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
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 platforms */
46 #error "Need GCC 3.1 or later to define type uint128_t"
53 /* The underlying field.
55 * P521 operates over GF(2^521-1). We can serialise an element of this field
56 * into 66 bytes where the most significant byte contains only a single bit. We
57 * call this an felem_bytearray. */
59 typedef u8 felem_bytearray[66];
61 /* These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
62 * These values are big-endian. */
63 static const felem_bytearray nistp521_curve_params[5] =
65 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
66 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
67 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
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,
74 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
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 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
84 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
85 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
86 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
87 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
88 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
89 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
90 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
92 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
93 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
94 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
95 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
96 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
97 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
98 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
99 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
101 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
102 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
103 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
104 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
105 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
106 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
107 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
108 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
113 * The representation of field elements.
114 * ------------------------------------
116 * We represent field elements with nine values. These values are either 64 or
117 * 128 bits and the field element represented is:
118 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
119 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
120 * 58 bits apart, but are greater than 58 bits in length, the most significant
121 * bits of each limb overlap with the least significant bits of the next.
123 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
128 typedef uint64_t limb;
129 typedef limb felem[NLIMBS];
130 typedef uint128_t largefelem[NLIMBS];
132 static const limb bottom57bits = 0x1ffffffffffffff;
133 static const limb bottom58bits = 0x3ffffffffffffff;
135 /* bin66_to_felem takes a little-endian byte array and converts it into felem
136 * form. This assumes that the CPU is little-endian. */
137 static void bin66_to_felem(felem out, const u8 in[66])
139 out[0] = (*((limb*) &in[0])) & bottom58bits;
140 out[1] = (*((limb*) &in[7]) >> 2) & bottom58bits;
141 out[2] = (*((limb*) &in[14]) >> 4) & bottom58bits;
142 out[3] = (*((limb*) &in[21]) >> 6) & bottom58bits;
143 out[4] = (*((limb*) &in[29])) & bottom58bits;
144 out[5] = (*((limb*) &in[36]) >> 2) & bottom58bits;
145 out[6] = (*((limb*) &in[43]) >> 4) & bottom58bits;
146 out[7] = (*((limb*) &in[50]) >> 6) & bottom58bits;
147 out[8] = (*((limb*) &in[58])) & bottom57bits;
150 /* felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
151 * array. This assumes that the CPU is little-endian. */
152 static void felem_to_bin66(u8 out[66], const felem in)
155 (*((limb*) &out[0])) = in[0];
156 (*((limb*) &out[7])) |= in[1] << 2;
157 (*((limb*) &out[14])) |= in[2] << 4;
158 (*((limb*) &out[21])) |= in[3] << 6;
159 (*((limb*) &out[29])) = in[4];
160 (*((limb*) &out[36])) |= in[5] << 2;
161 (*((limb*) &out[43])) |= in[6] << 4;
162 (*((limb*) &out[50])) |= in[7] << 6;
163 (*((limb*) &out[58])) = in[8];
166 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
167 static void flip_endian(u8 *out, const u8 *in, unsigned len)
170 for (i = 0; i < len; ++i)
171 out[i] = in[len-1-i];
174 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
175 static int BN_to_felem(felem out, const BIGNUM *bn)
177 felem_bytearray b_in;
178 felem_bytearray b_out;
181 /* BN_bn2bin eats leading zeroes */
182 memset(b_out, 0, sizeof b_out);
183 num_bytes = BN_num_bytes(bn);
184 if (num_bytes > sizeof b_out)
186 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
189 if (BN_is_negative(bn))
191 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
194 num_bytes = BN_bn2bin(bn, b_in);
195 flip_endian(b_out, b_in, num_bytes);
196 bin66_to_felem(out, b_out);
200 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
201 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
203 felem_bytearray b_in, b_out;
204 felem_to_bin66(b_in, in);
205 flip_endian(b_out, b_in, sizeof b_out);
206 return BN_bin2bn(b_out, sizeof b_out, out);
215 static void felem_one(felem out)
228 static void felem_assign(felem out, const felem in)
241 /* felem_sum64 sets out = out + in. */
242 static void felem_sum64(felem out, const felem in)
255 /* felem_scalar sets out = in * scalar */
256 static void felem_scalar(felem out, const felem in, limb scalar)
258 out[0] = in[0] * scalar;
259 out[1] = in[1] * scalar;
260 out[2] = in[2] * scalar;
261 out[3] = in[3] * scalar;
262 out[4] = in[4] * scalar;
263 out[5] = in[5] * scalar;
264 out[6] = in[6] * scalar;
265 out[7] = in[7] * scalar;
266 out[8] = in[8] * scalar;
269 /* felem_scalar64 sets out = out * scalar */
270 static void felem_scalar64(felem out, limb scalar)
283 /* felem_scalar128 sets out = out * scalar */
284 static void felem_scalar128(largefelem out, limb scalar)
298 * felem_neg sets |out| to |-in|
300 * in[i] < 2^59 + 2^14
304 static void felem_neg(felem out, const felem in)
306 /* In order to prevent underflow, we subtract from 0 mod p. */
307 static const limb two62m3 = (((limb)1) << 62) - (((limb)1) << 5);
308 static const limb two62m2 = (((limb)1) << 62) - (((limb)1) << 4);
310 out[0] = two62m3 - in[0];
