2 * Copyright 2011-2016 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication
29 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
30 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
31 * work which got its smarts from Daniel J. Bernstein's work on the same.
34 #include <openssl/opensslconf.h>
35 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
36 NON_EMPTY_TRANSLATION_UNIT
39 # ifndef OPENSSL_SYS_VMS
42 # include <inttypes.h>
46 # include <openssl/err.h>
49 # if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
50 /* even with gcc, the typedef won't work for 32-bit platforms */
51 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
54 # error "Need GCC 3.1 or later to define type uint128_t"
62 * The underlying field. P521 operates over GF(2^521-1). We can serialise an
63 * element of this field into 66 bytes where the most significant byte
64 * contains only a single bit. We call this an felem_bytearray.
67 typedef u8 felem_bytearray[66];
70 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
71 * These values are big-endian.
73 static const felem_bytearray nistp521_curve_params[5] = {
74 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
75 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
76 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
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,
83 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
84 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
85 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
86 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
87 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
88 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
89 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
90 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
92 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
93 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
94 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
95 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
96 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
97 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
98 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
99 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
101 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
102 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
103 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
104 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
105 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
106 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
107 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
108 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
110 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
111 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
112 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
113 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
114 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
115 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
116 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
117 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
122 * The representation of field elements.
123 * ------------------------------------
125 * We represent field elements with nine values. These values are either 64 or
126 * 128 bits and the field element represented is:
127 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
128 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
129 * 58 bits apart, but are greater than 58 bits in length, the most significant
130 * bits of each limb overlap with the least significant bits of the next.
132 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
137 typedef uint64_t limb;
138 typedef limb felem[NLIMBS];
139 typedef uint128_t largefelem[NLIMBS];
141 static const limb bottom57bits = 0x1ffffffffffffff;
142 static const limb bottom58bits = 0x3ffffffffffffff;
145 * bin66_to_felem takes a little-endian byte array and converts it into felem
146 * form. This assumes that the CPU is little-endian.
148 static void bin66_to_felem(felem out, const u8 in[66])
150 out[0] = (*((limb *) & in[0])) & bottom58bits;
151 out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits;
152 out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits;
153 out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits;
154 out[4] = (*((limb *) & in[29])) & bottom58bits;
155 out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits;
156 out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits;
157 out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits;
158 out[8] = (*((limb *) & in[58])) & bottom57bits;
162 * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
163 * array. This assumes that the CPU is little-endian.
165 static void felem_to_bin66(u8 out[66], const felem in)
168 (*((limb *) & out[0])) = in[0];
169 (*((limb *) & out[7])) |= in[1] << 2;
170 (*((limb *) & out[14])) |= in[2] << 4;
171 (*((limb *) & out[21])) |= in[3] << 6;
172 (*((limb *) & out[29])) = in[4];
173 (*((limb *) & out[36])) |= in[5] << 2;
174 (*((limb *) & out[43])) |= in[6] << 4;
175 (*((limb *) & out[50])) |= in[7] << 6;
176 (*((limb *) & out[58])) = in[8];
179 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
180 static void flip_endian(u8 *out, const u8 *in, unsigned len)
183 for (i = 0; i < len; ++i)
184 out[i] = in[len - 1 - i];
187 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
188 static int BN_to_felem(felem out, const BIGNUM *bn)
190 felem_bytearray b_in;
191 felem_bytearray b_out;
194 /* BN_bn2bin eats leading zeroes */
195 memset(b_out, 0, sizeof(b_out));
196 num_bytes = BN_num_bytes(bn);
197 if (num_bytes > sizeof b_out) {
198 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
201 if (BN_is_negative(bn)) {
202 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
205 num_bytes = BN_bn2bin(bn, b_in);
206 flip_endian(b_out, b_in, num_bytes);
207 bin66_to_felem(out, b_out);
211 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
212 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
214 felem_bytearray b_in, b_out;
215 felem_to_bin66(b_in, in);
216 flip_endian(b_out, b_in, sizeof b_out);
217 return BN_bin2bn(b_out, sizeof b_out, out);
225 static void felem_one(felem out)
238 static void felem_assign(felem out, const felem in)
251 /* felem_sum64 sets out = out + in. */
252 static void felem_sum64(felem out, const felem in)
265 /* felem_scalar sets out = in * scalar */
266 static void felem_scalar(felem out, const felem in, limb scalar)
268 out[0] = in[0] * scalar;
269 out[1] = in[1] * scalar;
270 out[2] = in[2] * scalar;
271 out[3] = in[3] * scalar;
272 out[4] = in[4] * scalar;
273 out[5] = in[5] * scalar;
274 out[6] = in[6] * scalar;
275 out[7] = in[7] * scalar;
276 out[8] = in[8] * scalar;
279 /* felem_scalar64 sets out = out * scalar */
280 static void felem_scalar64(felem out, limb scalar)
293 /* felem_scalar128 sets out = out * scalar */
294 static void felem_scalar128(largefelem out, limb scalar)
308 * felem_neg sets |out| to |-in|
310 * in[i] < 2^59 + 2^14
314 static void felem_neg(felem out, const felem in)
316 /* In order to prevent underflow, we subtract from 0 mod p. */
317 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
318 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
320 out[0] = two62m3 - in[0];
321 out[1] = two62m2 - in[1];
322 out[2] = two62m2 - in[2];
323 out[3] = two62m2 - in[3];
324 out[4] = two62m2 - in[4];
325 out[5] = two62m2 - in[5];
326 out[6] = two62m2 - in[6];
327 out[7] = two62m2 - in[7];
328 out[8] = two62m2 - in[8];
332 * felem_diff64 subtracts |in| from |out|
334 * in[i] < 2^59 + 2^14
336 * out[i] < out[i] + 2^62
338 static void felem_diff64(felem out, const felem in)
341 * In order to prevent underflow, we add 0 mod p before subtracting.
