2 * Copyright 2011-2018 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
29 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
30 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
31 * work which got its smarts from Daniel J. Bernstein's work on the same.
34 #include <openssl/opensslconf.h>
35 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
36 NON_EMPTY_TRANSLATION_UNIT
41 # include <openssl/err.h>
44 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
45 /* even with gcc, the typedef won't work for 32-bit platforms */
46 typedef __uint128_t uint128_t
; /* nonstandard; implemented by gcc on 64-bit
48 typedef __int128_t int128_t
;
50 # error "Your compiler doesn't appear to support 128-bit integer types"
58 * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
59 * can serialise an element of this field into 32 bytes. We call this an
63 typedef u8 felem_bytearray
[32];
66 * These are the parameters of P256, taken from FIPS 186-3, page 86. These
67 * values are big-endian.
69 static const felem_bytearray nistp256_curve_params
[5] = {
70 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
71 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
72 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
73 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
74 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
75 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
76 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
77 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */
78 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
79 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
80 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
81 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
82 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
83 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
84 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
85 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
86 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
87 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
88 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
89 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
93 * The representation of field elements.
94 * ------------------------------------
96 * We represent field elements with either four 128-bit values, eight 128-bit
97 * values, or four 64-bit values. The field element represented is:
98 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
100 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
102 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
103 * apart, but are 128-bits wide, the most significant bits of each limb overlap
104 * with the least significant bits of the next.
106 * A field element with four limbs is an 'felem'. One with eight limbs is a
109 * A field element with four, 64-bit values is called a 'smallfelem'. Small
110 * values are used as intermediate values before multiplication.
115 typedef uint128_t limb
;
116 typedef limb felem
[NLIMBS
];
117 typedef limb longfelem
[NLIMBS
* 2];
118 typedef u64 smallfelem
[NLIMBS
];
120 /* This is the value of the prime as four 64-bit words, little-endian. */
121 static const u64 kPrime
[4] =
122 { 0xfffffffffffffffful
, 0xffffffff, 0, 0xffffffff00000001ul
};
123 static const u64 bottom63bits
= 0x7ffffffffffffffful
;
126 * bin32_to_felem takes a little-endian byte array and converts it into felem
127 * form. This assumes that the CPU is little-endian.
129 static void bin32_to_felem(felem out
, const u8 in
[32])
131 out
[0] = *((u64
*)&in
[0]);
132 out
[1] = *((u64
*)&in
[8]);
133 out
[2] = *((u64
*)&in
[16]);
134 out
[3] = *((u64
*)&in
[24]);
138 * smallfelem_to_bin32 takes a smallfelem and serialises into a little
139 * endian, 32 byte array. This assumes that the CPU is little-endian.
141 static void smallfelem_to_bin32(u8 out
[32], const smallfelem in
)
143 *((u64
*)&out
[0]) = in
[0];
144 *((u64
*)&out
[8]) = in
[1];
145 *((u64
*)&out
[16]) = in
[2];
146 *((u64
*)&out
[24]) = in
[3];
149 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
150 static int BN_to_felem(felem out
, const BIGNUM
*bn
)
152 felem_bytearray b_out
;
155 if (BN_is_negative(bn
)) {
156 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
159 num_bytes
= BN_bn2lebinpad(bn
, b_out
, sizeof(b_out
));
161 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
164 bin32_to_felem(out
, b_out
);
168 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
169 static BIGNUM
*smallfelem_to_BN(BIGNUM
*out
, const smallfelem in
)
171 felem_bytearray b_out
;
172 smallfelem_to_bin32(b_out
, in
);
173 return BN_lebin2bn(b_out
, sizeof(b_out
), out
);
181 static void smallfelem_one(smallfelem out
)
189 static void smallfelem_assign(smallfelem out
, const smallfelem in
)
197 static void felem_assign(felem out
, const felem in
)
205 /* felem_sum sets out = out + in. */
206 static void felem_sum(felem out
, const felem in
)
214 /* felem_small_sum sets out = out + in. */
215 static void felem_small_sum(felem out
, const smallfelem in
)
223 /* felem_scalar sets out = out * scalar */
224 static void felem_scalar(felem out
, const u64 scalar
)
232 /* longfelem_scalar sets out = out * scalar */
233 static void longfelem_scalar(longfelem out
, const u64 scalar
)
245 # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
246 # define two105 (((limb)1) << 105)
247 # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
249 /* zero105 is 0 mod p */
250 static const felem zero105
=
251 { two105m41m9
, two105
, two105m41p9
, two105m41p9
};
254 * smallfelem_neg sets |out| to |-small|
256 * out[i] < out[i] + 2^105
258 static void smallfelem_neg(felem out
, const smallfelem small
)
260 /* In order to prevent underflow, we subtract from 0 mod p. */
261 out
[0] = zero105
[0] - small
[0];
262 out
[1] = zero105
[1] - small
[1];
263 out
[2] = zero105
[2] - small
[2];
264 out
[3] = zero105
[3] - small
[3];
268 * felem_diff subtracts |in| from |out|
272 * out[i] < out[i] + 2^105
274 static void felem_diff(felem out
, const felem in
)
277 * In order to prevent underflow, we add 0 mod p before subtracting.
279 out
[0] += zero105
[0];
280 out
[1] += zero105
[1];
281 out
[2] += zero105
[2];
282 out
[3] += zero105
[3];
290 # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
291 # define two107 (((limb)1) << 107)
292 # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
294 /* zero107 is 0 mod p */
295 static const felem zero107
=
296 { two107m43m11
, two107
, two107m43p11
, two107m43p11
};
299 * An alternative felem_diff for larger inputs |in|
300 * felem_diff_zero107 subtracts |in| from |out|
304 * out[i] < out[i] + 2^107
306 static void felem_diff_zero107(felem out
, const felem in
)
309 * In order to prevent underflow, we add 0 mod p before subtracting.
311 out
[0] += zero107
[0];
312 out
[1] += zero107
[1];
313 out
[2] += zero107
[2];
314 out
[3] += zero107
[3];
323 * longfelem_diff subtracts |in| from |out|
327 * out[i] < out[i] + 2^70 + 2^40
329 static void longfelem_diff(longfelem out
, const longfelem in
)
331 static const limb two70m8p6
=
332 (((limb
) 1) << 70) - (((limb
) 1) << 8) + (((limb
) 1) << 6);
333 static const limb two70p40
= (((limb
) 1) << 70) + (((limb
) 1) << 40);
334 static const limb two70
= (((limb
) 1) << 70);
335 static const limb two70m40m38p6
=
336 (((limb
) 1) << 70) - (((limb
) 1) << 40) - (((limb
) 1) << 38) +
338 static const limb two70m6
= (((limb
) 1) << 70) - (((limb
) 1) << 6);
340 /* add 0 mod p to avoid underflow */
344 out
[3] += two70m40m38p6
;
350 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
361 # define two64m0 (((limb)1) << 64) - 1
362 # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
363 # define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
364 # define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
366 /* zero110 is 0 mod p */
367 static const felem zero110
= { two64m0
, two110p32m0
, two64m46
, two64m32
};
370 * felem_shrink converts an felem into a smallfelem. The result isn't quite
371 * minimal as the value may be greater than p.
378 static void felem_shrink(smallfelem out
, const felem in
)
383 static const u64 kPrime3Test
= 0x7fffffff00000001ul
; /* 2^63 - 2^32 + 1 */
386 tmp
[3] = zero110
[3] + in
[3] + ((u64
)(in
[2] >> 64));
389 tmp
[2] = zero110
[2] + (u64
)in
[2];
390 tmp
[0] = zero110
[0] + in
[0];
391 tmp
[1] = zero110
[1] + in
[1];
392 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
395 * We perform two partial reductions where we eliminate the high-word of
396 * tmp[3]. We don't update the other words till the end.
398 a
= tmp
[3] >> 64; /* a < 2^46 */
399 tmp
[3] = (u64
)tmp
[3];
401 tmp
[3] += ((limb
) a
) << 32;
405 a
= tmp
[3] >> 64; /* a < 2^15 */
406 b
+= a
; /* b < 2^46 + 2^15 < 2^47 */
407 tmp
[3] = (u64
)tmp
[3];
409 tmp
[3] += ((limb
) a
) << 32;
410 /* tmp[3] < 2^64 + 2^47 */
413 * This adjusts the other two words to complete the two partial
417 tmp
[1] -= (((limb
) b
) << 32);
420 * In order to make space in tmp[3] for the carry from 2 -> 3, we
421 * conditionally subtract kPrime if tmp[3] is large enough.