311 out[1] = two62m2 - in[1];
312 out[2] = two62m2 - in[2];
313 out[3] = two62m2 - in[3];
314 out[4] = two62m2 - in[4];
315 out[5] = two62m2 - in[5];
316 out[6] = two62m2 - in[6];
317 out[7] = two62m2 - in[7];
318 out[8] = two62m2 - in[8];
322 * felem_diff64 subtracts |in| from |out|
324 * in[i] < 2^59 + 2^14
326 * out[i] < out[i] + 2^62
328 static void felem_diff64(felem out, const felem in)
330 /* In order to prevent underflow, we add 0 mod p before subtracting. */
331 static const limb two62m3 = (((limb)1) << 62) - (((limb)1) << 5);
332 static const limb two62m2 = (((limb)1) << 62) - (((limb)1) << 4);
334 out[0] += two62m3 - in[0];
335 out[1] += two62m2 - in[1];
336 out[2] += two62m2 - in[2];
337 out[3] += two62m2 - in[3];
338 out[4] += two62m2 - in[4];
339 out[5] += two62m2 - in[5];
340 out[6] += two62m2 - in[6];
341 out[7] += two62m2 - in[7];
342 out[8] += two62m2 - in[8];
346 * felem_diff_128_64 subtracts |in| from |out|
348 * in[i] < 2^62 + 2^17
350 * out[i] < out[i] + 2^63
352 static void felem_diff_128_64(largefelem out, const felem in)
354 /* In order to prevent underflow, we add 0 mod p before subtracting. */
355 static const limb two63m6 = (((limb)1) << 62) - (((limb)1) << 5);
356 static const limb two63m5 = (((limb)1) << 62) - (((limb)1) << 4);
358 out[0] += two63m6 - in[0];
359 out[1] += two63m5 - in[1];
360 out[2] += two63m5 - in[2];
361 out[3] += two63m5 - in[3];
362 out[4] += two63m5 - in[4];
363 out[5] += two63m5 - in[5];
364 out[6] += two63m5 - in[6];
365 out[7] += two63m5 - in[7];
366 out[8] += two63m5 - in[8];
370 * felem_diff_128_64 subtracts |in| from |out|
374 * out[i] < out[i] + 2^127 - 2^69
376 static void felem_diff128(largefelem out, const largefelem in)
378 /* In order to prevent underflow, we add 0 mod p before subtracting. */
379 static const uint128_t two127m70 = (((uint128_t)1) << 127) - (((uint128_t)1) << 70);
380 static const uint128_t two127m69 = (((uint128_t)1) << 127) - (((uint128_t)1) << 69);
382 out[0] += (two127m70 - in[0]);
383 out[1] += (two127m69 - in[1]);
384 out[2] += (two127m69 - in[2]);
385 out[3] += (two127m69 - in[3]);
386 out[4] += (two127m69 - in[4]);
387 out[5] += (two127m69 - in[5]);
388 out[6] += (two127m69 - in[6]);
389 out[7] += (two127m69 - in[7]);
390 out[8] += (two127m69 - in[8]);
394 * felem_square sets |out| = |in|^2
398 * out[i] < 17 * max(in[i]) * max(in[i])
400 static void felem_square(largefelem out, const felem in)
403 felem_scalar(inx2, in, 2);
404 felem_scalar(inx4, in, 4);
407 * We have many cases were we want to do
410 * This is obviously just
412 * However, rather than do the doubling on the 128 bit result, we
413 * double one of the inputs to the multiplication by reading from
416 out[0] = ((uint128_t) in[0]) * in[0];
417 out[1] = ((uint128_t) in[0]) * inx2[1];
418 out[2] = ((uint128_t) in[0]) * inx2[2] +
419 ((uint128_t) in[1]) * in[1];
420 out[3] = ((uint128_t) in[0]) * inx2[3] +
421 ((uint128_t) in[1]) * inx2[2];
422 out[4] = ((uint128_t) in[0]) * inx2[4] +
423 ((uint128_t) in[1]) * inx2[3] +
424 ((uint128_t) in[2]) * in[2];
425 out[5] = ((uint128_t) in[0]) * inx2[5] +
426 ((uint128_t) in[1]) * inx2[4] +
427 ((uint128_t) in[2]) * inx2[3];
428 out[6] = ((uint128_t) in[0]) * inx2[6] +
429 ((uint128_t) in[1]) * inx2[5] +
430 ((uint128_t) in[2]) * inx2[4] +
431 ((uint128_t) in[3]) * in[3];
432 out[7] = ((uint128_t) in[0]) * inx2[7] +
433 ((uint128_t) in[1]) * inx2[6] +
434 ((uint128_t) in[2]) * inx2[5] +
435 ((uint128_t) in[3]) * inx2[4];
436 out[8] = ((uint128_t) in[0]) * inx2[8] +
437 ((uint128_t) in[1]) * inx2[7] +
438 ((uint128_t) in[2]) * inx2[6] +
439 ((uint128_t) in[3]) * inx2[5] +
440 ((uint128_t) in[4]) * in[4];
442 /* The remaining limbs fall above 2^521, with the first falling at
443 * 2^522. They correspond to locations one bit up from the limbs
444 * produced above so we would have to multiply by two to align them.
445 * Again, rather than operate on the 128-bit result, we double one of
446 * the inputs to the multiplication. If we want to double for both this
447 * reason, and the reason above, then we end up multiplying by four. */
450 out[0] += ((uint128_t) in[1]) * inx4[8] +
451 ((uint128_t) in[2]) * inx4[7] +
452 ((uint128_t) in[3]) * inx4[6] +
453 ((uint128_t) in[4]) * inx4[5];
456 out[1] += ((uint128_t) in[2]) * inx4[8] +
457 ((uint128_t) in[3]) * inx4[7] +
458 ((uint128_t) in[4]) * inx4[6] +
459 ((uint128_t) in[5]) * inx2[5];
462 out[2] += ((uint128_t) in[3]) * inx4[8] +
463 ((uint128_t) in[4]) * inx4[7] +
464 ((uint128_t) in[5]) * inx4[6];
467 out[3] += ((uint128_t) in[4]) * inx4[8] +
468 ((uint128_t) in[5]) * inx4[7] +
469 ((uint128_t) in[6]) * inx2[6];
472 out[4] += ((uint128_t) in[5]) * inx4[8] +
473 ((uint128_t) in[6]) * inx4[7];
476 out[5] += ((uint128_t) in[6]) * inx4[8] +
477 ((uint128_t) in[7]) * inx2[7];
480 out[6] += ((uint128_t) in[7]) * inx4[8];
483 out[7] += ((uint128_t) in[8]) * inx2[8];
487 * felem_mul sets |out| = |in1| * |in2|
492 * out[i] < 17 * max(in1[i]) * max(in2[i])
494 static void felem_mul(largefelem out, const felem in1, const felem in2)
497 felem_scalar(in2x2, in2, 2);
499 out[0] = ((uint128_t) in1[0]) * in2[0];
501 out[1] = ((uint128_t) in1[0]) * in2[1] +
502 ((uint128_t) in1[1]) * in2[0];
504 out[2] = ((uint128_t) in1[0]) * in2[2] +
505 ((uint128_t) in1[1]) * in2[1] +
506 ((uint128_t) in1[2]) * in2[0];
508 out[3] = ((uint128_t) in1[0]) * in2[3] +
509 ((uint128_t) in1[1]) * in2[2] +
510 ((uint128_t) in1[2]) * in2[1] +
511 ((uint128_t) in1[3]) * in2[0];
513 out[4] = ((uint128_t) in1[0]) * in2[4] +
514 ((uint128_t) in1[1]) * in2[3] +
515 ((uint128_t) in1[2]) * in2[2] +
516 ((uint128_t) in1[3]) * in2[1] +
517 ((uint128_t) in1[4]) * in2[0];
519 out[5] = ((uint128_t) in1[0]) * in2[5] +
520 ((uint128_t) in1[1]) * in2[4] +
521 ((uint128_t) in1[2]) * in2[3] +
522 ((uint128_t) in1[3]) * in2[2] +
523 ((uint128_t) in1[4]) * in2[1] +
524 ((uint128_t) in1[5]) * in2[0];
526 out[6] = ((uint128_t) in1[0]) * in2[6] +
527 ((uint128_t) in1[1]) * in2[5] +
528 ((uint128_t) in1[2]) * in2[4] +
529 ((uint128_t) in1[3]) * in2[3] +
530 ((uint128_t) in1[4]) * in2[2] +
531 ((uint128_t) in1[5]) * in2[1] +
532 ((uint128_t) in1[6]) * in2[0];
534 out[7] = ((uint128_t) in1[0]) * in2[7] +
535 ((uint128_t) in1[1]) * in2[6] +
536 ((uint128_t) in1[2]) * in2[5] +
537 ((uint128_t) in1[3]) * in2[4] +
538 ((uint128_t) in1[4]) * in2[3] +
539 ((uint128_t) in1[5]) * in2[2] +
540 ((uint128_t) in1[6]) * in2[1] +
541 ((uint128_t) in1[7]) * in2[0];