343 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
344 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
346 out[0] += two62m3 - in[0];
347 out[1] += two62m2 - in[1];
348 out[2] += two62m2 - in[2];
349 out[3] += two62m2 - in[3];
350 out[4] += two62m2 - in[4];
351 out[5] += two62m2 - in[5];
352 out[6] += two62m2 - in[6];
353 out[7] += two62m2 - in[7];
354 out[8] += two62m2 - in[8];
358 * felem_diff_128_64 subtracts |in| from |out|
360 * in[i] < 2^62 + 2^17
362 * out[i] < out[i] + 2^63
364 static void felem_diff_128_64(largefelem out, const felem in)
367 * In order to prevent underflow, we add 0 mod p before subtracting.
369 static const limb two63m6 = (((limb) 1) << 62) - (((limb) 1) << 5);
370 static const limb two63m5 = (((limb) 1) << 62) - (((limb) 1) << 4);
372 out[0] += two63m6 - in[0];
373 out[1] += two63m5 - in[1];
374 out[2] += two63m5 - in[2];
375 out[3] += two63m5 - in[3];
376 out[4] += two63m5 - in[4];
377 out[5] += two63m5 - in[5];
378 out[6] += two63m5 - in[6];
379 out[7] += two63m5 - in[7];
380 out[8] += two63m5 - in[8];
384 * felem_diff_128_64 subtracts |in| from |out|
388 * out[i] < out[i] + 2^127 - 2^69
390 static void felem_diff128(largefelem out, const largefelem in)
393 * In order to prevent underflow, we add 0 mod p before subtracting.
395 static const uint128_t two127m70 =
396 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
397 static const uint128_t two127m69 =
398 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
400 out[0] += (two127m70 - in[0]);
401 out[1] += (two127m69 - in[1]);
402 out[2] += (two127m69 - in[2]);
403 out[3] += (two127m69 - in[3]);
404 out[4] += (two127m69 - in[4]);
405 out[5] += (two127m69 - in[5]);
406 out[6] += (two127m69 - in[6]);
407 out[7] += (two127m69 - in[7]);
408 out[8] += (two127m69 - in[8]);
412 * felem_square sets |out| = |in|^2
416 * out[i] < 17 * max(in[i]) * max(in[i])
418 static void felem_square(largefelem out, const felem in)
421 felem_scalar(inx2, in, 2);
422 felem_scalar(inx4, in, 4);
425 * We have many cases were we want to do
428 * This is obviously just
430 * However, rather than do the doubling on the 128 bit result, we
431 * double one of the inputs to the multiplication by reading from
435 out[0] = ((uint128_t) in[0]) * in[0];
436 out[1] = ((uint128_t) in[0]) * inx2[1];
437 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
438 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
439 out[4] = ((uint128_t) in[0]) * inx2[4] +
440 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
441 out[5] = ((uint128_t) in[0]) * inx2[5] +
442 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
443 out[6] = ((uint128_t) in[0]) * inx2[6] +
444 ((uint128_t) in[1]) * inx2[5] +
445 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
446 out[7] = ((uint128_t) in[0]) * inx2[7] +
447 ((uint128_t) in[1]) * inx2[6] +
448 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
449 out[8] = ((uint128_t) in[0]) * inx2[8] +
450 ((uint128_t) in[1]) * inx2[7] +
451 ((uint128_t) in[2]) * inx2[6] +
452 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
455 * The remaining limbs fall above 2^521, with the first falling at 2^522.
456 * They correspond to locations one bit up from the limbs produced above
457 * so we would have to multiply by two to align them. Again, rather than
458 * operate on the 128-bit result, we double one of the inputs to the
459 * multiplication. If we want to double for both this reason, and the
460 * reason above, then we end up multiplying by four.
464 out[0] += ((uint128_t) in[1]) * inx4[8] +
465 ((uint128_t) in[2]) * inx4[7] +
466 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
469 out[1] += ((uint128_t) in[2]) * inx4[8] +
470 ((uint128_t) in[3]) * inx4[7] +
471 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
474 out[2] += ((uint128_t) in[3]) * inx4[8] +
475 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
478 out[3] += ((uint128_t) in[4]) * inx4[8] +
479 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
482 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
485 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
488 out[6] += ((uint128_t) in[7]) * inx4[8];
491 out[7] += ((uint128_t) in[8]) * inx2[8];
495 * felem_mul sets |out| = |in1| * |in2|
500 * out[i] < 17 * max(in1[i]) * max(in2[i])
502 static void felem_mul(largefelem out, const felem in1, const felem in2)
505 felem_scalar(in2x2, in2, 2);
507 out[0] = ((uint128_t) in1[0]) * in2[0];
509 out[1] = ((uint128_t) in1[0]) * in2[1] +
510 ((uint128_t) in1[1]) * in2[0];
512 out[2] = ((uint128_t) in1[0]) * in2[2] +
513 ((uint128_t) in1[1]) * in2[1] +
514 ((uint128_t) in1[2]) * in2[0];
516 out[3] = ((uint128_t) in1[0]) * in2[3] +
517 ((uint128_t) in1[1]) * in2[2] +
518 ((uint128_t) in1[2]) * in2[1] +
519 ((uint128_t) in1[3]) * in2[0];
521 out[4] = ((uint128_t) in1[0]) * in2[4] +
522 ((uint128_t) in1[1]) * in2[3] +
523 ((uint128_t) in1[2]) * in2[2] +
524 ((uint128_t) in1[3]) * in2[1] +
525 ((uint128_t) in1[4]) * in2[0];
527 out[5] = ((uint128_t) in1[0]) * in2[5] +
528 ((uint128_t) in1[1]) * in2[4] +
529 ((uint128_t) in1[2]) * in2[3] +
530 ((uint128_t) in1[3]) * in2[2] +
