1 /* crypto/ec/ecp_nistp521.c */
3 * Written by Adam Langley (Google) for the OpenSSL project
5 /* Copyright 2011 Google Inc.
7 * Licensed under the Apache License, Version 2.0 (the "License");
9 * you may not use this file except in compliance with the License.
10 * You may obtain a copy of the License at
12 * http://www.apache.org/licenses/LICENSE-2.0
14 * Unless required by applicable law or agreed to in writing, software
15 * distributed under the License is distributed on an "AS IS" BASIS,
16 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
17 * See the License for the specific language governing permissions and
18 * limitations under the License.
22 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication
24 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
25 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
26 * work which got its smarts from Daniel J. Bernstein's work on the same.
29 #include <openssl/opensslconf.h>
30 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
32 # ifndef OPENSSL_SYS_VMS
35 # include <inttypes.h>
39 # include <openssl/err.h>
42 # if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
43 /* even with gcc, the typedef won't work for 32-bit platforms */
44 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
47 # error "Need GCC 3.1 or later to define type uint128_t"
55 * The underlying field. P521 operates over GF(2^521-1). We can serialise an
56 * element of this field into 66 bytes where the most significant byte
57 * contains only a single bit. We call this an felem_bytearray.
60 typedef u8 felem_bytearray[66];
63 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
64 * These values are big-endian.
66 static const felem_bytearray nistp521_curve_params[5] = {
67 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
68 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
69 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
70 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
71 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
72 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
73 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
74 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
76 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
77 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
78 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
79 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
80 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
81 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
82 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
83 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
85 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
86 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
87 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
88 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
89 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
90 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
91 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
92 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
94 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
95 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
96 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
97 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
98 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
99 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
100 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
101 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
103 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
104 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
105 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
106 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
107 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
108 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
109 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
110 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
115 * The representation of field elements.
116 * ------------------------------------
118 * We represent field elements with nine values. These values are either 64 or
119 * 128 bits and the field element represented is:
120 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
121 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
122 * 58 bits apart, but are greater than 58 bits in length, the most significant
123 * bits of each limb overlap with the least significant bits of the next.
125 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
130 typedef uint64_t limb;
131 typedef limb felem[NLIMBS];
132 typedef uint128_t largefelem[NLIMBS];
134 static const limb bottom57bits = 0x1ffffffffffffff;
135 static const limb bottom58bits = 0x3ffffffffffffff;
138 * bin66_to_felem takes a little-endian byte array and converts it into felem
139 * form. This assumes that the CPU is little-endian.
141 static void bin66_to_felem(felem out, const u8 in[66])
143 out[0] = (*((limb *) & in[0])) & bottom58bits;
144 out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits;
145 out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits;
146 out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits;
147 out[4] = (*((limb *) & in[29])) & bottom58bits;
148 out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits;
149 out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits;
150 out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits;
151 out[8] = (*((limb *) & in[58])) & bottom57bits;
155 * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
156 * array. This assumes that the CPU is little-endian.
158 static void felem_to_bin66(u8 out[66], const felem in)
161 (*((limb *) & out[0])) = in[0];
162 (*((limb *) & out[7])) |= in[1] << 2;
163 (*((limb *) & out[14])) |= in[2] << 4;
164 (*((limb *) & out[21])) |= in[3] << 6;
165 (*((limb *) & out[29])) = in[4];
166 (*((limb *) & out[36])) |= in[5] << 2;
167 (*((limb *) & out[43])) |= in[6] << 4;
168 (*((limb *) & out[50])) |= in[7] << 6;
169 (*((limb *) & out[58])) = in[8];
172 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
173 static void flip_endian(u8 *out, const u8 *in, unsigned len)
176 for (i = 0; i < len; ++i)
177 out[i] = in[len - 1 - i];
180 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
181 static int BN_to_felem(felem out, const BIGNUM *bn)
183 felem_bytearray b_in;
184 felem_bytearray b_out;
187 /* BN_bn2bin eats leading zeroes */
188 memset(b_out, 0, sizeof b_out);
189 num_bytes = BN_num_bytes(bn);
190 if (num_bytes > sizeof b_out) {
191 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
194 if (BN_is_negative(bn)) {
195 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
198 num_bytes = BN_bn2bin(bn, b_in);
199 flip_endian(b_out, b_in, num_bytes);
200 bin66_to_felem(out, b_out);
204 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
205 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
207 felem_bytearray b_in, b_out;
208 felem_to_bin66(b_in, in);
209 flip_endian(b_out, b_in, sizeof b_out);
210 return BN_bin2bn(b_out, sizeof b_out, out);
218 static void felem_one(felem out)
231 static void felem_assign(felem out, const felem in)
244 /* felem_sum64 sets out = out + in. */
245 static void felem_sum64(felem out, const felem in)
258 /* felem_scalar sets out = in * scalar */
259 static void felem_scalar(felem out, const felem in, limb scalar)
261 out[0] = in[0] * scalar;
262 out[1] = in[1] * scalar;
263 out[2] = in[2] * scalar;
264 out[3] = in[3] * scalar;
265 out[4] = in[4] * scalar;
266 out[5] = in[5] * scalar;
267 out[6] = in[6] * scalar;
268 out[7] = in[7] * scalar;
269 out[8] = in[8] * scalar;
272 /* felem_scalar64 sets out = out * scalar */
273 static void felem_scalar64(felem out, limb scalar)
286 /* felem_scalar128 sets out = out * scalar */
287 static void felem_scalar128(largefelem out, limb scalar)
301 * felem_neg sets |out| to |-in|
303 * in[i] < 2^59 + 2^14
307 static void felem_neg(felem out, const felem in)
309 /* In order to prevent underflow, we subtract from 0 mod p. */
310 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
311 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
313 out[0] = two62m3 - in[0];
314 out[1] = two62m2 - in[1];
315 out[2] = two62m2 - in[2];
316 out[3] = two62m2 - in[3];
317 out[4] = two62m2 - in[4];
318 out[5] = two62m2 - in[5];
319 out[6] = two62m2 - in[6];
320 out[7] = two62m2 - in[7];
321 out[8] = two62m2 - in[8];
325 * felem_diff64 subtracts |in| from |out|
327 * in[i] < 2^59 + 2^14
329 * out[i] < out[i] + 2^62
331 static void felem_diff64(felem out, const felem in)
334 * In order to prevent underflow, we add 0 mod p before subtracting.
336 static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
337 static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);
339 out[0] += two62m3 - in[0];
340 out[1] += two62m2 - in[1];
341 out[2] += two62m2 - in[2];
342 out[3] += two62m2 - in[3];
343 out[4] += two62m2 - in[4];
344 out[5] += two62m2 - in[5];
345 out[6] += two62m2 - in[6];
346 out[7] += two62m2 - in[7];
347 out[8] += two62m2 - in[8];
351 * felem_diff_128_64 subtracts |in| from |out|
353 * in[i] < 2^62 + 2^17
355 * out[i] < out[i] + 2^63
357 static void felem_diff_128_64(largefelem out, const felem in)
360 * In order to prevent underflow, we add 0 mod p before subtracting.