543 out[8] = ((uint128_t) in1[0]) * in2[8] +
544 ((uint128_t) in1[1]) * in2[7] +
545 ((uint128_t) in1[2]) * in2[6] +
546 ((uint128_t) in1[3]) * in2[5] +
547 ((uint128_t) in1[4]) * in2[4] +
548 ((uint128_t) in1[5]) * in2[3] +
549 ((uint128_t) in1[6]) * in2[2] +
550 ((uint128_t) in1[7]) * in2[1] +
551 ((uint128_t) in1[8]) * in2[0];
553 /* See comment in felem_square about the use of in2x2 here */
555 out[0] += ((uint128_t) in1[1]) * in2x2[8] +
556 ((uint128_t) in1[2]) * in2x2[7] +
557 ((uint128_t) in1[3]) * in2x2[6] +
558 ((uint128_t) in1[4]) * in2x2[5] +
559 ((uint128_t) in1[5]) * in2x2[4] +
560 ((uint128_t) in1[6]) * in2x2[3] +
561 ((uint128_t) in1[7]) * in2x2[2] +
562 ((uint128_t) in1[8]) * in2x2[1];
564 out[1] += ((uint128_t) in1[2]) * in2x2[8] +
565 ((uint128_t) in1[3]) * in2x2[7] +
566 ((uint128_t) in1[4]) * in2x2[6] +
567 ((uint128_t) in1[5]) * in2x2[5] +
568 ((uint128_t) in1[6]) * in2x2[4] +
569 ((uint128_t) in1[7]) * in2x2[3] +
570 ((uint128_t) in1[8]) * in2x2[2];
572 out[2] += ((uint128_t) in1[3]) * in2x2[8] +
573 ((uint128_t) in1[4]) * in2x2[7] +
574 ((uint128_t) in1[5]) * in2x2[6] +
575 ((uint128_t) in1[6]) * in2x2[5] +
576 ((uint128_t) in1[7]) * in2x2[4] +
577 ((uint128_t) in1[8]) * in2x2[3];
579 out[3] += ((uint128_t) in1[4]) * in2x2[8] +
580 ((uint128_t) in1[5]) * in2x2[7] +
581 ((uint128_t) in1[6]) * in2x2[6] +
582 ((uint128_t) in1[7]) * in2x2[5] +
583 ((uint128_t) in1[8]) * in2x2[4];
585 out[4] += ((uint128_t) in1[5]) * in2x2[8] +
586 ((uint128_t) in1[6]) * in2x2[7] +
587 ((uint128_t) in1[7]) * in2x2[6] +
588 ((uint128_t) in1[8]) * in2x2[5];
590 out[5] += ((uint128_t) in1[6]) * in2x2[8] +
591 ((uint128_t) in1[7]) * in2x2[7] +
592 ((uint128_t) in1[8]) * in2x2[6];
594 out[6] += ((uint128_t) in1[7]) * in2x2[8] +
595 ((uint128_t) in1[8]) * in2x2[7];
597 out[7] += ((uint128_t) in1[8]) * in2x2[8];
600 static const limb bottom52bits = 0xfffffffffffff;
603 * felem_reduce converts a largefelem to an felem.
607 * out[i] < 2^59 + 2^14
609 static void felem_reduce(felem out, const largefelem in)
611 u64 overflow1, overflow2;
613 out[0] = ((limb) in[0]) & bottom58bits;
614 out[1] = ((limb) in[1]) & bottom58bits;
615 out[2] = ((limb) in[2]) & bottom58bits;
616 out[3] = ((limb) in[3]) & bottom58bits;
617 out[4] = ((limb) in[4]) & bottom58bits;
618 out[5] = ((limb) in[5]) & bottom58bits;
619 out[6] = ((limb) in[6]) & bottom58bits;
620 out[7] = ((limb) in[7]) & bottom58bits;
621 out[8] = ((limb) in[8]) & bottom58bits;
625 out[1] += ((limb) in[0]) >> 58;
626 out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
628 * out[1] < 2^58 + 2^6 + 2^58
631 out[2] += ((limb) (in[0] >> 64)) >> 52;
633 out[2] += ((limb) in[1]) >> 58;
634 out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
635 out[3] += ((limb) (in[1] >> 64)) >> 52;
637 out[3] += ((limb) in[2]) >> 58;
638 out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
639 out[4] += ((limb) (in[2] >> 64)) >> 52;
641 out[4] += ((limb) in[3]) >> 58;
642 out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
643 out[5] += ((limb) (in[3] >> 64)) >> 52;
645 out[5] += ((limb) in[4]) >> 58;
646 out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
647 out[6] += ((limb) (in[4] >> 64)) >> 52;
649 out[6] += ((limb) in[5]) >> 58;
650 out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
651 out[7] += ((limb) (in[5] >> 64)) >> 52;
653 out[7] += ((limb) in[6]) >> 58;
654 out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
655 out[8] += ((limb) (in[6] >> 64)) >> 52;
657 out[8] += ((limb) in[7]) >> 58;
658 out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
660 * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
663 overflow1 = ((limb) (in[7] >> 64)) >> 52;
665 overflow1 += ((limb) in[8]) >> 58;
666 overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
667 overflow2 = ((limb) (in[8] >> 64)) >> 52;
669 overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
670 overflow2 <<= 1; /* overflow2 < 2^13 */
672 out[0] += overflow1; /* out[0] < 2^60 */
673 out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */
675 out[1] += out[0] >> 58; out[0] &= bottom58bits;
678 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
683 static void felem_square_reduce(felem out, const felem in)
686 felem_square(tmp, in);
687 felem_reduce(out, tmp);
690 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
693 felem_mul(tmp, in1, in2);
694 felem_reduce(out, tmp);
698 * felem_inv calculates |out| = |in|^{-1}
700 * Based on Fermat's Little Theorem:
702 * a^{p-1} = 1 (mod p)
703 * a^{p-2} = a^{-1} (mod p)
705 static void felem_inv(felem out, const felem in)
707 felem ftmp, ftmp2, ftmp3, ftmp4;
711 felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2^1 */
712 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
713 felem_assign(ftmp2, ftmp);
714 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
715 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */
716 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */
718 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */
719 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */
720 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */
722 felem_assign(ftmp2, ftmp3);
723 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */
724 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */
725 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */
726 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */
727 felem_assign(ftmp4, ftmp3);
728 felem_mul(tmp, ftmp3, ftmp); felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */
729 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */
730 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */
731 felem_assign(ftmp2, ftmp3);
733 for (i = 0; i < 8; i++)
735 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
737 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */
738 felem_assign(ftmp2, ftmp3);
740 for (i = 0; i < 16; i++)
742 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
744 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */
745 felem_assign(ftmp2, ftmp3);
747 for (i = 0; i < 32; i++)
749 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
751 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */
752 felem_assign(ftmp2, ftmp3);
754 for (i = 0; i < 64; i++)
756 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
758 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */
759 felem_assign(ftmp2, ftmp3);
761 for (i = 0; i < 128; i++)