423 high
= (u64
)(tmp
[3] >> 64);
424 /* As tmp[3] < 2^65, high is either 1 or 0 */
428 * all ones if the high word of tmp[3] is 1
429 * all zeros if the high word of tmp[3] if 0
432 mask
= 0 - (low
>> 63);
435 * all ones if the MSB of low is 1
436 * all zeros if the MSB of low if 0
440 /* if low was greater than kPrime3Test then the MSB is zero */
442 low
= 0 - (low
>> 63);
445 * all ones if low was > kPrime3Test
446 * all zeros if low was <= kPrime3Test
448 mask
= (mask
& low
) | high
;
449 tmp
[0] -= mask
& kPrime
[0];
450 tmp
[1] -= mask
& kPrime
[1];
451 /* kPrime[2] is zero, so omitted */
452 tmp
[3] -= mask
& kPrime
[3];
453 /* tmp[3] < 2**64 - 2**32 + 1 */
455 tmp
[1] += ((u64
)(tmp
[0] >> 64));
456 tmp
[0] = (u64
)tmp
[0];
457 tmp
[2] += ((u64
)(tmp
[1] >> 64));
458 tmp
[1] = (u64
)tmp
[1];
459 tmp
[3] += ((u64
)(tmp
[2] >> 64));
460 tmp
[2] = (u64
)tmp
[2];
469 /* smallfelem_expand converts a smallfelem to an felem */
470 static void smallfelem_expand(felem out
, const smallfelem in
)
479 * smallfelem_square sets |out| = |small|^2
483 * out[i] < 7 * 2^64 < 2^67
485 static void smallfelem_square(longfelem out
, const smallfelem small
)
490 a
= ((uint128_t
) small
[0]) * small
[0];
496 a
= ((uint128_t
) small
[0]) * small
[1];
503 a
= ((uint128_t
) small
[0]) * small
[2];
510 a
= ((uint128_t
) small
[0]) * small
[3];
516 a
= ((uint128_t
) small
[1]) * small
[2];
523 a
= ((uint128_t
) small
[1]) * small
[1];
529 a
= ((uint128_t
) small
[1]) * small
[3];
536 a
= ((uint128_t
) small
[2]) * small
[3];
544 a
= ((uint128_t
) small
[2]) * small
[2];
550 a
= ((uint128_t
) small
[3]) * small
[3];
558 * felem_square sets |out| = |in|^2
562 * out[i] < 7 * 2^64 < 2^67
564 static void felem_square(longfelem out
, const felem in
)
567 felem_shrink(small
, in
);
568 smallfelem_square(out
, small
);
572 * smallfelem_mul sets |out| = |small1| * |small2|
577 * out[i] < 7 * 2^64 < 2^67
579 static void smallfelem_mul(longfelem out
, const smallfelem small1
,
580 const smallfelem small2
)
585 a
= ((uint128_t
) small1
[0]) * small2
[0];
591 a
= ((uint128_t
) small1
[0]) * small2
[1];
597 a
= ((uint128_t
) small1
[1]) * small2
[0];
603 a
= ((uint128_t
) small1
[0]) * small2
[2];
609 a
= ((uint128_t
) small1
[1]) * small2
[1];
615 a
= ((uint128_t
) small1
[2]) * small2
[0];
621 a
= ((uint128_t
) small1
[0]) * small2
[3];
627 a
= ((uint128_t
) small1
[1]) * small2
[2];
633 a
= ((uint128_t
) small1
[2]) * small2
[1];
639 a
= ((uint128_t
) small1
[3]) * small2
[0];
645 a
= ((uint128_t
) small1
[1]) * small2
[3];
651 a
= ((uint128_t
) small1
[2]) * small2
[2];
657 a
= ((uint128_t
) small1
[3]) * small2
[1];
663 a
= ((uint128_t
) small1
[2]) * small2
[3];
669 a
= ((uint128_t
) small1
[3]) * small2
[2];
675 a
= ((uint128_t
) small1
[3]) * small2
[3];
683 * felem_mul sets |out| = |in1| * |in2|
688 * out[i] < 7 * 2^64 < 2^67
690 static void felem_mul(longfelem out
, const felem in1
, const felem in2
)
692 smallfelem small1
, small2
;
693 felem_shrink(small1
, in1
);
694 felem_shrink(small2
, in2
);
695 smallfelem_mul(out
, small1
, small2
);
699 * felem_small_mul sets |out| = |small1| * |in2|
704 * out[i] < 7 * 2^64 < 2^67
706 static void felem_small_mul(longfelem out
, const smallfelem small1
,
710 felem_shrink(small2
, in2
);
711 smallfelem_mul(out
, small1
, small2
);
714 # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
715 # define two100 (((limb)1) << 100)
716 # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
717 /* zero100 is 0 mod p */
718 static const felem zero100
=
719 { two100m36m4
, two100
, two100m36p4
, two100m36p4
};
722 * Internal function for the different flavours of felem_reduce.
723 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
725 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
726 * out[1] >= in[7] + 2^32*in[4]
727 * out[2] >= in[5] + 2^32*in[5]
728 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
730 * out[0] <= out[0] + in[4] + 2^32*in[5]
731 * out[1] <= out[1] + in[5] + 2^33*in[6]
732 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
733 * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
735 static void felem_reduce_(felem out
, const longfelem in
)
738 /* combine common terms from below */
739 c
= in
[4] + (in
[5] << 32);
747 /* the remaining terms */
748 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
749 out
[1] -= (in
[4] << 32);
750 out
[3] += (in
[4] << 32);
752 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
753 out
[2] -= (in
[5] << 32);
755 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
757 out
[0] -= (in
[6] << 32);
758 out
[1] += (in
[6] << 33);
759 out
[2] += (in
[6] * 2);
760 out
[3] -= (in
[6] << 32);
762 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
764 out
[0] -= (in
[7] << 32);
765 out
[2] += (in
[7] << 33);
766 out
[3] += (in
[7] * 3);
770 * felem_reduce converts a longfelem into an felem.
771 * To be called directly after felem_square or felem_mul.
773 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
774 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
778 static void felem_reduce(felem out
, const longfelem in
)
780 out
[0] = zero100
[0] + in
[0];
781 out
[1] = zero100
[1] + in
[1];
782 out
[2] = zero100
[2] + in
[2];
783 out
[3] = zero100
[3] + in
[3];
785 felem_reduce_(out
, in
);
788 * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
789 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
790 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
791 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
793 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
794 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
795 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
796 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
801 * felem_reduce_zero105 converts a larger longfelem into an felem.
807 static void felem_reduce_zero105(felem out
, const longfelem in
)
809 out
[0] = zero105
[0] + in
[0];
810 out
[1] = zero105
[1] + in
[1];
811 out
[2] = zero105
[2] + in
[2];
812 out
[3] = zero105
[3] + in
[3];
814 felem_reduce_(out
, in
);
817 * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
818 * out[1] > 2^105 - 2^71 - 2^103 > 0
819 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
820 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
822 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
823 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
824 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
825 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
830 * subtract_u64 sets *result = *result - v and *carry to one if the
831 * subtraction underflowed.
833 static void subtract_u64(u64
*result
, u64
*carry
, u64 v
)
835 uint128_t r
= *result
;
837 *carry
= (r
>> 64) & 1;
842 * felem_contract converts |in| to its unique, minimal representation. On
843 * entry: in[i] < 2^109
845 static void felem_contract(smallfelem out
, const felem in
)
848 u64 all_equal_so_far
= 0, result
= 0, carry
;
850 felem_shrink(out
, in
);
851 /* small is minimal except that the value might be > p */
855 * We are doing a constant time test if out >= kPrime. We need to compare
856 * each u64, from most-significant to least significant. For each one, if
857 * all words so far have been equal (m is all ones) then a non-equal
858 * result is the answer. Otherwise we continue.
860 for (i
= 3; i
< 4; i
--) {
862 uint128_t a
= ((uint128_t
) kPrime
[i
]) - out
[i
];
864 * if out[i] > kPrime[i] then a will underflow and the high 64-bits
867 result
|= all_equal_so_far
& ((u64
)(a
>> 64));
870 * if kPrime[i] == out[i] then |equal| will be all zeros and the
871 * decrement will make it all ones.
873 equal
= kPrime
[i
] ^ out
[i
];
875 equal
&= equal
<< 32;
876 equal
&= equal
<< 16;
881 equal
= 0 - (equal
>> 63);
883 all_equal_so_far
&= equal
;
887 * if all_equal_so_far is still all ones then the two values are equal
888 * and so out >= kPrime is true.