362 static const limb two63m6 = (((limb) 1) << 62) - (((limb) 1) << 5);
363 static const limb two63m5 = (((limb) 1) << 62) - (((limb) 1) << 4);
365 out[0] += two63m6 - in[0];
366 out[1] += two63m5 - in[1];
367 out[2] += two63m5 - in[2];
368 out[3] += two63m5 - in[3];
369 out[4] += two63m5 - in[4];
370 out[5] += two63m5 - in[5];
371 out[6] += two63m5 - in[6];
372 out[7] += two63m5 - in[7];
373 out[8] += two63m5 - in[8];
377 * felem_diff_128_64 subtracts |in| from |out|
381 * out[i] < out[i] + 2^127 - 2^69
383 static void felem_diff128(largefelem out, const largefelem in)
386 * In order to prevent underflow, we add 0 mod p before subtracting.
388 static const uint128_t two127m70 =
389 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
390 static const uint128_t two127m69 =
391 (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);
393 out[0] += (two127m70 - in[0]);
394 out[1] += (two127m69 - in[1]);
395 out[2] += (two127m69 - in[2]);
396 out[3] += (two127m69 - in[3]);
397 out[4] += (two127m69 - in[4]);
398 out[5] += (two127m69 - in[5]);
399 out[6] += (two127m69 - in[6]);
400 out[7] += (two127m69 - in[7]);
401 out[8] += (two127m69 - in[8]);
405 * felem_square sets |out| = |in|^2
409 * out[i] < 17 * max(in[i]) * max(in[i])
411 static void felem_square(largefelem out, const felem in)
414 felem_scalar(inx2, in, 2);
415 felem_scalar(inx4, in, 4);
418 * We have many cases were we want to do
421 * This is obviously just
423 * However, rather than do the doubling on the 128 bit result, we
424 * double one of the inputs to the multiplication by reading from
428 out[0] = ((uint128_t) in[0]) * in[0];
429 out[1] = ((uint128_t) in[0]) * inx2[1];
430 out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
431 out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
432 out[4] = ((uint128_t) in[0]) * inx2[4] +
433 ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
434 out[5] = ((uint128_t) in[0]) * inx2[5] +
435 ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
436 out[6] = ((uint128_t) in[0]) * inx2[6] +
437 ((uint128_t) in[1]) * inx2[5] +
438 ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
439 out[7] = ((uint128_t) in[0]) * inx2[7] +
440 ((uint128_t) in[1]) * inx2[6] +
441 ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
442 out[8] = ((uint128_t) in[0]) * inx2[8] +
443 ((uint128_t) in[1]) * inx2[7] +
444 ((uint128_t) in[2]) * inx2[6] +
445 ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];
448 * The remaining limbs fall above 2^521, with the first falling at 2^522.
449 * They correspond to locations one bit up from the limbs produced above
450 * so we would have to multiply by two to align them. Again, rather than
451 * operate on the 128-bit result, we double one of the inputs to the
452 * multiplication. If we want to double for both this reason, and the
453 * reason above, then we end up multiplying by four.
457 out[0] += ((uint128_t) in[1]) * inx4[8] +
458 ((uint128_t) in[2]) * inx4[7] +
459 ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];
462 out[1] += ((uint128_t) in[2]) * inx4[8] +
463 ((uint128_t) in[3]) * inx4[7] +
464 ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];
467 out[2] += ((uint128_t) in[3]) * inx4[8] +
468 ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];
471 out[3] += ((uint128_t) in[4]) * inx4[8] +
472 ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];
475 out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];
478 out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];
481 out[6] += ((uint128_t) in[7]) * inx4[8];
484 out[7] += ((uint128_t) in[8]) * inx2[8];
488 * felem_mul sets |out| = |in1| * |in2|
493 * out[i] < 17 * max(in1[i]) * max(in2[i])
495 static void felem_mul(largefelem out, const felem in1, const felem in2)
498 felem_scalar(in2x2, in2, 2);
500 out[0] = ((uint128_t) in1[0]) * in2[0];
502 out[1] = ((uint128_t) in1[0]) * in2[1] +
503 ((uint128_t) in1[1]) * in2[0];
505 out[2] = ((uint128_t) in1[0]) * in2[2] +
506 ((uint128_t) in1[1]) * in2[1] +
507 ((uint128_t) in1[2]) * in2[0];
509 out[3] = ((uint128_t) in1[0]) * in2[3] +
510 ((uint128_t) in1[1]) * in2[2] +
511 ((uint128_t) in1[2]) * in2[1] +
512 ((uint128_t) in1[3]) * in2[0];
514 out[4] = ((uint128_t) in1[0]) * in2[4] +
515 ((uint128_t) in1[1]) * in2[3] +
516 ((uint128_t) in1[2]) * in2[2] +
517 ((uint128_t) in1[3]) * in2[1] +
518 ((uint128_t) in1[4]) * in2[0];
520 out[5] = ((uint128_t) in1[0]) * in2[5] +
521 ((uint128_t) in1[1]) * in2[4] +
522 ((uint128_t) in1[2]) * in2[3] +
523 ((uint128_t) in1[3]) * in2[2] +
524 ((uint128_t) in1[4]) * in2[1] +
525 ((uint128_t) in1[5]) * in2[0];
527 out[6] = ((uint128_t) in1[0]) * in2[6] +
528 ((uint128_t) in1[1]) * in2[5] +
529 ((uint128_t) in1[2]) * in2[4] +
530 ((uint128_t) in1[3]) * in2[3] +
531 ((uint128_t) in1[4]) * in2[2] +
532 ((uint128_t) in1[5]) * in2[1] +
533 ((uint128_t) in1[6]) * in2[0];
535 out[7] = ((uint128_t) in1[0]) * in2[7] +
536 ((uint128_t) in1[1]) * in2[6] +
537 ((uint128_t) in1[2]) * in2[5] +
538 ((uint128_t) in1[3]) * in2[4] +
539 ((uint128_t) in1[4]) * in2[3] +
540 ((uint128_t) in1[5]) * in2[2] +
541 ((uint128_t) in1[6]) * in2[1] +
542 ((uint128_t) in1[7]) * in2[0];
544 out[8] = ((uint128_t) in1[0]) * in2[8] +
545 ((uint128_t) in1[1]) * in2[7] +
546 ((uint128_t) in1[2]) * in2[6] +
547 ((uint128_t) in1[3]) * in2[5] +
548 ((uint128_t) in1[4]) * in2[4] +
549 ((uint128_t) in1[5]) * in2[3] +
550 ((uint128_t) in1[6]) * in2[2] +
551 ((uint128_t) in1[7]) * in2[1] +
552 ((uint128_t) in1[8]) * in2[0];
554 /* See comment in felem_square about the use of in2x2 here */
556 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
557 ((uint128_t) in1[2]) * in2x2[7] +
558 ((uint128_t) in1[3]) * in2x2[6] +
559 ((uint128_t) in1[4]) * in2x2[5] +
560 ((uint128_t) in1[5]) * in2x2[4] +
561 ((uint128_t) in1[6]) * in2x2[3] +
562 ((uint128_t) in1[7]) * in2x2[2] +
563 ((uint128_t) in1[8]) * in2x2[1];
565 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
566 ((uint128_t) in1[3]) * in2x2[7] +
567 ((uint128_t) in1[4]) * in2x2[6] +
568 ((uint128_t) in1[5]) * in2x2[5] +
569 ((uint128_t) in1[6]) * in2x2[4] +
570 ((uint128_t) in1[7]) * in2x2[3] +
571 ((uint128_t) in1[8]) * in2x2[2];
573 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
574 ((uint128_t) in1[4]) * in2x2[7] +
575 ((uint128_t) in1[5]) * in2x2[6] +
576 ((uint128_t) in1[6]) * in2x2[5] +
577 ((uint128_t) in1[7]) * in2x2[4] +
578 ((uint128_t) in1[8]) * in2x2[3];
580 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
581 ((uint128_t) in1[5]) * in2x2[7] +
582 ((uint128_t) in1[6]) * in2x2[6] +
583 ((uint128_t) in1[7]) * in2x2[5] +
584 ((uint128_t) in1[8]) * in2x2[4];
586 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
587 ((uint128_t) in1[6]) * in2x2[7] +
588 ((uint128_t) in1[7]) * in2x2[6] +
589 ((uint128_t) in1[8]) * in2x2[5];
591 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
592 ((uint128_t) in1[7]) * in2x2[7] +
593 ((uint128_t) in1[8]) * in2x2[6];
595 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
596 ((uint128_t) in1[8]) * in2x2[7];
598 out[7] += ((uint128_t) in1[8]) * in2x2[8];
601 static const limb bottom52bits = 0xfffffffffffff;
604 * felem_reduce converts a largefelem to an felem.
