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 #ifdef EC_NISTP_64_GCC_128
33 #include <openssl/err.h>
36 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
37 /* even with gcc, the typedef won't work for 32-bit platforms */
38 typedef __uint128_t uint128_t
; /* nonstandard; implemented by gcc on 64-bit platforms */
40 #error "Need GCC 3.1 or later to define type uint128_t"
47 /* The underlying field.
49 * P521 operates over GF(2^521-1). We can serialise an element of this field
50 * into 66 bytes where the most significant byte contains only a single bit. We
51 * call this an felem_bytearray. */
53 typedef u8 felem_bytearray
[66];
55 /* These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
56 * These values are big-endian. */
57 static const felem_bytearray nistp521_curve_params
[5] =
59 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
60 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
61 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
62 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
63 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
64 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
65 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
66 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
68 {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
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,
75 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
77 {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
78 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
79 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
80 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
81 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
82 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
83 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
84 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
86 {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
87 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
88 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
89 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
90 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
91 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
92 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
93 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
95 {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
96 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
97 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
98 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
99 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
100 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
101 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
102 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
106 /* The representation of field elements.
107 * ------------------------------------
109 * We represent field elements with nine values. These values are either 64 or
110 * 128 bits and the field element represented is:
111 * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p)
112 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
113 * 58 bits apart, but are greater than 58 bits in length, the most significant
114 * bits of each limb overlap with the least significant bits of the next.
116 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
121 typedef uint64_t limb
;
122 typedef limb felem
[NLIMBS
];
123 typedef uint128_t largefelem
[NLIMBS
];
125 static const limb bottom57bits
= 0x1ffffffffffffff;
126 static const limb bottom58bits
= 0x3ffffffffffffff;
128 /* bin66_to_felem takes a little-endian byte array and converts it into felem
129 * form. This assumes that the CPU is little-endian. */
130 static void bin66_to_felem(felem out
, const u8 in
[66])
132 out
[0] = (*((limb
*) &in
[0])) & bottom58bits
;
133 out
[1] = (*((limb
*) &in
[7]) >> 2) & bottom58bits
;
134 out
[2] = (*((limb
*) &in
[14]) >> 4) & bottom58bits
;
135 out
[3] = (*((limb
*) &in
[21]) >> 6) & bottom58bits
;
136 out
[4] = (*((limb
*) &in
[29])) & bottom58bits
;
137 out
[5] = (*((limb
*) &in
[36]) >> 2) & bottom58bits
;
138 out
[6] = (*((limb
*) &in
[43]) >> 4) & bottom58bits
;
139 out
[7] = (*((limb
*) &in
[50]) >> 6) & bottom58bits
;
140 out
[8] = (*((limb
*) &in
[58])) & bottom57bits
;
143 /* felem_to_bin66 takes an felem and serialises into a little endian, 66 byte
144 * array. This assumes that the CPU is little-endian. */
145 static void felem_to_bin66(u8 out
[66], const felem in
)
148 (*((limb
*) &out
[0])) = in
[0];
149 (*((limb
*) &out
[7])) |= in
[1] << 2;
150 (*((limb
*) &out
[14])) |= in
[2] << 4;
151 (*((limb
*) &out
[21])) |= in
[3] << 6;
152 (*((limb
*) &out
[29])) = in
[4];
153 (*((limb
*) &out
[36])) |= in
[5] << 2;
154 (*((limb
*) &out
[43])) |= in
[6] << 4;
155 (*((limb
*) &out
[50])) |= in
[7] << 6;
156 (*((limb
*) &out
[58])) = in
[8];
159 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
160 static void flip_endian(u8
*out
, const u8
*in
, unsigned len
)
163 for (i
= 0; i
< len
; ++i
)
164 out
[i
] = in
[len
-1-i
];
167 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
168 static int BN_to_felem(felem out
, const BIGNUM
*bn
)
170 felem_bytearray b_in
;
171 felem_bytearray b_out
;
174 /* BN_bn2bin eats leading zeroes */
175 memset(b_out
, 0, sizeof b_out
);
176 num_bytes
= BN_num_bytes(bn
);
177 if (num_bytes
> sizeof b_out
)
179 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
182 if (BN_is_negative(bn
))
184 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
187 num_bytes
= BN_bn2bin(bn
, b_in
);
188 flip_endian(b_out
, b_in
, num_bytes
);
189 bin66_to_felem(out
, b_out
);
193 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
194 static BIGNUM
*felem_to_BN(BIGNUM
*out
, const felem in
)
196 felem_bytearray b_in
, b_out
;
197 felem_to_bin66(b_in
, in
);
198 flip_endian(b_out
, b_in
, sizeof b_out
);
199 return BN_bin2bn(b_out
, sizeof b_out
, out
);
204 * ---------------- */
206 static void felem_one(felem out
)
219 static void felem_assign(felem out
, const felem in
)
232 /* felem_sum64 sets out = out + in. */
233 static void felem_sum64(felem out
, const felem in
)
246 /* felem_scalar sets out = in * scalar */
247 static void felem_scalar(felem out
, const felem in
, limb scalar
)
249 out
[0] = in
[0] * scalar
;
250 out
[1] = in
[1] * scalar
;
251 out
[2] = in
[2] * scalar
;
252 out
[3] = in
[3] * scalar
;
253 out
[4] = in
[4] * scalar
;
254 out
[5] = in
[5] * scalar
;
255 out
[6] = in
[6] * scalar
;
256 out
[7] = in
[7] * scalar
;
257 out
[8] = in
[8] * scalar
;
260 /* felem_scalar64 sets out = out * scalar */
261 static void felem_scalar64(felem out
, limb scalar
)
274 /* felem_scalar128 sets out = out * scalar */
275 static void felem_scalar128(largefelem out
, limb scalar
)
288 /* felem_neg sets |out| to |-in|
290 * in[i] < 2^59 + 2^14
294 static void felem_neg(felem out
, const felem in
)
296 /* In order to prevent underflow, we subtract from 0 mod p. */
297 static const limb two62m3
= (((limb
)1) << 62) - (((limb
)1) << 5);
298 static const limb two62m2
= (((limb
)1) << 62) - (((limb
)1) << 4);
300 out
[0] = two62m3
- in
[0];
301 out
[1] = two62m2
- in
[1];
302 out
[2] = two62m2
- in
[2];
303 out
[3] = two62m2
- in
[3];
304 out
[4] = two62m2
- in
[4];
305 out
[5] = two62m2
- in
[5];
306 out
[6] = two62m2
- in
[6];
307 out
[7] = two62m2
- in
[7];
308 out
[8] = two62m2
- in
[8];
311 /* felem_diff64 subtracts |in| from |out|
313 * in[i] < 2^59 + 2^14
315 * out[i] < out[i] + 2^62
317 static void felem_diff64(felem out
, const felem in
)
319 /* In order to prevent underflow, we add 0 mod p before subtracting. */
320 static const limb two62m3
= (((limb
)1) << 62) - (((limb
)1) << 5);
321 static const limb two62m2
= (((limb
)1) << 62) - (((limb
)1) << 4);
323 out
[0] += two62m3
- in
[0];
324 out
[1] += two62m2
- in
[1];
325 out
[2] += two62m2
- in
[2];
326 out
[3] += two62m2
- in
[3];
327 out
[4] += two62m2
- in
[4];
328 out
[5] += two62m2
- in
[5];
329 out
[6] += two62m2
- in
[6];
330 out
[7] += two62m2
- in
[7];
331 out
[8] += two62m2
- in
[8];
334 /* felem_diff_128_64 subtracts |in| from |out|
336 * in[i] < 2^62 + 2^17
338 * out[i] < out[i] + 2^63
340 static void felem_diff_128_64(largefelem out
, const felem in
)
342 // In order to prevent underflow, we add 0 mod p before subtracting.
343 static const limb two63m6
= (((limb
)1) << 62) - (((limb
)1) << 5);
344 static const limb two63m5
= (((limb
)1) << 62) - (((limb
)1) << 4);
346 out
[0] += two63m6
- in
[0];
347 out
[1] += two63m5
- in
[1];
348 out
[2] += two63m5
- in
[2];
349 out
[3] += two63m5
- in
[3];
350 out
[4] += two63m5
- in
[4];
351 out
[5] += two63m5
- in
[5];
352 out
[6] += two63m5
- in
[6];
353 out
[7] += two63m5
- in
[7];
354 out
[8] += two63m5
- in
[8];
357 /* felem_diff_128_64 subtracts |in| from |out|
361 * out[i] < out[i] + 2^127 - 2^69
363 static void felem_diff128(largefelem out
, const largefelem in
)
365 // In order to prevent underflow, we add 0 mod p before subtracting.