763 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
765 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */
766 felem_assign(ftmp2, ftmp3);
768 for (i = 0; i < 256; i++)
770 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
772 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */
774 for (i = 0; i < 9; i++)
776 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
778 felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */
779 felem_mul(tmp, ftmp3, in); felem_reduce(out, tmp); /* 2^512 - 3 */
782 /* This is 2^521-1, expressed as an felem */
783 static const felem kPrime =
785 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
786 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
787 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
791 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
794 * in[i] < 2^59 + 2^14
796 static limb felem_is_zero(const felem in)
800 felem_assign(ftmp, in);
802 ftmp[0] += ftmp[8] >> 57; ftmp[8] &= bottom57bits;
804 ftmp[1] += ftmp[0] >> 58; ftmp[0] &= bottom58bits;
805 ftmp[2] += ftmp[1] >> 58; ftmp[1] &= bottom58bits;
806 ftmp[3] += ftmp[2] >> 58; ftmp[2] &= bottom58bits;
807 ftmp[4] += ftmp[3] >> 58; ftmp[3] &= bottom58bits;
808 ftmp[5] += ftmp[4] >> 58; ftmp[4] &= bottom58bits;
809 ftmp[6] += ftmp[5] >> 58; ftmp[5] &= bottom58bits;
810 ftmp[7] += ftmp[6] >> 58; ftmp[6] &= bottom58bits;
811 ftmp[8] += ftmp[7] >> 58; ftmp[7] &= bottom58bits;
812 /* ftmp[8] < 2^57 + 4 */
814 /* The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is
815 * greater than our bound for ftmp[8]. Therefore we only have to check
816 * if the zero is zero or 2^521-1. */
830 /* We know that ftmp[i] < 2^63, therefore the only way that the top bit
831 * can be set is if is_zero was 0 before the decrement. */
832 is_zero = ((s64) is_zero) >> 63;
834 is_p = ftmp[0] ^ kPrime[0];
835 is_p |= ftmp[1] ^ kPrime[1];
836 is_p |= ftmp[2] ^ kPrime[2];
837 is_p |= ftmp[3] ^ kPrime[3];
838 is_p |= ftmp[4] ^ kPrime[4];
839 is_p |= ftmp[5] ^ kPrime[5];
840 is_p |= ftmp[6] ^ kPrime[6];
841 is_p |= ftmp[7] ^ kPrime[7];
842 is_p |= ftmp[8] ^ kPrime[8];
845 is_p = ((s64) is_p) >> 63;
851 static int felem_is_zero_int(const felem in)
853 return (int) (felem_is_zero(in) & ((limb)1));
857 * felem_contract converts |in| to its unique, minimal representation.
859 * in[i] < 2^59 + 2^14
861 static void felem_contract(felem out, const felem in)
863 limb is_p, is_greater, sign;
864 static const limb two58 = ((limb)1) << 58;
866 felem_assign(out, in);
868 out[0] += out[8] >> 57; out[8] &= bottom57bits;
870 out[1] += out[0] >> 58; out[0] &= bottom58bits;
871 out[2] += out[1] >> 58; out[1] &= bottom58bits;
872 out[3] += out[2] >> 58; out[2] &= bottom58bits;
873 out[4] += out[3] >> 58; out[3] &= bottom58bits;
874 out[5] += out[4] >> 58; out[4] &= bottom58bits;
875 out[6] += out[5] >> 58; out[5] &= bottom58bits;
876 out[7] += out[6] >> 58; out[6] &= bottom58bits;
877 out[8] += out[7] >> 58; out[7] &= bottom58bits;
878 /* out[8] < 2^57 + 4 */
880 /* If the value is greater than 2^521-1 then we have to subtract
881 * 2^521-1 out. See the comments in felem_is_zero regarding why we
882 * don't test for other multiples of the prime. */
884 /* First, if |out| is equal to 2^521-1, we subtract it out to get zero. */
886 is_p = out[0] ^ kPrime[0];
887 is_p |= out[1] ^ kPrime[1];
888 is_p |= out[2] ^ kPrime[2];
889 is_p |= out[3] ^ kPrime[3];
890 is_p |= out[4] ^ kPrime[4];
891 is_p |= out[5] ^ kPrime[5];
892 is_p |= out[6] ^ kPrime[6];
893 is_p |= out[7] ^ kPrime[7];
894 is_p |= out[8] ^ kPrime[8];
903 is_p = ((s64) is_p) >> 63;
906 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
918 /* In order to test that |out| >= 2^521-1 we need only test if out[8]
919 * >> 57 is greater than zero as (2^521-1) + x >= 2^522 */
920 is_greater = out[8] >> 57;
921 is_greater |= is_greater << 32;
922 is_greater |= is_greater << 16;
923 is_greater |= is_greater << 8;
924 is_greater |= is_greater << 4;
925 is_greater |= is_greater << 2;
926 is_greater |= is_greater << 1;
927 is_greater = ((s64) is_greater) >> 63;
929 out[0] -= kPrime[0] & is_greater;
930 out[1] -= kPrime[1] & is_greater;
931 out[2] -= kPrime[2] & is_greater;
932 out[3] -= kPrime[3] & is_greater;
933 out[4] -= kPrime[4] & is_greater;
934 out[5] -= kPrime[5] & is_greater;
935 out[6] -= kPrime[6] & is_greater;
936 out[7] -= kPrime[7] & is_greater;
937 out[8] -= kPrime[8] & is_greater;
939 /* Eliminate negative coefficients */
940 sign = -(out[0] >> 63); out[0] += (two58 & sign); out[1] -= (1 & sign);
941 sign = -(out[1] >> 63); out[1] += (two58 & sign); out[2] -= (1 & sign);
942 sign = -(out[2] >> 63); out[2] += (two58 & sign); out[3] -= (1 & sign);
943 sign = -(out[3] >> 63); out[3] += (two58 & sign); out[4] -= (1 & sign);
944 sign = -(out[4] >> 63); out[4] += (two58 & sign); out[5] -= (1 & sign);
945 sign = -(out[0] >> 63); out[5] += (two58 & sign); out[6] -= (1 & sign);
946 sign = -(out[6] >> 63); out[6] += (two58 & sign); out[7] -= (1 & sign);
947 sign = -(out[7] >> 63); out[7] += (two58 & sign); out[8] -= (1 & sign);
948 sign = -(out[5] >> 63); out[5] += (two58 & sign); out[6] -= (1 & sign);
949 sign = -(out[6] >> 63); out[6] += (two58 & sign); out[7] -= (1 & sign);
950 sign = -(out[7] >> 63); out[7] += (two58 & sign); out[8] -= (1 & sign);
957 * Building on top of the field operations we have the operations on the
958 * elliptic curve group itself. Points on the curve are represented in Jacobian
962 * point_double calcuates 2*(x_in, y_in, z_in)
964 * The method is taken from:
965 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
967 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
968 * while x_out == y_in is not (maybe this works, but it's not tested). */
970 point_double(felem x_out, felem y_out, felem z_out,
971 const felem x_in, const felem y_in, const felem z_in)
973 largefelem tmp, tmp2;
974 felem delta, gamma, beta, alpha, ftmp, ftmp2;
976 felem_assign(ftmp, x_in);
977 felem_assign(ftmp2, x_in);
980 felem_square(tmp, z_in);
981 felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */
984 felem_square(tmp, y_in);
985 felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */
988 felem_mul(tmp, x_in, gamma);
989 felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */
991 /* alpha = 3*(x-delta)*(x+delta) */
992 felem_diff64(ftmp, delta);
994 felem_sum64(ftmp2, delta);
995 /* ftmp2[i] < 2^60 + 2^15 */
996 felem_scalar64(ftmp2, 3);
997 /* ftmp2[i] < 3*2^60 + 3*2^15 */
998 felem_mul(tmp, ftmp, ftmp2);