608 * out[i] < 2^59 + 2^14
610 static void felem_reduce(felem out, const largefelem in)
612 u64 overflow1, overflow2;
614 out[0] = ((limb) in[0]) & bottom58bits;
615 out[1] = ((limb) in[1]) & bottom58bits;
616 out[2] = ((limb) in[2]) & bottom58bits;
617 out[3] = ((limb) in[3]) & bottom58bits;
618 out[4] = ((limb) in[4]) & bottom58bits;
619 out[5] = ((limb) in[5]) & bottom58bits;
620 out[6] = ((limb) in[6]) & bottom58bits;
621 out[7] = ((limb) in[7]) & bottom58bits;
622 out[8] = ((limb) in[8]) & bottom58bits;
626 out[1] += ((limb) in[0]) >> 58;
627 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
629 * out[1] < 2^58 + 2^6 + 2^58
632 out[2] += ((limb) (in[0] >> 64)) >> 52;
634 out[2] += ((limb) in[1]) >> 58;
635 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
636 out[3] += ((limb) (in[1] >> 64)) >> 52;
638 out[3] += ((limb) in[2]) >> 58;
639 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
640 out[4] += ((limb) (in[2] >> 64)) >> 52;
642 out[4] += ((limb) in[3]) >> 58;
643 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
644 out[5] += ((limb) (in[3] >> 64)) >> 52;
646 out[5] += ((limb) in[4]) >> 58;
647 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
648 out[6] += ((limb) (in[4] >> 64)) >> 52;
650 out[6] += ((limb) in[5]) >> 58;
651 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
652 out[7] += ((limb) (in[5] >> 64)) >> 52;
654 out[7] += ((limb) in[6]) >> 58;
655 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
656 out[8] += ((limb) (in[6] >> 64)) >> 52;
658 out[8] += ((limb) in[7]) >> 58;
659 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
661 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
664 overflow1 = ((limb) (in[7] >> 64)) >> 52;
666 overflow1 += ((limb) in[8]) >> 58;
667 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
668 overflow2 = ((limb) (in[8] >> 64)) >> 52;
670 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
671 overflow2 <<= 1; /* overflow2 < 2^13 */
673 out[0] += overflow1; /* out[0] < 2^60 */
674 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
676 out[1] += out[0] >> 58;
677 out[0] &= bottom58bits;
680 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
685 static void felem_square_reduce(felem out, const felem in)
688 felem_square(tmp, in);
689 felem_reduce(out, tmp);
692 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
695 felem_mul(tmp, in1, in2);
696 felem_reduce(out, tmp);
700 * felem_inv calculates |out| = |in|^{-1}
702 * Based on Fermat's Little Theorem:
704 * a^{p-1} = 1 (mod p)
705 * a^{p-2} = a^{-1} (mod p)
707 static void felem_inv(felem out, const felem in)
709 felem ftmp, ftmp2, ftmp3, ftmp4;
713 felem_square(tmp, in);
714 felem_reduce(ftmp, tmp); /* 2^1 */
715 felem_mul(tmp, in, ftmp);
716 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
717 felem_assign(ftmp2, ftmp);
718 felem_square(tmp, ftmp);
719 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
720 felem_mul(tmp, in, ftmp);
721 felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
722 felem_square(tmp, ftmp);
723 felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
725 felem_square(tmp, ftmp2);
726 felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
727 felem_square(tmp, ftmp3);
728 felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
729 felem_mul(tmp, ftmp3, ftmp2);
730 felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
732 felem_assign(ftmp2, ftmp3);
733 felem_square(tmp, ftmp3);
734 felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
735 felem_square(tmp, ftmp3);
736 felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
737 felem_square(tmp, ftmp3);
738 felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
739 felem_square(tmp, ftmp3);
740 felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
741 felem_assign(ftmp4, ftmp3);
742 felem_mul(tmp, ftmp3, ftmp);
743 felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
744 felem_square(tmp, ftmp4);
745 felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
746 felem_mul(tmp, ftmp3, ftmp2);
747 felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
748 felem_assign(ftmp2, ftmp3);
750 for (i = 0; i < 8; i++) {
751 felem_square(tmp, ftmp3);
752 felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
754 felem_mul(tmp, ftmp3, ftmp2);
755 felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
756 felem_assign(ftmp2, ftmp3);
758 for (i = 0; i < 16; i++) {
759 felem_square(tmp, ftmp3);
760 felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
762 felem_mul(tmp, ftmp3, ftmp2);
763 felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
764 felem_assign(ftmp2, ftmp3);
766 for (i = 0; i < 32; i++) {
767 felem_square(tmp, ftmp3);
768 felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
770 felem_mul(tmp, ftmp3, ftmp2);
771 felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
772 felem_assign(ftmp2, ftmp3);
774 for (i = 0; i < 64; i++) {
775 felem_square(tmp, ftmp3);
776 felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
778 felem_mul(tmp, ftmp3, ftmp2);
779 felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
780 felem_assign(ftmp2, ftmp3);
782 for (i = 0; i < 128; i++) {
783 felem_square(tmp, ftmp3);
784 felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
786 felem_mul(tmp, ftmp3, ftmp2);
787 felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
788 felem_assign(ftmp2, ftmp3);
790 for (i = 0; i < 256; i++) {
791 felem_square(tmp, ftmp3);
792 felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
794 felem_mul(tmp, ftmp3, ftmp2);
795 felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
797 for (i = 0; i < 9; i++) {
798 felem_square(tmp, ftmp3);
799 felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
801 felem_mul(tmp, ftmp3, ftmp4);
802 felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
803 felem_mul(tmp, ftmp3, in);
804 felem_reduce(out, tmp); /* 2^512 - 3 */
807 /* This is 2^521-1, expressed as an felem */
808 static const felem kPrime = {
809 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
810 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
811 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
815 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
818 * in[i] < 2^59 + 2^14
820 static limb felem_is_zero(const felem in)
824 felem_assign(ftmp, in);
826 ftmp[0] += ftmp[8] >> 57;
827 ftmp[8] &= bottom57bits;
829 ftmp[1] += ftmp[0] >> 58;
830 ftmp[0] &= bottom58bits;
831 ftmp[2] += ftmp[1] >> 58;
832 ftmp[1] &= bottom58bits;
833 ftmp[3] += ftmp[2] >> 58;
834 ftmp[2] &= bottom58bits;
835 ftmp[4] += ftmp[3] >> 58;
836 ftmp[3] &= bottom58bits;
837 ftmp[5] += ftmp[4] >> 58;
838 ftmp[4] &= bottom58bits;
839 ftmp[6] += ftmp[5] >> 58;
840 ftmp[5] &= bottom58bits;
841 ftmp[7] += ftmp[6] >> 58;
842 ftmp[6] &= bottom58bits;
843 ftmp[8] += ftmp[7] >> 58;
844 ftmp[7] &= bottom58bits;
845 /* ftmp[8] < 2^57 + 4 */
848 * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
849 * than our bound for ftmp[8]. Therefore we only have to check if the
850 * zero is zero or 2^521-1.