890 result
|= all_equal_so_far
;
892 /* if out >= kPrime then we subtract kPrime. */
893 subtract_u64(&out
[0], &carry
, result
& kPrime
[0]);
894 subtract_u64(&out
[1], &carry
, carry
);
895 subtract_u64(&out
[2], &carry
, carry
);
896 subtract_u64(&out
[3], &carry
, carry
);
898 subtract_u64(&out
[1], &carry
, result
& kPrime
[1]);
899 subtract_u64(&out
[2], &carry
, carry
);
900 subtract_u64(&out
[3], &carry
, carry
);
902 subtract_u64(&out
[2], &carry
, result
& kPrime
[2]);
903 subtract_u64(&out
[3], &carry
, carry
);
905 subtract_u64(&out
[3], &carry
, result
& kPrime
[3]);
908 static void smallfelem_square_contract(smallfelem out
, const smallfelem in
)
913 smallfelem_square(longtmp
, in
);
914 felem_reduce(tmp
, longtmp
);
915 felem_contract(out
, tmp
);
918 static void smallfelem_mul_contract(smallfelem out
, const smallfelem in1
,
919 const smallfelem in2
)
924 smallfelem_mul(longtmp
, in1
, in2
);
925 felem_reduce(tmp
, longtmp
);
926 felem_contract(out
, tmp
);
930 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
935 static limb
smallfelem_is_zero(const smallfelem small
)
940 u64 is_zero
= small
[0] | small
[1] | small
[2] | small
[3];
942 is_zero
&= is_zero
<< 32;
943 is_zero
&= is_zero
<< 16;
944 is_zero
&= is_zero
<< 8;
945 is_zero
&= is_zero
<< 4;
946 is_zero
&= is_zero
<< 2;
947 is_zero
&= is_zero
<< 1;
948 is_zero
= 0 - (is_zero
>> 63);
950 is_p
= (small
[0] ^ kPrime
[0]) |
951 (small
[1] ^ kPrime
[1]) |
952 (small
[2] ^ kPrime
[2]) | (small
[3] ^ kPrime
[3]);
960 is_p
= 0 - (is_p
>> 63);
965 result
|= ((limb
) is_zero
) << 64;
969 static int smallfelem_is_zero_int(const void *small
)
971 return (int)(smallfelem_is_zero(small
) & ((limb
) 1));
975 * felem_inv calculates |out| = |in|^{-1}
977 * Based on Fermat's Little Theorem:
979 * a^{p-1} = 1 (mod p)
980 * a^{p-2} = a^{-1} (mod p)
982 static void felem_inv(felem out
, const felem in
)
985 /* each e_I will hold |in|^{2^I - 1} */
986 felem e2
, e4
, e8
, e16
, e32
, e64
;
990 felem_square(tmp
, in
);
991 felem_reduce(ftmp
, tmp
); /* 2^1 */
992 felem_mul(tmp
, in
, ftmp
);
993 felem_reduce(ftmp
, tmp
); /* 2^2 - 2^0 */
994 felem_assign(e2
, ftmp
);
995 felem_square(tmp
, ftmp
);
996 felem_reduce(ftmp
, tmp
); /* 2^3 - 2^1 */
997 felem_square(tmp
, ftmp
);
998 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^2 */
999 felem_mul(tmp
, ftmp
, e2
);
1000 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^0 */
1001 felem_assign(e4
, ftmp
);
1002 felem_square(tmp
, ftmp
);
1003 felem_reduce(ftmp
, tmp
); /* 2^5 - 2^1 */
1004 felem_square(tmp
, ftmp
);
1005 felem_reduce(ftmp
, tmp
); /* 2^6 - 2^2 */
1006 felem_square(tmp
, ftmp
);
1007 felem_reduce(ftmp
, tmp
); /* 2^7 - 2^3 */
1008 felem_square(tmp
, ftmp
);
1009 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^4 */
1010 felem_mul(tmp
, ftmp
, e4
);
1011 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^0 */
1012 felem_assign(e8
, ftmp
);
1013 for (i
= 0; i
< 8; i
++) {
1014 felem_square(tmp
, ftmp
);
1015 felem_reduce(ftmp
, tmp
);
1017 felem_mul(tmp
, ftmp
, e8
);
1018 felem_reduce(ftmp
, tmp
); /* 2^16 - 2^0 */
1019 felem_assign(e16
, ftmp
);
1020 for (i
= 0; i
< 16; i
++) {
1021 felem_square(tmp
, ftmp
);
1022 felem_reduce(ftmp
, tmp
);
1024 felem_mul(tmp
, ftmp
, e16
);
1025 felem_reduce(ftmp
, tmp
); /* 2^32 - 2^0 */
1026 felem_assign(e32
, ftmp
);
1027 for (i
= 0; i
< 32; i
++) {
1028 felem_square(tmp
, ftmp
);
1029 felem_reduce(ftmp
, tmp
);
1031 felem_assign(e64
, ftmp
);
1032 felem_mul(tmp
, ftmp
, in
);
1033 felem_reduce(ftmp
, tmp
); /* 2^64 - 2^32 + 2^0 */
1034 for (i
= 0; i
< 192; i
++) {
1035 felem_square(tmp
, ftmp
);
1036 felem_reduce(ftmp
, tmp
);
1037 } /* 2^256 - 2^224 + 2^192 */
1039 felem_mul(tmp
, e64
, e32
);
1040 felem_reduce(ftmp2
, tmp
); /* 2^64 - 2^0 */
1041 for (i
= 0; i
< 16; i
++) {
1042 felem_square(tmp
, ftmp2
);
1043 felem_reduce(ftmp2
, tmp
);
1045 felem_mul(tmp
, ftmp2
, e16
);
1046 felem_reduce(ftmp2
, tmp
); /* 2^80 - 2^0 */
1047 for (i
= 0; i
< 8; i
++) {
1048 felem_square(tmp
, ftmp2
);
1049 felem_reduce(ftmp2
, tmp
);
1051 felem_mul(tmp
, ftmp2
, e8
);
1052 felem_reduce(ftmp2
, tmp
); /* 2^88 - 2^0 */
1053 for (i
= 0; i
< 4; i
++) {
1054 felem_square(tmp
, ftmp2
);
1055 felem_reduce(ftmp2
, tmp
);
1057 felem_mul(tmp
, ftmp2
, e4
);
1058 felem_reduce(ftmp2
, tmp
); /* 2^92 - 2^0 */
1059 felem_square(tmp
, ftmp2
);
1060 felem_reduce(ftmp2
, tmp
); /* 2^93 - 2^1 */
1061 felem_square(tmp
, ftmp2
);
1062 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^2 */
1063 felem_mul(tmp
, ftmp2
, e2
);
1064 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^0 */
1065 felem_square(tmp
, ftmp2
);
1066 felem_reduce(ftmp2
, tmp
); /* 2^95 - 2^1 */
1067 felem_square(tmp
, ftmp2
);
1068 felem_reduce(ftmp2
, tmp
); /* 2^96 - 2^2 */
1069 felem_mul(tmp
, ftmp2
, in
);
1070 felem_reduce(ftmp2
, tmp
); /* 2^96 - 3 */
1072 felem_mul(tmp
, ftmp2
, ftmp
);
1073 felem_reduce(out
, tmp
); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1076 static void smallfelem_inv_contract(smallfelem out
, const smallfelem in
)
1080 smallfelem_expand(tmp
, in
);
1081 felem_inv(tmp
, tmp
);
1082 felem_contract(out
, tmp
);
1089 * Building on top of the field operations we have the operations on the
1090 * elliptic curve group itself. Points on the curve are represented in Jacobian
1095 * point_double calculates 2*(x_in, y_in, z_in)
1097 * The method is taken from:
1098 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1100 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1101 * while x_out == y_in is not (maybe this works, but it's not tested).