366 static const uint128_t two127m70
= (((uint128_t
)1) << 127) - (((uint128_t
)1) << 70);
367 static const uint128_t two127m69
= (((uint128_t
)1) << 127) - (((uint128_t
)1) << 69);
369 out
[0] += (two127m70
- in
[0]);
370 out
[1] += (two127m69
- in
[1]);
371 out
[2] += (two127m69
- in
[2]);
372 out
[3] += (two127m69
- in
[3]);
373 out
[4] += (two127m69
- in
[4]);
374 out
[5] += (two127m69
- in
[5]);
375 out
[6] += (two127m69
- in
[6]);
376 out
[7] += (two127m69
- in
[7]);
377 out
[8] += (two127m69
- in
[8]);
380 /* felem_square sets |out| = |in|^2
384 * out[i] < 17 * max(in[i]) * max(in[i])
386 static void felem_square(largefelem out
, const felem in
)
389 felem_scalar(inx2
, in
, 2);
390 felem_scalar(inx4
, in
, 4);
392 /* We have many cases were we want to do
395 * This is obviously just
397 * However, rather than do the doubling on the 128 bit result, we
398 * double one of the inputs to the multiplication by reading from
401 out
[0] = ((uint128_t
) in
[0]) * in
[0];
402 out
[1] = ((uint128_t
) in
[0]) * inx2
[1];
403 out
[2] = ((uint128_t
) in
[0]) * inx2
[2] +
404 ((uint128_t
) in
[1]) * in
[1];
405 out
[3] = ((uint128_t
) in
[0]) * inx2
[3] +
406 ((uint128_t
) in
[1]) * inx2
[2];
407 out
[4] = ((uint128_t
) in
[0]) * inx2
[4] +
408 ((uint128_t
) in
[1]) * inx2
[3] +
409 ((uint128_t
) in
[2]) * in
[2];
410 out
[5] = ((uint128_t
) in
[0]) * inx2
[5] +
411 ((uint128_t
) in
[1]) * inx2
[4] +
412 ((uint128_t
) in
[2]) * inx2
[3];
413 out
[6] = ((uint128_t
) in
[0]) * inx2
[6] +
414 ((uint128_t
) in
[1]) * inx2
[5] +
415 ((uint128_t
) in
[2]) * inx2
[4] +
416 ((uint128_t
) in
[3]) * in
[3];
417 out
[7] = ((uint128_t
) in
[0]) * inx2
[7] +
418 ((uint128_t
) in
[1]) * inx2
[6] +
419 ((uint128_t
) in
[2]) * inx2
[5] +
420 ((uint128_t
) in
[3]) * inx2
[4];
421 out
[8] = ((uint128_t
) in
[0]) * inx2
[8] +
422 ((uint128_t
) in
[1]) * inx2
[7] +
423 ((uint128_t
) in
[2]) * inx2
[6] +
424 ((uint128_t
) in
[3]) * inx2
[5] +
425 ((uint128_t
) in
[4]) * in
[4];
427 /* The remaining limbs fall above 2^521, with the first falling at
428 * 2^522. They correspond to locations one bit up from the limbs
429 * produced above so we would have to multiply by two to align them.
430 * Again, rather than operate on the 128-bit result, we double one of
431 * the inputs to the multiplication. If we want to double for both this
432 * reason, and the reason above, then we end up multiplying by four. */
435 out
[0] += ((uint128_t
) in
[1]) * inx4
[8] +
436 ((uint128_t
) in
[2]) * inx4
[7] +
437 ((uint128_t
) in
[3]) * inx4
[6] +
438 ((uint128_t
) in
[4]) * inx4
[5];
441 out
[1] += ((uint128_t
) in
[2]) * inx4
[8] +
442 ((uint128_t
) in
[3]) * inx4
[7] +
443 ((uint128_t
) in
[4]) * inx4
[6] +
444 ((uint128_t
) in
[5]) * inx2
[5];
447 out
[2] += ((uint128_t
) in
[3]) * inx4
[8] +
448 ((uint128_t
) in
[4]) * inx4
[7] +
449 ((uint128_t
) in
[5]) * inx4
[6];
452 out
[3] += ((uint128_t
) in
[4]) * inx4
[8] +
453 ((uint128_t
) in
[5]) * inx4
[7] +
454 ((uint128_t
) in
[6]) * inx2
[6];
457 out
[4] += ((uint128_t
) in
[5]) * inx4
[8] +
458 ((uint128_t
) in
[6]) * inx4
[7];
461 out
[5] += ((uint128_t
) in
[6]) * inx4
[8] +
462 ((uint128_t
) in
[7]) * inx2
[7];
465 out
[6] += ((uint128_t
) in
[7]) * inx4
[8];
468 out
[7] += ((uint128_t
) in
[8]) * inx2
[8];
471 /* felem_mul sets |out| = |in1| * |in2|
476 * out[i] < 17 * max(in1[i]) * max(in2[i])
478 static void felem_mul(largefelem out
, const felem in1
, const felem in2
)
481 felem_scalar(in2x2
, in2
, 2);
483 out
[0] = ((uint128_t
) in1
[0]) * in2
[0];
485 out
[1] = ((uint128_t
) in1
[0]) * in2
[1] +
486 ((uint128_t
) in1
[1]) * in2
[0];
488 out
[2] = ((uint128_t
) in1
[0]) * in2
[2] +
489 ((uint128_t
) in1
[1]) * in2
[1] +
490 ((uint128_t
) in1
[2]) * in2
[0];
492 out
[3] = ((uint128_t
) in1
[0]) * in2
[3] +
493 ((uint128_t
) in1
[1]) * in2
[2] +
494 ((uint128_t
) in1
[2]) * in2
[1] +
495 ((uint128_t
) in1
[3]) * in2
[0];
497 out
[4] = ((uint128_t
) in1
[0]) * in2
[4] +
498 ((uint128_t
) in1
[1]) * in2
[3] +
499 ((uint128_t
) in1
[2]) * in2
[2] +
500 ((uint128_t
) in1
[3]) * in2
[1] +
501 ((uint128_t
) in1
[4]) * in2
[0];
503 out
[5] = ((uint128_t
) in1
[0]) * in2
[5] +
504 ((uint128_t
) in1
[1]) * in2
[4] +
505 ((uint128_t
) in1
[2]) * in2
[3] +
506 ((uint128_t
) in1
[3]) * in2
[2] +
507 ((uint128_t
) in1
[4]) * in2
[1] +
508 ((uint128_t
) in1
[5]) * in2
[0];
510 out
[6] = ((uint128_t
) in1
[0]) * in2
[6] +
511 ((uint128_t
) in1
[1]) * in2
[5] +
512 ((uint128_t
) in1
[2]) * in2
[4] +
513 ((uint128_t
) in1
[3]) * in2
[3] +
514 ((uint128_t
) in1
[4]) * in2
[2] +
515 ((uint128_t
) in1
[5]) * in2
[1] +
516 ((uint128_t
) in1
[6]) * in2
[0];
518 out
[7] = ((uint128_t
) in1
[0]) * in2
[7] +
519 ((uint128_t
) in1
[1]) * in2
[6] +
520 ((uint128_t
) in1
[2]) * in2
[5] +
521 ((uint128_t
) in1
[3]) * in2
[4] +
522 ((uint128_t
) in1
[4]) * in2
[3] +
523 ((uint128_t
) in1
[5]) * in2
[2] +
524 ((uint128_t
) in1
[6]) * in2
[1] +
525 ((uint128_t
) in1
[7]) * in2
[0];
527 out
[8] = ((uint128_t
) in1
[0]) * in2
[8] +
528 ((uint128_t
) in1
[1]) * in2
[7] +
529 ((uint128_t
) in1
[2]) * in2
[6] +
530 ((uint128_t
) in1
[3]) * in2
[5] +
531 ((uint128_t
) in1
[4]) * in2
[4] +
532 ((uint128_t
) in1
[5]) * in2
[3] +
533 ((uint128_t
) in1
[6]) * in2
[2] +
534 ((uint128_t
) in1
[7]) * in2
[1] +
535 ((uint128_t
) in1
[8]) * in2
[0];
537 /* See comment in felem_square about the use of in2x2 here */
539 out
[0] += ((uint128_t
) in1
[1]) * in2x2
[8] +
540 ((uint128_t
) in1
[2]) * in2x2
[7] +
541 ((uint128_t
) in1
[3]) * in2x2
[6] +
542 ((uint128_t
) in1
[4]) * in2x2
[5] +
543 ((uint128_t
) in1
[5]) * in2x2
[4] +
544 ((uint128_t
) in1
[6]) * in2x2
[3] +
545 ((uint128_t
) in1
[7]) * in2x2
[2] +
546 ((uint128_t
) in1
[8]) * in2x2
[1];
548 out
[1] += ((uint128_t
) in1
[2]) * in2x2
[8] +
549 ((uint128_t
) in1
[3]) * in2x2
[7] +
550 ((uint128_t
) in1
[4]) * in2x2
[6] +
551 ((uint128_t
) in1
[5]) * in2x2
[5] +
552 ((uint128_t
) in1
[6]) * in2x2
[4] +
553 ((uint128_t
) in1
[7]) * in2x2
[3] +
554 ((uint128_t
) in1
[8]) * in2x2
[2];
556 out
[2] += ((uint128_t
) in1
[3]) * in2x2
[8] +
557 ((uint128_t
) in1
[4]) * in2x2
[7] +
558 ((uint128_t
) in1
[5]) * in2x2
[6] +
559 ((uint128_t
) in1
[6]) * in2x2
[5] +
560 ((uint128_t
) in1
[7]) * in2x2
[4] +
561 ((uint128_t
) in1
[8]) * in2x2
[3];
563 out
[3] += ((uint128_t
) in1
[4]) * in2x2
[8] +
564 ((uint128_t
) in1
[5]) * in2x2
[7] +
565 ((uint128_t
) in1
[6]) * in2x2
[6] +
566 ((uint128_t
) in1
[7]) * in2x2
[5] +
567 ((uint128_t
) in1
[8]) * in2x2
[4];
569 out
[4] += ((uint128_t
) in1
[5]) * in2x2
[8] +
570 ((uint128_t
) in1
[6]) * in2x2
[7] +
571 ((uint128_t
) in1
[7]) * in2x2
[6] +
572 ((uint128_t
) in1
[8]) * in2x2
[5];
574 out
[5] += ((uint128_t
) in1
[6]) * in2x2
[8] +
575 ((uint128_t
) in1
[7]) * in2x2
[7] +
576 ((uint128_t
) in1
[8]) * in2x2
[6];
578 out
[6] += ((uint128_t
) in1
[7]) * in2x2
[8] +
579 ((uint128_t
) in1
[8]) * in2x2
[7];
581 out
[7] += ((uint128_t
) in1
[8]) * in2x2
[8];
584 static const limb bottom52bits
= 0xfffffffffffff;
586 /* felem_reduce converts a largefelem to an felem.