531 ((uint128_t) in1[4]) * in2[1] +
532 ((uint128_t) in1[5]) * in2[0];
534 out[6] = ((uint128_t) in1[0]) * in2[6] +
535 ((uint128_t) in1[1]) * in2[5] +
536 ((uint128_t) in1[2]) * in2[4] +
537 ((uint128_t) in1[3]) * in2[3] +
538 ((uint128_t) in1[4]) * in2[2] +
539 ((uint128_t) in1[5]) * in2[1] +
540 ((uint128_t) in1[6]) * in2[0];
542 out[7] = ((uint128_t) in1[0]) * in2[7] +
543 ((uint128_t) in1[1]) * in2[6] +
544 ((uint128_t) in1[2]) * in2[5] +
545 ((uint128_t) in1[3]) * in2[4] +
546 ((uint128_t) in1[4]) * in2[3] +
547 ((uint128_t) in1[5]) * in2[2] +
548 ((uint128_t) in1[6]) * in2[1] +
549 ((uint128_t) in1[7]) * in2[0];
551 out[8] = ((uint128_t) in1[0]) * in2[8] +
552 ((uint128_t) in1[1]) * in2[7] +
553 ((uint128_t) in1[2]) * in2[6] +
554 ((uint128_t) in1[3]) * in2[5] +
555 ((uint128_t) in1[4]) * in2[4] +
556 ((uint128_t) in1[5]) * in2[3] +
557 ((uint128_t) in1[6]) * in2[2] +
558 ((uint128_t) in1[7]) * in2[1] +
559 ((uint128_t) in1[8]) * in2[0];
561 /* See comment in felem_square about the use of in2x2 here */
563 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
564 ((uint128_t) in1[2]) * in2x2[7] +
565 ((uint128_t) in1[3]) * in2x2[6] +
566 ((uint128_t) in1[4]) * in2x2[5] +
567 ((uint128_t) in1[5]) * in2x2[4] +
568 ((uint128_t) in1[6]) * in2x2[3] +
569 ((uint128_t) in1[7]) * in2x2[2] +
570 ((uint128_t) in1[8]) * in2x2[1];
572 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
573 ((uint128_t) in1[3]) * in2x2[7] +
574 ((uint128_t) in1[4]) * in2x2[6] +
575 ((uint128_t) in1[5]) * in2x2[5] +
576 ((uint128_t) in1[6]) * in2x2[4] +
577 ((uint128_t) in1[7]) * in2x2[3] +
578 ((uint128_t) in1[8]) * in2x2[2];
580 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
581 ((uint128_t) in1[4]) * in2x2[7] +
582 ((uint128_t) in1[5]) * in2x2[6] +
583 ((uint128_t) in1[6]) * in2x2[5] +
584 ((uint128_t) in1[7]) * in2x2[4] +
585 ((uint128_t) in1[8]) * in2x2[3];
587 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
588 ((uint128_t) in1[5]) * in2x2[7] +
589 ((uint128_t) in1[6]) * in2x2[6] +
590 ((uint128_t) in1[7]) * in2x2[5] +
591 ((uint128_t) in1[8]) * in2x2[4];
593 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
594 ((uint128_t) in1[6]) * in2x2[7] +
595 ((uint128_t) in1[7]) * in2x2[6] +
596 ((uint128_t) in1[8]) * in2x2[5];
598 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
599 ((uint128_t) in1[7]) * in2x2[7] +
600 ((uint128_t) in1[8]) * in2x2[6];
602 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
603 ((uint128_t) in1[8]) * in2x2[7];
605 out[7] += ((uint128_t) in1[8]) * in2x2[8];
608 static const limb bottom52bits = 0xfffffffffffff;
611 * felem_reduce converts a largefelem to an felem.
615 * out[i] < 2^59 + 2^14
617 static void felem_reduce(felem out, const largefelem in)
619 u64 overflow1, overflow2;
621 out[0] = ((limb) in[0]) & bottom58bits;
622 out[1] = ((limb) in[1]) & bottom58bits;
623 out[2] = ((limb) in[2]) & bottom58bits;
624 out[3] = ((limb) in[3]) & bottom58bits;
625 out[4] = ((limb) in[4]) & bottom58bits;
626 out[5] = ((limb) in[5]) & bottom58bits;
627 out[6] = ((limb) in[6]) & bottom58bits;
628 out[7] = ((limb) in[7]) & bottom58bits;
629 out[8] = ((limb) in[8]) & bottom58bits;
633 out[1] += ((limb) in[0]) >> 58;
634 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
636 * out[1] < 2^58 + 2^6 + 2^58
639 out[2] += ((limb) (in[0] >> 64)) >> 52;
641 out[2] += ((limb) in[1]) >> 58;
642 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
643 out[3] += ((limb) (in[1] >> 64)) >> 52;
645 out[3] += ((limb) in[2]) >> 58;
646 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
647 out[4] += ((limb) (in[2] >> 64)) >> 52;
649 out[4] += ((limb) in[3]) >> 58;
650 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
651 out[5] += ((limb) (in[3] >> 64)) >> 52;
653 out[5] += ((limb) in[4]) >> 58;
654 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
655 out[6] += ((limb) (in[4] >> 64)) >> 52;
657 out[6] += ((limb) in[5]) >> 58;
658 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
659 out[7] += ((limb) (in[5] >> 64)) >> 52;
661 out[7] += ((limb) in[6]) >> 58;
662 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
663 out[8] += ((limb) (in[6] >> 64)) >> 52;
665 out[8] += ((limb) in[7]) >> 58;
666 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
668 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
671 overflow1 = ((limb) (in[7] >> 64)) >> 52;
673 overflow1 += ((limb) in[8]) >> 58;
674 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
675 overflow2 = ((limb) (in[8] >> 64)) >> 52;
677 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
678 overflow2 <<= 1; /* overflow2 < 2^13 */
680 out[0] += overflow1; /* out[0] < 2^60 */
681 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
683 out[1] += out[0] >> 58;
684 out[0] &= bottom58bits;
687 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
692 static void felem_square_reduce(felem out, const felem in)
695 felem_square(tmp, in);
696 felem_reduce(out, tmp);
699 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
702 felem_mul(tmp, in1, in2);
703 felem_reduce(out, tmp);
707 * felem_inv calculates |out| = |in|^{-1}