1104 point_double(felem x_out
, felem y_out
, felem z_out
,
1105 const felem x_in
, const felem y_in
, const felem z_in
)
1107 longfelem tmp
, tmp2
;
1108 felem delta
, gamma
, beta
, alpha
, ftmp
, ftmp2
;
1109 smallfelem small1
, small2
;
1111 felem_assign(ftmp
, x_in
);
1112 /* ftmp[i] < 2^106 */
1113 felem_assign(ftmp2
, x_in
);
1114 /* ftmp2[i] < 2^106 */
1117 felem_square(tmp
, z_in
);
1118 felem_reduce(delta
, tmp
);
1119 /* delta[i] < 2^101 */
1122 felem_square(tmp
, y_in
);
1123 felem_reduce(gamma
, tmp
);
1124 /* gamma[i] < 2^101 */
1125 felem_shrink(small1
, gamma
);
1127 /* beta = x*gamma */
1128 felem_small_mul(tmp
, small1
, x_in
);
1129 felem_reduce(beta
, tmp
);
1130 /* beta[i] < 2^101 */
1132 /* alpha = 3*(x-delta)*(x+delta) */
1133 felem_diff(ftmp
, delta
);
1134 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1135 felem_sum(ftmp2
, delta
);
1136 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1137 felem_scalar(ftmp2
, 3);
1138 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1139 felem_mul(tmp
, ftmp
, ftmp2
);
1140 felem_reduce(alpha
, tmp
);
1141 /* alpha[i] < 2^101 */
1142 felem_shrink(small2
, alpha
);
1144 /* x' = alpha^2 - 8*beta */
1145 smallfelem_square(tmp
, small2
);
1146 felem_reduce(x_out
, tmp
);
1147 felem_assign(ftmp
, beta
);
1148 felem_scalar(ftmp
, 8);
1149 /* ftmp[i] < 8 * 2^101 = 2^104 */
1150 felem_diff(x_out
, ftmp
);
1151 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1153 /* z' = (y + z)^2 - gamma - delta */
1154 felem_sum(delta
, gamma
);
1155 /* delta[i] < 2^101 + 2^101 = 2^102 */
1156 felem_assign(ftmp
, y_in
);
1157 felem_sum(ftmp
, z_in
);
1158 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1159 felem_square(tmp
, ftmp
);
1160 felem_reduce(z_out
, tmp
);
1161 felem_diff(z_out
, delta
);
1162 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1164 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1165 felem_scalar(beta
, 4);
1166 /* beta[i] < 4 * 2^101 = 2^103 */
1167 felem_diff_zero107(beta
, x_out
);
1168 /* beta[i] < 2^107 + 2^103 < 2^108 */
1169 felem_small_mul(tmp
, small2
, beta
);
1170 /* tmp[i] < 7 * 2^64 < 2^67 */
1171 smallfelem_square(tmp2
, small1
);
1172 /* tmp2[i] < 7 * 2^64 */
1173 longfelem_scalar(tmp2
, 8);
1174 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1175 longfelem_diff(tmp
, tmp2
);
1176 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1177 felem_reduce_zero105(y_out
, tmp
);
1178 /* y_out[i] < 2^106 */
1182 * point_double_small is the same as point_double, except that it operates on
1186 point_double_small(smallfelem x_out
, smallfelem y_out
, smallfelem z_out
,
1187 const smallfelem x_in
, const smallfelem y_in
,
1188 const smallfelem z_in
)
1190 felem felem_x_out
, felem_y_out
, felem_z_out
;
1191 felem felem_x_in
, felem_y_in
, felem_z_in
;
1193 smallfelem_expand(felem_x_in
, x_in
);
1194 smallfelem_expand(felem_y_in
, y_in
);
1195 smallfelem_expand(felem_z_in
, z_in
);
1196 point_double(felem_x_out
, felem_y_out
, felem_z_out
,
1197 felem_x_in
, felem_y_in
, felem_z_in
);
1198 felem_shrink(x_out
, felem_x_out
);
1199 felem_shrink(y_out
, felem_y_out
);
1200 felem_shrink(z_out
, felem_z_out
);
1203 /* copy_conditional copies in to out iff mask is all ones. */
1204 static void copy_conditional(felem out
, const felem in
, limb mask
)
1207 for (i
= 0; i
< NLIMBS
; ++i
) {
1208 const limb tmp
= mask
& (in
[i
] ^ out
[i
]);
1213 /* copy_small_conditional copies in to out iff mask is all ones. */
1214 static void copy_small_conditional(felem out
, const smallfelem in
, limb mask
)
1217 const u64 mask64
= mask
;
1218 for (i
= 0; i
< NLIMBS
; ++i
) {
1219 out
[i
] = ((limb
) (in
[i
] & mask64
)) | (out
[i
] & ~mask
);
1224 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1226 * The method is taken from:
1227 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1228 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1230 * This function includes a branch for checking whether the two input points
1231 * are equal, (while not equal to the point at infinity). This case never
1232 * happens during single point multiplication, so there is no timing leak for
1233 * ECDH or ECDSA signing.
1235 static void point_add(felem x3
, felem y3
, felem z3
,
1236 const felem x1
, const felem y1
, const felem z1
,
1237 const int mixed
, const smallfelem x2
,
1238 const smallfelem y2
, const smallfelem z2
)
1240 felem ftmp
, ftmp2
, ftmp3
, ftmp4
, ftmp5
, ftmp6
, x_out
, y_out
, z_out
;
1241 longfelem tmp
, tmp2
;
1242 smallfelem small1
, small2
, small3
, small4
, small5
;
1243 limb x_equal
, y_equal
, z1_is_zero
, z2_is_zero
;
1245 felem_shrink(small3
, z1
);
1247 z1_is_zero
= smallfelem_is_zero(small3
);
1248 z2_is_zero
= smallfelem_is_zero(z2
);
1250 /* ftmp = z1z1 = z1**2 */
1251 smallfelem_square(tmp
, small3
);
1252 felem_reduce(ftmp
, tmp
);
1253 /* ftmp[i] < 2^101 */
1254 felem_shrink(small1
, ftmp
);
1257 /* ftmp2 = z2z2 = z2**2 */
1258 smallfelem_square(tmp
, z2
);
1259 felem_reduce(ftmp2
, tmp
);
1260 /* ftmp2[i] < 2^101 */
1261 felem_shrink(small2
, ftmp2
);
1263 felem_shrink(small5
, x1
);
1265 /* u1 = ftmp3 = x1*z2z2 */
1266 smallfelem_mul(tmp
, small5
, small2
);
1267 felem_reduce(ftmp3
, tmp
);
1268 /* ftmp3[i] < 2^101 */
1270 /* ftmp5 = z1 + z2 */
1271 felem_assign(ftmp5
, z1
);
1272 felem_small_sum(ftmp5
, z2
);
1273 /* ftmp5[i] < 2^107 */
1275 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1276 felem_square(tmp
, ftmp5
);
1277 felem_reduce(ftmp5
, tmp
);
1278 /* ftmp2 = z2z2 + z1z1 */
1279 felem_sum(ftmp2
, ftmp
);
1280 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1281 felem_diff(ftmp5
, ftmp2
);
1282 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1284 /* ftmp2 = z2 * z2z2 */
1285 smallfelem_mul(tmp
, small2
, z2
);
1286 felem_reduce(ftmp2
, tmp
);
1288 /* s1 = ftmp2 = y1 * z2**3 */
1289 felem_mul(tmp
, y1
, ftmp2
);
1290 felem_reduce(ftmp6
, tmp
);
1291 /* ftmp6[i] < 2^101 */
1294 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1297 /* u1 = ftmp3 = x1*z2z2 */
1298 felem_assign(ftmp3
, x1
);
1299 /* ftmp3[i] < 2^106 */
1302 felem_assign(ftmp5
, z1
);
1303 felem_scalar(ftmp5
, 2);
1304 /* ftmp5[i] < 2*2^106 = 2^107 */
1306 /* s1 = ftmp2 = y1 * z2**3 */
1307 felem_assign(ftmp6
, y1
);
1308 /* ftmp6[i] < 2^106 */
1312 smallfelem_mul(tmp
, x2
, small1
);
1313 felem_reduce(ftmp4
, tmp
);
1315 /* h = ftmp4 = u2 - u1 */
1316 felem_diff_zero107(ftmp4
, ftmp3
);
1317 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1318 felem_shrink(small4
, ftmp4
);
1320 x_equal
= smallfelem_is_zero(small4
);
1322 /* z_out = ftmp5 * h */
1323 felem_small_mul(tmp
, small4
, ftmp5
);
1324 felem_reduce(z_out
, tmp
);
1325 /* z_out[i] < 2^101 */
1327 /* ftmp = z1 * z1z1 */
1328 smallfelem_mul(tmp
, small1
, small3
);
1329 felem_reduce(ftmp
, tmp
);
1331 /* s2 = tmp = y2 * z1**3 */
1332 felem_small_mul(tmp
, y2
, ftmp
);
1333 felem_reduce(ftmp5
, tmp
);
1335 /* r = ftmp5 = (s2 - s1)*2 */
1336 felem_diff_zero107(ftmp5
, ftmp6
);
1337 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1338 felem_scalar(ftmp5
, 2);
1339 /* ftmp5[i] < 2^109 */
1340 felem_shrink(small1
, ftmp5
);
1341 y_equal
= smallfelem_is_zero(small1
);
1343 if (x_equal
&& y_equal
&& !z1_is_zero
&& !z2_is_zero
) {
1344 point_double(x3
, y3
, z3
, x1
, y1
, z1
);
1348 /* I = ftmp = (2h)**2 */
1349 felem_assign(ftmp
, ftmp4
);
1350 felem_scalar(ftmp
, 2);
1351 /* ftmp[i] < 2*2^108 = 2^109 */
1352 felem_square(tmp
, ftmp
);
1353 felem_reduce(ftmp
, tmp
);
1355 /* J = ftmp2 = h * I */
1356 felem_mul(tmp
, ftmp4
, ftmp
);
1357 felem_reduce(ftmp2
, tmp
);
1359 /* V = ftmp4 = U1 * I */
1360 felem_mul(tmp
, ftmp3
, ftmp
);
1361 felem_reduce(ftmp4
, tmp
);
1363 /* x_out = r**2 - J - 2V */
1364 smallfelem_square(tmp
, small1
);
1365 felem_reduce(x_out
, tmp
);
1366 felem_assign(ftmp3
, ftmp4
);
1367 felem_scalar(ftmp4
, 2);
1368 felem_sum(ftmp4
, ftmp2
);
1369 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1370 felem_diff(x_out
, ftmp4
);
1371 /* x_out[i] < 2^105 + 2^101 */
1373 /* y_out = r(V-x_out) - 2 * s1 * J */
1374 felem_diff_zero107(ftmp3
, x_out
);
1375 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1376 felem_small_mul(tmp
, small1
, ftmp3
);
1377 felem_mul(tmp2
, ftmp6
, ftmp2
);
1378 longfelem_scalar(tmp2
, 2);
1379 /* tmp2[i] < 2*2^67 = 2^68 */
1380 longfelem_diff(tmp
, tmp2
);
1381 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1382 felem_reduce_zero105(y_out
, tmp
);
1383 /* y_out[i] < 2^106 */
1385 copy_small_conditional(x_out
, x2
, z1_is_zero
);
1386 copy_conditional(x_out