1000 * tmp[i] < 17(3*2^121 + 3*2^76)
1001 * = 61*2^121 + 61*2^76
1002 * < 64*2^121 + 64*2^76
1006 felem_reduce(alpha, tmp);
1008 /* x' = alpha^2 - 8*beta */
1009 felem_square(tmp, alpha);
1010 /* tmp[i] < 17*2^120
1012 felem_assign(ftmp, beta);
1013 felem_scalar64(ftmp, 8);
1014 /* ftmp[i] < 2^62 + 2^17 */
1015 felem_diff_128_64(tmp, ftmp);
1016 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
1017 felem_reduce(x_out, tmp);
1019 /* z' = (y + z)^2 - gamma - delta */
1020 felem_sum64(delta, gamma);
1021 /* delta[i] < 2^60 + 2^15 */
1022 felem_assign(ftmp, y_in);
1023 felem_sum64(ftmp, z_in);
1024 /* ftmp[i] < 2^60 + 2^15 */
1025 felem_square(tmp, ftmp);
1026 /* tmp[i] < 17(2^122)
1028 felem_diff_128_64(tmp, delta);
1029 /* tmp[i] < 2^127 + 2^63 */
1030 felem_reduce(z_out, tmp);
1032 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1033 felem_scalar64(beta, 4);
1034 /* beta[i] < 2^61 + 2^16 */
1035 felem_diff64(beta, x_out);
1036 /* beta[i] < 2^61 + 2^60 + 2^16 */
1037 felem_mul(tmp, alpha, beta);
1039 * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1040 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1041 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1044 felem_square(tmp2, gamma);
1046 * tmp2[i] < 17*(2^59 + 2^14)^2
1047 * = 17*(2^118 + 2^74 + 2^28)
1049 felem_scalar128(tmp2, 8);
1051 * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1052 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1055 felem_diff128(tmp, tmp2);
1057 * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1058 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1059 * 2^74 + 2^69 + 2^34 + 2^30
1062 felem_reduce(y_out, tmp);
1065 /* copy_conditional copies in to out iff mask is all ones. */
1067 copy_conditional(felem out, const felem in, limb mask)
1070 for (i = 0; i < NLIMBS; ++i)
1072 const limb tmp = mask & (in[i] ^ out[i]);
1078 * point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1080 * The method is taken from
1081 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1082 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1084 * This function includes a branch for checking whether the two input points
1085 * are equal (while not equal to the point at infinity). This case never
1086 * happens during single point multiplication, so there is no timing leak for
1087 * ECDH or ECDSA signing. */
1088 static void point_add(felem x3, felem y3, felem z3,
1089 const felem x1, const felem y1, const felem z1,
1090 const int mixed, const felem x2, const felem y2, const felem z2)
1092 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1093 largefelem tmp, tmp2;
1094 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1096 z1_is_zero = felem_is_zero(z1);
1097 z2_is_zero = felem_is_zero(z2);
1099 /* ftmp = z1z1 = z1**2 */
1100 felem_square(tmp, z1);
1101 felem_reduce(ftmp, tmp);
1105 /* ftmp2 = z2z2 = z2**2 */
1106 felem_square(tmp, z2);
1107 felem_reduce(ftmp2, tmp);
1109 /* u1 = ftmp3 = x1*z2z2 */
1110 felem_mul(tmp, x1, ftmp2);
1111 felem_reduce(ftmp3, tmp);
1113 /* ftmp5 = z1 + z2 */
1114 felem_assign(ftmp5, z1);
1115 felem_sum64(ftmp5, z2);
1116 /* ftmp5[i] < 2^61 */
1118 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1119 felem_square(tmp, ftmp5);
1120 /* tmp[i] < 17*2^122 */
1121 felem_diff_128_64(tmp, ftmp);
1122 /* tmp[i] < 17*2^122 + 2^63 */
1123 felem_diff_128_64(tmp, ftmp2);
1124 /* tmp[i] < 17*2^122 + 2^64 */
1125 felem_reduce(ftmp5, tmp);
1127 /* ftmp2 = z2 * z2z2 */
1128 felem_mul(tmp, ftmp2, z2);
1129 felem_reduce(ftmp2, tmp);
1131 /* s1 = ftmp6 = y1 * z2**3 */
1132 felem_mul(tmp, y1, ftmp2);
1133 felem_reduce(ftmp6, tmp);
1137 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
1139 /* u1 = ftmp3 = x1*z2z2 */
1140 felem_assign(ftmp3, x1);
1142 /* ftmp5 = 2*z1z2 */
1143 felem_scalar(ftmp5, z1, 2);
1145 /* s1 = ftmp6 = y1 * z2**3 */
1146 felem_assign(ftmp6, y1);
1150 felem_mul(tmp, x2, ftmp);
1151 /* tmp[i] < 17*2^120 */
1153 /* h = ftmp4 = u2 - u1 */
1154 felem_diff_128_64(tmp, ftmp3);
1155 /* tmp[i] < 17*2^120 + 2^63 */
1156 felem_reduce(ftmp4, tmp);
1158 x_equal = felem_is_zero(ftmp4);
1160 /* z_out = ftmp5 * h */
1161 felem_mul(tmp, ftmp5, ftmp4);
1162 felem_reduce(z_out, tmp);
1164 /* ftmp = z1 * z1z1 */
1165 felem_mul(tmp, ftmp, z1);
1166 felem_reduce(ftmp, tmp);
1168 /* s2 = tmp = y2 * z1**3 */
1169 felem_mul(tmp, y2, ftmp);
1170 /* tmp[i] < 17*2^120 */
1172 /* r = ftmp5 = (s2 - s1)*2 */
1173 felem_diff_128_64(tmp, ftmp6);
1174 /* tmp[i] < 17*2^120 + 2^63 */
1175 felem_reduce(ftmp5, tmp);
1176 y_equal = felem_is_zero(ftmp5);
1177 felem_scalar64(ftmp5, 2);
1178 /* ftmp5[i] < 2^61 */
1180 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
1182 point_double(x3, y3, z3, x1, y1, z1);
1186 /* I = ftmp = (2h)**2 */
1187 felem_assign(ftmp, ftmp4);
1188 felem_scalar64(ftmp, 2);
1189 /* ftmp[i] < 2^61 */
1190 felem_square(tmp, ftmp);
1191 /* tmp[i] < 17*2^122 */
1192 felem_reduce(ftmp, tmp);
1194 /* J = ftmp2 = h * I */
1195 felem_mul(tmp, ftmp4, ftmp);
1196 felem_reduce(ftmp2, tmp);
1198 /* V = ftmp4 = U1 * I */
1199 felem_mul(tmp, ftmp3, ftmp);
1200 felem_reduce(ftmp4, tmp);
1202 /* x_out = r**2 - J - 2V */
1203 felem_square(tmp, ftmp5);
1204 /* tmp[i] < 17*2^122 */
1205 felem_diff_128_64(tmp, ftmp2);
1206 /* tmp[i] < 17*2^122 + 2^63 */
1207 felem_assign(ftmp3, ftmp4);
1208 felem_scalar64(ftmp4, 2);
1209 /* ftmp4[i] < 2^61 */
1210 felem_diff_128_64(tmp, ftmp4);
1211 /* tmp[i] < 17*2^122 + 2^64 */
1212 felem_reduce(x_out, tmp);
1214 /* y_out = r(V-x_out) - 2 * s1 * J */
1215 felem_diff64(ftmp3, x_out);
1216 /* ftmp3[i] < 2^60 + 2^60
1218 felem_mul(tmp, ftmp5, ftmp3);
1219 /* tmp[i] < 17*2^122 */
1220 felem_mul(tmp2, ftmp6, ftmp2);
1221 /* tmp2[i] < 17*2^120 */
1222 felem_scalar128(tmp2, 2);
1223 /* tmp2[i] < 17*2^121 */
1224 felem_diff128(tmp, tmp2);
1226 * tmp[i] < 2^127 - 2^69 + 17*2^122
1227 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1230 felem_reduce(y_out, tmp);
1232 copy_conditional(x_out, x2, z1_is_zero);
1233 copy_conditional(x_out, x1, z2_is_zero);
1234 copy_conditional(y_out, y2, z1_is_zero);
1235 copy_conditional(y_out, y1, z2_is_zero);
1236 copy_conditional(z_out, z2, z1_is_zero);
1237 copy_conditional(z_out, z1, z2_is_zero);
1238 felem_assign(x3, x_out);
1239 felem_assign(y3, y_out);
1240 felem_assign(z3, z_out);
1244 * Base point pre computation
1245 * --------------------------
1247 * Two different sorts of precomputed tables are used in the following code.
