2 * Written by Adam Langley (Google) for the OpenSSL project
4 /* Copyright 2011 Google Inc.
6 * Licensed under the Apache License, Version 2.0 (the "License");
8 * you may not use this file except in compliance with the License.
9 * You may obtain a copy of the License at
11 * http://www.apache.org/licenses/LICENSE-2.0
13 * Unless required by applicable law or agreed to in writing, software
14 * distributed under the License is distributed on an "AS IS" BASIS,
15 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
16 * See the License for the specific language governing permissions and
17 * limitations under the License.
21 * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
23 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
24 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
25 * work which got its smarts from Daniel J. Bernstein's work on the same.
28 #include <openssl/opensslconf.h>
29 #ifndef OPENSSL_NO_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
40 typedef __int128_t int128_t
;
42 # error "Need GCC 3.1 or later to define type uint128_t"
51 * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
52 * can serialise an element of this field into 32 bytes. We call this an
56 typedef u8 felem_bytearray
[32];
59 * These are the parameters of P256, taken from FIPS 186-3, page 86. These
60 * values are big-endian.
62 static const felem_bytearray nistp256_curve_params
[5] = {
63 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
64 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
65 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
66 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
67 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
68 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
69 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
70 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */
71 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
72 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
73 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
74 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
75 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
76 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
77 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
78 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
79 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
80 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
81 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
82 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
86 * The representation of field elements.
87 * ------------------------------------
89 * We represent field elements with either four 128-bit values, eight 128-bit
90 * values, or four 64-bit values. The field element represented is:
91 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
93 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
95 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
96 * apart, but are 128-bits wide, the most significant bits of each limb overlap
97 * with the least significant bits of the next.
99 * A field element with four limbs is an 'felem'. One with eight limbs is a
102 * A field element with four, 64-bit values is called a 'smallfelem'. Small
103 * values are used as intermediate values before multiplication.
108 typedef uint128_t limb
;
109 typedef limb felem
[NLIMBS
];
110 typedef limb longfelem
[NLIMBS
* 2];
111 typedef u64 smallfelem
[NLIMBS
];
113 /* This is the value of the prime as four 64-bit words, little-endian. */
114 static const u64 kPrime
[4] =
115 { 0xfffffffffffffffful
, 0xffffffff, 0, 0xffffffff00000001ul
};
116 static const u64 bottom63bits
= 0x7ffffffffffffffful
;
119 * bin32_to_felem takes a little-endian byte array and converts it into felem
120 * form. This assumes that the CPU is little-endian.
122 static void bin32_to_felem(felem out
, const u8 in
[32])
124 out
[0] = *((u64
*)&in
[0]);
125 out
[1] = *((u64
*)&in
[8]);
126 out
[2] = *((u64
*)&in
[16]);
127 out
[3] = *((u64
*)&in
[24]);
131 * smallfelem_to_bin32 takes a smallfelem and serialises into a little
132 * endian, 32 byte array. This assumes that the CPU is little-endian.
134 static void smallfelem_to_bin32(u8 out
[32], const smallfelem in
)
136 *((u64
*)&out
[0]) = in
[0];
137 *((u64
*)&out
[8]) = in
[1];
138 *((u64
*)&out
[16]) = in
[2];
139 *((u64
*)&out
[24]) = in
[3];
142 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
143 static void flip_endian(u8
*out
, const u8
*in
, unsigned len
)
146 for (i
= 0; i
< len
; ++i
)
147 out
[i
] = in
[len
- 1 - i
];
150 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
151 static int BN_to_felem(felem out
, const BIGNUM
*bn
)
153 felem_bytearray b_in
;
154 felem_bytearray b_out
;
157 /* BN_bn2bin eats leading zeroes */
158 memset(b_out
, 0, sizeof(b_out
));
159 num_bytes
= BN_num_bytes(bn
);
160 if (num_bytes
> sizeof b_out
) {
161 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
164 if (BN_is_negative(bn
)) {
165 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
168 num_bytes
= BN_bn2bin(bn
, b_in
);
169 flip_endian(b_out
, b_in
, num_bytes
);
170 bin32_to_felem(out
, b_out
);
174 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
175 static BIGNUM
*smallfelem_to_BN(BIGNUM
*out
, const smallfelem in
)
177 felem_bytearray b_in
, b_out
;
178 smallfelem_to_bin32(b_in
, in
);
179 flip_endian(b_out
, b_in
, sizeof b_out
);
180 return BN_bin2bn(b_out
, sizeof b_out
, out
);
188 static void smallfelem_one(smallfelem out
)
196 static void smallfelem_assign(smallfelem out
, const smallfelem in
)
204 static void felem_assign(felem out
, const felem in
)
212 /* felem_sum sets out = out + in. */
213 static void felem_sum(felem out
, const felem in
)
221 /* felem_small_sum sets out = out + in. */
222 static void felem_small_sum(felem out
, const smallfelem in
)
230 /* felem_scalar sets out = out * scalar */
231 static void felem_scalar(felem out
, const u64 scalar
)
239 /* longfelem_scalar sets out = out * scalar */
240 static void longfelem_scalar(longfelem out
, const u64 scalar
)
252 # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
253 # define two105 (((limb)1) << 105)
254 # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
256 /* zero105 is 0 mod p */
257 static const felem zero105
=
258 { two105m41m9
, two105
, two105m41p9
, two105m41p9
};
261 * smallfelem_neg sets |out| to |-small|
263 * out[i] < out[i] + 2^105
265 static void smallfelem_neg(felem out
, const smallfelem small
)
267 /* In order to prevent underflow, we subtract from 0 mod p. */
268 out
[0] = zero105
[0] - small
[0];
269 out
[1] = zero105
[1] - small
[1];
270 out
[2] = zero105
[2] - small
[2];
271 out
[3] = zero105
[3] - small
[3];
275 * felem_diff subtracts |in| from |out|
279 * out[i] < out[i] + 2^105
281 static void felem_diff(felem out
, const felem in
)
284 * In order to prevent underflow, we add 0 mod p before subtracting.
