmp[1], tmp[2]);
1583 memcpy(nq, tmp, 3 * sizeof(felem));
1589 felem_assign(x_out, nq[0]);
1590 felem_assign(y_out, nq[1]);
1591 felem_assign(z_out, nq[2]);
1594 /* Precomputation for the group generator. */
1595 struct nistp521_pre_comp_st {
1596 felem g_pre_comp[16][3];
1597 CRYPTO_REF_COUNT references;
1598 CRYPTO_RWLOCK *lock;
1601 const EC_METHOD *EC_GFp_nistp521_method(void)
1603 static const EC_METHOD ret = {
1604 EC_FLAGS_DEFAULT_OCT,
1605 NID_X9_62_prime_field,
1606 ec_GFp_nistp521_group_init,
1607 ec_GFp_simple_group_finish,
1608 ec_GFp_simple_group_clear_finish,
1609 ec_GFp_nist_group_copy,
1610 ec_GFp_nistp521_group_set_curve,
1611 ec_GFp_simple_group_get_curve,
1612 ec_GFp_simple_group_get_degree,
1613 ec_group_simple_order_bits,
1614 ec_GFp_simple_group_check_discriminant,
1615 ec_GFp_simple_point_init,
1616 ec_GFp_simple_point_finish,
1617 ec_GFp_simple_point_clear_finish,
1618 ec_GFp_simple_point_copy,
1619 ec_GFp_simple_point_set_to_infinity,
1620 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1621 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1622 ec_GFp_simple_point_set_affine_coordinates,
1623 ec_GFp_nistp521_point_get_affine_coordinates,
1624 0 /* point_set_compressed_coordinates */ ,
1629 ec_GFp_simple_invert,
1630 ec_GFp_simple_is_at_infinity,
1631 ec_GFp_simple_is_on_curve,
1633 ec_GFp_simple_make_affine,
1634 ec_GFp_simple_points_make_affine,
1635 ec_GFp_nistp521_points_mul,
1636 ec_GFp_nistp521_precompute_mult,
1637 ec_GFp_nistp521_have_precompute_mult,
1638 ec_GFp_nist_field_mul,
1639 ec_GFp_nist_field_sqr,
1641 ec_GFp_simple_field_inv,
1642 0 /* field_encode */ ,
1643 0 /* field_decode */ ,
1644 0, /* field_set_to_one */
1645 ec_key_simple_priv2oct,
1646 ec_key_simple_oct2priv,
1647 0, /* set private */
1648 ec_key_simple_generate_key,
1649 ec_key_simple_check_key,
1650 ec_key_simple_generate_public_key,
1653 ecdh_simple_compute_key,
1654 ecdsa_simple_sign_setup,
1655 ecdsa_simple_sign_sig,
1656 ecdsa_simple_verify_sig,
1657 0, /* field_inverse_mod_ord */
1658 0, /* blind_coordinates */
1660 0, /* ladder_step */
1667 /******************************************************************************/
1669 * FUNCTIONS TO MANAGE PRECOMPUTATION
1672 static NISTP521_PRE_COMP *nistp521_pre_comp_new(void)
1674 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1677 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1681 ret->references = 1;
1683 ret->lock = CRYPTO_THREAD_lock_new();
1684 if (ret->lock == NULL) {
1685 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1692 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1696 CRYPTO_UP_REF(&p->references, &i, p->lock);
1700 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1707 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1708 REF_PRINT_COUNT("EC_nistp521", x);
1711 REF_ASSERT_ISNT(i < 0);
1713 CRYPTO_THREAD_lock_free(p->lock);
1717 /******************************************************************************/
1719 * OPENSSL EC_METHOD FUNCTIONS
1722 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1725 ret = ec_GFp_simple_group_init(group);
1726 group->a_is_minus3 = 1;
1730 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1731 const BIGNUM *a, const BIGNUM *b,
1735 BIGNUM *curve_p, *curve_a, *curve_b;
1737 BN_CTX *new_ctx = NULL;
1740 ctx = new_ctx = BN_CTX_new();
1746 curve_p = BN_CTX_get(ctx);
1747 curve_a = BN_CTX_get(ctx);
1748 curve_b = BN_CTX_get(ctx);
1749 if (curve_b == NULL)
1751 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1752 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1753 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1754 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1755 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1756 EC_R_WRONG_CURVE_PARAMETERS);
1759 group->field_mod_func = BN_nist_mod_521;
1760 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1764 BN_CTX_free(new_ctx);
1770 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1773 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1774 const EC_POINT *point,
1775 BIGNUM *x, BIGNUM *y,
1778 felem z1, z2, x_in, y_in, x_out, y_out;
1781 if (EC_POINT_is_at_infinity(group, point)) {
1782 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1783 EC_R_POINT_AT_INFINITY);
1786 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1787 (!BN_to_felem(z1, point->Z)))
1790 felem_square(tmp, z2);
1791 felem_reduce(z1, tmp);
1792 felem_mul(tmp, x_in, z1);
1793 felem_reduce(x_in, tmp);
1794 felem_contract(x_out, x_in);
1796 if (!felem_to_BN(x, x_out)) {
1797 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1802 felem_mul(tmp, z1, z2);
1803 felem_reduce(z1, tmp);
1804 felem_mul(tmp, y_in, z1);
1805 felem_reduce(y_in, tmp);
1806 felem_contract(y_out, y_in);
1808 if (!felem_to_BN(y, y_out)) {
1809 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1817 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1818 static void make_points_affine(size_t num, felem points[][3],
1822 * Runs in constant time, unless an input is the point at infinity (which
1823 * normally shouldn't happen).