1248 * Each contain various points on the curve, where each point is three field
1249 * elements (x, y, z).
1251 * For the base point table, z is usually 1 (0 for the point at infinity).
1252 * This table has 16 elements:
1253 * index | bits | point
1254 * ------+---------+------------------------------
1257 * 2 | 0 0 1 0 | 2^130G
1258 * 3 | 0 0 1 1 | (2^130 + 1)G
1259 * 4 | 0 1 0 0 | 2^260G
1260 * 5 | 0 1 0 1 | (2^260 + 1)G
1261 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1262 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1263 * 8 | 1 0 0 0 | 2^390G
1264 * 9 | 1 0 0 1 | (2^390 + 1)G
1265 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1266 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1267 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1268 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1269 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1270 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1272 * The reason for this is so that we can clock bits into four different
1273 * locations when doing simple scalar multiplies against the base point.
1275 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1277 /* gmul is the table of precomputed base points */
1278 static const felem gmul[16][3] =
1279 {{{0, 0, 0, 0, 0, 0, 0, 0, 0},
1280 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1281 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1282 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1283 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1284 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1285 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1286 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1287 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1288 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1289 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1290 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1291 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1292 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1293 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1294 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1295 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1296 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1297 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1298 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1299 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1300 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1301 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1302 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1303 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1304 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1305 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1306 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1307 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1308 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1309 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1310 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1311 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1312 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1313 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1314 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1315 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1316 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1317 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1318 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1319 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1320 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1321 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1322 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1323 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1324 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1325 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1326 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1327 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1328 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1329 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1330 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1331 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1332 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1333 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1334 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1335 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1336 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1337 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1338 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1339 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1340 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1341 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1342 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1343 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1344 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1345 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1346 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1347 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1348 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1349 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1350 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1351 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1352 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1353 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1354 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1355 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1356 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1357 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1358 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1359 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1360 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1361 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1362 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1363 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1364 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1365 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1366 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1367 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1368 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1369 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1370 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1371 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1372 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1373 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1374 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1375 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1376 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1377 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1378 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1379 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1380 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1381 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1382 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1383 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1384 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1385 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1386 {1, 0, 0, 0, 0, 0, 0, 0, 0}}};
1388 /* select_point selects the |idx|th point from a precomputation table and
1389 * copies it to out. */
1390 static void select_point(const limb idx, unsigned int size, const felem pre_comp[/* size */][3],
1394 limb *outlimbs = &out[0][0];
1395 memset(outlimbs, 0, 3 * sizeof(felem));
1397 for (i = 0; i < size; i++)
1399 const limb *inlimbs = &pre_comp[i][0][0];
1400 limb mask = i ^ idx;
1406 for (j = 0; j < NLIMBS * 3; j++)
1407 outlimbs[j] |= inlimbs[j] & mask;
1411 /* get_bit returns the |i|th bit in |in| */
1412 static char get_bit(const felem_bytearray in, int i)
1416 return (in[i >> 3] >> (i & 7)) & 1;
1419 /* Interleaved point multiplication using precomputed point multiples:
1420 * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
1421 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1422 * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1423 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1424 static void batch_mul(felem x_out, felem y_out, felem z_out,
1425 const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
1426 const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[16][3])
1429 unsigned num, gen_mul = (g_scalar != NULL);
1430 felem nq[3], tmp[4];
1434 /* set nq to the point at infinity */
1435 memset(nq, 0, 3 * sizeof(felem));
1437 /* Loop over all scalars msb-to-lsb, interleaving additions
1438 * of multiples of the generator (last quarter of rounds)
1439 * and additions of other points multiples (every 5th round).