866 * We know that ftmp[i] < 2^63, therefore the only way that the top bit
867 * can be set is if is_zero was 0 before the decrement.
869 is_zero = ((s64) is_zero) >> 63;
871 is_p = ftmp[0] ^ kPrime[0];
872 is_p |= ftmp[1] ^ kPrime[1];
873 is_p |= ftmp[2] ^ kPrime[2];
874 is_p |= ftmp[3] ^ kPrime[3];
875 is_p |= ftmp[4] ^ kPrime[4];
876 is_p |= ftmp[5] ^ kPrime[5];
877 is_p |= ftmp[6] ^ kPrime[6];
878 is_p |= ftmp[7] ^ kPrime[7];
879 is_p |= ftmp[8] ^ kPrime[8];
882 is_p = ((s64) is_p) >> 63;
888 static int felem_is_zero_int(const felem in)
890 return (int)(felem_is_zero(in) & ((limb) 1));
894 * felem_contract converts |in| to its unique, minimal representation.
896 * in[i] < 2^59 + 2^14
898 static void felem_contract(felem out, const felem in)
900 limb is_p, is_greater, sign;
901 static const limb two58 = ((limb) 1) << 58;
903 felem_assign(out, in);
905 out[0] += out[8] >> 57;
906 out[8] &= bottom57bits;
908 out[1] += out[0] >> 58;
909 out[0] &= bottom58bits;
910 out[2] += out[1] >> 58;
911 out[1] &= bottom58bits;
912 out[3] += out[2] >> 58;
913 out[2] &= bottom58bits;
914 out[4] += out[3] >> 58;
915 out[3] &= bottom58bits;
916 out[5] += out[4] >> 58;
917 out[4] &= bottom58bits;
918 out[6] += out[5] >> 58;
919 out[5] &= bottom58bits;
920 out[7] += out[6] >> 58;
921 out[6] &= bottom58bits;
922 out[8] += out[7] >> 58;
923 out[7] &= bottom58bits;
924 /* out[8] < 2^57 + 4 */
927 * If the value is greater than 2^521-1 then we have to subtract 2^521-1
928 * out. See the comments in felem_is_zero regarding why we don't test for
929 * other multiples of the prime.
933 * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
936 is_p = out[0] ^ kPrime[0];
937 is_p |= out[1] ^ kPrime[1];
938 is_p |= out[2] ^ kPrime[2];
939 is_p |= out[3] ^ kPrime[3];
940 is_p |= out[4] ^ kPrime[4];
941 is_p |= out[5] ^ kPrime[5];
942 is_p |= out[6] ^ kPrime[6];
943 is_p |= out[7] ^ kPrime[7];
944 is_p |= out[8] ^ kPrime[8];
953 is_p = ((s64) is_p) >> 63;
956 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
969 * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
970 * 57 is greater than zero as (2^521-1) + x >= 2^522
972 is_greater = out[8] >> 57;
973 is_greater |= is_greater << 32;
974 is_greater |= is_greater << 16;
975 is_greater |= is_greater << 8;
976 is_greater |= is_greater << 4;
977 is_greater |= is_greater << 2;
978 is_greater |= is_greater << 1;
979 is_greater = ((s64) is_greater) >> 63;
981 out[0] -= kPrime[0] & is_greater;
982 out[1] -= kPrime[1] & is_greater;
983 out[2] -= kPrime[2] & is_greater;
984 out[3] -= kPrime[3] & is_greater;
985 out[4] -= kPrime[4] & is_greater;
986 out[5] -= kPrime[5] & is_greater;
987 out[6] -= kPrime[6] & is_greater;
988 out[7] -= kPrime[7] & is_greater;
989 out[8] -= kPrime[8] & is_greater;
991 /* Eliminate negative coefficients */
992 sign = -(out[0] >> 63);
993 out[0] += (two58 & sign);
994 out[1] -= (1 & sign);
995 sign = -(out[1] >> 63);
996 out[1] += (two58 & sign);
997 out[2] -= (1 & sign);
998 sign = -(out[2] >> 63);
999 out[2] += (two58 & sign);
1000 out[3] -= (1 & sign);
1001 sign = -(out[3] >> 63);
1002 out[3] += (two58 & sign);
1003 out[4] -= (1 & sign);
1004 sign = -(out[4] >> 63);
1005 out[4] += (two58 & sign);
1006 out[5] -= (1 & sign);
1007 sign = -(out[0] >> 63);
1008 out[5] += (two58 & sign);
1009 out[6] -= (1 & sign);
1010 sign = -(out[6] >> 63);
1011 out[6] += (two58 & sign);
1012 out[7] -= (1 & sign);
1013 sign = -(out[7] >> 63);
1014 out[7] += (two58 & sign);
1015 out[8] -= (1 & sign);
1016 sign = -(out[5] >> 63);
1017 out[5] += (two58 & sign);
1018 out[6] -= (1 & sign);
1019 sign = -(out[6] >> 63);
1020 out[6] += (two58 & sign);
1021 out[7] -= (1 & sign);
1022 sign = -(out[7] >> 63);
1023 out[7] += (two58 & sign);
1024 out[8] -= (1 & sign);
1031 * Building on top of the field operations we have the operations on the
1032 * elliptic curve group itself. Points on the curve are represented in Jacobian
1036 * point_double calcuates 2*(x_in, y_in, z_in)
1038 * The method is taken from:
1039 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1041 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1042 * while x_out == y_in is not (maybe this works, but it's not tested). */
1044 point_double(felem x_out, felem y_out, felem z_out,
1045 const felem x_in, const felem y_in, const felem z_in)
1047 largefelem tmp, tmp2;
1048 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1050 felem_assign(ftmp, x_in);
1051 felem_assign(ftmp2, x_in);
1054 felem_square(tmp, z_in);
1055 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
1058 felem_square(tmp, y_in);
1059 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
1061 /* beta = x*gamma */
1062 felem_mul(tmp, x_in, gamma);
1063 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
1065 /* alpha = 3*(x-delta)*(x+delta) */
1066 felem_diff64(ftmp, delta);
1067 /* ftmp[i] < 2^61 */
1068 felem_sum64(ftmp2, delta);
1069 /* ftmp2[i] < 2^60 + 2^15 */
1070 felem_scalar64(ftmp2, 3);
1071 /* ftmp2[i] < 3*2^60 + 3*2^15 */
1072 felem_mul(tmp, ftmp, ftmp2);
1074 * tmp[i] < 17(3*2^121 + 3*2^76)
1075 * = 61*2^121 + 61*2^76
1076 * < 64*2^121 + 64*2^76
1080 felem_reduce(alpha, tmp);
1082 /* x' = alpha^2 - 8*beta */
1083 felem_square(tmp, alpha);
1085 * tmp[i] < 17*2^120 < 2^125
1087 felem_assign(ftmp, beta);
1088 felem_scalar64(ftmp, 8);
1089 /* ftmp[i] < 2^62 + 2^17 */
1090 felem_diff_128_64(tmp, ftmp);
1091 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1092 felem_reduce(x_out, tmp);
1094 /* z' = (y + z)^2 - gamma - delta */
1095 felem_sum64(delta, gamma);
1096 /* delta[i] < 2^60 + 2^15 */
1097 felem_assign(ftmp, y_in);
1098 felem_sum64(ftmp, z_in);
1099 /* ftmp[i] < 2^60 + 2^15 */
1100 felem_square(tmp, ftmp);
1102 * tmp[i] < 17(2^122) < 2^127
1104 felem_diff_128_64(tmp, delta);
1105 /* tmp[i] < 2^127 + 2^63 */
1106 felem_reduce(z_out, tmp);
1108 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1109 felem_scalar64(beta, 4);
1110 /* beta[i] < 2^61 + 2^16 */
1111 felem_diff64(beta, x_out);
1112 /* beta[i] < 2^61 + 2^60 + 2^16 */
1113 felem_mul(tmp, alpha, beta);
1115 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1116 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1117 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1120 felem_square(tmp2, gamma);
1122 * tmp2[i] < 17*(2^59 + 2^14)^2
1123 * = 17*(2^118 + 2^74 + 2^28)
1125 felem_scalar128(tmp2, 8);
1127 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1128 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1131 felem_diff128(tmp, tmp2);
1133 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1134 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1135 * 2^74 + 2^69 + 2^34 + 2^30
1138 felem_reduce(y_out, tmp);
1141 /* copy_conditional copies in to out iff mask is all ones. */
1142 static void copy_conditional(felem out, const felem in, limb mask)
1145 for (i = 0; i < NLIMBS; ++i) {