286 out
[0] += zero105
[0];
287 out
[1] += zero105
[1];
288 out
[2] += zero105
[2];
289 out
[3] += zero105
[3];
297 # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
298 # define two107 (((limb)1) << 107)
299 # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
301 /* zero107 is 0 mod p */
302 static const felem zero107
=
303 { two107m43m11
, two107
, two107m43p11
, two107m43p11
};
306 * An alternative felem_diff for larger inputs |in|
307 * felem_diff_zero107 subtracts |in| from |out|
311 * out[i] < out[i] + 2^107
313 static void felem_diff_zero107(felem out
, const felem in
)
316 * In order to prevent underflow, we add 0 mod p before subtracting.
318 out
[0] += zero107
[0];
319 out
[1] += zero107
[1];
320 out
[2] += zero107
[2];
321 out
[3] += zero107
[3];
330 * longfelem_diff subtracts |in| from |out|
334 * out[i] < out[i] + 2^70 + 2^40
336 static void longfelem_diff(longfelem out
, const longfelem in
)
338 static const limb two70m8p6
=
339 (((limb
) 1) << 70) - (((limb
) 1) << 8) + (((limb
) 1) << 6);
340 static const limb two70p40
= (((limb
) 1) << 70) + (((limb
) 1) << 40);
341 static const limb two70
= (((limb
) 1) << 70);
342 static const limb two70m40m38p6
=
343 (((limb
) 1) << 70) - (((limb
) 1) << 40) - (((limb
) 1) << 38) +
345 static const limb two70m6
= (((limb
) 1) << 70) - (((limb
) 1) << 6);
347 /* add 0 mod p to avoid underflow */
351 out
[3] += two70m40m38p6
;
357 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
368 # define two64m0 (((limb)1) << 64) - 1
369 # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
370 # define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
371 # define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
373 /* zero110 is 0 mod p */
374 static const felem zero110
= { two64m0
, two110p32m0
, two64m46
, two64m32
};
377 * felem_shrink converts an felem into a smallfelem. The result isn't quite
378 * minimal as the value may be greater than p.
385 static void felem_shrink(smallfelem out
, const felem in
)
390 static const u64 kPrime3Test
= 0x7fffffff00000001ul
; /* 2^63 - 2^32 + 1 */
393 tmp
[3] = zero110
[3] + in
[3] + ((u64
)(in
[2] >> 64));
396 tmp
[2] = zero110
[2] + (u64
)in
[2];
397 tmp
[0] = zero110
[0] + in
[0];
398 tmp
[1] = zero110
[1] + in
[1];
399 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
402 * We perform two partial reductions where we eliminate the high-word of
403 * tmp[3]. We don't update the other words till the end.
405 a
= tmp
[3] >> 64; /* a < 2^46 */
406 tmp
[3] = (u64
)tmp
[3];
408 tmp
[3] += ((limb
) a
) << 32;
412 a
= tmp
[3] >> 64; /* a < 2^15 */
413 b
+= a
; /* b < 2^46 + 2^15 < 2^47 */
414 tmp
[3] = (u64
)tmp
[3];
416 tmp
[3] += ((limb
) a
) << 32;
417 /* tmp[3] < 2^64 + 2^47 */
420 * This adjusts the other two words to complete the two partial
424 tmp
[1] -= (((limb
) b
) << 32);
427 * In order to make space in tmp[3] for the carry from 2 -> 3, we
428 * conditionally subtract kPrime if tmp[3] is large enough.