1441 skip = 1; /* save two point operations in the first round */
1442 for (i = (num_points ? 520 : 130); i >= 0; --i)
1446 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1448 /* add multiples of the generator */
1449 if (gen_mul && (i <= 130))
1451 bits = get_bit(g_scalar, i + 390) << 3;
1454 bits |= get_bit(g_scalar, i + 260) << 2;
1455 bits |= get_bit(g_scalar, i + 130) << 1;
1456 bits |= get_bit(g_scalar, i);
1458 /* select the point to add, in constant time */
1459 select_point(bits, 16, g_pre_comp, tmp);
1462 point_add(nq[0], nq[1], nq[2],
1463 nq[0], nq[1], nq[2],
1464 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1468 memcpy(nq, tmp, 3 * sizeof(felem));
1473 /* do other additions every 5 doublings */
1474 if (num_points && (i % 5 == 0))
1476 /* loop over all scalars */
1477 for (num = 0; num < num_points; ++num)
1479 bits = get_bit(scalars[num], i + 4) << 5;
1480 bits |= get_bit(scalars[num], i + 3) << 4;
1481 bits |= get_bit(scalars[num], i + 2) << 3;
1482 bits |= get_bit(scalars[num], i + 1) << 2;
1483 bits |= get_bit(scalars[num], i) << 1;
1484 bits |= get_bit(scalars[num], i - 1);
1485 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1487 /* select the point to add or subtract, in constant time */
1488 select_point(digit, 17, pre_comp[num], tmp);
1489 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
1490 copy_conditional(tmp[1], tmp[3], (-(limb) sign));
1494 point_add(nq[0], nq[1], nq[2],
1495 nq[0], nq[1], nq[2],
1496 mixed, tmp[0], tmp[1], tmp[2]);
1500 memcpy(nq, tmp, 3 * sizeof(felem));
1506 felem_assign(x_out, nq[0]);
1507 felem_assign(y_out, nq[1]);
1508 felem_assign(z_out, nq[2]);
1512 /* Precomputation for the group generator. */
1514 felem g_pre_comp[16][3];
1516 } NISTP521_PRE_COMP;
1518 const EC_METHOD *EC_GFp_nistp521_method(void)
1520 static const EC_METHOD ret = {
1521 EC_FLAGS_DEFAULT_OCT,
1522 NID_X9_62_prime_field,
1523 ec_GFp_nistp521_group_init,
1524 ec_GFp_simple_group_finish,
1525 ec_GFp_simple_group_clear_finish,
1526 ec_GFp_nist_group_copy,
1527 ec_GFp_nistp521_group_set_curve,
1528 ec_GFp_simple_group_get_curve,
1529 ec_GFp_simple_group_get_degree,
1530 ec_GFp_simple_group_check_discriminant,
1531 ec_GFp_simple_point_init,
1532 ec_GFp_simple_point_finish,
1533 ec_GFp_simple_point_clear_finish,
1534 ec_GFp_simple_point_copy,
1535 ec_GFp_simple_point_set_to_infinity,
1536 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1537 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1538 ec_GFp_simple_point_set_affine_coordinates,
1539 ec_GFp_nistp521_point_get_affine_coordinates,
1540 0 /* point_set_compressed_coordinates */,
1545 ec_GFp_simple_invert,
1546 ec_GFp_simple_is_at_infinity,
1547 ec_GFp_simple_is_on_curve,
1549 ec_GFp_simple_make_affine,
1550 ec_GFp_simple_points_make_affine,
1551 ec_GFp_nistp521_points_mul,
1552 ec_GFp_nistp521_precompute_mult,
1553 ec_GFp_nistp521_have_precompute_mult,
1554 ec_GFp_nist_field_mul,
1555 ec_GFp_nist_field_sqr,
1557 0 /* field_encode */,
1558 0 /* field_decode */,
1559 0 /* field_set_to_one */ };
1565 /******************************************************************************/
1566 /* FUNCTIONS TO MANAGE PRECOMPUTATION
1569 static NISTP521_PRE_COMP *nistp521_pre_comp_new()
1571 NISTP521_PRE_COMP *ret = NULL;
1572 ret = (NISTP521_PRE_COMP *)OPENSSL_malloc(sizeof(NISTP521_PRE_COMP));
1575 ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1578 memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1579 ret->references = 1;
1583 static void *nistp521_pre_comp_dup(void *src_)
1585 NISTP521_PRE_COMP *src = src_;
1587 /* no need to actually copy, these objects never change! */
1588 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1593 static void nistp521_pre_comp_free(void *pre_)
1596 NISTP521_PRE_COMP *pre = pre_;
1601 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1608 static void nistp521_pre_comp_clear_free(void *pre_)
1611 NISTP521_PRE_COMP *pre = pre_;
1616 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1620 OPENSSL_cleanse(pre, sizeof(*pre));
1624 /******************************************************************************/
1625 /* OPENSSL EC_METHOD FUNCTIONS
1628 int ec_GFp_nistp521_group_init(EC_GROUP *group)
1631 ret = ec_GFp_simple_group_init(group);
1632 group->a_is_minus3 = 1;
1636 int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1637 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1640 BN_CTX *new_ctx = NULL;
1641 BIGNUM *curve_p, *curve_a, *curve_b;
1644 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1646 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1647 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1648 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1649 BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
1650 BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
1651 BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
1652 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1653 (BN_cmp(curve_b, b)))
1655 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE,
1656 EC_R_WRONG_CURVE_PARAMETERS);
1659 group->field_mod_func = BN_nist_mod_521;
1660 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1663 if (new_ctx != NULL)
1664 BN_CTX_free(new_ctx);
1668 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1669 * (X', Y') = (X/Z^2, Y/Z^3) */
1670 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
1671 const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1673 felem z1, z2, x_in, y_in, x_out, y_out;
1676 if (EC_POINT_is_at_infinity(group, point))
1678 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES,
1679 EC_R_POINT_AT_INFINITY);
1682 if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1683 (!BN_to_felem(z1, &point->Z))) return 0;
1685 felem_square(tmp, z2); felem_reduce(z1, tmp);
1686 felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1687 felem_contract(x_out, x_in);
1690 if (!felem_to_BN(x, x_out))
1692 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
1696 felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1697 felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1698 felem_contract(y_out, y_in);
1701 if (!felem_to_BN(y, y_out))
1703 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
1710 static void make_points_affine(size_t num, felem points[/* num */][3], felem tmp_felems[/* num+1 */])
1712 /* Runs in constant time, unless an input is the point at infinity
1713 * (which normally shouldn't happen). */
1714 ec_GFp_nistp_points_make_affine_internal(
1719 (void (*)(void *)) felem_one,
1720 (int (*)(const void *)) felem_is_zero_int,
1721 (void (*)(void *, const void *)) felem_assign,
1722 (void (*)(void *, const void *)) felem_square_reduce,
1723 (void (*)(void *, const void *, const void *)) felem_mul_reduce,
1724 (void (*)(void *, const void *)) felem_inv,
1725 (void (*)(void *, const void *)) felem_contract);
1728 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1729 * Result is stored in r (r can equal one of the inputs). */
1730 int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
1731 const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1732 const BIGNUM *scalars[], BN_CTX *ctx)
1737 BN_CTX *new_ctx = NULL;
1738 BIGNUM *x, *y, *z, *tmp_scalar;
1739 felem_bytearray g_secret;
1740 felem_bytearray *secrets = NULL;
1741 felem (*pre_comp)[17][3] = NULL;
1742 felem *tmp_felems = NULL;
1743 felem_bytearray tmp;
1744 unsigned i, num_bytes;
1745 int have_pre_comp = 0;
1746 size_t num_points = num;
1747 felem x_in, y_in, z_in, x_out, y_out, z_out;
1748 NISTP521_PRE_COMP *pre = NULL;
1749 felem (*g_pre_comp)[3] = NULL;
1750 EC_POINT *generator = NULL;
1751 const EC_POINT *p = NULL;
1752 const BIGNUM *p_scalar = NULL;
1755 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1757 if (((x = BN_CTX_get(ctx)) == NULL) ||
1758 ((y = BN_CTX_get(ctx)) == NULL) ||
1759 ((z = BN_CTX_get(ctx)) == NULL) ||
1760 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1765 pre = EC_EX_DATA_get_data(group->extra_data,
1766 nistp521_pre_comp_dup, nistp521_pre_comp_free,
1767 nistp521_pre_comp_clear_free);
1769 /* we have precomputation, try to use it */
1770 g_pre_comp = &pre->g_pre_comp[0];
1772 /* try to use the standard precomputation */
1773 g_pre_comp = (felem (*)[3]) gmul;
1774 generator = EC_POINT_new(group);
1775 if (generator == NULL)
1777 /* get the generator from precomputation */