590 * out[i] < 2^59 + 2^14
592 static void felem_reduce(felem out
, const largefelem in
)
594 out
[0] = ((limb
) in
[0]) & bottom58bits
;
595 out
[1] = ((limb
) in
[1]) & bottom58bits
;
596 out
[2] = ((limb
) in
[2]) & bottom58bits
;
597 out
[3] = ((limb
) in
[3]) & bottom58bits
;
598 out
[4] = ((limb
) in
[4]) & bottom58bits
;
599 out
[5] = ((limb
) in
[5]) & bottom58bits
;
600 out
[6] = ((limb
) in
[6]) & bottom58bits
;
601 out
[7] = ((limb
) in
[7]) & bottom58bits
;
602 out
[8] = ((limb
) in
[8]) & bottom58bits
;
606 out
[1] += ((limb
) in
[0]) >> 58;
607 out
[1] += (((limb
) (in
[0] >> 64)) & bottom52bits
) << 6;
608 /* out[1] < 2^58 + 2^6 + 2^58
610 out
[2] += ((limb
) (in
[0] >> 64)) >> 52;
612 out
[2] += ((limb
) in
[1]) >> 58;
613 out
[2] += (((limb
) (in
[1] >> 64)) & bottom52bits
) << 6;
614 out
[3] += ((limb
) (in
[1] >> 64)) >> 52;
616 out
[3] += ((limb
) in
[2]) >> 58;
617 out
[3] += (((limb
) (in
[2] >> 64)) & bottom52bits
) << 6;
618 out
[4] += ((limb
) (in
[2] >> 64)) >> 52;
620 out
[4] += ((limb
) in
[3]) >> 58;
621 out
[4] += (((limb
) (in
[3] >> 64)) & bottom52bits
) << 6;
622 out
[5] += ((limb
) (in
[3] >> 64)) >> 52;
624 out
[5] += ((limb
) in
[4]) >> 58;
625 out
[5] += (((limb
) (in
[4] >> 64)) & bottom52bits
) << 6;
626 out
[6] += ((limb
) (in
[4] >> 64)) >> 52;
628 out
[6] += ((limb
) in
[5]) >> 58;
629 out
[6] += (((limb
) (in
[5] >> 64)) & bottom52bits
) << 6;
630 out
[7] += ((limb
) (in
[5] >> 64)) >> 52;
632 out
[7] += ((limb
) in
[6]) >> 58;
633 out
[7] += (((limb
) (in
[6] >> 64)) & bottom52bits
) << 6;
634 out
[8] += ((limb
) (in
[6] >> 64)) >> 52;
636 out
[8] += ((limb
) in
[7]) >> 58;
637 out
[8] += (((limb
) (in
[7] >> 64)) & bottom52bits
) << 6;
638 /* out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
640 u64 overflow1
= ((limb
) (in
[7] >> 64)) >> 52;
642 overflow1
+= ((limb
) in
[8]) >> 58;
643 overflow1
+= (((limb
) (in
[8] >> 64)) & bottom52bits
) << 6;
644 u64 overflow2
= ((limb
) (in
[8] >> 64)) >> 52;
646 overflow1
<<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */
647 overflow2
<<= 1; /* overflow2 < 2^13 */
649 out
[0] += overflow1
; /* out[0] < 2^60 */
650 out
[1] += overflow2
; /* out[1] < 2^59 + 2^6 + 2^13 */
652 out
[1] += out
[0] >> 58; out
[0] &= bottom58bits
;
654 * out[1] < 2^59 + 2^6 + 2^13 + 2^2
658 static void felem_square_reduce(felem out
, const felem in
)
661 felem_square(tmp
, in
);
662 felem_reduce(out
, tmp
);
665 static void felem_mul_reduce(felem out
, const felem in1
, const felem in2
)
668 felem_mul(tmp
, in1
, in2
);
669 felem_reduce(out
, tmp
);
672 /* felem_inv calculates |out| = |in|^{-1}
674 * Based on Fermat's Little Theorem:
676 * a^{p-1} = 1 (mod p)
677 * a^{p-2} = a^{-1} (mod p)
679 static void felem_inv(felem out
, const felem in
)
681 felem ftmp
, ftmp2
, ftmp3
, ftmp4
;
685 felem_square(tmp
, in
); felem_reduce(ftmp
, tmp
); /* 2^1 */
686 felem_mul(tmp
, in
, ftmp
); felem_reduce(ftmp
, tmp
); /* 2^2 - 2^0 */
687 felem_assign(ftmp2
, ftmp
);
688 felem_square(tmp
, ftmp
); felem_reduce(ftmp
, tmp
); /* 2^3 - 2^1 */
689 felem_mul(tmp
, in
, ftmp
); felem_reduce(ftmp
, tmp
); /* 2^3 - 2^0 */
690 felem_square(tmp
, ftmp
); felem_reduce(ftmp
, tmp
); /* 2^4 - 2^1 */
692 felem_square(tmp
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^3 - 2^1 */
693 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^4 - 2^2 */
694 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^4 - 2^0 */
696 felem_assign(ftmp2
, ftmp3
);
697 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^5 - 2^1 */
698 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^6 - 2^2 */
699 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^7 - 2^3 */
700 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^8 - 2^4 */
701 felem_assign(ftmp4
, ftmp3
);
702 felem_mul(tmp
, ftmp3
, ftmp
); felem_reduce(ftmp4
, tmp
); /* 2^8 - 2^1 */
703 felem_square(tmp
, ftmp4
); felem_reduce(ftmp4
, tmp
); /* 2^9 - 2^2 */
704 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^8 - 2^0 */
705 felem_assign(ftmp2
, ftmp3
);
707 for (i
= 0; i
< 8; i
++)
709 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^16 - 2^8 */
711 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^16 - 2^0 */
712 felem_assign(ftmp2
, ftmp3
);
714 for (i
= 0; i
< 16; i
++)
716 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^32 - 2^16 */
718 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^32 - 2^0 */
719 felem_assign(ftmp2
, ftmp3
);
721 for (i
= 0; i
< 32; i
++)
723 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^64 - 2^32 */
725 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^64 - 2^0 */
726 felem_assign(ftmp2
, ftmp3
);
728 for (i
= 0; i
< 64; i
++)
730 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^128 - 2^64 */
732 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^128 - 2^0 */
733 felem_assign(ftmp2
, ftmp3
);
735 for (i
= 0; i
< 128; i
++)
737 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^256 - 2^128 */
739 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^256 - 2^0 */
740 felem_assign(ftmp2
, ftmp3
);
742 for (i
= 0; i
< 256; i
++)
744 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^512 - 2^256 */
746 felem_mul(tmp
, ftmp3
, ftmp2
); felem_reduce(ftmp3
, tmp
); /* 2^512 - 2^0 */
748 for (i
= 0; i
< 9; i
++)
750 felem_square(tmp
, ftmp3
); felem_reduce(ftmp3
, tmp
); /* 2^521 - 2^9 */
752 felem_mul(tmp
, ftmp3
, ftmp4
); felem_reduce(ftmp3
, tmp
); /* 2^512 - 2^2 */
753 felem_mul(tmp
, ftmp3
, in
); felem_reduce(out
, tmp
); /* 2^512 - 3 */
756 /* This is 2^521-1, expressed as an felem */
757 static const felem kPrime
=
759 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
760 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
761 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
764 /* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
767 * in[i] < 2^59 + 2^14
769 static limb
felem_is_zero(const felem in
)
773 felem_assign(ftmp
, in
);
775 ftmp
[0] += ftmp
[8] >> 57; ftmp
[8] &= bottom57bits
;
777 ftmp
[1] += ftmp
[0] >> 58; ftmp
[0] &= bottom58bits
;
778 ftmp
[2] += ftmp
[1] >> 58; ftmp
[1] &= bottom58bits
;
779 ftmp
[3] += ftmp
[2] >> 58; ftmp
[2] &= bottom58bits
;
780 ftmp
[4] += ftmp
[3] >> 58; ftmp
[3] &= bottom58bits
;
781 ftmp
[5] += ftmp
[4] >> 58; ftmp
[4] &= bottom58bits
;
782 ftmp
[6] += ftmp
[5] >> 58; ftmp
[5] &= bottom58bits
;
783 ftmp
[7] += ftmp
[6] >> 58; ftmp
[6] &= bottom58bits
;
784 ftmp
[8] += ftmp
[7] >> 58; ftmp
[7] &= bottom58bits
;
785 /* ftmp[8] < 2^57 + 4 */
787 /* The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is
788 * greater than our bound for ftmp[8]. Therefore we only have to check
789 * if the zero is zero or 2^521-1. */
803 // We know that ftmp[i] < 2^63, therefore the only way that the top bit
804 // can be set is if is_zero was 0 before the decrement.