431 /* As tmp[3] < 2^65, high is either 1 or 0 */
436 * all ones if the high word of tmp[3] is 1
437 * all zeros if the high word of tmp[3] if 0 */
442 * all ones if the MSB of low is 1
443 * all zeros if the MSB of low if 0 */
446 /* if low was greater than kPrime3Test then the MSB is zero */
451 * all ones if low was > kPrime3Test
452 * all zeros if low was <= kPrime3Test */
453 mask
= (mask
& low
) | high
;
454 tmp
[0] -= mask
& kPrime
[0];
455 tmp
[1] -= mask
& kPrime
[1];
456 /* kPrime[2] is zero, so omitted */
457 tmp
[3] -= mask
& kPrime
[3];
458 /* tmp[3] < 2**64 - 2**32 + 1 */
460 tmp
[1] += ((u64
)(tmp
[0] >> 64));
461 tmp
[0] = (u64
)tmp
[0];
462 tmp
[2] += ((u64
)(tmp
[1] >> 64));
463 tmp
[1] = (u64
)tmp
[1];
464 tmp
[3] += ((u64
)(tmp
[2] >> 64));
465 tmp
[2] = (u64
)tmp
[2];
474 /* smallfelem_expand converts a smallfelem to an felem */
475 static void smallfelem_expand(felem out
, const smallfelem in
)
484 * smallfelem_square sets |out| = |small|^2
488 * out[i] < 7 * 2^64 < 2^67
490 static void smallfelem_square(longfelem out
, const smallfelem small
)
495 a
= ((uint128_t
) small
[0]) * small
[0];
501 a
= ((uint128_t
) small
[0]) * small
[1];
508 a
= ((uint128_t
) small
[0]) * small
[2];
515 a
= ((uint128_t
) small
[0]) * small
[3];
521 a
= ((uint128_t
) small
[1]) * small
[2];
528 a
= ((uint128_t
) small
[1]) * small
[1];
534 a
= ((uint128_t
) small
[1]) * small
[3];
541 a
= ((uint128_t
) small
[2]) * small
[3];
549 a
= ((uint128_t
) small
[2]) * small
[2];
555 a
= ((uint128_t
) small
[3]) * small
[3];
563 * felem_square sets |out| = |in|^2
567 * out[i] < 7 * 2^64 < 2^67
569 static void felem_square(longfelem out
, const felem in
)
572 felem_shrink(small
, in
);
573 smallfelem_square(out
, small
);
577 * smallfelem_mul sets |out| = |small1| * |small2|
582 * out[i] < 7 * 2^64 < 2^67
584 static void smallfelem_mul(longfelem out
, const smallfelem small1
,
585 const smallfelem small2
)
590 a
= ((uint128_t
) small1
[0]) * small2
[0];
596 a
= ((uint128_t
) small1
[0]) * small2
[1];
602 a
= ((uint128_t
) small1
[1]) * small2
[0];
608 a
= ((uint128_t
) small1
[0]) * small2
[2];
614 a
= ((uint128_t
) small1
[1]) * small2
[1];
620 a
= ((uint128_t
) small1
[2]) * small2
[0];
626 a
= ((uint128_t
) small1
[0]) * small2
[3];
632 a
= ((uint128_t
) small1
[1]) * small2
[2];
638 a
= ((uint128_t
) small1
[2]) * small2
[1];
644 a
= ((uint128_t
) small1
[3]) * small2
[0];
650 a
= ((uint128_t
) small1
[1]) * small2
[3];
656 a
= ((uint128_t
) small1
[2]) * small2
[2];
662 a
= ((uint128_t
) small1
[3]) * small2
[1];
668 a
= ((uint128_t
) small1
[2]) * small2
[3];
674 a
= ((uint128_t
) small1
[3]) * small2
[2];
680 a
= ((uint128_t
) small1
[3]) * small2
[3];
688 * felem_mul sets |out| = |in1| * |in2|
693 * out[i] < 7 * 2^64 < 2^67
695 static void felem_mul(longfelem out
, const felem in1
, const felem in2
)
697 smallfelem small1
, small2
;
698 felem_shrink(small1
, in1
);
699 felem_shrink(small2
, in2
);
700 smallfelem_mul(out
, small1
, small2
);
704 * felem_small_mul sets |out| = |small1| * |in2|
709 * out[i] < 7 * 2^64 < 2^67
711 static void felem_small_mul(longfelem out
, const smallfelem small1
,
715 felem_shrink(small2
, in2
);
716 smallfelem_mul(out
, small1
, small2
);
719 # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
720 # define two100 (((limb)1) << 100)
721 # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
722 /* zero100 is 0 mod p */
723 static const felem zero100
=
724 { two100m36m4
, two100
, two100m36p4
, two100m36p4
};
727 * Internal function for the different flavours of felem_reduce.