709 * Based on Fermat's Little Theorem:
711 * a^{p-1} = 1 (mod p)
712 * a^{p-2} = a^{-1} (mod p)
714 static void felem_inv(felem out, const felem in)
716 felem ftmp, ftmp2, ftmp3, ftmp4;
720 felem_square(tmp, in);
721 felem_reduce(ftmp, tmp); /* 2^1 */
722 felem_mul(tmp, in, ftmp);
723 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
724 felem_assign(ftmp2, ftmp);
725 felem_square(tmp, ftmp);
726 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
727 felem_mul(tmp, in, ftmp);
728 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
729 felem_square(tmp, ftmp);
730 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
732 felem_square(tmp, ftmp2);
733 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
734 felem_square(tmp, ftmp3);
735 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
736 felem_mul(tmp, ftmp3, ftmp2);
737 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
739 felem_assign(ftmp2, ftmp3);
740 felem_square(tmp, ftmp3);
741 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
742 felem_square(tmp, ftmp3);
743 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
744 felem_square(tmp, ftmp3);
745 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
746 felem_square(tmp, ftmp3);
747 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
748 felem_assign(ftmp4, ftmp3);
749 felem_mul(tmp, ftmp3, ftmp);
750 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
751 felem_square(tmp, ftmp4);
752 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
753 felem_mul(tmp, ftmp3, ftmp2);
754 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
755 felem_assign(ftmp2, ftmp3);
757 for (i = 0; i < 8; i++) {
758 felem_square(tmp, ftmp3);
759 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
761 felem_mul(tmp, ftmp3, ftmp2);
762 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
763 felem_assign(ftmp2, ftmp3);
765 for (i = 0; i < 16; i++) {
766 felem_square(tmp, ftmp3);
767 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
769 felem_mul(tmp, ftmp3, ftmp2);
770 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
771 felem_assign(ftmp2, ftmp3);
773 for (i = 0; i < 32; i++) {
774 felem_square(tmp, ftmp3);
775 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
777 felem_mul(tmp, ftmp3, ftmp2);
778 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
779 felem_assign(ftmp2, ftmp3);
781 for (i = 0; i < 64; i++) {
782 felem_square(tmp, ftmp3);
783 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
785 felem_mul(tmp, ftmp3, ftmp2);
786 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
787 felem_assign(ftmp2, ftmp3);
789 for (i = 0; i < 128; i++) {
790 felem_square(tmp, ftmp3);
791 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
793 felem_mul(tmp, ftmp3, ftmp2);
794 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
795 felem_assign(ftmp2, ftmp3);
797 for (i = 0; i < 256; i++) {
798 felem_square(tmp, ftmp3);
799 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
801 felem_mul(tmp, ftmp3, ftmp2);
802 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
804 for (i = 0; i < 9; i++) {
805 felem_square(tmp, ftmp3);
806 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
808 felem_mul(tmp, ftmp3, ftmp4);
809 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
810 felem_mul(tmp, ftmp3, in);
811 felem_reduce(out, tmp); /* 2^512 - 3 */
814 /* This is 2^521-1, expressed as an felem */
815 static const felem kPrime = {
816 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
817 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
818 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
822 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
825 * in[i] < 2^59 + 2^14
827 static limb felem_is_zero(const felem in)
831 felem_assign(ftmp, in);
833 ftmp[0] += ftmp[8] >> 57;
834 ftmp[8] &= bottom57bits;
836 ftmp[1] += ftmp[0] >> 58;
837 ftmp[0] &= bottom58bits;
838 ftmp[2] += ftmp[1] >> 58;
839 ftmp[1] &= bottom58bits;
840 ftmp[3] += ftmp[2] >> 58;
841 ftmp[2] &= bottom58bits;
842 ftmp[4] += ftmp[3] >> 58;
843 ftmp[3] &= bottom58bits;
844 ftmp[5] += ftmp[4] >> 58;
845 ftmp[4] &= bottom58bits;
846 ftmp[6] += ftmp[5] >> 58;
847 ftmp[5] &= bottom58bits;
848 ftmp[7] += ftmp[6] >> 58;
849 ftmp[6] &= bottom58bits;
850 ftmp[8] += ftmp[7] >> 58;
851 ftmp[7] &= bottom58bits;
852 /* ftmp[8] < 2^57 + 4 */
855 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
856 * than our bound for ftmp[8]. Therefore we only have to check if the
857 * zero is zero or 2^521-1.
873 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
874 * can be set is if is_zero was 0 before the decrement.
876 is_zero = ((s64) is_zero) >> 63;
878 is_p = ftmp[0] ^ kPrime[0];
879 is_p |= ftmp[1] ^ kPrime[1];
880 is_p |= ftmp[2] ^ kPrime[2];
881 is_p |= ftmp[3] ^ kPrime[3];
882 is_p |= ftmp[4] ^ kPrime[4];
883 is_p |= ftmp[5] ^ kPrime[5];
884 is_p |= ftmp[6] ^ kPrime[6];
885 is_p |= ftmp[7] ^ kPrime[7];
886 is_p |= ftmp[8] ^ kPrime[8];
889 is_p = ((s64) is_p) >> 63;
895 static int felem_is_zero_int(const felem in)
897 return (int)(felem_is_zero(in) & ((limb) 1));
901 * felem_contract converts |in| to its unique, minimal representation.