805 is_zero
= ((s64
) is_zero
) >> 63;
807 is_p
= ftmp
[0] ^ kPrime
[0];
808 is_p
|= ftmp
[1] ^ kPrime
[1];
809 is_p
|= ftmp
[2] ^ kPrime
[2];
810 is_p
|= ftmp
[3] ^ kPrime
[3];
811 is_p
|= ftmp
[4] ^ kPrime
[4];
812 is_p
|= ftmp
[5] ^ kPrime
[5];
813 is_p
|= ftmp
[6] ^ kPrime
[6];
814 is_p
|= ftmp
[7] ^ kPrime
[7];
815 is_p
|= ftmp
[8] ^ kPrime
[8];
818 is_p
= ((s64
) is_p
) >> 63;
824 static int felem_is_zero_int(const felem in
)
826 return (int) (felem_is_zero(in
) & ((limb
)1));
829 /* felem_contract converts |in| to its unique, minimal representation.
831 * in[i] < 2^59 + 2^14
833 static void felem_contract(felem out
, const felem in
)
835 limb is_p
, is_greater
, sign
;
836 static const limb two58
= ((limb
)1) << 58;
838 felem_assign(out
, in
);
840 out
[0] += out
[8] >> 57; out
[8] &= bottom57bits
;
842 out
[1] += out
[0] >> 58; out
[0] &= bottom58bits
;
843 out
[2] += out
[1] >> 58; out
[1] &= bottom58bits
;
844 out
[3] += out
[2] >> 58; out
[2] &= bottom58bits
;
845 out
[4] += out
[3] >> 58; out
[3] &= bottom58bits
;
846 out
[5] += out
[4] >> 58; out
[4] &= bottom58bits
;
847 out
[6] += out
[5] >> 58; out
[5] &= bottom58bits
;
848 out
[7] += out
[6] >> 58; out
[6] &= bottom58bits
;
849 out
[8] += out
[7] >> 58; out
[7] &= bottom58bits
;
850 /* out[8] < 2^57 + 4 */
852 /* If the value is greater than 2^521-1 then we have to subtract
853 * 2^521-1 out. See the comments in felem_is_zero regarding why we
854 * don't test for other multiples of the prime. */
856 /* First, if |out| is equal to 2^521-1, we subtract it out to get zero. */
858 is_p
= out
[0] ^ kPrime
[0];
859 is_p
|= out
[1] ^ kPrime
[1];
860 is_p
|= out
[2] ^ kPrime
[2];
861 is_p
|= out
[3] ^ kPrime
[3];
862 is_p
|= out
[4] ^ kPrime
[4];
863 is_p
|= out
[5] ^ kPrime
[5];
864 is_p
|= out
[6] ^ kPrime
[6];
865 is_p
|= out
[7] ^ kPrime
[7];
866 is_p
|= out
[8] ^ kPrime
[8];
875 is_p
= ((s64
) is_p
) >> 63;
878 /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */
890 /* In order to test that |out| >= 2^521-1 we need only test if out[8]
891 * >> 57 is greater than zero as (2^521-1) + x >= 2^522 */
892 is_greater
= out
[8] >> 57;
893 is_greater
|= is_greater
<< 32;
894 is_greater
|= is_greater
<< 16;
895 is_greater
|= is_greater
<< 8;
896 is_greater
|= is_greater
<< 4;
897 is_greater
|= is_greater
<< 2;
898 is_greater
|= is_greater
<< 1;
899 is_greater
= ((s64
) is_greater
) >> 63;
901 out
[0] -= kPrime
[0] & is_greater
;
902 out
[1] -= kPrime
[1] & is_greater
;
903 out
[2] -= kPrime
[2] & is_greater
;
904 out
[3] -= kPrime
[3] & is_greater
;
905 out
[4] -= kPrime
[4] & is_greater
;
906 out
[5] -= kPrime
[5] & is_greater
;
907 out
[6] -= kPrime
[6] & is_greater
;
908 out
[7] -= kPrime
[7] & is_greater
;
909 out
[8] -= kPrime
[8] & is_greater
;
911 /* Eliminate negative coefficients */
912 sign
= -(out
[0] >> 63); out
[0] += (two58
& sign
); out
[1] -= (1 & sign
);
913 sign
= -(out
[1] >> 63); out
[1] += (two58
& sign
); out
[2] -= (1 & sign
);
914 sign
= -(out
[2] >> 63); out
[2] += (two58
& sign
); out
[3] -= (1 & sign
);
915 sign
= -(out
[3] >> 63); out
[3] += (two58
& sign
); out
[4] -= (1 & sign
);
916 sign
= -(out
[4] >> 63); out
[4] += (two58
& sign
); out
[5] -= (1 & sign
);
917 sign
= -(out
[0] >> 63); out
[5] += (two58
& sign
); out
[6] -= (1 & sign
);
918 sign
= -(out
[6] >> 63); out
[6] += (two58
& sign
); out
[7] -= (1 & sign
);
919 sign
= -(out
[7] >> 63); out
[7] += (two58
& sign
); out
[8] -= (1 & sign
);
920 sign
= -(out
[5] >> 63); out
[5] += (two58
& sign
); out
[6] -= (1 & sign
);
921 sign
= -(out
[6] >> 63); out
[6] += (two58
& sign
); out
[7] -= (1 & sign
);
922 sign
= -(out
[7] >> 63); out
[7] += (two58
& sign
); out
[8] -= (1 & sign
);
928 * Building on top of the field operations we have the operations on the
929 * elliptic curve group itself. Points on the curve are represented in Jacobian
932 /* point_double calcuates 2*(x_in, y_in, z_in)
934 * The method is taken from:
935 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
937 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
938 * while x_out == y_in is not (maybe this works, but it's not tested). */
940 point_double(felem x_out
, felem y_out
, felem z_out
,
941 const felem x_in
, const felem y_in
, const felem z_in
)
943 largefelem tmp
, tmp2
;
944 felem delta
, gamma
, beta
, alpha
, ftmp
, ftmp2
;
946 felem_assign(ftmp
, x_in
);
947 felem_assign(ftmp2
, x_in
);
950 felem_square(tmp
, z_in
);
951 felem_reduce(delta
, tmp
); /* delta[i] < 2^59 + 2^14 */
954 felem_square(tmp
, y_in
);
955 felem_reduce(gamma
, tmp
); /* gamma[i] < 2^59 + 2^14 */
958 felem_mul(tmp
, x_in
, gamma
);
959 felem_reduce(beta
, tmp
); /* beta[i] < 2^59 + 2^14 */
961 /* alpha = 3*(x-delta)*(x+delta) */
962 felem_diff64(ftmp
, delta
);
964 felem_sum64(ftmp2
, delta
);
965 /* ftmp2[i] < 2^60 + 2^15 */
966 felem_scalar64(ftmp2
, 3);
967 /* ftmp2[i] < 3*2^60 + 3*2^15 */
968 felem_mul(tmp
, ftmp
, ftmp2
);
969 /* tmp[i] < 17(3*2^121 + 3*2^76)
970 * = 61*2^121 + 61*2^76
971 * < 64*2^121 + 64*2^76
974 felem_reduce(alpha
, tmp
);
976 /* x' = alpha^2 - 8*beta */
977 felem_square(tmp
, alpha
);
980 felem_assign(ftmp
, beta
);
981 felem_scalar64(ftmp
, 8);
982 /* ftmp[i] < 2^62 + 2^17 */
983 felem_diff_128_64(tmp
, ftmp
);
984 /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
985 felem_reduce(x_out
, tmp
);
987 /* z' = (y + z)^2 - gamma - delta */
988 felem_sum64(delta
, gamma
);
989 /* delta[i] < 2^60 + 2^15 */
990 felem_assign(ftmp
, y_in
);
991 felem_sum64(ftmp
, z_in
);
992 /* ftmp[i] < 2^60 + 2^15 */
993 felem_square(tmp
, ftmp
);
994 /* tmp[i] < 17(2^122)
996 felem_diff_128_64(tmp
, delta
);
997 /* tmp[i] < 2^127 + 2^63 */
998 felem_reduce(z_out
, tmp
);
1000 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1001 felem_scalar64(beta
, 4);
1002 /* beta[i] < 2^61 + 2^16 */
1003 felem_diff64(beta
, x_out
);
1004 /* beta[i] < 2^61 + 2^60 + 2^16 */
1005 felem_mul(tmp
, alpha
, beta
);
1006 /* tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
1007 * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
1008 * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1010 felem_square(tmp2
, gamma
);
1011 /* tmp2[i] < 17*(2^59 + 2^14)^2
1012 * = 17*(2^118 + 2^74 + 2^28) */
1013 felem_scalar128(tmp2
, 8);
1014 /* tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