728 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
730 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
731 * out[1] >= in[7] + 2^32*in[4]
732 * out[2] >= in[5] + 2^32*in[5]
733 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
735 * out[0] <= out[0] + in[4] + 2^32*in[5]
736 * out[1] <= out[1] + in[5] + 2^33*in[6]
737 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
738 * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
740 static void felem_reduce_(felem out
, const longfelem in
)
743 /* combine common terms from below */
744 c
= in
[4] + (in
[5] << 32);
752 /* the remaining terms */
753 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
754 out
[1] -= (in
[4] << 32);
755 out
[3] += (in
[4] << 32);
757 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
758 out
[2] -= (in
[5] << 32);
760 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
762 out
[0] -= (in
[6] << 32);
763 out
[1] += (in
[6] << 33);
764 out
[2] += (in
[6] * 2);
765 out
[3] -= (in
[6] << 32);
767 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
769 out
[0] -= (in
[7] << 32);
770 out
[2] += (in
[7] << 33);
771 out
[3] += (in
[7] * 3);
775 * felem_reduce converts a longfelem into an felem.
776 * To be called directly after felem_square or felem_mul.
778 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
779 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
783 static void felem_reduce(felem out
, const longfelem in
)
785 out
[0] = zero100
[0] + in
[0];
786 out
[1] = zero100
[1] + in
[1];
787 out
[2] = zero100
[2] + in
[2];
788 out
[3] = zero100
[3] + in
[3];
790 felem_reduce_(out
, in
);
793 * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
794 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
795 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
796 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
798 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
799 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
800 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
801 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
806 * felem_reduce_zero105 converts a larger longfelem into an felem.
812 static void felem_reduce_zero105(felem out
, const longfelem in
)
814 out
[0] = zero105
[0] + in
[0];
815 out
[1] = zero105
[1] + in
[1];
816 out
[2] = zero105
[2] + in
[2];
817 out
[3] = zero105
[3] + in
[3];
819 felem_reduce_(out
, in
);
822 * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
823 * out[1] > 2^105 - 2^71 - 2^103 > 0
824 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
825 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
827 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
828 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
829 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
830 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
835 * subtract_u64 sets *result = *result - v and *carry to one if the
836 * subtraction underflowed.
838 static void subtract_u64(u64
*result
, u64
*carry
, u64 v
)
840 uint128_t r
= *result
;
842 *carry
= (r
>> 64) & 1;
847 * felem_contract converts |in| to its unique, minimal representation. On
848 * entry: in[i] < 2^109
850 static void felem_contract(smallfelem out
, const felem in
)
853 u64 all_equal_so_far
= 0, result
= 0, carry
;
855 felem_shrink(out
, in
);
856 /* small is minimal except that the value might be > p */
860 * We are doing a constant time test if out >= kPrime. We need to compare
861 * each u64, from most-significant to least significant. For each one, if
862 * all words so far have been equal (m is all ones) then a non-equal
863 * result is the answer. Otherwise we continue.
865 for (i
= 3; i
< 4; i
--) {
867 uint128_t a
= ((uint128_t
) kPrime
[i
]) - out
[i
];
869 * if out[i] > kPrime[i] then a will underflow and the high 64-bits
872 result
|= all_equal_so_far
& ((u64
)(a
>> 64));
875 * if kPrime[i] == out[i] then |equal| will be all zeros and the
876 * decrement will make it all ones.
878 equal
= kPrime
[i
] ^ out
[i
];
880 equal
&= equal
<< 32;
881 equal
&= equal
<< 16;
886 equal
= ((s64
) equal
) >> 63;
888 all_equal_so_far
&= equal
;
892 * if all_equal_so_far is still all ones then the two values are equal
893 * and so out >= kPrime is true.
895 result
|= all_equal_so_far
;
897 /* if out >= kPrime then we subtract kPrime. */
898 subtract_u64(&out
[0], &carry
, result
& kPrime
[0]);
899 subtract_u64(&out
[1], &carry
, carry
);
900 subtract_u64(&out
[2], &carry
, carry
);
901 subtract_u64(&out
[3], &carry
, carry
);
903 subtract_u64(&out
[1], &carry
, result
& kPrime
[1]);
904 subtract_u64(&out
[2], &carry
, carry
);
905 subtract_u64(&out
[3], &carry
, carry
);
907 subtract_u64(&out
[2], &carry
, result
& kPrime
[2]);
908 subtract_u64(&out
[3], &carry
, carry
);
910 subtract_u64(&out
[3], &carry
, result
& kPrime
[3]);
913 static void smallfelem_square_contract(smallfelem out
, const smallfelem in
)
918 smallfelem_square(longtmp