903 * in[i] < 2^59 + 2^14
905 static void felem_contract(felem out, const felem in)
907 limb is_p, is_greater, sign;
908 static const limb two58 = ((limb) 1) << 58;
910 felem_assign(out, in);
912 out[0] += out[8] >> 57;
913 out[8] &= bottom57bits;
915 out[1] += out[0] >> 58;
916 out[0] &= bottom58bits;
917 out[2] += out[1] >> 58;
918 out[1] &= bottom58bits;
919 out[3] += out[2] >> 58;
920 out[2] &= bottom58bits;
921 out[4] += out[3] >> 58;
922 out[3] &= bottom58bits;
923 out[5] += out[4] >> 58;
924 out[4] &= bottom58bits;
925 out[6] += out[5] >> 58;
926 out[5] &= bottom58bits;
927 out[7] += out[6] >> 58;
928 out[6] &= bottom58bits;
929 out[8] += out[7] >> 58;
930 out[7] &= bottom58bits;
931 /* out[8] < 2^57 + 4 */
934 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
935 * out. See the comments in felem_is_zero regarding why we don't test for
936 * other multiples of the prime.
940 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
943 is_p = out[0] ^ kPrime[0];
944 is_p |= out[1] ^ kPrime[1];
945 is_p |= out[2] ^ kPrime[2];
946 is_p |= out[3] ^ kPrime[3];
947 is_p |= out[4] ^ kPrime[4];
948 is_p |= out[5] ^ kPrime[5];
949 is_p |= out[6] ^ kPrime[6];
950 is_p |= out[7] ^ kPrime[7];
951 is_p |= out[8] ^ kPrime[8];
960 is_p = ((s64) is_p) >> 63;
963 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
976 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
977 * 57 is greater than zero as (2^521-1) + x >= 2^522
979 is_greater = out[8] >> 57;
980 is_greater |= is_greater << 32;
981 is_greater |= is_greater << 16;
982 is_greater |= is_greater << 8;
983 is_greater |= is_greater << 4;
984 is_greater |= is_greater << 2;
985 is_greater |= is_greater << 1;
986 is_greater = ((s64) is_greater) >> 63;
988 out[0] -= kPrime[0] & is_greater;
989 out[1] -= kPrime[1] & is_greater;
990 out[2] -= kPrime[2] & is_greater;
991 out[3] -= kPrime[3] & is_greater;
992 out[4] -= kPrime[4] & is_greater;
993 out[5] -= kPrime[5] & is_greater;
994 out[6] -= kPrime[6] & is_greater;
995 out[7] -= kPrime[7] & is_greater;
996 out[8] -= kPrime[8] & is_greater;
998 /* Eliminate negative coefficients */
999 sign = -(out[0] >> 63);
1000 out[0] += (two58 & sign);
1001 out[1] -= (1 & sign);
1002 sign = -(out[1] >> 63);
1003 out[1] += (two58 & sign);
1004 out[2] -= (1 & sign);
1005 sign = -(out[2] >> 63);
1006 out[2] += (two58 & sign);
1007 out[3] -= (1 & sign);
1008 sign = -(out[3] >> 63);
1009 out[3] += (two58 & sign);
1010 out[4] -= (1 & sign);
1011 sign = -(out[4] >> 63);
1012 out[4] += (two58 & sign);
1013 out[5] -= (1 & sign);
1014 sign = -(out[0] >> 63);
1015 out[5] += (two58 & sign);
1016 out[6] -= (1 & sign);
1017 sign = -(out[6] >> 63);
1018 out[6] += (two58 & sign);
1019 out[7] -= (1 & sign);
1020 sign = -(out[7] >> 63);
1021 out[7] += (two58 & sign);
1022 out[8] -= (1 & sign);
1023 sign = -(out[5] >> 63);
1024 out[5] += (two58 & sign);
1025 out[6] -= (1 & sign);
1026 sign = -(out[6] >> 63);
1027 out[6] += (two58 & sign);
1028 out[7] -= (1 & sign);
1029 sign = -(out[7] >> 63);
1030 out[7] += (two58 & sign);
1031 out[8] -= (1 & sign);
1038 * Building on top of the field operations we have the operations on the
1039 * elliptic curve group itself. Points on the curve are represented in Jacobian
1043 * point_double calculates 2*(x_in, y_in, z_in)
1045 * The method is taken from:
1046 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1048 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1049 * while x_out == y_in is not (maybe this works, but it's not tested). */
1051 point_double(felem x_out, felem y_out, felem z_out,
1052 const felem x_in, const felem y_in, const felem z_in)
1054 largefelem tmp, tmp2;
1055 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1057 felem_assign(ftmp, x_in);
1058 felem_assign(ftmp2, x_in);
1061 felem_square(tmp, z_in);
1062 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1065 felem_square(tmp, y_in);
1066 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1068 /* beta = x*gamma */
1069 felem_mul(tmp, x_in, gamma);
1070 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1072 /* alpha = 3*(x-delta)*(x+delta) */
1073 felem_diff64(ftmp, delta);
1074 /* ftmp[i] < 2^61 */
1075 felem_sum64(ftmp2, delta);
1076 /* ftmp2[i] < 2^60 + 2^15 */
1077 felem_scalar64(ftmp2, 3);
1078 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1079 felem_mul(tmp, ftmp, ftmp2);
1081 * tmp[i] < 17(3*2^121 + 3*2^76)
1082 * = 61*2^121 + 61*2^76
1083 * < 64*2^121 + 64*2^76
1087 felem_reduce(alpha, tmp);
1089 /* x' = alpha^2 - 8*beta */
1090 felem_square(tmp, alpha);
1092 * tmp[i] < 17*2^120 < 2^125
1094 felem_assign(ftmp, beta);
1095 felem_scalar64(ftmp, 8);
1096 /* ftmp[i] < 2^62 + 2^17 */
1097 felem_diff_128_64(tmp, ftmp);
1098 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1099 felem_reduce(x_out, tmp);
1101 /* z' = (y + z)^2 - gamma - delta */
1102 felem_sum64(delta, gamma);
1103 /* delta[i] < 2^60 + 2^15 */
1104 felem_assign(ftmp, y_in);
1105 felem_sum64(ftmp, z_in);
1106 /* ftmp[i] < 2^60 + 2^15 */
1107 felem_square(tmp, ftmp);
1109 * tmp[i] < 17(2^122) < 2^127
1111 felem_diff_128_64(tmp, delta);
1112 /* tmp[i] < 2^127 + 2^63 */
1113 felem_reduce(z_out, tmp);
1115 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1116 felem_scalar64(beta, 4);
1117 /* beta[i] < 2^61 + 2^16 */
1118 felem_diff64(beta, x_out);
1119 /* beta[i] < 2^61 + 2^60 + 2^16 */
1120 felem_mul(tmp, alpha, beta);