1015 * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
1017 felem_diff128(tmp
, tmp2
);
1018 /* tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
1019 * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
1020 * 2^74 + 2^69 + 2^34 + 2^30
1022 felem_reduce(y_out
, tmp
);
1025 /* copy_conditional copies in to out iff mask is all ones. */
1027 copy_conditional(felem out
, const felem in
, limb mask
)
1030 for (i
= 0; i
< NLIMBS
; ++i
)
1032 const limb tmp
= mask
& (in
[i
] ^ out
[i
]);
1037 /* point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1039 * The method is taken from
1040 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1041 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1043 * This function includes a branch for checking whether the two input points
1044 * are equal (while not equal to the point at infinity). This case never
1045 * happens during single point multiplication, so there is no timing leak for
1046 * ECDH or ECDSA signing. */
1047 static void point_add(felem x3
, felem y3
, felem z3
,
1048 const felem x1
, const felem y1
, const felem z1
,
1049 const int mixed
, const felem x2
, const felem y2
, const felem z2
)
1051 felem ftmp
, ftmp2
, ftmp3
, ftmp4
, ftmp5
, ftmp6
, x_out
, y_out
, z_out
;
1052 largefelem tmp
, tmp2
;
1053 limb x_equal
, y_equal
, z1_is_zero
, z2_is_zero
;
1055 z1_is_zero
= felem_is_zero(z1
);
1056 z2_is_zero
= felem_is_zero(z2
);
1058 /* ftmp = z1z1 = z1**2 */
1059 felem_square(tmp
, z1
);
1060 felem_reduce(ftmp
, tmp
);
1064 /* ftmp2 = z2z2 = z2**2 */
1065 felem_square(tmp
, z2
);
1066 felem_reduce(ftmp2
, tmp
);
1068 /* u1 = ftmp3 = x1*z2z2 */
1069 felem_mul(tmp
, x1
, ftmp2
);
1070 felem_reduce(ftmp3
, tmp
);
1072 /* ftmp5 = z1 + z2 */
1073 felem_assign(ftmp5
, z1
);
1074 felem_sum64(ftmp5
, z2
);
1075 /* ftmp5[i] < 2^61 */
1077 /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
1078 felem_square(tmp
, ftmp5
);
1079 /* tmp[i] < 17*2^122 */
1080 felem_diff_128_64(tmp
, ftmp
);
1081 /* tmp[i] < 17*2^122 + 2^63 */
1082 felem_diff_128_64(tmp
, ftmp2
);
1083 /* tmp[i] < 17*2^122 + 2^64 */
1084 felem_reduce(ftmp5
, tmp
);
1086 /* ftmp2 = z2 * z2z2 */
1087 felem_mul(tmp
, ftmp2
, z2
);
1088 felem_reduce(ftmp2
, tmp
);
1090 /* s1 = ftmp6 = y1 * z2**3 */
1091 felem_mul(tmp
, y1
, ftmp2
);
1092 felem_reduce(ftmp6
, tmp
);
1096 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
1098 /* u1 = ftmp3 = x1*z2z2 */
1099 felem_assign(ftmp3
, x1
);
1101 /* ftmp5 = 2*z1z2 */
1102 felem_scalar(ftmp5
, z1
, 2);
1104 /* s1 = ftmp6 = y1 * z2**3 */
1105 felem_assign(ftmp6
, y1
);
1109 felem_mul(tmp
, x2
, ftmp
);
1110 /* tmp[i] < 17*2^120 */
1112 /* h = ftmp4 = u2 - u1 */
1113 felem_diff_128_64(tmp
, ftmp3
);
1114 /* tmp[i] < 17*2^120 + 2^63 */
1115 felem_reduce(ftmp4
, tmp
);
1117 x_equal
= felem_is_zero(ftmp4
);
1119 /* z_out = ftmp5 * h */
1120 felem_mul(tmp
, ftmp5
, ftmp4
);
1121 felem_reduce(z_out
, tmp
);
1123 /* ftmp = z1 * z1z1 */
1124 felem_mul(tmp
, ftmp
, z1
);
1125 felem_reduce(ftmp
, tmp
);
1127 /* s2 = tmp = y2 * z1**3 */
1128 felem_mul(tmp
, y2
, ftmp
);
1129 /* tmp[i] < 17*2^120 */
1131 /* r = ftmp5 = (s2 - s1)*2 */
1132 felem_diff_128_64(tmp
, ftmp6
);
1133 /* tmp[i] < 17*2^120 + 2^63 */
1134 felem_reduce(ftmp5
, tmp
);
1135 y_equal
= felem_is_zero(ftmp5
);
1136 felem_scalar64(ftmp5
, 2);
1137 /* ftmp5[i] < 2^61 */
1139 if (x_equal
&& y_equal
&& !z1_is_zero
&& !z2_is_zero
)
1141 point_double(x3
, y3
, z3
, x1
, y1
, z1
);
1145 /* I = ftmp = (2h)**2 */
1146 felem_assign(ftmp
, ftmp4
);
1147 felem_scalar64(ftmp
, 2);
1148 /* ftmp[i] < 2^61 */
1149 felem_square(tmp
, ftmp
);
1150 /* tmp[i] < 17*2^122 */
1151 felem_reduce(ftmp
, tmp
);
1153 /* J = ftmp2 = h * I */
1154 felem_mul(tmp
, ftmp4
, ftmp
);
1155 felem_reduce(ftmp2
, tmp
);
1157 /* V = ftmp4 = U1 * I */
1158 felem_mul(tmp
, ftmp3
, ftmp
);
1159 felem_reduce(ftmp4
, tmp
);
1161 /* x_out = r**2 - J - 2V */
1162 felem_square(tmp
, ftmp5
);
1163 /* tmp[i] < 17*2^122 */
1164 felem_diff_128_64(tmp
, ftmp2
);
1165 /* tmp[i] < 17*2^122 + 2^63 */
1166 felem_assign(ftmp3
, ftmp4
);
1167 felem_scalar64(ftmp4
, 2);
1168 /* ftmp4[i] < 2^61 */
1169 felem_diff_128_64(tmp
, ftmp4
);
1170 /* tmp[i] < 17*2^122 + 2^64 */
1171 felem_reduce(x_out
, tmp
);
1173 /* y_out = r(V-x_out) - 2 * s1 * J */
1174 felem_diff64(ftmp3
, x_out
);
1175 /* ftmp3[i] < 2^60 + 2^60
1177 felem_mul(tmp
, ftmp5
, ftmp3
);
1178 /* tmp[i] < 17*2^122 */
1179 felem_mul(tmp2
, ftmp6
, ftmp2
);
1180 /* tmp2[i] < 17*2^120 */
1181 felem_scalar128(tmp2
, 2);
1182 /* tmp2[i] < 17*2^121 */
1183 felem_diff128(tmp
, tmp2
);
1184 /* tmp[i] < 2^127 - 2^69 + 17*2^122
1185 * = 2^126 - 2^122 - 2^6 - 2^2 - 1
1187 felem_reduce(y_out
, tmp
);
1189 copy_conditional(x_out
, x2
, z1_is_zero
);
1190 copy_conditional(x_out
, x1
, z2_is_zero
);
1191 copy_conditional(y_out
, y2
, z1_is_zero
);
1192 copy_conditional(y_out
, y1
, z2_is_zero
);
1193 copy_conditional(z_out
, z2
, z1_is_zero
);
1194 copy_conditional(z_out
, z1
, z2_is_zero
);
1195 felem_assign(x3
, x_out
);
1196 felem_assign(y3
, y_out
);
1197 felem_assign(z3
, z_out
);
1200 /* Base point pre computation
1201 * --------------------------
1203 * Two different sorts of precomputed tables are used in the following code.
1204 * Each contain various points on the curve, where each point is three field
1205 * elements (x, y, z).
1207 * For the base point table, z is usually 1 (0 for the point at infinity).
1208 * This table has 16 elements:
1209 * index | bits | point
1210 * ------+---------+------------------------------
1213 * 2 | 0 0 1 0 | 2^130G
1214 * 3 | 0 0 1 1 | (2^130 + 1)G
1215 * 4 | 0 1 0 0 | 2^260G
1216 * 5 | 0 1 0 1 | (2^260 + 1)G
1217 * 6 | 0 1 1 0 | (2^260 + 2^130)G
1218 * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
1219 * 8 | 1 0 0 0 | 2^390G
1220 * 9 | 1 0 0 1 | (2^390 + 1)G
1221 * 10 | 1 0 1 0 | (2^390 + 2^130)G
1222 * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
1223 * 12 | 1 1 0 0 | (2^390 + 2^260)G
1224 * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
1225 * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
1226 * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
1228 * The reason for this is so that we can clock bits into four different
1229 * locations when doing simple scalar multiplies against the base point.