1146 const limb tmp = mask & (in[i] ^ out[i]);
1152 * point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1154 * The method is taken from
1155 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1156 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1158 * This function includes a branch for checking whether the two input points
1159 * are equal (while not equal to the point at infinity). This case never
1160 * happens during single point multiplication, so there is no timing leak for
1161 * ECDH or ECDSA signing. */
1162 static void point_add(felem x3, felem y3, felem z3,
1163 const felem x1, const felem y1, const felem z1,
1164 const int mixed, const felem x2, const felem y2,
1167 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1168 largefelem tmp, tmp2;
1169 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1171 z1_is_zero = felem_is_zero(z1);
1172 z2_is_zero = felem_is_zero(z2);
1174 /* ftmp = z1z1 = z1**2 */
1175 felem_square(tmp, z1);
1176 felem_reduce(ftmp, tmp);
1179 /* ftmp2 = z2z2 = z2**2 */
1180 felem_square(tmp, z2);
1181 felem_reduce(ftmp2, tmp);
1183 /* u1 = ftmp3 = x1*z2z2 */
1184 felem_mul(tmp, x1, ftmp2);
1185 felem_reduce(ftmp3, tmp);
1187 /* ftmp5 = z1 + z2 */
1188 felem_assign(ftmp5, z1);
1189 felem_sum64(ftmp5, z2);
1190 /* ftmp5[i] < 2^61 */
1192 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1193 felem_square(tmp, ftmp5);
1194 /* tmp[i] < 17*2^122 */
1195 felem_diff_128_64(tmp, ftmp);
1196 /* tmp[i] < 17*2^122 + 2^63 */
1197 felem_diff_128_64(tmp, ftmp2);
1198 /* tmp[i] < 17*2^122 + 2^64 */
1199 felem_reduce(ftmp5, tmp);
1201 /* ftmp2 = z2 * z2z2 */
1202 felem_mul(tmp, ftmp2, z2);
1203 felem_reduce(ftmp2, tmp);
1205 /* s1 = ftmp6 = y1 * z2**3 */
1206 felem_mul(tmp, y1, ftmp2);
1207 felem_reduce(ftmp6, tmp);
1210 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1213 /* u1 = ftmp3 = x1*z2z2 */
1214 felem_assign(ftmp3, x1);
1216 /* ftmp5 = 2*z1z2 */
1217 felem_scalar(ftmp5, z1, 2);
1219 /* s1 = ftmp6 = y1 * z2**3 */
1220 felem_assign(ftmp6, y1);
1224 felem_mul(tmp, x2, ftmp);
1225 /* tmp[i] < 17*2^120 */
1227 /* h = ftmp4 = u2 - u1 */
1228 felem_diff_128_64(tmp, ftmp3);
1229 /* tmp[i] < 17*2^120 + 2^63 */
1230 felem_reduce(ftmp4, tmp);
1232 x_equal = felem_is_zero(ftmp4);
1234 /* z_out = ftmp5 * h */
1235 felem_mul(tmp, ftmp5, ftmp4);
1236 felem_reduce(z_out, tmp);
1238 /* ftmp = z1 * z1z1 */
1239 felem_mul(tmp, ftmp, z1);
1240 felem_reduce(ftmp, tmp);
1242 /* s2 = tmp = y2 * z1**3 */
1243 felem_mul(tmp, y2, ftmp);
1244 /* tmp[i] < 17*2^120 */
1246 /* r = ftmp5 = (s2 - s1)*2 */
1247 felem_diff_128_64(tmp, ftmp6);
1248 /* tmp[i] < 17*2^120 + 2^63 */
1249 felem_reduce(ftmp5, tmp);
1250 y_equal = felem_is_zero(ftmp5);
1251 felem_scalar64(ftmp5, 2);
1252 /* ftmp5[i] < 2^61 */
1254 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1255 point_double(x3, y3, z3, x1, y1, z1);
1259 /* I = ftmp = (2h)**2 */
1260 felem_assign(ftmp, ftmp4);
1261 felem_scalar64(ftmp, 2);
1262 /* ftmp[i] < 2^61 */
1263 felem_square(tmp, ftmp);
1264 /* tmp[i] < 17*2^122 */
1265 felem_reduce(ftmp, tmp);
1267 /* J = ftmp2 = h * I */
1268 felem_mul(tmp, ftmp4, ftmp);
1269 felem_reduce(ftmp2, tmp);
1271 /* V = ftmp4 = U1 * I */
1272 felem_mul(tmp, ftmp3, ftmp);
1273 felem_reduce(ftmp4, tmp);
1275 /* x_out = r**2 - J - 2V */
1276 felem_square(tmp, ftmp5);
1277 /* tmp[i] < 17*2^122 */
1278 felem_diff_128_64(tmp, ftmp2);
1279 /* tmp[i] < 17*2^122 + 2^63 */
1280 felem_assign(ftmp3, ftmp4);
1281 felem_scalar64(ftmp4, 2);
1282 /* ftmp4[i] < 2^61 */
1283 felem_diff_128_64(tmp, ftmp4);
1284 /* tmp[i] < 17*2^122 + 2^64 */
1285 felem_reduce(x_out, tmp);
1287 /* y_out = r(V-x_out) - 2 * s1 * J */
1288 felem_diff64(ftmp3, x_out);
1290 * ftmp3[i] < 2^60 + 2^60 = 2^61
1292 felem_mul(tmp, ftmp5, ftmp3);
1293 /* tmp[i] < 17*2^122 */
1294 felem_mul(tmp2, ftmp6, ftmp2);
1295 /* tmp2[i] < 17*2^120 */
1296 felem_scalar128(tmp2, 2);
1297 /* tmp2[i] < 17*2^121 */
1298 felem_diff128(tmp, tmp2);
1300 * tmp[i] < 2^127 - 2^69 + 17*2^122
1301 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1304 felem_reduce(y_out, tmp);
1306 copy_conditional(x_out, x2, z1_is_zero);
1307 copy_conditional(x_out, x1, z2_is_zero);
1308 copy_conditional(y_out, y2, z1_is_zero);
1309 copy_conditional(y_out, y1, z2_is_zero);
1310 copy_conditional(z_out, z2, z1_is_zero);
1311 copy_conditional(z_out, z1, z2_is_zero);
1312 felem_assign(x3, x_out);
1313 felem_assign(y3, y_out);
1314 felem_assign(z3, z_out);
1318 * Base point pre computation
1319 * --------------------------
1321 * Two different sorts of precomputed tables are used in the following code.
1322 * Each contain various points on the curve, where each point is three field
1323 * elements (x, y, z).
1325 * For the base point table, z is usually 1 (0 for the point at infinity).
1326 * This table has 16 elements:
1327 * index | bits | point
1328 * ------+---------+------------------------------
1331 * 2 | 0 0 1 0 | 2^130G
1332 * 3 | 0 0 1 1 | (2^130 + 1)G
1333 * 4 | 0 1 0 0 | 2^260G
1334 * 5 | 0 1 0 1 | (2^260 + 1)G
1335 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1336 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1337 * 8 | 1 0 0 0 | 2^390G
1338 * 9 | 1 0 0 1 | (2^390 + 1)G
1339 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1340 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1341 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1342 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1343 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1344 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1346 * The reason for this is so that we can clock bits into four different
1347 * locations when doing simple scalar multiplies against the base point.
1349 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1351 /* gmul is the table of precomputed base points */
1352 static const felem gmul[16][3] = {
1353 {{0, 0, 0, 0, 0, 0, 0, 0, 0},