, x1
, z2_is_zero
);
1387 copy_small_conditional(y_out
, y2
, z1_is_zero
);
1388 copy_conditional(y_out
, y1
, z2_is_zero
);
1389 copy_small_conditional(z_out
, z2
, z1_is_zero
);
1390 copy_conditional(z_out
, z1
, z2_is_zero
);
1391 felem_assign(x3
, x_out
);
1392 felem_assign(y3
, y_out
);
1393 felem_assign(z3
, z_out
);
1397 * point_add_small is the same as point_add, except that it operates on
1400 static void point_add_small(smallfelem x3
, smallfelem y3
, smallfelem z3
,
1401 smallfelem x1
, smallfelem y1
, smallfelem z1
,
1402 smallfelem x2
, smallfelem y2
, smallfelem z2
)
1404 felem felem_x3
, felem_y3
, felem_z3
;
1405 felem felem_x1
, felem_y1
, felem_z1
;
1406 smallfelem_expand(felem_x1
, x1
);
1407 smallfelem_expand(felem_y1
, y1
);
1408 smallfelem_expand(felem_z1
, z1
);
1409 point_add(felem_x3
, felem_y3
, felem_z3
, felem_x1
, felem_y1
, felem_z1
, 0,
1411 felem_shrink(x3
, felem_x3
);
1412 felem_shrink(y3
, felem_y3
);
1413 felem_shrink(z3
, felem_z3
);
1417 * Base point pre computation
1418 * --------------------------
1420 * Two different sorts of precomputed tables are used in the following code.
1421 * Each contain various points on the curve, where each point is three field
1422 * elements (x, y, z).
1424 * For the base point table, z is usually 1 (0 for the point at infinity).
1425 * This table has 2 * 16 elements, starting with the following:
1426 * index | bits | point
1427 * ------+---------+------------------------------
1430 * 2 | 0 0 1 0 | 2^64G
1431 * 3 | 0 0 1 1 | (2^64 + 1)G
1432 * 4 | 0 1 0 0 | 2^128G
1433 * 5 | 0 1 0 1 | (2^128 + 1)G
1434 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1435 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1436 * 8 | 1 0 0 0 | 2^192G
1437 * 9 | 1 0 0 1 | (2^192 + 1)G
1438 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1439 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1440 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1441 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1442 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1443 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1444 * followed by a copy of this with each element multiplied by 2^32.
1446 * The reason for this is so that we can clock bits into four different
1447 * locations when doing simple scalar multiplies against the base point,
1448 * and then another four locations using the second 16 elements.
1450 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1452 /* gmul is the table of precomputed base points */
1453 static const smallfelem gmul
[2][16][3] = {
1457 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1458 0x6b17d1f2e12c4247},
1459 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1460 0x4fe342e2fe1a7f9b},
1462 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1463 0x0fa822bc2811aaa5},
1464 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1465 0xbff44ae8f5dba80d},
1467 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1468 0x300a4bbc89d6726f},
1469 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1470 0x72aac7e0d09b4644},
1472 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1473 0x447d739beedb5e67},
1474 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1475 0x2d4825ab834131ee},
1477 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1478 0xef9519328a9c72ff},
1479 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1480 0x611e9fc37dbb2c9b},
1482 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1483 0x550663797b51f5d8},
1484 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1485 0x157164848aecb851},
1487 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1488 0xeb5d7745b21141ea},
1489 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1490 0xeafd72ebdbecc17b},
1492 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1493 0xa6d39677a7849276},
1494 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1495 0x674f84749b0b8816},
1497 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1498 0x4e769e7672c9ddad},
1499 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1500 0x42b99082de830663},
1502 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1503 0x78878ef61c6ce04d},
1504 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1505 0xb6cb3f5d7b72c321},
1507 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1508 0x0c88bc4d716b1287},
1509 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1510 0xdd5ddea3f3901dc6},
1512 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1513 0x68f344af6b317466},
1514 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1515 0x31b9c405f8540a20},
1517 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1518 0x4052bf4b6f461db9},
1519 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1520 0xfecf4d5190b0fc61},
1522 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1523 0x1eddbae2c802e41a},
1524 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1525 0x43104d86560ebcfc},
1527 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1528 0xb48e26b484f7a21c},
1529 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1530 0xfac015404d4d3dab},
1535 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1536 0x7fe36b40af22af89},
1537 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1538 0xe697d45825b63624},
1540 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1541 0x4a5b506612a677a6},
1542 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1543 0xeb13461ceac089f1},
1545 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1546 0x0781b8291c6a220a},
1547 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1548 0x690cde8df0151593},
1550 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1551 0x8a535f566ec73617},
1552 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1553 0x0455c08468b08bd7},
1555 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1556 0x06bada7ab77f8276},
1557 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1558 0x5b476dfd0e6cb18a},
1560 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1561 0x3e29864e8a2ec908},
1562 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1563 0x239b90ea3dc31e7e},
1565 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1566 0x820f4dd949f72ff7},
1567 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1568 0x140406ec783a05ec},
1570 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1571 0x68f6b8542783dfee},
1572 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1573 0xcbe1feba92e40ce6},
1575 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1576 0xd0b2f94d2f420109},
1577 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1578 0x971459828b0719e5},
1580 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1581 0x961610004a866aba},
1582 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1583 0x7acb9fadcee75e44},
1585 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1586 0x24eb9acca333bf5b},
1587 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1588 0x69f891c5acd079cc},
1590 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1591 0xe51f547c5972a107},
1592 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1593 0x1c309a2b25bb1387},
1595 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1596 0x20b87b8aa2c4e503},
1597 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1598 0xf5c6fa49919776be},
1600 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1601 0x1ed7d1b9332010b9},
1602 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1603 0x3a2b03f03217257a},
1605 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1606 0x15fee545c78dd9f6},
1607 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1608 0x4ab5b6b2b8753f81},
1613 * select_point selects the |idx|th point from a precomputation table and
1616 static void select_point(const u64 idx
, unsigned int size
,
1617 const smallfelem pre_comp
[16][3], smallfelem out
[3])
1620 u64
*outlimbs
= &out
[0][0];
1622 memset(out
, 0, sizeof(*out
) * 3);
1624 for (i
= 0; i
< size
; i
++) {
1625 const u64
*inlimbs
= (u64
*)&pre_comp
[i
][0][0];
1632 for (j
= 0; j
< NLIMBS
* 3; j
++)
1633 outlimbs
[j
] |= inlimbs
[j
] & mask
;
1637 /* get_bit returns the |i|th bit in |in| */
1638 static char get_bit(const felem_bytearray in
, int i
)
1640 if ((i
< 0) || (i
>= 256))
1642 return (in
[i
>> 3] >> (i
& 7)) & 1;
1646 * Interleaved point multiplication using precomputed point multiples: The
1647 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1648 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1649 * generator, using certain (large) precomputed multiples in g_pre_comp.