1231 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1233 /* gmul is the table of precomputed base points */
1234 static const felem gmul
[16][3] =
1235 {{{0, 0, 0, 0, 0, 0, 0, 0, 0},
1236 {0, 0, 0, 0, 0, 0, 0, 0, 0},
1237 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
1238 {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
1239 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
1240 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
1241 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
1242 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
1243 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
1244 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1245 {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
1246 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
1247 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
1248 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
1249 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
1250 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
1251 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1252 {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
1253 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
1254 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
1255 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
1256 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
1257 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
1258 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1259 {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
1260 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
1261 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
1262 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
1263 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
1264 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
1265 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1266 {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
1267 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
1268 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
1269 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
1270 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
1271 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
1272 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1273 {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
1274 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
1275 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
1276 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
1277 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
1278 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
1279 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1280 {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
1281 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
1282 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
1283 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
1284 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
1285 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
1286 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1287 {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
1288 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
1289 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
1290 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
1291 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
1292 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
1293 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1294 {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
1295 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
1296 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
1297 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
1298 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
1299 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
1300 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1301 {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
1302 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
1303 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
1304 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
1305 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
1306 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
1307 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1308 {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
1309 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
1310 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
1311 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
1312 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
1313 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
1314 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1315 {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
1316 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
1317 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
1318 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
1319 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
1320 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
1321 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1322 {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
1323 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
1324 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
1325 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
1326 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
1327 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
1328 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1329 {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
1330 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
1331 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
1332 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
1333 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
1334 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
1335 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
1336 {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
1337 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
1338 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
1339 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
1340 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
1341 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
1342 {1, 0, 0, 0, 0, 0, 0, 0, 0}}};
1344 /* select_point selects the |index|th point from a precomputation table and
1345 * copies it to out. */
1346 static void select_point(const limb index
, unsigned int size
, const felem pre_comp
[size
][3],
1350 limb
*outlimbs
= &out
[0][0];
1351 memset(outlimbs
, 0, 3 * sizeof(felem
));
1353 for (i
= 0; i
< size
; i
++)
1355 const limb
*inlimbs
= &pre_comp
[i
][0][0];
1356 limb mask
= i
^ index
;
1362 for (j
= 0; j
< NLIMBS
* 3; j
++)
1363 outlimbs
[j
] |= inlimbs
[j
] & mask
;
1367 /* get_bit returns the |i|th bit in |in| */
1368 static char get_bit(const felem_bytearray in
, int i
)
1372 return (in
[i
>> 3] >> (i
& 7)) & 1;
1375 /* Interleaved point multiplication using precomputed point multiples:
1376 * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
1377 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1378 * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1379 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1380 static void batch_mul(felem x_out
, felem y_out
, felem z_out
,
1381 const felem_bytearray scalars
[], const unsigned num_points
, const u8
*g_scalar
,
1382 const int mixed
, const felem pre_comp
[][17][3], const felem g_pre_comp
[16][3])
1385 unsigned num
, gen_mul
= (g_scalar
!= NULL
);
1386 felem nq
[3], tmp
[4];
1390 /* set nq to the point at infinity */
1391 memset(nq
, 0, 3 * sizeof(felem
));
1393 /* Loop over all scalars msb-to-lsb, interleaving additions
1394 * of multiples of the generator (last quarter of rounds)
1395 * and additions of other points multiples (every 5th round).
1397 skip
= 1; /* save two point operations in the first round */
1398 for (i
= (num_points
? 520 : 130); i
>= 0; --i
)
1402 point_double(nq
[0], nq
[1], nq
[2], nq
[0], nq
[1], nq
[2]);
1404 /* add multiples of the generator */
1405 if (gen_mul
&& (i
<= 130))
1407 bits
= get_bit(g_scalar
, i
+ 390) << 3;
1410 bits
|= get_bit(g_scalar
, i
+ 260) << 2;
1411 bits
|= get_bit(g_scalar
, i
+ 130) << 1;
1412 bits
|= get_bit(g_scalar
, i
);
1414 /* select the point to add, in constant time */
1415 select_point(bits
, 16, g_pre_comp
, tmp
);