1122 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1123 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1124 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1127 felem_square(tmp2, gamma);
1129 * tmp2[i] < 17*(2^59 + 2^14)^2
1130 * = 17*(2^118 + 2^74 + 2^28)
1132 felem_scalar128(tmp2, 8);
1134 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1135 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1138 felem_diff128(tmp, tmp2);
1140 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1141 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1142 * 2^74 + 2^69 + 2^34 + 2^30
1145 felem_reduce(y_out, tmp);
1148 /* copy_conditional copies in to out iff mask is all ones. */
1149 static void copy_conditional(felem out, const felem in, limb mask)
1152 for (i = 0; i < NLIMBS; ++i) {
1153 const limb tmp = mask & (in[i] ^ out[i]);
1159 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1161 * The method is taken from
1162 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1163 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1165 * This function includes a branch for checking whether the two input points
1166 * are equal (while not equal to the point at infinity). This case never
1167 * happens during single point multiplication, so there is no timing leak for
1168 * ECDH or ECDSA signing. */
1169 static void point_add(felem x3, felem y3, felem z3,
1170 const felem x1, const felem y1, const felem z1,
1171 const int mixed, const felem x2, const felem y2,
1174 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1175 largefelem tmp, tmp2;
1176 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1178 z1_is_zero = felem_is_zero(z1);
1179 z2_is_zero = felem_is_zero(z2);
1181 /* ftmp = z1z1 = z1**2 */
1182 felem_square(tmp, z1);
1183 felem_reduce(ftmp, tmp);
1186 /* ftmp2 = z2z2 = z2**2 */
1187 felem_square(tmp, z2);
1188 felem_reduce(ftmp2, tmp);
1190 /* u1 = ftmp3 = x1*z2z2 */
1191 felem_mul(tmp, x1, ftmp2);
1192 felem_reduce(ftmp3, tmp);
1194 /* ftmp5 = z1 + z2 */
1195 felem_assign(ftmp5, z1);
1196 felem_sum64(ftmp5, z2);
1197 /* ftmp5[i] < 2^61 */
1199 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1200 felem_square(tmp, ftmp5);
1201 /* tmp[i] < 17*2^122 */
1202 felem_diff_128_64(tmp, ftmp);
1203 /* tmp[i] < 17*2^122 + 2^63 */
1204 felem_diff_128_64(tmp, ftmp2);
1205 /* tmp[i] < 17*2^122 + 2^64 */
1206 felem_reduce(ftmp5, tmp);
1208 /* ftmp2 = z2 * z2z2 */
1209 felem_mul(tmp, ftmp2, z2);
1210 felem_reduce(ftmp2, tmp);
1212 /* s1 = ftmp6 = y1 * z2**3 */
1213 felem_mul(tmp, y1, ftmp2);
1214 felem_reduce(ftmp6, tmp);
1217 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1220 /* u1 = ftmp3 = x1*z2z2 */
1221 felem_assign(ftmp3, x1);
1223 /* ftmp5 = 2*z1z2 */
1224 felem_scalar(ftmp5, z1, 2);
1226 /* s1 = ftmp6 = y1 * z2**3 */
1227 felem_assign(ftmp6, y1);
1231 felem_mul(tmp, x2, ftmp);
1232 /* tmp[i] < 17*2^120 */
1234 /* h = ftmp4 = u2 - u1 */
1235 felem_diff_128_64(tmp, ftmp3);
1236 /* tmp[i] < 17*2^120 + 2^63 */
1237 felem_reduce(ftmp4, tmp);
1239 x_equal = felem_is_zero(ftmp4);
1241 /* z_out = ftmp5 * h */
1242 felem_mul(tmp, ftmp5, ftmp4);
1243 felem_reduce(z_out, tmp);
1245 /* ftmp = z1 * z1z1 */
1246 felem_mul(tmp, ftmp, z1);
1247 felem_reduce(ftmp, tmp);
1249 /* s2 = tmp = y2 * z1**3 */
1250 felem_mul(tmp, y2, ftmp);
1251 /* tmp[i] < 17*2^120 */
1253 /* r = ftmp5 = (s2 - s1)*2 */
1254 felem_diff_128_64(tmp, ftmp6);
1255 /* tmp[i] < 17*2^120 + 2^63 */
1256 felem_reduce(ftmp5, tmp);
1257 y_equal = felem_is_zero(ftmp5);
1258 felem_scalar64(ftmp5, 2);
1259 /* ftmp5[i] < 2^61 */
1261 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
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];
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 0 /* field_encode */ ,
1642 0 /* field_decode */ ,
1643 0, /* field_set_to_one */
1644 ec_key_simple_priv2oct,
1645 ec_key_simple_oct2priv,
1646 0, /* set private */
1647 ec_key_simple_generate_key,
1648 ec_key_simple_check_key,
1649 ec_key_simple_generate_public_key,
1652 ecdh_simple_compute_key
1658 /******************************************************************************/
1660 * FUNCTIONS TO MANAGE PRECOMPUTATION
1663 static NISTP521_PRE_COMP *nistp521_pre_comp_new()
1665 NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1668 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1672 ret->references = 1;
1674 ret->lock = CRYPTO_THREAD_lock_new();
1675 if (ret->lock == NULL) {
1676 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1683 NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
1687 CRYPTO_atomic_add(&p->references, 1, &i, p->lock);
1691 void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
1698 CRYPTO_atomic_add(&p->references, -1, &i, p->lock);
1699 REF_PRINT_COUNT("EC_nistp521", x);
1702 REF_ASSERT_ISNT(i < 0);
1704 CRYPTO_THREAD_lock_free(p->lock);
1708 /******************************************************************************/
1710 * OPENSSL EC_METHOD FUNCTIONS
1713 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1716 ret = ec_GFp_simple_group_init(group);
1717 group->a_is_minus3 = 1;
1721 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1722 const BIGNUM *a, const BIGNUM *b,
1726 BN_CTX *new_ctx = NULL;
1727 BIGNUM *curve_p, *curve_a, *curve_b;
1730 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1733 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1734 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1735 ((curve_b = BN_CTX_get(ctx)) == NULL))