1825 ec_GFp_nistp_points_make_affine_internal(num,
1829 (void (*)(void *))felem_one,
1831 (void (*)(void *, const void *))
1833 (void (*)(void *, const void *))
1834 felem_square_reduce, (void (*)
1841 (void (*)(void *, const void *))
1843 (void (*)(void *, const void *))
1848 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1849 * values Result is stored in r (r can equal one of the inputs).
1851 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1852 const BIGNUM *scalar, size_t num,
1853 const EC_POINT *points[],
1854 const BIGNUM *scalars[], BN_CTX *ctx)
1859 BIGNUM *x, *y, *z, *tmp_scalar;
1860 felem_bytearray g_secret;
1861 felem_bytearray *secrets = NULL;
1862 felem (*pre_comp)[17][3] = NULL;
1863 felem *tmp_felems = NULL;
1866 int have_pre_comp = 0;
1867 size_t num_points = num;
1868 felem x_in, y_in, z_in, x_out, y_out, z_out;
1869 NISTP521_PRE_COMP *pre = NULL;
1870 felem(*g_pre_comp)[3] = NULL;
1871 EC_POINT *generator = NULL;
1872 const EC_POINT *p = NULL;
1873 const BIGNUM *p_scalar = NULL;
1876 x = BN_CTX_get(ctx);
1877 y = BN_CTX_get(ctx);
1878 z = BN_CTX_get(ctx);
1879 tmp_scalar = BN_CTX_get(ctx);
1880 if (tmp_scalar == NULL)
1883 if (scalar != NULL) {
1884 pre = group->pre_comp.nistp521;
1886 /* we have precomputation, try to use it */
1887 g_pre_comp = &pre->g_pre_comp[0];
1889 /* try to use the standard precomputation */
1890 g_pre_comp = (felem(*)[3]) gmul;
1891 generator = EC_POINT_new(group);
1892 if (generator == NULL)
1894 /* get the generator from precomputation */
1895 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1896 !felem_to_BN(y, g_pre_comp[1][1]) ||
1897 !felem_to_BN(z, g_pre_comp[1][2])) {
1898 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1901 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1905 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1906 /* precomputation matches generator */
1910 * we don't have valid precomputation: treat the generator as a
1916 if (num_points > 0) {
1917 if (num_points >= 2) {
1919 * unless we precompute multiples for just one point, converting
1920 * those into affine form is time well spent
1924 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1925 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1928 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1929 if ((secrets == NULL) || (pre_comp == NULL)
1930 || (mixed && (tmp_felems == NULL))) {
1931 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1936 * we treat NULL scalars as 0, and NULL points as points at infinity,
1937 * i.e., they contribute nothing to the linear combination
1939 for (i = 0; i < num_points; ++i) {
1942 * we didn't have a valid precomputation, so we pick the
1945 p = EC_GROUP_get0_generator(group);
1948 /* the i^th point */
1950 p_scalar = scalars[i];
1952 if ((p_scalar != NULL) && (p != NULL)) {
1953 /* reduce scalar to 0 <= scalar < 2^521 */
1954 if ((BN_num_bits(p_scalar) > 521)
1955 || (BN_is_negative(p_scalar))) {
1957 * this is an unusual input, and we don't guarantee
1960 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1961 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1964 num_bytes = BN_bn2lebinpad(tmp_scalar,
1965 secrets[i], sizeof(secrets[i]));
1967 num_bytes = BN_bn2lebinpad(p_scalar,
1968 secrets[i], sizeof(secrets[i]));
1970 if (num_bytes < 0) {
1971 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1974 /* precompute multiples */
1975 if ((!BN_to_felem(x_out, p->X)) ||
1976 (!BN_to_felem(y_out, p->Y)) ||
1977 (!BN_to_felem(z_out, p->Z)))
1979 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1980 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1981 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1982 for (j = 2; j <= 16; ++j) {
1984 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1985 pre_comp[i][j][2], pre_comp[i][1][0],
1986 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1987 pre_comp[i][j - 1][0],
1988 pre_comp[i][j - 1][1],
1989 pre_comp[i][j - 1][2]);
1991 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1992 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1993 pre_comp[i][j / 2][1],
1994 pre_comp[i][j / 2][2]);
2000 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
2003 /* the scalar for the generator */
2004 if ((scalar != NULL) && (have_pre_comp)) {
2005 memset(g_secret, 0, sizeof(g_secret));
2006 /* reduce scalar to 0 <= scalar < 2^521 */
2007 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
2009 * this is an unusual input, and we don't guarantee
2012 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