1650 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1652 static void batch_mul(felem x_out
, felem y_out
, felem z_out
,
1653 const felem_bytearray scalars
[],
1654 const unsigned num_points
, const u8
*g_scalar
,
1655 const int mixed
, const smallfelem pre_comp
[][17][3],
1656 const smallfelem g_pre_comp
[2][16][3])
1659 unsigned num
, gen_mul
= (g_scalar
!= NULL
);
1665 /* set nq to the point at infinity */
1666 memset(nq
, 0, sizeof(nq
));
1669 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1670 * of the generator (two in each of the last 32 rounds) and additions of
1671 * other points multiples (every 5th round).
1673 skip
= 1; /* save two point operations in the first
1675 for (i
= (num_points
? 255 : 31); i
>= 0; --i
) {
1678 point_double(nq
[0], nq
[1], nq
[2], nq
[0], nq
[1], nq
[2]);
1680 /* add multiples of the generator */
1681 if (gen_mul
&& (i
<= 31)) {
1682 /* first, look 32 bits upwards */
1683 bits
= get_bit(g_scalar
, i
+ 224) << 3;
1684 bits
|= get_bit(g_scalar
, i
+ 160) << 2;
1685 bits
|= get_bit(g_scalar
, i
+ 96) << 1;
1686 bits
|= get_bit(g_scalar
, i
+ 32);
1687 /* select the point to add, in constant time */
1688 select_point(bits
, 16, g_pre_comp
[1], tmp
);
1691 /* Arg 1 below is for "mixed" */
1692 point_add(nq
[0], nq
[1], nq
[2],
1693 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1695 smallfelem_expand(nq
[0], tmp
[0]);
1696 smallfelem_expand(nq
[1], tmp
[1]);
1697 smallfelem_expand(nq
[2], tmp
[2]);
1701 /* second, look at the current position */
1702 bits
= get_bit(g_scalar
, i
+ 192) << 3;
1703 bits
|= get_bit(g_scalar
, i
+ 128) << 2;
1704 bits
|= get_bit(g_scalar
, i
+ 64) << 1;
1705 bits
|= get_bit(g_scalar
, i
);
1706 /* select the point to add, in constant time */
1707 select_point(bits
, 16, g_pre_comp
[0], tmp
);
1708 /* Arg 1 below is for "mixed" */
1709 point_add(nq
[0], nq
[1], nq
[2],
1710 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1713 /* do other additions every 5 doublings */
1714 if (num_points
&& (i
% 5 == 0)) {
1715 /* loop over all scalars */
1716 for (num
= 0; num
< num_points
; ++num
) {
1717 bits
= get_bit(scalars
[num
], i
+ 4) << 5;
1718 bits
|= get_bit(scalars
[num
], i
+ 3) << 4;
1719 bits
|= get_bit(scalars
[num
], i
+ 2) << 3;
1720 bits
|= get_bit(scalars
[num
], i
+ 1) << 2;
1721 bits
|= get_bit(scalars
[num
], i
) << 1;
1722 bits
|= get_bit(scalars
[num
], i
- 1);
1723 ec_GFp_nistp_recode_scalar_bits(&sign
, &digit
, bits
);
1726 * select the point to add or subtract, in constant time
1728 select_point(digit
, 17, pre_comp
[num
], tmp
);
1729 smallfelem_neg(ftmp
, tmp
[1]); /* (X, -Y, Z) is the negative
1731 copy_small_conditional(ftmp
, tmp
[1], (((limb
) sign
) - 1));
1732 felem_contract(tmp
[1], ftmp
);
1735 point_add(nq
[0], nq
[1], nq
[2],
1736 nq
[0], nq
[1], nq
[2],
1737 mixed
, tmp
[0], tmp
[1], tmp
[2]);
1739 smallfelem_expand(nq
[0], tmp
[0]);
1740 smallfelem_expand(nq
[1], tmp
[1]);
1741 smallfelem_expand(nq
[2], tmp
[2]);
1747 felem_assign(x_out
, nq
[0]);
1748 felem_assign(y_out
, nq
[1]);
1749 felem_assign(z_out
, nq
[2]);
1752 /* Precomputation for the group generator. */
1753 struct nistp256_pre_comp_st
{
1754 smallfelem g_pre_comp
[2][16][3];
1755 CRYPTO_REF_COUNT references
;
1756 CRYPTO_RWLOCK
*lock
;
1759 const EC_METHOD
*EC_GFp_nistp256_method(void)
1761 static const EC_METHOD ret
= {
1762 EC_FLAGS_DEFAULT_OCT
,
1763 NID_X9_62_prime_field
,
1764 ec_GFp_nistp256_group_init
,
1765 ec_GFp_simple_group_finish
,
1766 ec_GFp_simple_group_clear_finish
,
1767 ec_GFp_nist_group_copy
,
1768 ec_GFp_nistp256_group_set_curve
,
1769 ec_GFp_simple_group_get_curve
,
1770 ec_GFp_simple_group_get_degree
,
1771 ec_group_simple_order_bits
,
1772 ec_GFp_simple_group_check_discriminant
,
1773 ec_GFp_simple_point_init
,
1774 ec_GFp_simple_point_finish
,
1775 ec_GFp_simple_point_clear_finish
,
1776 ec_GFp_simple_point_copy
,
1777 ec_GFp_simple_point_set_to_infinity
,
1778 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
1779 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
1780 ec_GFp_simple_point_set_affine_coordinates
,
1781 ec_GFp_nistp256_point_get_affine_coordinates
,
1782 0 /* point_set_compressed_coordinates */ ,
1787 ec_GFp_simple_invert
,
1788 ec_GFp_simple_is_at_infinity
,
1789 ec_GFp_simple_is_on_curve
,
1791 ec_GFp_simple_make_affine
,
1792 ec_GFp_simple_points_make_affine
,
1793 ec_GFp_nistp256_points_mul
,
1794 ec_GFp_nistp256_precompute_mult
,
1795 ec_GFp_nistp256_have_precompute_mult
,
1796 ec_GFp_nist_field_mul
,
1797 ec_GFp_nist_field_sqr
,
1799 ec_GFp_simple_field_inv
,
1800 0 /* field_encode */ ,
1801 0 /* field_decode */ ,
1802 0, /* field_set_to_one */
1803 ec_key_simple_priv2oct
,
1804 ec_key_simple_oct2priv
,
1805 0, /* set private */
1806 ec_key_simple_generate_key
,
1807 ec_key_simple_check_key
,
1808 ec_key_simple_generate_public_key
,
1811 ecdh_simple_compute_key
,
1812 ecdsa_simple_sign_setup
,
1813 ecdsa_simple_sign_sig
,
1814 ecdsa_simple_verify_sig
,
1815 0, /* field_inverse_mod_ord */
1816 0, /* blind_coordinates */
1818 0, /* ladder_step */
1825 /******************************************************************************/
1827 * FUNCTIONS TO MANAGE PRECOMPUTATION
1830 static NISTP256_PRE_COMP
*nistp256_pre_comp_new(void)
1832 NISTP256_PRE_COMP
*ret
= OPENSSL_zalloc(sizeof(*ret
));
1835 ECerr(EC_F_NISTP256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1839 ret
->references
= 1;
1841 ret
->lock
= CRYPTO_THREAD_lock_new();
1842 if (ret
->lock
== NULL
) {
1843 ECerr(EC_F_NISTP256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1850 NISTP256_PRE_COMP
*EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP
*p
)
1854 CRYPTO_UP_REF(&p
->references
, &i
, p
->lock
);
1858 void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP
*pre
)
1865 CRYPTO_DOWN_REF(&pre
->references
, &i
, pre
->lock
);
1866 REF_PRINT_COUNT("EC_nistp256", x
);
1869 REF_ASSERT_ISNT(i
< 0);
1871 CRYPTO_THREAD_lock_free(pre
->lock
);
1875 /******************************************************************************/
1877 * OPENSSL EC_METHOD FUNCTIONS
1880 int ec_GFp_nistp256_group_init(EC_GROUP
*group
)
1883 ret
= ec_GFp_simple_group_init(group
);
1884 group
->a_is_minus3
= 1;
1888 int ec_GFp_nistp256_group_set_curve(EC_GROUP
*group
, const BIGNUM
*p
,
1889 const BIGNUM
*a
, const BIGNUM
*b
,
1893 BIGNUM
*curve_p
, *curve_a
, *curve_b
;
1895 BN_CTX
*new_ctx
= NULL
;
1898 ctx
= new_ctx
= BN_CTX_new();
1904 curve_p
= BN_CTX_get(ctx
);
1905 curve_a
= BN_CTX_get(ctx
);
1906 curve_b
= BN_CTX_get(ctx