1418 point_add(nq
[0], nq
[1], nq
[2],
1419 nq
[0], nq
[1], nq
[2],
1420 1 /* mixed */, tmp
[0], tmp
[1], tmp
[2]);
1424 memcpy(nq
, tmp
, 3 * sizeof(felem
));
1429 /* do other additions every 5 doublings */
1430 if (num_points
&& (i
% 5 == 0))
1432 /* loop over all scalars */
1433 for (num
= 0; num
< num_points
; ++num
)
1435 bits
= get_bit(scalars
[num
], i
+ 4) << 5;
1436 bits
|= get_bit(scalars
[num
], i
+ 3) << 4;
1437 bits
|= get_bit(scalars
[num
], i
+ 2) << 3;
1438 bits
|= get_bit(scalars
[num
], i
+ 1) << 2;
1439 bits
|= get_bit(scalars
[num
], i
) << 1;
1440 bits
|= get_bit(scalars
[num
], i
- 1);
1441 ec_GFp_nistp_recode_scalar_bits(&sign
, &digit
, bits
);
1443 /* select the point to add or subtract, in constant time */
1444 select_point(digit
, 17, pre_comp
[num
], tmp
);
1445 felem_neg(tmp
[3], tmp
[1]); /* (X, -Y, Z) is the negative point */
1446 copy_conditional(tmp
[1], tmp
[3], (-(limb
) sign
));
1450 point_add(nq
[0], nq
[1], nq
[2],
1451 nq
[0], nq
[1], nq
[2],
1452 mixed
, tmp
[0], tmp
[1], tmp
[2]);
1456 memcpy(nq
, tmp
, 3 * sizeof(felem
));
1462 felem_assign(x_out
, nq
[0]);
1463 felem_assign(y_out
, nq
[1]);
1464 felem_assign(z_out
, nq
[2]);
1468 /* Precomputation for the group generator. */
1470 felem g_pre_comp
[16][3];
1472 } NISTP521_PRE_COMP
;
1474 const EC_METHOD
*EC_GFp_nistp521_method(void)
1476 static const EC_METHOD ret
= {
1477 EC_FLAGS_DEFAULT_OCT
,
1478 NID_X9_62_prime_field
,
1479 ec_GFp_nistp521_group_init
,
1480 ec_GFp_simple_group_finish
,
1481 ec_GFp_simple_group_clear_finish
,
1482 ec_GFp_nist_group_copy
,
1483 ec_GFp_nistp521_group_set_curve
,
1484 ec_GFp_simple_group_get_curve
,
1485 ec_GFp_simple_group_get_degree
,
1486 ec_GFp_simple_group_check_discriminant
,
1487 ec_GFp_simple_point_init
,
1488 ec_GFp_simple_point_finish
,
1489 ec_GFp_simple_point_clear_finish
,
1490 ec_GFp_simple_point_copy
,
1491 ec_GFp_simple_point_set_to_infinity
,
1492 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
1493 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
1494 ec_GFp_simple_point_set_affine_coordinates
,
1495 ec_GFp_nistp521_point_get_affine_coordinates
,
1496 0 /* point_set_compressed_coordinates */,
1501 ec_GFp_simple_invert
,
1502 ec_GFp_simple_is_at_infinity
,
1503 ec_GFp_simple_is_on_curve
,
1505 ec_GFp_simple_make_affine
,
1506 ec_GFp_simple_points_make_affine
,
1507 ec_GFp_nistp521_points_mul
,
1508 ec_GFp_nistp521_precompute_mult
,
1509 ec_GFp_nistp521_have_precompute_mult
,
1510 ec_GFp_nist_field_mul
,
1511 ec_GFp_nist_field_sqr
,
1513 0 /* field_encode */,
1514 0 /* field_decode */,
1515 0 /* field_set_to_one */ };
1521 /******************************************************************************/
1522 /* FUNCTIONS TO MANAGE PRECOMPUTATION
1525 static NISTP521_PRE_COMP
*nistp521_pre_comp_new()
1527 NISTP521_PRE_COMP
*ret
= NULL
;
1528 ret
= (NISTP521_PRE_COMP
*)OPENSSL_malloc(sizeof(NISTP521_PRE_COMP
));
1531 ECerr(EC_F_NISTP521_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1534 memset(ret
->g_pre_comp
, 0, sizeof(ret
->g_pre_comp
));
1535 ret
->references
= 1;
1539 static void *nistp521_pre_comp_dup(void *src_
)
1541 NISTP521_PRE_COMP
*src
= src_
;
1543 /* no need to actually copy, these objects never change! */
1544 CRYPTO_add(&src
->references
, 1, CRYPTO_LOCK_EC_PRE_COMP
);
1549 static void nistp521_pre_comp_free(void *pre_
)
1552 NISTP521_PRE_COMP
*pre
= pre_
;
1557 i
= CRYPTO_add(&pre
->references
, -1, CRYPTO_LOCK_EC_PRE_COMP
);
1564 static void nistp521_pre_comp_clear_free(void *pre_
)
1567 NISTP521_PRE_COMP
*pre
= pre_
;
1572 i
= CRYPTO_add(&pre
->references
, -1, CRYPTO_LOCK_EC_PRE_COMP
);
1576 OPENSSL_cleanse(pre
, sizeof(*pre
));
1580 /******************************************************************************/
1581 /* OPENSSL EC_METHOD FUNCTIONS
1584 int ec_GFp_nistp521_group_init(EC_GROUP
*group
)
1587 ret
= ec_GFp_simple_group_init(group
);
1588 group
->a_is_minus3
= 1;
1592 int ec_GFp_nistp521_group_set_curve(EC_GROUP
*group
, const BIGNUM
*p
,
1593 const BIGNUM
*a
, const BIGNUM
*b
, BN_CTX
*ctx
)
1596 BN_CTX
*new_ctx
= NULL
;
1597 BIGNUM
*curve_p
, *curve_a
, *curve_b
;
1600 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
) return 0;
1602 if (((curve_p
= BN_CTX_get(ctx
)) == NULL
) ||
1603 ((curve_a
= BN_CTX_get(ctx
)) == NULL
) ||
1604 ((curve_b
= BN_CTX_get(ctx
)) == NULL
)) goto err
;
1605 BN_bin2bn(nistp521_curve_params
[0], sizeof(felem_bytearray
), curve_p
);
1606 BN_bin2bn(nistp521_curve_params
[1], sizeof(felem_bytearray
), curve_a
);
1607 BN_bin2bn(nistp521_curve_params
[2], sizeof(felem_bytearray
), curve_b
);
1608 if ((BN_cmp(curve_p
, p
)) || (BN_cmp(curve_a
, a
)) ||
1609 (BN_cmp(curve_b
, b
)))
1611 ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE
,
1612 EC_R_WRONG_CURVE_PARAMETERS
);
1615 group
->field_mod_func
= BN_nist_mod_521
;
1616 ret
= ec_GFp_simple_group_set_curve(group
, p
, a
, b
, ctx
);
1619 if (new_ctx
!= NULL
)
1620 BN_CTX_free(new_ctx
);
1624 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1625 * (X', Y') = (X/Z^2, Y/Z^3) */
1626 int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP
*group
,
1627 const EC_POINT
*point
, BIGNUM
*x
, BIGNUM
*y
, BN_CTX
*ctx
)
1629 felem z1
, z2
, x_in
, y_in
, x_out
, y_out
;
1632 if (EC_POINT_is_at_infinity(group
, point
))
1634 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES
,
1635 EC_R_POINT_AT_INFINITY
);
1638 if ((!BN_to_felem(x_in
, &point
->X
)) || (!BN_to_felem(y_in
, &point
->Y
)) ||
1639 (!BN_to_felem(z1
, &point
->Z
))) return 0;
1641 felem_square(tmp
, z2
); felem_reduce(z1
, tmp
);
1642 felem_mul(tmp
, x_in
, z1
); felem_reduce(x_in
, tmp
);
1643 felem_contract(x_out
, x_in
);
1646 if (!felem_to_BN(x
, x_out
))
1648 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES
, ERR_R_BN_LIB
);
1652 felem_mul(tmp
, z1
, z2
); felem_reduce(z1
, tmp
);
1653 felem_mul(tmp
, y_in
, z1
); felem_reduce(y_in
, tmp
);
1654 felem_contract(y_out
, y_in
);
1657 if (!felem_to_BN(y
, y_out
))
1659 ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES
, ERR_R_BN_LIB
);
1666 static void make_points_affine(size_t num
, felem points
[num
][3], felem tmp_felems
[num
+1])
1668 /* Runs in constant time, unless an input is the point at infinity
1669 * (which normally shouldn't happen). */
1670 ec_GFp_nistp_points_make_affine_internal(
1675 (void (*)(void *)) felem_one
,
1676 (int (*)(const void *)) felem_is_zero_int
,
1677 (void (*)(void *, const void *)) felem_assign
,
1678 (void (*)(void *, const void *)) felem_square_reduce
,
1679 (void (*)(void *, const void *, const void *)) felem_mul_reduce
,
1680 (void (*)(void *, const void *)) felem_inv
,
1681 (void (*)(void *, const void *)) felem_contract
);
1684 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1685 * Result is stored in r (r can equal one of the inputs). */
1686 int ec_GFp_nistp521_points_mul(const EC_GROUP
*group
, EC_POINT
*r
,
1687 const BIGNUM
*scalar
, size_t num
, const EC_POINT
*points
[],
1688 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
1693 BN_CTX
*new_ctx
= NULL
;
1694 BIGNUM
*x
, *y
, *z
, *tmp_scalar
;
1695 felem_bytearray g_secret
;
1696 felem_bytearray
*secrets
= NULL
;
1697 felem (*pre_comp
)[17][3] = NULL
;
1698 felem
*tmp_felems
= NULL
;
1699 felem_bytearray tmp
;
1700 unsigned i
, num_bytes
;
1701 int have_pre_comp
= 0;
1702 size_t num_points
= num
;
1703 felem x_in
, y_in
, z_in
, x_out
, y_out
, z_out
;
1704 NISTP521_PRE_COMP
*pre
= NULL
;
1705 felem (*g_pre_comp
)[3] = NULL
;
1706 EC_POINT
*generator
= NULL
;
1707 const EC_POINT
*p
= NULL
;
1708 const BIGNUM
*p_scalar
= NULL
;
1711 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
) return 0;
1713 if (((x
= BN_CTX_get(ctx
)) == NULL
) ||
1714 ((y
= BN_CTX_get(ctx
)) == NULL
) ||
1715 ((z
= BN_CTX_get(ctx
)) == NULL
) ||
1716 ((tmp_scalar
= BN_CTX_get(ctx
)) == NULL
))
1721 pre
= EC_EX_DATA_get_data(group
->extra_data
,
1722 nistp521_pre_comp_dup
, nistp521_pre_comp_free
,
1723 nistp521_pre_comp_clear_free
);
1725 /* we have precomputation, try to use it */
1726 g_pre_comp
= &pre
->g_pre_comp
[0];
1728 /* try to use the standard precomputation */
1729 g_pre_comp
= (felem (*)[3]) gmul
;
1730 generator
= EC_POINT_new(group
);
1731 if (generator
== NULL
)
1733 /* get the generator from precomputation */
1734 if (!felem_to_BN(x
, g_pre_comp
[1][0]) ||
1735 !felem_to_BN(y
, g_pre_comp
[1][1]) ||
1736 !felem_to_BN(z
, g_pre_comp
[1][2]))
1738 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL
, ERR_R_BN_LIB
);
1741 if (!EC_POINT_set_Jprojective_coordinates_GFp(group
,
1742 generator
, x