2013 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2016 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
2018 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
2020 /* do the multiplication with generator precomputation */
2021 batch_mul(x_out, y_out, z_out,
2022 (const felem_bytearray(*))secrets, num_points,
2024 mixed, (const felem(*)[17][3])pre_comp,
2025 (const felem(*)[3])g_pre_comp);
2027 /* do the multiplication without generator precomputation */
2028 batch_mul(x_out, y_out, z_out,
2029 (const felem_bytearray(*))secrets, num_points,
2030 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2032 /* reduce the output to its unique minimal representation */
2033 felem_contract(x_in, x_out);
2034 felem_contract(y_in, y_out);
2035 felem_contract(z_in, z_out);
2036 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2037 (!felem_to_BN(z, z_in))) {
2038 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2041 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2045 EC_POINT_free(generator);
2046 OPENSSL_free(secrets);
2047 OPENSSL_free(pre_comp);
2048 OPENSSL_free(tmp_felems);
2052 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2055 NISTP521_PRE_COMP *pre = NULL;
2058 EC_POINT *generator = NULL;
2059 felem tmp_felems[16];
2061 BN_CTX *new_ctx = NULL;
2064 /* throw away old precomputation */
2065 EC_pre_comp_free(group);
2069 ctx = new_ctx = BN_CTX_new();
2075 x = BN_CTX_get(ctx);
2076 y = BN_CTX_get(ctx);
2079 /* get the generator */
2080 if (group->generator == NULL)
2082 generator = EC_POINT_new(group);
2083 if (generator == NULL)
2085 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2086 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2087 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
2089 if ((pre = nistp521_pre_comp_new()) == NULL)
2092 * if the generator is the standard one, use built-in precomputation
2094 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2095 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2098 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2099 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2100 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2102 /* compute 2^130*G, 2^260*G, 2^390*G */
2103 for (i = 1; i <= 4; i <<= 1) {
2104 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2105 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2106 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2107 for (j = 0; j < 129; ++j) {
2108 point_double(pre->g_pre_comp[2 * i][0],
2109 pre->g_pre_comp[2 * i][1],
2110 pre->g_pre_comp[2 * i][2],
2111 pre->g_pre_comp[2 * i][0],
2112 pre->g_pre_comp[2 * i][1],
2113 pre->g_pre_comp[2 * i][2]);
2116 /* g_pre_comp[0] is the point at infinity */
2117 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2118 /* the remaining multiples */
2119 /* 2^130*G + 2^260*G */
2120 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2121 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2122 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2123 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2124 pre->g_pre_comp[2][2]);
2125 /* 2^130*G + 2^390*G */
2126 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2127 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2128 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2129 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2130 pre->g_pre_comp[2][2]);
2131 /* 2^260*G + 2^390*G */
2132 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2133 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2134 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2135 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2136 pre->g_pre_comp[4][2]);
2137 /* 2^130*G + 2^260*G + 2^390*G */
2138 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2139 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2140 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2141 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2142 pre->g_pre_comp[2][2]);
2143 for (i = 1; i < 8; ++i) {
2144 /* odd multiples: add G */
2145 point_add(pre->g_pre_comp[2 * i + 1][0],
2146 pre->g_pre_comp[2 * i + 1][1],
2147 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2148 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2149 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2150 pre->g_pre_comp[1][2]);
2152 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2155 SETPRECOMP(group, nistp521, pre);
2160 EC_POINT_free(generator);
2162 BN_CTX_free(new_ctx);
2164 EC_nistp521_pre_comp_free(pre);
2168 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2170 return HAVEPRECOMP(group, nistp521);