, in
);
919 felem_reduce(tmp
, longtmp
);
920 felem_contract(out
, tmp
);
923 static void smallfelem_mul_contract(smallfelem out
, const smallfelem in1
,
924 const smallfelem in2
)
929 smallfelem_mul(longtmp
, in1
, in2
);
930 felem_reduce(tmp
, longtmp
);
931 felem_contract(out
, tmp
);
935 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
940 static limb
smallfelem_is_zero(const smallfelem small
)
945 u64 is_zero
= small
[0] | small
[1] | small
[2] | small
[3];
947 is_zero
&= is_zero
<< 32;
948 is_zero
&= is_zero
<< 16;
949 is_zero
&= is_zero
<< 8;
950 is_zero
&= is_zero
<< 4;
951 is_zero
&= is_zero
<< 2;
952 is_zero
&= is_zero
<< 1;
953 is_zero
= ((s64
) is_zero
) >> 63;
955 is_p
= (small
[0] ^ kPrime
[0]) |
956 (small
[1] ^ kPrime
[1]) |
957 (small
[2] ^ kPrime
[2]) | (small
[3] ^ kPrime
[3]);
965 is_p
= ((s64
) is_p
) >> 63;
970 result
|= ((limb
) is_zero
) << 64;
974 static int smallfelem_is_zero_int(const smallfelem small
)
976 return (int)(smallfelem_is_zero(small
) & ((limb
) 1));
980 * felem_inv calculates |out| = |in|^{-1}
982 * Based on Fermat's Little Theorem:
984 * a^{p-1} = 1 (mod p)
985 * a^{p-2} = a^{-1} (mod p)
987 static void felem_inv(felem out
, const felem in
)
990 /* each e_I will hold |in|^{2^I - 1} */
991 felem e2
, e4
, e8
, e16
, e32
, e64
;
995 felem_square(tmp
, in
);
996 felem_reduce(ftmp
, tmp
); /* 2^1 */
997 felem_mul(tmp
, in
, ftmp
);
998 felem_reduce(ftmp
, tmp
); /* 2^2 - 2^0 */
999 felem_assign(e2
, ftmp
);
1000 felem_square(tmp
, ftmp
);
1001 felem_reduce(ftmp
, tmp
); /* 2^3 - 2^1 */
1002 felem_square(tmp
, ftmp
);
1003 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^2 */
1004 felem_mul(tmp
, ftmp
, e2
);
1005 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^0 */
1006 felem_assign(e4
, ftmp
);
1007 felem_square(tmp
, ftmp
);
1008 felem_reduce(ftmp
, tmp
); /* 2^5 - 2^1 */
1009 felem_square(tmp
, ftmp
);
1010 felem_reduce(ftmp
, tmp
); /* 2^6 - 2^2 */
1011 felem_square(tmp
, ftmp
);
1012 felem_reduce(ftmp
, tmp
); /* 2^7 - 2^3 */
1013 felem_square(tmp
, ftmp
);
1014 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^4 */
1015 felem_mul(tmp
, ftmp
, e4
);
1016 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^0 */
1017 felem_assign(e8
, ftmp
);
1018 for (i
= 0; i
< 8; i
++) {
1019 felem_square(tmp
, ftmp
);
1020 felem_reduce(ftmp
, tmp
);
1022 felem_mul(tmp
, ftmp
, e8
);
1023 felem_reduce(ftmp
, tmp
); /* 2^16 - 2^0 */
1024 felem_assign(e16
, ftmp
);
1025 for (i
= 0; i
< 16; i
++) {
1026 felem_square(tmp
, ftmp
);
1027 felem_reduce(ftmp
, tmp
);
1029 felem_mul(tmp
, ftmp
, e16
);
1030 felem_reduce(ftmp
, tmp
); /* 2^32 - 2^0 */
1031 felem_assign(e32
, ftmp
);
1032 for (i
= 0; i
< 32; i
++) {
1033 felem_square(tmp
, ftmp
);
1034 felem_reduce(ftmp
, tmp
);
1036 felem_assign(e64
, ftmp
);
1037 felem_mul(tmp
, ftmp
, in
);
1038 felem_reduce(ftmp
, tmp
); /* 2^64 - 2^32 + 2^0 */
1039 for (i
= 0; i
< 192; i
++) {
1040 felem_square(tmp
, ftmp
);
1041 felem_reduce(ftmp
, tmp
);
1042 } /* 2^256 - 2^224 + 2^192 */
1044 felem_mul(tmp
, e64
, e32
);
1045 felem_reduce(ftmp2
, tmp
); /* 2^64 - 2^0 */
1046 for (i
= 0; i
< 16; i
++) {
1047 felem_square(tmp
, ftmp2
);
1048 felem_reduce(ftmp2
, tmp
);
1050 felem_mul(tmp
, ftmp2
, e16
);
1051 felem_reduce(ftmp2
, tmp
); /* 2^80 - 2^0 */
1052 for (i
= 0; i
< 8; i
++) {
1053 felem_square(tmp
, ftmp2
);
1054 felem_reduce(ftmp2
, tmp
);
1056 felem_mul(tmp
, ftmp2
, e8
);
1057 felem_reduce(ftmp2
, tmp
); /* 2^88 - 2^0 */
1058 for (i
= 0; i
< 4; i
++) {
1059 felem_square(tmp
, ftmp2
);
1060 felem_reduce(ftmp2
, tmp
);
1062 felem_mul(tmp
, ftmp2
, e4
);
1063 felem_reduce(ftmp2
, tmp
); /* 2^92 - 2^0 */
1064 felem_square(tmp
, ftmp2
);
1065 felem_reduce(ftmp2
, tmp
); /* 2^93 - 2^1 */
1066 felem_square(tmp
, ftmp2
);
1067 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^2 */
1068 felem_mul(tmp
, ftmp2
, e2
);
1069 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^0 */
1070 felem_square(tmp
, ftmp2
);
1071 felem_reduce(ftmp2
, tmp
); /* 2^95 - 2^1 */
1072 felem_square(tmp
, ftmp2
);
1073 felem_reduce(ftmp2
, tmp
); /* 2^96 - 2^2 */
1074 felem_mul(tmp
, ftmp2
, in
);
1075 felem_reduce(ftmp2
, tmp
); /* 2^96 - 3 */
1077 felem_mul(tmp
, ftmp2
, ftmp
);
1078 felem_reduce(out
, tmp
); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1081 static void smallfelem_inv_contract(smallfelem out
, const smallfelem in
)
1085 smallfelem_expand(tmp
, in
);
1086 felem_inv(tmp
, tmp
);
1087 felem_contract(out
, tmp
);
1094 * Building on top of the field operations we have the operations on the
1095 * elliptic curve group itself. Points on the curve are represented in Jacobian
1100 * point_double calculates 2*(x_in, y_in, z_in)
1102 * The method is taken from:
1103 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1105 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1106 * while x_out == y_in is not (maybe this works, but it's not tested).