1737 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1738 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1739 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1740 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1741 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1742 EC_R_WRONG_CURVE_PARAMETERS);
1745 group->field_mod_func = BN_nist_mod_521;
1746 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1749 BN_CTX_free(new_ctx);
1754 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1757 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1758 const EC_POINT *point,
1759 BIGNUM *x, BIGNUM *y,
1762 felem z1, z2, x_in, y_in, x_out, y_out;
1765 if (EC_POINT_is_at_infinity(group, point)) {
1766 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1767 EC_R_POINT_AT_INFINITY);
1770 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1771 (!BN_to_felem(z1, point->Z)))
1774 felem_square(tmp, z2);
1775 felem_reduce(z1, tmp);
1776 felem_mul(tmp, x_in, z1);
1777 felem_reduce(x_in, tmp);
1778 felem_contract(x_out, x_in);
1780 if (!felem_to_BN(x, x_out)) {
1781 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1786 felem_mul(tmp, z1, z2);
1787 felem_reduce(z1, tmp);
1788 felem_mul(tmp, y_in, z1);
1789 felem_reduce(y_in, tmp);
1790 felem_contract(y_out, y_in);
1792 if (!felem_to_BN(y, y_out)) {
1793 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1801 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1802 static void make_points_affine(size_t num, felem points[][3],
1806 * Runs in constant time, unless an input is the point at infinity (which
1807 * normally shouldn't happen).
1809 ec_GFp_nistp_points_make_affine_internal(num,
1813 (void (*)(void *))felem_one,
1814 (int (*)(const void *))
1816 (void (*)(void *, const void *))
1818 (void (*)(void *, const void *))
1819 felem_square_reduce, (void (*)
1826 (void (*)(void *, const void *))
1828 (void (*)(void *, const void *))
1833 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1834 * values Result is stored in r (r can equal one of the inputs).
1836 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1837 const BIGNUM *scalar, size_t num,
1838 const EC_POINT *points[],
1839 const BIGNUM *scalars[], BN_CTX *ctx)
1844 BN_CTX *new_ctx = NULL;
1845 BIGNUM *x, *y, *z, *tmp_scalar;
1846 felem_bytearray g_secret;
1847 felem_bytearray *secrets = NULL;
1848 felem (*pre_comp)[17][3] = NULL;
1849 felem *tmp_felems = NULL;
1850 felem_bytearray tmp;
1851 unsigned i, num_bytes;
1852 int have_pre_comp = 0;
1853 size_t num_points = num;
1854 felem x_in, y_in, z_in, x_out, y_out, z_out;
1855 NISTP521_PRE_COMP *pre = NULL;
1856 felem(*g_pre_comp)[3] = NULL;
1857 EC_POINT *generator = NULL;
1858 const EC_POINT *p = NULL;
1859 const BIGNUM *p_scalar = NULL;
1862 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1865 if (((x = BN_CTX_get(ctx)) == NULL) ||
1866 ((y = BN_CTX_get(ctx)) == NULL) ||
1867 ((z = BN_CTX_get(ctx)) == NULL) ||
1868 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1871 if (scalar != NULL) {
1872 pre = group->pre_comp.nistp521;
1874 /* we have precomputation, try to use it */
1875 g_pre_comp = &pre->g_pre_comp[0];
1877 /* try to use the standard precomputation */
1878 g_pre_comp = (felem(*)[3]) gmul;
1879 generator = EC_POINT_new(group);
1880 if (generator == NULL)
1882 /* get the generator from precomputation */
1883 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1884 !felem_to_BN(y, g_pre_comp[1][1]) ||
1885 !felem_to_BN(z, g_pre_comp[1][2])) {
1886 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1889 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1893 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1894 /* precomputation matches generator */
1898 * we don't have valid precomputation: treat the generator as a
1904 if (num_points > 0) {
1905 if (num_points >= 2) {
1907 * unless we precompute multiples for just one point, converting
1908 * those into affine form is time well spent
1912 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1913 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1916 OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
1917 if ((secrets == NULL) || (pre_comp == NULL)
1918 || (mixed && (tmp_felems == NULL))) {
1919 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1924 * we treat NULL scalars as 0, and NULL points as points at infinity,
1925 * i.e., they contribute nothing to the linear combination
1927 for (i = 0; i < num_points; ++i) {
1930 * we didn't have a valid precomputation, so we pick the
1934 p = EC_GROUP_get0_generator(group);
1937 /* the i^th point */
1940 p_scalar = scalars[i];
1942 if ((p_scalar != NULL) && (p != NULL)) {
1943 /* reduce scalar to 0 <= scalar < 2^521 */
1944 if ((BN_num_bits(p_scalar) > 521)
1945 || (BN_is_negative(p_scalar))) {
1947 * this is an unusual input, and we don't guarantee
1950 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1951 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1954 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1956 num_bytes = BN_bn2bin(p_scalar, tmp);
1957 flip_endian(secrets[i], tmp, num_bytes);
1958 /* precompute multiples */
1959 if ((!BN_to_felem(x_out, p->X)) ||
1960 (!BN_to_felem(y_out, p->Y)) ||
1961 (!BN_to_felem(z_out, p->Z)))
1963 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1964 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1965 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1966 for (j = 2; j <= 16; ++j) {
1968 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1969 pre_comp[i][j][2], pre_comp[i][1][0],