);
1907 if (curve_b
== NULL
)
1909 BN_bin2bn(nistp256_curve_params
[0], sizeof(felem_bytearray
), curve_p
);
1910 BN_bin2bn(nistp256_curve_params
[1], sizeof(felem_bytearray
), curve_a
);
1911 BN_bin2bn(nistp256_curve_params
[2], sizeof(felem_bytearray
), curve_b
);
1912 if ((BN_cmp(curve_p
, p
)) || (BN_cmp(curve_a
, a
)) || (BN_cmp(curve_b
, b
))) {
1913 ECerr(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE
,
1914 EC_R_WRONG_CURVE_PARAMETERS
);
1917 group
->field_mod_func
= BN_nist_mod_256
;
1918 ret
= ec_GFp_simple_group_set_curve(group
, p
, a
, b
, ctx
);
1922 BN_CTX_free(new_ctx
);
1928 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1931 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP
*group
,
1932 const EC_POINT
*point
,
1933 BIGNUM
*x
, BIGNUM
*y
,
1936 felem z1
, z2
, x_in
, y_in
;
1937 smallfelem x_out
, y_out
;
1940 if (EC_POINT_is_at_infinity(group
, point
)) {
1941 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1942 EC_R_POINT_AT_INFINITY
);
1945 if ((!BN_to_felem(x_in
, point
->X
)) || (!BN_to_felem(y_in
, point
->Y
)) ||
1946 (!BN_to_felem(z1
, point
->Z
)))
1949 felem_square(tmp
, z2
);
1950 felem_reduce(z1
, tmp
);
1951 felem_mul(tmp
, x_in
, z1
);
1952 felem_reduce(x_in
, tmp
);
1953 felem_contract(x_out
, x_in
);
1955 if (!smallfelem_to_BN(x
, x_out
)) {
1956 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1961 felem_mul(tmp
, z1
, z2
);
1962 felem_reduce(z1
, tmp
);
1963 felem_mul(tmp
, y_in
, z1
);
1964 felem_reduce(y_in
, tmp
);
1965 felem_contract(y_out
, y_in
);
1967 if (!smallfelem_to_BN(y
, y_out
)) {
1968 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1976 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
1977 static void make_points_affine(size_t num
, smallfelem points
[][3],
1978 smallfelem tmp_smallfelems
[])
1981 * Runs in constant time, unless an input is the point at infinity (which
1982 * normally shouldn't happen).
1984 ec_GFp_nistp_points_make_affine_internal(num
,
1988 (void (*)(void *))smallfelem_one
,
1989 smallfelem_is_zero_int
,
1990 (void (*)(void *, const void *))
1992 (void (*)(void *, const void *))
1993 smallfelem_square_contract
,
1995 (void *, const void *,
1997 smallfelem_mul_contract
,
1998 (void (*)(void *, const void *))
1999 smallfelem_inv_contract
,
2000 /* nothing to contract */
2001 (void (*)(void *, const void *))
2006 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
2007 * values Result is stored in r (r can equal one of the inputs).
2009 int ec_GFp_nistp256_points_mul(const EC_GROUP
*group
, EC_POINT
*r
,
2010 const BIGNUM
*scalar
, size_t num
,
2011 const EC_POINT
*points
[],
2012 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
2017 BIGNUM
*x
, *y
, *z
, *tmp_scalar
;
2018 felem_bytearray g_secret
;
2019 felem_bytearray
*secrets
= NULL
;
2020 smallfelem (*pre_comp
)[17][3] = NULL
;
2021 smallfelem
*tmp_smallfelems
= NULL
;
2024 int have_pre_comp
= 0;
2025 size_t num_points
= num
;
2026 smallfelem x_in
, y_in
, z_in
;
2027 felem x_out
, y_out
, z_out
;
2028 NISTP256_PRE_COMP
*pre
= NULL
;
2029 const smallfelem(*g_pre_comp
)[16][3] = NULL
;
2030 EC_POINT
*generator
= NULL
;
2031 const EC_POINT
*p
= NULL
;
2032 const BIGNUM
*p_scalar
= NULL
;
2035 x
= BN_CTX_get(ctx
);
2036 y
= BN_CTX_get(ctx
);
2037 z
= BN_CTX_get(ctx
);
2038 tmp_scalar
= BN_CTX_get(ctx
);
2039 if (tmp_scalar
== NULL
)
2042 if (scalar
!= NULL
) {
2043 pre
= group
->pre_comp
.nistp256
;
2045 /* we have precomputation, try to use it */
2046 g_pre_comp
= (const smallfelem(*)[16][3])pre
->g_pre_comp
;
2048 /* try to use the standard precomputation */
2049 g_pre_comp
= &gmul
[0];
2050 generator
= EC_POINT_new(group
);
2051 if (generator
== NULL
)
2053 /* get the generator from precomputation */
2054 if (!smallfelem_to_BN(x
, g_pre_comp
[0][1][0]) ||
2055 !smallfelem_to_BN(y
, g_pre_comp
[0][1][1]) ||
2056 !smallfelem_to_BN(z
, g_pre_comp
[0][1][2])) {
2057 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2060 if (!EC_POINT_set_Jprojective_coordinates_GFp(group
,
2064 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
2065 /* precomputation matches generator */
2069 * we don't have valid precomputation: treat the generator as a
2074 if (num_points
> 0) {
2075 if (num_points
>= 3) {
2077 * unless we precompute multiples for just one or two points,
2078 * converting those into affine form is time well spent
2082 secrets
= OPENSSL_malloc(sizeof(*secrets
) * num_points
);
2083 pre_comp
= OPENSSL_malloc(sizeof(*pre_comp
) * num_points
);
2086 OPENSSL_malloc(sizeof(*tmp_smallfelems
) * (num_points
* 17 + 1));
2087 if ((secrets
== NULL
) || (pre_comp
== NULL
)
2088 || (mixed
&& (tmp_smallfelems
== NULL
))) {
2089 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
2094 * we treat NULL scalars as 0, and NULL points as points at infinity,
2095 * i.e., they contribute nothing to the linear combination
2097 memset(secrets
, 0, sizeof(*secrets
) * num_points
);
2098 memset(pre_comp
, 0, sizeof(*pre_comp
) * num_points
);
2099 for (i
= 0; i
< num_points
; ++i
) {
2102 * we didn't have a valid precomputation, so we pick the
2105 p
= EC_GROUP_get0_generator(group
);
2108 /* the i^th point */
2110 p_scalar
= scalars
[i
];
2112 if ((p_scalar
!= NULL
) && (p
!= NULL
)) {
2113 /* reduce scalar to 0 <= scalar < 2^256 */
2114 if ((BN_num_bits(p_scalar
) > 256)
2115 || (BN_is_negative(p_scalar
))) {
2117 * this is an unusual input, and we don't guarantee
2120 if (!BN_nnmod(tmp_scalar
, p_scalar
, group
->order
, ctx
)) {
2121 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2124 num_bytes
= BN_bn2lebinpad(tmp_scalar
,
2125 secrets
[i
], sizeof(secrets
[i
]));
2127 num_bytes
= BN_bn2lebinpad(p_scalar
,
2128 secrets
[i
], sizeof(secrets
[i
]));
2130 if (num_bytes
< 0) {
2131 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2134 /* precompute multiples */
2135 if ((!BN_to_felem(x_out
, p
->X
)) ||
2136 (!BN_to_felem(y_out
, p
->Y
)) ||
2137 (!BN_to_felem(z_out
, p
->Z
)))
2139 felem_shrink(pre_comp
[i
][1][0], x_out
);
2140 felem_shrink(pre_comp
[i
][1][1], y_out
);
2141 felem_shrink(pre_comp
[i
][1][2], z_out
);
2142 for (j
= 2; j
<= 16; ++j
) {
2144 point_add_small(pre_comp
[i
][j
][0], pre_comp
[i
][j
][1],
2145 pre_comp
[i
][j
][2], pre_comp
[i
][1][0],
2146 pre_comp
[i
][1][1], pre_comp
[i
][1][2],
2147 pre_comp
[i
][j
- 1][0],
2148 pre_comp
[i
][j
- 1][1],
2149 pre_comp
[i
][j
- 1][2]);
2151 point_double_small(pre_comp
[i
][j
][0],
2154 pre_comp
[i
][j
/ 2][0],
2155 pre_comp
[i
][j
/ 2][1],
2156 pre_comp
[i
][j
/ 2][2]);