1109 point_double(felem x_out
, felem y_out
, felem z_out
,
1110 const felem x_in
, const felem y_in
, const felem z_in
)
1112 longfelem tmp
, tmp2
;
1113 felem delta
, gamma
, beta
, alpha
, ftmp
, ftmp2
;
1114 smallfelem small1
, small2
;
1116 felem_assign(ftmp
, x_in
);
1117 /* ftmp[i] < 2^106 */
1118 felem_assign(ftmp2
, x_in
);
1119 /* ftmp2[i] < 2^106 */
1122 felem_square(tmp
, z_in
);
1123 felem_reduce(delta
, tmp
);
1124 /* delta[i] < 2^101 */
1127 felem_square(tmp
, y_in
);
1128 felem_reduce(gamma
, tmp
);
1129 /* gamma[i] < 2^101 */
1130 felem_shrink(small1
, gamma
);
1132 /* beta = x*gamma */
1133 felem_small_mul(tmp
, small1
, x_in
);
1134 felem_reduce(beta
, tmp
);
1135 /* beta[i] < 2^101 */
1137 /* alpha = 3*(x-delta)*(x+delta) */
1138 felem_diff(ftmp
, delta
);
1139 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1140 felem_sum(ftmp2
, delta
);
1141 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1142 felem_scalar(ftmp2
, 3);
1143 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1144 felem_mul(tmp
, ftmp
, ftmp2
);
1145 felem_reduce(alpha
, tmp
);
1146 /* alpha[i] < 2^101 */
1147 felem_shrink(small2
, alpha
);
1149 /* x' = alpha^2 - 8*beta */
1150 smallfelem_square(tmp
, small2
);
1151 felem_reduce(x_out
, tmp
);
1152 felem_assign(ftmp
, beta
);
1153 felem_scalar(ftmp
, 8);
1154 /* ftmp[i] < 8 * 2^101 = 2^104 */
1155 felem_diff(x_out
, ftmp
);
1156 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1158 /* z' = (y + z)^2 - gamma - delta */
1159 felem_sum(delta
, gamma
);
1160 /* delta[i] < 2^101 + 2^101 = 2^102 */
1161 felem_assign(ftmp
, y_in
);
1162 felem_sum(ftmp
, z_in
);
1163 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1164 felem_square(tmp
, ftmp
);
1165 felem_reduce(z_out
, tmp
);
1166 felem_diff(z_out
, delta
);
1167 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1169 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1170 felem_scalar(beta
, 4);
1171 /* beta[i] < 4 * 2^101 = 2^103 */
1172 felem_diff_zero107(beta
, x_out
);
1173 /* beta[i] < 2^107 + 2^103 < 2^108 */
1174 felem_small_mul(tmp
, small2
, beta
);
1175 /* tmp[i] < 7 * 2^64 < 2^67 */
1176 smallfelem_square(tmp2
, small1
);
1177 /* tmp2[i] < 7 * 2^64 */
1178 longfelem_scalar(tmp2
, 8);
1179 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1180 longfelem_diff(tmp
, tmp2
);
1181 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1182 felem_reduce_zero105(y_out
, tmp
);
1183 /* y_out[i] < 2^106 */
1187 * point_double_small is the same as point_double, except that it operates on
1191 point_double_small(smallfelem x_out
, smallfelem y_out
, smallfelem z_out
,
1192 const smallfelem x_in
, const smallfelem y_in
,
1193 const smallfelem z_in
)
1195 felem felem_x_out
, felem_y_out
, felem_z_out
;
1196 felem felem_x_in
, felem_y_in
, felem_z_in
;
1198 smallfelem_expand(felem_x_in
, x_in
);
1199 smallfelem_expand(felem_y_in
, y_in
);
1200 smallfelem_expand(felem_z_in
, z_in
);
1201 point_double(felem_x_out
, felem_y_out
, felem_z_out
,
1202 felem_x_in
, felem_y_in
, felem_z_in
);
1203 felem_shrink(x_out
, felem_x_out
);
1204 felem_shrink(y_out
, felem_y_out
);
1205 felem_shrink(z_out
, felem_z_out
);
1208 /* copy_conditional copies in to out iff mask is all ones. */
1209 static void copy_conditional(felem out
, const felem in
, limb mask
)
1212 for (i
= 0; i