1354 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1355 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1356 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1357 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1358 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1359 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1360 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1361 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1362 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1363 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1364 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1365 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1366 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1367 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1368 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1369 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1370 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1371 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1372 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1373 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1374 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1375 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1376 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1377 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1378 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1379 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1380 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1381 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1382 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1383 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1384 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1385 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1386 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1387 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1388 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1389 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1390 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1391 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1392 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1393 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1394 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1395 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1396 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1397 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1398 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1399 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1400 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1401 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1402 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1403 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1404 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1405 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1406 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1407 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1408 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1409 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1410 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1411 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1412 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1413 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1414 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1415 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1416 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1417 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1418 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1419 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1420 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1421 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1422 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1423 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1424 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1425 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1426 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1427 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1428 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1429 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1430 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1431 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1432 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1433 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1434 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1435 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1436 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1437 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1438 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1439 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1440 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1441 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1442 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1443 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1444 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1445 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1446 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1447 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1448 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1449 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1450 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1451 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1452 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1453 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1454 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1455 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1456 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1457 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1458 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1459 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1460 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
1464 * select_point selects the |idx|th point from a precomputation table and
1467 /* pre_comp below is of the size provided in |size| */
1468 static void select_point(const limb idx, unsigned int size,
1469 const felem pre_comp[][3], felem out[3])
1472 limb *outlimbs = &out[0][0];
1473 memset(outlimbs, 0, 3 * sizeof(felem));
1475 for (i = 0; i < size; i++) {
1476 const limb *inlimbs = &pre_comp[i][0][0];
1477 limb mask = i ^ idx;
1483 for (j = 0; j < NLIMBS * 3; j++)
1484 outlimbs[j] |= inlimbs[j] & mask;
1488 /* get_bit returns the |i|th bit in |in| */
1489 static char get_bit(const felem_bytearray in, int i)
1493 return (in[i >> 3] >> (i & 7)) & 1;
1497 * Interleaved point multiplication using precomputed point multiples: The
1498 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1499 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1500 * generator, using certain (large) precomputed multiples in g_pre_comp.
1501 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1503 static void batch_mul(felem x_out, felem y_out, felem z_out,
1504 const felem_bytearray scalars[],
1505 const unsigned num_points, const u8 *g_scalar,
1506 const int mixed, const felem pre_comp[][17][3],
1507 const felem g_pre_comp[16][3])
1510 unsigned num, gen_mul = (g_scalar != NULL);
1511 felem nq[3], tmp[4];
1515 /* set nq to the point at infinity */
1516 memset(nq, 0, 3 * sizeof(felem));
1519 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1520 * of the generator (last quarter of rounds) and additions of other
1521 * points multiples (every 5th round).