2162 make_points_affine(num_points
* 17, pre_comp
[0], tmp_smallfelems
);
2165 /* the scalar for the generator */
2166 if ((scalar
!= NULL
) && (have_pre_comp
)) {
2167 memset(g_secret
, 0, sizeof(g_secret
));
2168 /* reduce scalar to 0 <= scalar < 2^256 */
2169 if ((BN_num_bits(scalar
) > 256) || (BN_is_negative(scalar
))) {
2171 * this is an unusual input, and we don't guarantee
2174 if (!BN_nnmod(tmp_scalar
, scalar
, group
->order
, ctx
)) {
2175 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2178 num_bytes
= BN_bn2lebinpad(tmp_scalar
, g_secret
, sizeof(g_secret
));
2180 num_bytes
= BN_bn2lebinpad(scalar
, g_secret
, sizeof(g_secret
));
2182 /* do the multiplication with generator precomputation */
2183 batch_mul(x_out
, y_out
, z_out
,
2184 (const felem_bytearray(*))secrets
, num_points
,
2186 mixed
, (const smallfelem(*)[17][3])pre_comp
, g_pre_comp
);
2188 /* do the multiplication without generator precomputation */
2189 batch_mul(x_out
, y_out
, z_out
,
2190 (const felem_bytearray(*))secrets
, num_points
,
2191 NULL
, mixed
, (const smallfelem(*)[17][3])pre_comp
, NULL
);
2193 /* reduce the output to its unique minimal representation */
2194 felem_contract(x_in
, x_out
);
2195 felem_contract(y_in
, y_out
);
2196 felem_contract(z_in
, z_out
);
2197 if ((!smallfelem_to_BN(x
, x_in
)) || (!smallfelem_to_BN(y
, y_in
)) ||
2198 (!smallfelem_to_BN(z
, z_in
))) {
2199 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2202 ret
= EC_POINT_set_Jprojective_coordinates_GFp(group
, r
, x
, y
, z
, ctx
);
2206 EC_POINT_free(generator
);
2207 OPENSSL_free(secrets
);
2208 OPENSSL_free(pre_comp
);
2209 OPENSSL_free(tmp_smallfelems
);
2213 int ec_GFp_nistp256_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
2216 NISTP256_PRE_COMP
*pre
= NULL
;
2219 EC_POINT
*generator
= NULL
;
2220 smallfelem tmp_smallfelems
[32];
2221 felem x_tmp
, y_tmp
, z_tmp
;
2223 BN_CTX
*new_ctx
= NULL
;
2226 /* throw away old precomputation */
2227 EC_pre_comp_free(group
);
2231 ctx
= new_ctx
= BN_CTX_new();
2237 x
= BN_CTX_get(ctx
);
2238 y
= BN_CTX_get(ctx
);
2241 /* get the generator */
2242 if (group
->generator
== NULL
)
2244 generator
= EC_POINT_new(group
);
2245 if (generator
== NULL
)
2247 BN_bin2bn(nistp256_curve_params
[3], sizeof(felem_bytearray
), x
);
2248 BN_bin2bn(nistp256_curve_params
[4], sizeof(felem_bytearray
), y
);
2249 if (!EC_POINT_set_affine_coordinates(group
, generator
, x
, y
, ctx
))
2251 if ((pre
= nistp256_pre_comp_new()) == NULL
)
2254 * if the generator is the standard one, use built-in precomputation
2256 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
)) {
2257 memcpy(pre
->g_pre_comp
, gmul
, sizeof(pre
->g_pre_comp
));
2260 if ((!BN_to_felem(x_tmp
, group
->generator
->X
)) ||
2261 (!BN_to_felem(y_tmp
, group
->generator
->Y
)) ||
2262 (!BN_to_felem(z_tmp
, group
->generator
->Z
)))
2264 felem_shrink(pre
->g_pre_comp
[0][1][0], x_tmp
);
2265 felem_shrink(pre
->g_pre_comp
[0][1][1], y_tmp
);
2266 felem_shrink(pre
->g_pre_comp
[0][1][2], z_tmp
);
2268 * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
2269 * 2^160*G, 2^224*G for the second one
2271 for (i
= 1; i
<= 8; i
<<= 1) {
2272 point_double_small(pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2273 pre
->g_pre_comp
[1][i
][2], pre
->g_pre_comp
[0][i
][0],
2274 pre
->g_pre_comp
[0][i
][1],
2275 pre
->g_pre_comp
[0][i
][2]);
2276 for (j
= 0; j
< 31; ++j
) {
2277 point_double_small(pre
->g_pre_comp
[1][i
][0],
2278 pre
->g_pre_comp
[1][i
][1],
2279 pre
->g_pre_comp
[1][i
][2],
2280 pre
->g_pre_comp
[1][i
][0],
2281 pre
->g_pre_comp
[1][i
][1],
2282 pre
->g_pre_comp
[1][i
][2]);
2286 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2287 pre
->g_pre_comp
[0][2 * i
][1],
2288 pre
->g_pre_comp
[0][2 * i
][2],
2289 pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2290 pre
->g_pre_comp
[1][i
][2]);
2291 for (j
= 0; j
< 31; ++j
) {
2292 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2293 pre
->g_pre_comp
[0][2 * i
][1],
2294 pre
->g_pre_comp
[0][2 * i
][2],
2295 pre
->g_pre_comp
[0][2 * i
][0],
2296 pre
->g_pre_comp
[0][2 * i
][1],
2297 pre
->g_pre_comp
[0][2 * i
][2]);
2300 for (i
= 0; i
< 2; i
++) {
2301 /* g_pre_comp[i][0] is the point at infinity */
2302 memset(pre
->g_pre_comp
[i
][0], 0, sizeof(pre
->g_pre_comp
[i
][0]));
2303 /* the remaining multiples */
2304 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2305 point_add_small(pre
->g_pre_comp
[i
][6][0], pre
->g_pre_comp
[i
][6][1],
2306 pre
->g_pre_comp
[i
][6][2], pre
->g_pre_comp
[i
][4][0],
2307 pre
->g_pre_comp
[i
][4][1], pre
->g_pre_comp
[i
][4][2],
2308 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2309 pre
->g_pre_comp
[i
][2][2]);
2310 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2311 point_add_small(pre
->g_pre_comp
[i
][10][0], pre
->g_pre_comp
[i
][10][1],
2312 pre
->g_pre_comp
[i
][10][2], pre
->g_pre_comp
[i
][8][0],
2313 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2314 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2315 pre
->g_pre_comp
[i
][2][2]);
2316 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2317 point_add_small(pre
->g_pre_comp
[i
][12][0], pre
->g_pre_comp
[i
][12][1],
2318 pre
->g_pre_comp
[i
][12][2], pre
->g_pre_comp
[i
][8][0],
2319 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2320 pre
->g_pre_comp
[i
][4][0], pre
->g_pre_comp
[i
][4][1],
2321 pre
->g_pre_comp
[i
][4][2]);
2323 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
2325 point_add_small(pre
->g_pre_comp
[i
][14][0], pre
->g_pre_comp
[i
][14][1],
2326 pre
->g_pre_comp
[i
][14][2], pre
->g_pre_comp
[i
][12][0],
2327 pre
->g_pre_comp
[i
][12][1], pre
->g_pre_comp
[i
][12][2],
2328 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2329 pre
->g_pre_comp
[i
][2][2]);
2330 for (j
= 1; j
< 8; ++j
) {
2331 /* odd multiples: add G resp. 2^32*G */
2332 point_add_small(pre
->g_pre_comp
[i
][2 * j
+ 1][0],
2333 pre
->g_pre_comp
[i
][2 * j
+ 1][1],
2334 pre
->g_pre_comp
[i
][2 * j
+ 1][2],
2335 pre
->g_pre_comp
[i
][2 * j
][0],
2336 pre
->g_pre_comp
[i
][2 * j
][1],
2337 pre
->g_pre_comp
[i
][2 * j
][2],
2338 pre
->g_pre_comp
[i
][1][0],
2339 pre
->g_pre_comp
[i
][1][1],
2340 pre
->g_pre_comp
[i
][1][2]);
2343 make_points_affine(31, &(pre
->g_pre_comp
[0][1]), tmp_smallfelems
);
2346 SETPRECOMP(group
, nistp256
, pre
);
2352 EC_POINT_free(generator
);
2354 BN_CTX_free(new_ctx
);
2356 EC_nistp256_pre_comp_free(pre
);
2360 int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP
*group
)
2362 return HAVEPRECOMP(group
, nistp256
);