< NLIMBS
; ++i
) {
1213 const limb tmp
= mask
& (in
[i
] ^ out
[i
]);
1218 /* copy_small_conditional copies in to out iff mask is all ones. */
1219 static void copy_small_conditional(felem out
, const smallfelem in
, limb mask
)
1222 const u64 mask64
= mask
;
1223 for (i
= 0; i
< NLIMBS
; ++i
) {
1224 out
[i
] = ((limb
) (in
[i
] & mask64
)) | (out
[i
] & ~mask
);
1229 * point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1231 * The method is taken from:
1232 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1233 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1235 * This function includes a branch for checking whether the two input points
1236 * are equal, (while not equal to the point at infinity). This case never
1237 * happens during single point multiplication, so there is no timing leak for
1238 * ECDH or ECDSA signing.
1240 static void point_add(felem x3
, felem y3
, felem z3
,
1241 const felem x1
, const felem y1
, const felem z1
,
1242 const int mixed
, const smallfelem x2
,
1243 const smallfelem y2
, const smallfelem z2
)
1245 felem ftmp
, ftmp2
, ftmp3
, ftmp4
, ftmp5
, ftmp6
, x_out
, y_out
, z_out
;
1246 longfelem tmp
, tmp2
;
1247 smallfelem small1
, small2
, small3
, small4
, small5
;
1248 limb x_equal
, y_equal
, z1_is_zero
, z2_is_zero
;
1250 felem_shrink(small3
, z1
);
1252 z1_is_zero
= smallfelem_is_zero(small3
);
1253 z2_is_zero
= smallfelem_is_zero(z2
);
1255 /* ftmp = z1z1 = z1**2 */
1256 smallfelem_square(tmp
, small3
);
1257 felem_reduce(ftmp
, tmp
);
1258 /* ftmp[i] < 2^101 */
1259 felem_shrink(small1
, ftmp
);
1262 /* ftmp2 = z2z2 = z2**2 */
1263 smallfelem_square(tmp
, z2
);
1264 felem_reduce(ftmp2
, tmp
);
1265 /* ftmp2[i] < 2^101 */
1266 felem_shrink(small2
, ftmp2
);
1268 felem_shrink(small5
, x1
);
1270 /* u1 = ftmp3 = x1*z2z2 */
1271 smallfelem_mul(tmp
, small5
, small2
);
1272 felem_reduce(ftmp3
, tmp
);
1273 /* ftmp3[i] < 2^101 */
1275 /* ftmp5 = z1 + z2 */
1276 felem_assign(ftmp5
, z1
);
1277 felem_small_sum(ftmp5
, z2
);
1278 /* ftmp5[i] < 2^107 */
1280 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1281 felem_square(tmp
, ftmp5
);
1282 felem_reduce(ftmp5
, tmp
);
1283 /* ftmp2 = z2z2 + z1z1 */
1284 felem_sum(ftmp2
, ftmp
);
1285 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1286 felem_diff(ftmp5
, ftmp2
);
1287 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1289 /* ftmp2 = z2 * z2z2 */
1290 smallfelem_mul(tmp
, small2
, z2
);
1291 felem_reduce(ftmp2
, tmp
);
1293 /* s1 = ftmp2 = y1 * z2**3 */
1294 felem_mul(tmp
, y1
, ftmp2
);
1295 felem_reduce(ftmp6
, tmp
);
1296 /* ftmp6[i] < 2^101 */
1299 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1302 /* u1 = ftmp3 = x1*z2z2 */
1303 felem_assign(ftmp3
, x1
);
1304 /* ftmp3[i] < 2^106 */
1307 felem_assign(ftmp5
, z1
);
1308 felem_scalar(ftmp5
, 2);
1309 /* ftmp5[i] < 2*2^106 = 2^107 */
1311 /* s1 = ftmp2 = y1 * z2**3 */
1312 felem_assign(ftmp6
, y1
);
1313 /* ftmp6[i] < 2^106 */
1317 smallfelem_mul(tmp
, x2
, small1
);
1318 felem_reduce(ftmp4
, tmp
);
1320 /* h = ftmp4 = u2 - u1 */
1321 felem_diff_zero107(ftmp4
, ftmp3
);
1322 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1323 felem_shrink(small4
, ftmp4
);
1325 x_equal
= smallfelem_is_zero(small4
);
1327 /* z_out = ftmp5 * h */
1328 felem_small_mul(tmp
, small4
, ftmp5
);
1329 felem_reduce(z_out
, tmp