1778 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1779 !felem_to_BN(y, g_pre_comp[1][1]) ||
1780 !felem_to_BN(z, g_pre_comp[1][2]))
1782 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1785 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1786 generator, x, y, z, ctx))
1788 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1789 /* precomputation matches generator */
1792 /* we don't have valid precomputation:
1793 * treat the generator as a random point */
1799 if (num_points >= 2)
1801 /* unless we precompute multiples for just one point,
1802 * converting those into affine form is time well spent */
1805 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1806 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
1808 tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1809 if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL)))
1811 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1815 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1816 * i.e., they contribute nothing to the linear combination */
1817 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1818 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1819 for (i = 0; i < num_points; ++i)
1822 /* we didn't have a valid precomputation, so we pick
1825 p = EC_GROUP_get0_generator(group);
1829 /* the i^th point */
1832 p_scalar = scalars[i];
1834 if ((p_scalar != NULL) && (p != NULL))
1836 /* reduce scalar to 0 <= scalar < 2^521 */
1837 if ((BN_num_bits(p_scalar) > 521) || (BN_is_negative(p_scalar)))
1839 /* this is an unusual input, and we don't guarantee
1840 * constant-timeness */
1841 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1843 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1846 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1849 num_bytes = BN_bn2bin(p_scalar, tmp);
1850 flip_endian(secrets[i], tmp, num_bytes);
1851 /* precompute multiples */
1852 if ((!BN_to_felem(x_out, &p->X)) ||
1853 (!BN_to_felem(y_out, &p->Y)) ||
1854 (!BN_to_felem(z_out, &p->Z))) goto err;
1855 memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
1856 memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
1857 memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
1858 for (j = 2; j <= 16; ++j)
1863 pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1864 pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
1865 0, pre_comp[i][j-1][0], pre_comp[i][j-1][1], pre_comp[i][j-1][2]);
1870 pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1871 pre_comp[i][j/2][0], pre_comp[i][j/2][1], pre_comp[i][j/2][2]);
1877 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1880 /* the scalar for the generator */
1881 if ((scalar != NULL) && (have_pre_comp))
1883 memset(g_secret, 0, sizeof(g_secret));
1884 /* reduce scalar to 0 <= scalar < 2^521 */
1885 if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar)))
1887 /* this is an unusual input, and we don't guarantee
1888 * constant-timeness */
1889 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1891 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1894 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1897 num_bytes = BN_bn2bin(scalar, tmp);
1898 flip_endian(g_secret, tmp, num_bytes);
1899 /* do the multiplication with generator precomputation*/
1900 batch_mul(x_out, y_out, z_out,
1901 (const felem_bytearray (*)) secrets, num_points,
1903 mixed, (const felem (*)[17][3]) pre_comp,
1904 (const felem (*)[3]) g_pre_comp);
1907 /* do the multiplication without generator precomputation */
1908 batch_mul(x_out, y_out, z_out,
1909 (const felem_bytearray (*)) secrets, num_points,
1910 NULL, mixed, (const felem (*)[17][3]) pre_comp, NULL);
1911 /* reduce the output to its unique minimal representation */
1912 felem_contract(x_in, x_out);
1913 felem_contract(y_in, y_out);
1914 felem_contract(z_in, z_out);
1915 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1916 (!felem_to_BN(z, z_in)))
1918 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB);
1921 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1925 if (generator != NULL)
1926 EC_POINT_free(generator);
1927 if (new_ctx != NULL)
1928 BN_CTX_free(new_ctx);
1929 if (secrets != NULL)
1930 OPENSSL_free(secrets);
1931 if (pre_comp != NULL)
1932 OPENSSL_free(pre_comp);
1933 if (tmp_felems != NULL)
1934 OPENSSL_free(tmp_felems);
1938 int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1941 NISTP521_PRE_COMP *pre = NULL;
1943 BN_CTX *new_ctx = NULL;
1945 EC_POINT *generator = NULL;
1946 felem tmp_felems[16];
1948 /* throw away old precomputation */
1949 EC_EX_DATA_free_data(&group->extra_data, nistp521_pre_comp_dup,
1950 nistp521_pre_comp_free, nistp521_pre_comp_clear_free);
1952 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1954 if (((x = BN_CTX_get(ctx)) == NULL) ||
1955 ((y = BN_CTX_get(ctx)) == NULL))
1957 /* get the generator */
1958 if (group->generator == NULL) goto err;
1959 generator = EC_POINT_new(group);
1960 if (generator == NULL)
1962 BN_bin2bn(nistp521_curve_params[3], sizeof (felem_bytearray), x);
1963 BN_bin2bn(nistp521_curve_params[4], sizeof (felem_bytearray), y);
1964 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1966 if ((pre = nistp521_pre_comp_new()) == NULL)
1968 /* if the generator is the standard one, use built-in precomputation */
1969 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1971 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1975 if ((!BN_to_felem(pre->g_pre_comp[1][0], &group->generator->X)) ||
1976 (!BN_to_felem(pre->g_pre_comp[1][1], &group->generator->Y)) ||
1977 (!BN_to_felem(pre->g_pre_comp[1][2], &group->generator->Z)))
1979 /* compute 2^130*G, 2^260*G, 2^390*G */
1980 for (i = 1; i <= 4; i <<= 1)
1982 point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1],
1983 pre->g_pre_comp[2*i][2], pre->g_pre_comp[i][0],
1984 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
1985 for (j = 0; j < 129; ++j)
1987 point_double(pre->g_pre_comp[2*i][0],
1988 pre->g_pre_comp[2*i][1],
1989 pre->g_pre_comp[2*i][2],
1990 pre->g_pre_comp[2*i][0],
1991 pre->g_pre_comp[2*i][1],
1992 pre->g_pre_comp[2*i][2]);
1995 /* g_pre_comp[0] is the point at infinity */
1996 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
1997 /* the remaining multiples */
1998 /* 2^130*G + 2^260*G */
1999 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
2000 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
2001 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
2002 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2003 pre->g_pre_comp[2][2]);
2004 /* 2^130*G + 2^390*G */
2005 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
2006 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
2007 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2008 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2009 pre->g_pre_comp[2][2]);
2010 /* 2^260*G + 2^390*G */
2011 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
2012 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
2013 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
2014 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
2015 pre->g_pre_comp[4][2]);
2016 /* 2^130*G + 2^260*G + 2^390*G */
2017 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
2018 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
2019 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
2020 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
2021 pre->g_pre_comp[2][2]);
2022 for (i = 1; i < 8; ++i)
2024 /* odd multiples: add G */
2025 point_add(pre->g_pre_comp[2*i+1][0], pre->g_pre_comp[2*i+1][1],
2026 pre->g_pre_comp[2*i+1][2], pre->g_pre_comp[2*i][0],
2027 pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2],
2028 0, pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
2029 pre->g_pre_comp[1][2]);
2031 make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);
2033 if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp521_pre_comp_dup,
2034 nistp521_pre_comp_free, nistp521_pre_comp_clear_free))
2040 if (generator != NULL)
2041 EC_POINT_free(generator);
2042 if (new_ctx != NULL)
2043 BN_CTX_free(new_ctx);
2045 nistp521_pre_comp_free(pre);
2049 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
2051 if (EC_EX_DATA_get_data(group->extra_data, nistp521_pre_comp_dup,
2052 nistp521_pre_comp_free, nistp521_pre_comp_clear_free)
2060 static void *dummy=&dummy;