1523 skip = 1; /* save two point operations in the first
1525 for (i = (num_points ? 520 : 130); i >= 0; --i) {
1528 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1530 /* add multiples of the generator */
1531 if (gen_mul && (i <= 130)) {
1532 bits = get_bit(g_scalar, i + 390) << 3;
1534 bits |= get_bit(g_scalar, i + 260) << 2;
1535 bits |= get_bit(g_scalar, i + 130) << 1;
1536 bits |= get_bit(g_scalar, i);
1538 /* select the point to add, in constant time */
1539 select_point(bits, 16, g_pre_comp, tmp);
1541 /* The 1 argument below is for "mixed" */
1542 point_add(nq[0], nq[1], nq[2],
1543 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1545 memcpy(nq, tmp, 3 * sizeof(felem));
1550 /* do other additions every 5 doublings */
1551 if (num_points && (i % 5 == 0)) {
1552 /* loop over all scalars */
1553 for (num = 0; num < num_points; ++num) {
1554 bits = get_bit(scalars[num], i + 4) << 5;
1555 bits |= get_bit(scalars[num], i + 3) << 4;
1556 bits |= get_bit(scalars[num], i + 2) << 3;
1557 bits |= get_bit(scalars[num], i + 1) << 2;
1558 bits |= get_bit(scalars[num], i) << 1;
1559 bits |= get_bit(scalars[num], i - 1);
1560 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1563 * select the point to add or subtract, in constant time
1565 select_point(digit, 17, pre_comp[num], tmp);
1566 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1568 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1571 point_add(nq[0], nq[1], nq[2],
1572 nq[0], nq[1], nq[2],
1573 mixed, tmp[0], tmp[1], tmp[2]);
1575 memcpy(nq, tmp, 3 * sizeof(felem));
1581 felem_assign(x_out, nq[0]);
1582 felem_assign(y_out, nq[1]);
1583 felem_assign(z_out, nq[2]);
1586 /* Precomputation for the group generator. */
1588 felem g_pre_comp[16][3];
1590 } NISTP521_PRE_COMP;
1592 const EC_METHOD *EC_GFp_nistp521_method(void)
1594 static const EC_METHOD ret = {
1595 EC_FLAGS_DEFAULT_OCT,
1596 NID_X9_62_prime_field,
1597 ec_GFp_nistp521_group_init,
1598 ec_GFp_simple_group_finish,
1599 ec_GFp_simple_group_clear_finish,
1600 ec_GFp_nist_group_copy,
1601 ec_GFp_nistp521_group_set_curve,
1602 ec_GFp_simple_group_get_curve,
1603 ec_GFp_simple_group_get_degree,
1604 ec_GFp_simple_group_check_discriminant,
1605 ec_GFp_simple_point_init,
1606 ec_GFp_simple_point_finish,
1607 ec_GFp_simple_point_clear_finish,
1608 ec_GFp_simple_point_copy,
1609 ec_GFp_simple_point_set_to_infinity,
1610 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1611 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1612 ec_GFp_simple_point_set_affine_coordinates,
1613 ec_GFp_nistp521_point_get_affine_coordinates,
1614 0 /* point_set_compressed_coordinates */ ,
1619 ec_GFp_simple_invert,
1620 ec_GFp_simple_is_at_infinity,
1621 ec_GFp_simple_is_on_curve,
1623 ec_GFp_simple_make_affine,
1624 ec_GFp_simple_points_make_affine,
1625 ec_GFp_nistp521_points_mul,
1626 ec_GFp_nistp521_precompute_mult,
1627 ec_GFp_nistp521_have_precompute_mult,
1628 ec_GFp_nist_field_mul,
1629 ec_GFp_nist_field_sqr,
1631 0 /* field_encode */ ,
1632 0 /* field_decode */ ,
1633 0 /* field_set_to_one */
1639 /******************************************************************************/
1641 * FUNCTIONS TO MANAGE PRECOMPUTATION
1644 static NISTP521_PRE_COMP *nistp521_pre_comp_new()
1646 NISTP521_PRE_COMP *ret = NULL;
1647 ret = (NISTP521_PRE_COMP *) OPENSSL_malloc(sizeof(NISTP521_PRE_COMP));
1649 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1652 memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1653 ret->references = 1;
1657 static void *nistp521_pre_comp_dup(void *src_)
1659 NISTP521_PRE_COMP *src = src_;
1661 /* no need to actually copy, these objects never change! */
1662 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1667 static void nistp521_pre_comp_free(void *pre_)
1670 NISTP521_PRE_COMP *pre = pre_;
1675 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1682 static void nistp521_pre_comp_clear_free(void *pre_)
1685 NISTP521_PRE_COMP *pre = pre_;
1690 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1694 OPENSSL_cleanse(pre, sizeof(*pre));
1698 /******************************************************************************/
1700 * OPENSSL EC_METHOD FUNCTIONS
1703 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1706 ret = ec_GFp_simple_group_init(group);
1707 group->a_is_minus3 = 1;
1711 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1712 const BIGNUM *a, const BIGNUM *b,
1716 BN_CTX *new_ctx = NULL;
1717 BIGNUM *curve_p, *curve_a, *curve_b;
1720 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1723 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1724 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1725 ((curve_b = BN_CTX_get(ctx)) == NULL))
1727 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1728 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1729 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1730 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1731 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1732 EC_R_WRONG_CURVE_PARAMETERS);
1735 group->field_mod_func = BN_nist_mod_521;
1736 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1739 if (new_ctx != NULL)
1740 BN_CTX_free(new_ctx);
1745 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1748 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1749 const EC_POINT *point,
1750 BIGNUM *x, BIGNUM *y,
1753 felem z1, z2, x_in, y_in, x_out, y_out;
1756 if (EC_POINT_is_at_infinity(group, point)) {
1757 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1758 EC_R_POINT_AT_INFINITY);
1761 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1762 (!BN_to_felem(z1, point->Z)))
1765 felem_square(tmp, z2);
1766 felem_reduce(z1, tmp);
1767 felem_mul(tmp, x_in, z1);
1768 felem_reduce(x_in, tmp);
1769 felem_contract(x_out, x_in);
1771 if (!felem_to_BN(x, x_out)) {
1772 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1777 felem_mul(tmp, z1, z2);
1778 felem_reduce(z1, tmp);
1779 felem_mul(tmp, y_in, z1);
1780 felem_reduce(y_in, tmp);
1781 felem_contract(y_out, y_in);
1783 if (!felem_to_BN(y, y_out)) {
1784 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1792 /* points below is of size |num|, and tmp_felems is of size |num+1/ */
1793 static void make_points_affine(size_t num, felem points[][3],
1797 * Runs in constant time, unless an input is the point at infinity (which
1798 * normally shouldn't happen).
1800 ec_GFp_nistp_points_make_affine_internal(num,
1804 (void (*)(void *))felem_one,
1805 (int (*)(const void *))
1807 (void (*)(void *, const void *))
1809 (void (*)(void *, const void *))
1810 felem_square_reduce, (void (*)
1817 (void (*)(void *, const void *))
1819 (void (*)(void *, const void *))
1824 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1825 * values Result is stored in r (r can equal one of the inputs).