, y
, z
, ctx
))
1744 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
1745 /* precomputation matches generator */
1748 /* we don't have valid precomputation:
1749 * treat the generator as a random point */
1755 if (num_points
>= 2)
1757 /* unless we precompute multiples for just one point,
1758 * converting those into affine form is time well spent */
1761 secrets
= OPENSSL_malloc(num_points
* sizeof(felem_bytearray
));
1762 pre_comp
= OPENSSL_malloc(num_points
* 17 * 3 * sizeof(felem
));
1764 tmp_felems
= OPENSSL_malloc((num_points
* 17 + 1) * sizeof(felem
));
1765 if ((secrets
== NULL
) || (pre_comp
== NULL
) || (mixed
&& (tmp_felems
== NULL
)))
1767 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
1771 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1772 * i.e., they contribute nothing to the linear combination */
1773 memset(secrets
, 0, num_points
* sizeof(felem_bytearray
));
1774 memset(pre_comp
, 0, num_points
* 17 * 3 * sizeof(felem
));
1775 for (i
= 0; i
< num_points
; ++i
)
1778 /* we didn't have a valid precomputation, so we pick
1781 p
= EC_GROUP_get0_generator(group
);
1785 /* the i^th point */
1788 p_scalar
= scalars
[i
];
1790 if ((p_scalar
!= NULL
) && (p
!= NULL
))
1792 /* reduce scalar to 0 <= scalar < 2^521 */
1793 if ((BN_num_bits(p_scalar
) > 521) || (BN_is_negative(p_scalar
)))
1795 /* this is an unusual input, and we don't guarantee
1796 * constant-timeness */
1797 if (!BN_nnmod(tmp_scalar
, p_scalar
, &group
->order
, ctx
))
1799 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL
, ERR_R_BN_LIB
);
1802 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
1805 num_bytes
= BN_bn2bin(p_scalar
, tmp
);
1806 flip_endian(secrets
[i
], tmp
, num_bytes
);
1807 /* precompute multiples */
1808 if ((!BN_to_felem(x_out
, &p
->X
)) ||
1809 (!BN_to_felem(y_out
, &p
->Y
)) ||
1810 (!BN_to_felem(z_out
, &p
->Z
))) goto err
;
1811 memcpy(pre_comp
[i
][1][0], x_out
, sizeof(felem
));
1812 memcpy(pre_comp
[i
][1][1], y_out
, sizeof(felem
));
1813 memcpy(pre_comp
[i
][1][2], z_out
, sizeof(felem
));
1814 for (j
= 2; j
<= 16; ++j
)
1819 pre_comp
[i
][j
][0], pre_comp
[i
][j
][1], pre_comp
[i
][j
][2],
1820 pre_comp
[i
][1][0], pre_comp
[i
][1][1], pre_comp
[i
][1][2],
1821 0, pre_comp
[i
][j
-1][0], pre_comp
[i
][j
-1][1], pre_comp
[i
][j
-1][2]);
1826 pre_comp
[i
][j
][0], pre_comp
[i
][j
][1], pre_comp
[i
][j
][2],
1827 pre_comp
[i
][j
/2][0], pre_comp
[i
][j
/2][1], pre_comp
[i
][j
/2][2]);
1833 make_points_affine(num_points
* 17, pre_comp
[0], tmp_felems
);
1836 /* the scalar for the generator */
1837 if ((scalar
!= NULL
) && (have_pre_comp
))
1839 memset(g_secret
, 0, sizeof(g_secret
));
1840 /* reduce scalar to 0 <= scalar < 2^521 */
1841 if ((BN_num_bits(scalar
) > 521) || (BN_is_negative(scalar
)))
1843 /* this is an unusual input, and we don't guarantee
1844 * constant-timeness */
1845 if (!BN_nnmod(tmp_scalar
, scalar
, &group
->order
, ctx
))
1847 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL
, ERR_R_BN_LIB
);
1850 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
1853 num_bytes
= BN_bn2bin(scalar
, tmp
);
1854 flip_endian(g_secret
, tmp
, num_bytes
);
1855 /* do the multiplication with generator precomputation*/
1856 batch_mul(x_out
, y_out
, z_out
,
1857 (const felem_bytearray (*)) secrets
, num_points
,
1859 mixed
, (const felem (*)[17][3]) pre_comp
,
1860 (const felem (*)[3]) g_pre_comp
);
1863 /* do the multiplication without generator precomputation */
1864 batch_mul(x_out
, y_out
, z_out
,
1865 (const felem_bytearray (*)) secrets
, num_points
,
1866 NULL
, mixed
, (const felem (*)[17][3]) pre_comp
, NULL
);
1867 /* reduce the output to its unique minimal representation */
1868 felem_contract(x_in
, x_out
);
1869 felem_contract(y_in
, y_out
);
1870 felem_contract(z_in
, z_out
);
1871 if ((!felem_to_BN(x
, x_in
)) || (!felem_to_BN(y
, y_in
)) ||
1872 (!felem_to_BN(z
, z_in
)))
1874 ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL
, ERR_R_BN_LIB
);
1877 ret
= EC_POINT_set_Jprojective_coordinates_GFp(group
, r
, x
, y
, z
, ctx
);
1881 if (generator
!= NULL
)
1882 EC_POINT_free(generator
);
1883 if (new_ctx
!= NULL
)
1884 BN_CTX_free(new_ctx
);
1885 if (secrets
!= NULL
)
1886 OPENSSL_free(secrets
);
1887 if (pre_comp
!= NULL
)
1888 OPENSSL_free(pre_comp
);
1889 if (tmp_felems
!= NULL
)
1890 OPENSSL_free(tmp_felems
);
1894 int ec_GFp_nistp521_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
1897 NISTP521_PRE_COMP
*pre
= NULL
;
1899 BN_CTX
*new_ctx
= NULL
;
1901 EC_POINT
*generator
= NULL
;
1902 felem tmp_felems
[16];
1904 /* throw away old precomputation */
1905 EC_EX_DATA_free_data(&group
->extra_data
, nistp521_pre_comp_dup
,
1906 nistp521_pre_comp_free
, nistp521_pre_comp_clear_free
);
1908 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
) return 0;
1910 if (((x
= BN_CTX_get(ctx
)) == NULL
) ||
1911 ((y
= BN_CTX_get(ctx
)) == NULL
))
1913 /* get the generator */
1914 if (group
->generator
== NULL
) goto err
;
1915 generator
= EC_POINT_new(group
);
1916 if (generator
== NULL
)
1918 BN_bin2bn(nistp521_curve_params
[3], sizeof (felem_bytearray
), x
);
1919 BN_bin2bn(nistp521_curve_params
[4], sizeof (felem_bytearray
), y
);
1920 if (!EC_POINT_set_affine_coordinates_GFp(group
, generator
, x
, y
, ctx
))
1922 if ((pre
= nistp521_pre_comp_new()) == NULL
)
1924 /* if the generator is the standard one, use built-in precomputation */
1925 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
1927 memcpy(pre
->g_pre_comp
, gmul
, sizeof(pre
->g_pre_comp
));
1931 if ((!BN_to_felem(pre
->g_pre_comp
[1][0], &group
->generator
->X
)) ||
1932 (!BN_to_felem(pre
->g_pre_comp
[1][1], &group
->generator
->Y
)) ||
1933 (!BN_to_felem(pre
->g_pre_comp
[1][2], &group
->generator
->Z
)))
1935 /* compute 2^130*G, 2^260*G, 2^390*G */
1936 for (i
= 1; i
<= 4; i
<<= 1)
1938 point_double(pre
->g_pre_comp
[2*i
][0], pre
->g_pre_comp
[2*i
][1],
1939 pre
->g_pre_comp
[2*i
][2], pre
->g_pre_comp
[i
][0],
1940 pre
->g_pre_comp
[i
][1], pre
->g_pre_comp
[i
][2]);
1941 for (j
= 0; j
< 129; ++j
)
1943 point_double(pre
->g_pre_comp
[2*i
][0],
1944 pre
->g_pre_comp
[2*i
][1],
1945 pre
->g_pre_comp
[2*i
][2],
1946 pre
->g_pre_comp
[2*i
][0],
1947 pre
->g_pre_comp
[2*i
][1],
1948 pre
->g_pre_comp
[2*i
][2]);
1951 /* g_pre_comp[0] is the point at infinity */
1952 memset(pre
->g_pre_comp
[0], 0, sizeof(pre
->g_pre_comp
[0]));
1953 /* the remaining multiples */
1954 /* 2^130*G + 2^260*G */
1955 point_add(pre
->g_pre_comp
[6][0], pre
->g_pre_comp
[6][1],
1956 pre
->g_pre_comp
[6][2], pre
->g_pre_comp
[4][0],
1957 pre
->g_pre_comp
[4][1], pre
->g_pre_comp
[4][2],
1958 0, pre
->g_pre_comp
[2][0], pre
->g_pre_comp
[2][1],
1959 pre
->g_pre_comp
[2][2]);
1960 /* 2^130*G + 2^390*G */
1961 point_add(pre
->g_pre_comp
[10][0], pre
->g_pre_comp
[10][1],
1962 pre
->g_pre_comp
[10][2], pre
->g_pre_comp
[8][0],
1963 pre
->g_pre_comp
[8][1], pre
->g_pre_comp
[8][2],
1964 0, pre
->g_pre_comp
[2][0], pre
->g_pre_comp
[2][1],
1965 pre
->g_pre_comp
[2][2]);
1966 /* 2^260*G + 2^390*G */
1967 point_add(pre
->g_pre_comp
[12][0], pre
->g_pre_comp
[12][1],
1968 pre
->g_pre_comp
[12][2], pre
->g_pre_comp
[8][0],
1969 pre
->g_pre_comp
[8][1], pre
->g_pre_comp
[8][2],
1970 0, pre
->g_pre_comp
[4][0], pre
->g_pre_comp
[4][1],
1971 pre
->g_pre_comp
[4][2]);
1972 /* 2^130*G + 2^260*G + 2^390*G */
1973 point_add(pre
->g_pre_comp
[14][0], pre
->g_pre_comp
[14][1],
1974 pre
->g_pre_comp
[14][2], pre
->g_pre_comp
[12][0],
1975 pre
->g_pre_comp
[12][1], pre
->g_pre_comp
[12][2],
1976 0, pre
->g_pre_comp
[2][0], pre
->g_pre_comp
[2][1],
1977 pre
->g_pre_comp
[2][2]);
1978 for (i
= 1; i
< 8; ++i
)
1980 /* odd multiples: add G */
1981 point_add(pre
->g_pre_comp
[2*i
+1][0], pre
->g_pre_comp
[2*i
+1][1],
1982 pre
->g_pre_comp
[2*i
+1][2], pre
->g_pre_comp
[2*i
][0],
1983 pre
->g_pre_comp
[2*i
][1], pre
->g_pre_comp
[2*i
][2],
1984 0, pre
->g_pre_comp
[1][0], pre
->g_pre_comp
[1][1],
1985 pre
->g_pre_comp
[1][2]);
1987 make_points_affine(15, &(pre
->g_pre_comp
[1]), tmp_felems
);
1989 if (!EC_EX_DATA_set_data(&group
->extra_data
, pre
, nistp521_pre_comp_dup
,
1990 nistp521_pre_comp_free
, nistp521_pre_comp_clear_free
))
1996 if (generator
!= NULL
)
1997 EC_POINT_free(generator
);
1998 if (new_ctx
!= NULL
)
1999 BN_CTX_free(new_ctx
);
2001 nistp521_pre_comp_free(pre
);
2005 int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP
*group
)
2007 if (EC_EX_DATA_get_data(group
->extra_data
, nistp521_pre_comp_dup
,
2008 nistp521_pre_comp_free
, nistp521_pre_comp_clear_free
)
2016 static void *dummy
=&dummy
;