1970 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1971 pre_comp[i][j - 1][0],
1972 pre_comp[i][j - 1][1],
1973 pre_comp[i][j - 1][2]);
1975 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1976 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1977 pre_comp[i][j / 2][1],
1978 pre_comp[i][j / 2][2]);
1984 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1987 /* the scalar for the generator */
1988 if ((scalar != NULL) && (have_pre_comp)) {
1989 memset(g_secret, 0, sizeof(g_secret));
1990 /* reduce scalar to 0 <= scalar < 2^521 */
1991 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
1993 * this is an unusual input, and we don't guarantee
1996 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1997 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2000 num_bytes = BN_bn2bin(tmp_scalar, tmp);
2002 num_bytes = BN_bn2bin(scalar, tmp);
2003 flip_endian(g_secret, tmp, num_bytes);
2004 /* do the multiplication with generator precomputation */
2005 batch_mul(x_out, y_out, z_out,
2006 (const felem_bytearray(*))secrets, num_points,
2008 mixed, (const felem(*)[17][3])pre_comp,
2009 (const felem(*)[3])g_pre_comp);
2011 /* do the multiplication without generator precomputation */
2012 batch_mul(x_out, y_out, z_out,
2013 (const felem_bytearray(*))secrets, num_points,
2014 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2015 /* reduce the output to its unique minimal representation */
2016 felem_contract(x_in, x_out);
2017 felem_contract(y_in, y_out);
2018 felem_contract(z_in, z_out);
2019 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2020 (!felem_to_BN(z, z_in))) {
2021 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2024 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2028 EC_POINT_free(generator);
2029 BN_CTX_free(new_ctx);
2030 OPENSSL_free(secrets);
2031 OPENSSL_free(pre_comp);
2032 OPENSSL_free(tmp_felems);
2036 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2039 NISTP521_PRE_COMP *pre = NULL;
2041 BN_CTX *new_ctx = NULL;
2043 EC_POINT *generator = NULL;
2044 felem tmp_felems[16];
2046 /* throw away old precomputation */
2047 EC_pre_comp_free(group);
2049 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2052 if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
2054 /* get the generator */
2055 if (group->generator == NULL)
2057 generator = EC_POINT_new(group);
2058 if (generator == NULL)
2060 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2061 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2062 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
2064 if ((pre = nistp521_pre_comp_new()) == NULL)
2067 * if the generator is the standard one, use built-in precomputation
2069 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2070 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2073 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2074 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2075 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2077 /* compute 2^130*G, 2^260*G, 2^390*G */
2078 for (i = 1; i <= 4; i <<= 1) {
2079 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2080 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2081 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2082 for (j = 0; j < 129; ++j) {
2083 point_double(pre->g_pre_comp[2 * i][0],
2084 pre->g_pre_comp[2 * i][1],
2085 pre->g_pre_comp[2 * i][2],
2086 pre->g_pre_comp[2 * i][0],
2087 pre->g_pre_comp[2 * i][1],
2088 pre->g_pre_comp[2 * i][2]);
2091 /* g_pre_comp[0] is the point at infinity */
2092 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2093 /* the remaining multiples */
2094 /* 2^130*G + 2^260*G */
2095 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2096 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2097 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2098 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2099 pre->g_pre_comp[2][2]);
2100 /* 2^130*G + 2^390*G */
2101 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2102 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2103 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2104 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2105 pre->g_pre_comp[2][2]);
2106 /* 2^260*G + 2^390*G */
2107 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2108 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2109 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2110 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2111 pre->g_pre_comp[4][2]);
2112 /* 2^130*G + 2^260*G + 2^390*G */
2113 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2114 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2115 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2116 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2117 pre->g_pre_comp[2][2]);
2118 for (i = 1; i < 8; ++i) {
2119 /* odd multiples: add G */
2120 point_add(pre->g_pre_comp[2 * i + 1][0],
2121 pre->g_pre_comp[2 * i + 1][1],
2122 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2123 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2124 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2125 pre->g_pre_comp[1][2]);
2127 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2130 SETPRECOMP(group, nistp521, pre);
2135 EC_POINT_free(generator);
2136 BN_CTX_free(new_ctx);
2137 EC_nistp521_pre_comp_free(pre);
2141 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2143 return HAVEPRECOMP(group, nistp521);