1827 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1828 const BIGNUM *scalar, size_t num,
1829 const EC_POINT *points[],
1830 const BIGNUM *scalars[], BN_CTX *ctx)
1835 BN_CTX *new_ctx = NULL;
1836 BIGNUM *x, *y, *z, *tmp_scalar;
1837 felem_bytearray g_secret;
1838 felem_bytearray *secrets = NULL;
1839 felem(*pre_comp)[17][3] = NULL;
1840 felem *tmp_felems = NULL;
1841 felem_bytearray tmp;
1842 unsigned i, num_bytes;
1843 int have_pre_comp = 0;
1844 size_t num_points = num;
1845 felem x_in, y_in, z_in, x_out, y_out, z_out;
1846 NISTP521_PRE_COMP *pre = NULL;
1847 felem(*g_pre_comp)[3] = NULL;
1848 EC_POINT *generator = NULL;
1849 const EC_POINT *p = NULL;
1850 const BIGNUM *p_scalar = NULL;
1853 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1856 if (((x = BN_CTX_get(ctx)) == NULL) ||
1857 ((y = BN_CTX_get(ctx)) == NULL) ||
1858 ((z = BN_CTX_get(ctx)) == NULL) ||
1859 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1862 if (scalar != NULL) {
1863 pre = EC_EX_DATA_get_data(group->extra_data,
1864 nistp521_pre_comp_dup,
1865 nistp521_pre_comp_free,
1866 nistp521_pre_comp_clear_free);
1868 /* we have precomputation, try to use it */
1869 g_pre_comp = &pre->g_pre_comp[0];
1871 /* try to use the standard precomputation */
1872 g_pre_comp = (felem(*)[3]) gmul;
1873 generator = EC_POINT_new(group);
1874 if (generator == NULL)
1876 /* get the generator from precomputation */
1877 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1878 !felem_to_BN(y, g_pre_comp[1][1]) ||
1879 !felem_to_BN(z, g_pre_comp[1][2])) {
1880 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1883 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1887 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1888 /* precomputation matches generator */
1892 * we don't have valid precomputation: treat the generator as a
1898 if (num_points > 0) {
1899 if (num_points >= 2) {
1901 * unless we precompute multiples for just one point, converting
1902 * those into affine form is time well spent
1906 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1907 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
1910 OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1911 if ((secrets == NULL) || (pre_comp == NULL)
1912 || (mixed && (tmp_felems == NULL))) {
1913 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1918 * we treat NULL scalars as 0, and NULL points as points at infinity,
1919 * i.e., they contribute nothing to the linear combination
1921 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1922 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1923 for (i = 0; i < num_points; ++i) {
1926 * we didn't have a valid precomputation, so we pick the
1930 p = EC_GROUP_get0_generator(group);
1933 /* the i^th point */
1936 p_scalar = scalars[i];
1938 if ((p_scalar != NULL) && (p != NULL)) {
1939 /* reduce scalar to 0 <= scalar < 2^521 */
1940 if ((BN_num_bits(p_scalar) > 521)
1941 || (BN_is_negative(p_scalar))) {
1943 * this is an unusual input, and we don't guarantee
1946 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1947 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1950 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1952 num_bytes = BN_bn2bin(p_scalar, tmp);
1953 flip_endian(secrets[i], tmp, num_bytes);
1954 /* precompute multiples */
1955 if ((!BN_to_felem(x_out, p->X)) ||
1956 (!BN_to_felem(y_out, p->Y)) ||
1957 (!BN_to_felem(z_out, p->Z)))
1959 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1960 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1961 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1962 for (j = 2; j <= 16; ++j) {
1964 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1965 pre_comp[i][j][2], pre_comp[i][1][0],
1966 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1967 pre_comp[i][j - 1][0],
1968 pre_comp[i][j - 1][1],
1969 pre_comp[i][j - 1][2]);
1971 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1972 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1973 pre_comp[i][j / 2][1],
1974 pre_comp[i][j / 2][2]);
1980 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1983 /* the scalar for the generator */
1984 if ((scalar != NULL) && (have_pre_comp)) {
1985 memset(g_secret, 0, sizeof(g_secret));
1986 /* reduce scalar to 0 <= scalar < 2^521 */
1987 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
1989 * this is an unusual input, and we don't guarantee
1992 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1993 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1996 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1998 num_bytes = BN_bn2bin(scalar, tmp);
1999 flip_endian(g_secret, tmp, num_bytes);
2000 /* do the multiplication with generator precomputation */
2001 batch_mul(x_out, y_out, z_out,
2002 (const felem_bytearray(*))secrets, num_points,
2004 mixed, (const felem(*)[17][3])pre_comp,
2005 (const felem(*)[3])g_pre_comp);
2007 /* do the multiplication without generator precomputation */
2008 batch_mul(x_out, y_out, z_out,
2009 (const felem_bytearray(*))secrets, num_points,
2010 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
2011 /* reduce the output to its unique minimal representation */
2012 felem_contract(x_in, x_out);
2013 felem_contract(y_in, y_out);
2014 felem_contract(z_in, z_out);
2015 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
2016 (!felem_to_BN(z, z_in))) {
2017 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
2020 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2024 EC_POINT_free(generator);
2025 if (new_ctx != NULL)
2026 BN_CTX_free(new_ctx);
2027 if (secrets != NULL)
2028 OPENSSL_free(secrets);
2029 if (pre_comp != NULL)
2030 OPENSSL_free(pre_comp);
2031 if (tmp_felems != NULL)
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_EX_DATA_free_data(&group->extra_data, nistp521_pre_comp_dup,
2048 nistp521_pre_comp_free,
2049 nistp521_pre_comp_clear_free);
2051 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2054 if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
2056 /* get the generator */
2057 if (group->generator == NULL)
2059 generator = EC_POINT_new(group);
2060 if (generator == NULL)
2062 BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
2063 BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
2064 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
2066 if ((pre = nistp521_pre_comp_new()) == NULL)
2069 * if the generator is the standard one, use built-in precomputation
2071 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2072 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2076 if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
2077 (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
2078 (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
2080 /* compute 2^130*G, 2^260*G, 2^390*G */
2081 for (i = 1; i <= 4; i <<= 1) {
2082 point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
2083 pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
2084 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
2085 for (j = 0; j < 129; ++j) {
2086 point_double(pre->g_pre_comp[2 * i][0],
2087 pre->g_pre_comp[2 * i][1],
2088 pre->g_pre_comp[2 * i][2],
2089 pre->g_pre_comp[2 * i][0],
2090 pre->g_pre_comp[2 * i][1],
2091 pre->g_pre_comp[2 * i][2]);
2094 /* g_pre_comp[0] is the point at infinity */
2095 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
2096 /* the remaining multiples */
2097 /* 2^130*G + 2^260*G */
2098 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2099 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2100 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2101 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2102 pre->g_pre_comp[2][2]);
2103 /* 2^130*G + 2^390*G */
2104 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2105 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2106 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2107 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2108 pre->g_pre_comp[2][2]);
2109 /* 2^260*G + 2^390*G */
2110 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2111 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2112 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2113 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2114 pre->g_pre_comp[4][2]);
2115 /* 2^130*G + 2^260*G + 2^390*G */
2116 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2117 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2118 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2119 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2120 pre->g_pre_comp[2][2]);
2121 for (i = 1; i < 8; ++i) {
2122 /* odd multiples: add G */
2123 point_add(pre->g_pre_comp[2 * i + 1][0],
2124 pre->g_pre_comp[2 * i + 1][1],
2125 pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
2126 pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
2127 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2128 pre->g_pre_comp[1][2]);
2130 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2132 if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp521_pre_comp_dup,
2133 nistp521_pre_comp_free,
2134 nistp521_pre_comp_clear_free))
2140 EC_POINT_free(generator);
2141 if (new_ctx != NULL)
2142 BN_CTX_free(new_ctx);
2143 nistp521_pre_comp_free(pre);
2147 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2149 if (EC_EX_DATA_get_data(group->extra_data, nistp521_pre_comp_dup,
2150 nistp521_pre_comp_free,
2151 nistp521_pre_comp_clear_free)
2159 static void *dummy = &dummy;