);
1330 /* z_out[i] < 2^101 */
1332 /* ftmp = z1 * z1z1 */
1333 smallfelem_mul(tmp
, small1
, small3
);
1334 felem_reduce(ftmp
, tmp
);
1336 /* s2 = tmp = y2 * z1**3 */
1337 felem_small_mul(tmp
, y2
, ftmp
);
1338 felem_reduce(ftmp5
, tmp
);
1340 /* r = ftmp5 = (s2 - s1)*2 */
1341 felem_diff_zero107(ftmp5
, ftmp6
);
1342 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1343 felem_scalar(ftmp5
, 2);
1344 /* ftmp5[i] < 2^109 */
1345 felem_shrink(small1
, ftmp5
);
1346 y_equal
= smallfelem_is_zero(small1
);
1348 if (x_equal
&& y_equal
&& !z1_is_zero
&& !z2_is_zero
) {
1349 point_double(x3
, y3
, z3
, x1
, y1
, z1
);
1353 /* I = ftmp = (2h)**2 */
1354 felem_assign(ftmp
, ftmp4
);
1355 felem_scalar(ftmp
, 2);
1356 /* ftmp[i] < 2*2^108 = 2^109 */
1357 felem_square(tmp
, ftmp
);
1358 felem_reduce(ftmp
, tmp
);
1360 /* J = ftmp2 = h * I */
1361 felem_mul(tmp
, ftmp4
, ftmp
);
1362 felem_reduce(ftmp2
, tmp
);
1364 /* V = ftmp4 = U1 * I */
1365 felem_mul(tmp
, ftmp3
, ftmp
);
1366 felem_reduce(ftmp4
, tmp
);
1368 /* x_out = r**2 - J - 2V */
1369 smallfelem_square(tmp
, small1
);
1370 felem_reduce(x_out
, tmp
);
1371 felem_assign(ftmp3
, ftmp4
);
1372 felem_scalar(ftmp4
, 2);
1373 felem_sum(ftmp4
, ftmp2
);
1374 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1375 felem_diff(x_out
, ftmp4
);
1376 /* x_out[i] < 2^105 + 2^101 */
1378 /* y_out = r(V-x_out) - 2 * s1 * J */
1379 felem_diff_zero107(ftmp3
, x_out
);
1380 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1381 felem_small_mul(tmp
, small1
, ftmp3
);
1382 felem_mul(tmp2
, ftmp6
, ftmp2
);
1383 longfelem_scalar(tmp2
, 2);
1384 /* tmp2[i] < 2*2^67 = 2^68 */
1385 longfelem_diff(tmp
, tmp2
);
1386 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1387 felem_reduce_zero105(y_out
, tmp
);
1388 /* y_out[i] < 2^106 */
1390 copy_small_conditional(x_out
, x2
, z1_is_zero
);
1391 copy_conditional(x_out
, x1
, z2_is_zero
);
1392 copy_small_conditional(y_out
, y2
, z1_is_zero
);
1393 copy_conditional(y_out
, y1
, z2_is_zero
);
1394 copy_small_conditional(z_out
, z2
, z1_is_zero
);
1395 copy_conditional(z_out
, z1
, z2_is_zero
);
1396 felem_assign(x3
, x_out
);
1397 felem_assign(y3
, y_out
);
1398 felem_assign(z3
, z_out
);
1402 * point_add_small is the same as point_add, except that it operates on
1405 static void point_add_small(smallfelem x3
, smallfelem y3
, smallfelem z3
,
1406 smallfelem x1
, smallfelem y1
, smallfelem z1
,
1407 smallfelem x2
, smallfelem y2
, smallfelem z2
)
1409 felem felem_x3
, felem_y3
, felem_z3
;
1410 felem felem_x1
, felem_y1
, felem_z1
;
1411 smallfelem_expand(felem_x1
, x1
);
1412 smallfelem_expand(felem_y1
, y1
);
1413 smallfelem_expand(felem_z1
, z1
);
1414 point_add(felem_x3
, felem_y3
, felem_z3
, felem_x1
, felem_y1
, felem_z1
, 0,
1416 felem_shrink(x3
, felem_x3
);
1417 felem_shrink(y3
, felem_y3
);
1418 felem_shrink(z3
, felem_z3
);
1422 * Base point pre computation
1423 * --------------------------
1425 * Two different sorts of precomputed tables are used in the following code.
1426 * Each contain various points on the curve, where each point is three field
1427 * elements (x, y, z).
1429 * For the base point table, z is usually 1 (0 for the point at infinity).
1430 * This table has 2 * 16 elements, starting with the following:
1431 * index | bits | point
1432 * ------+---------+------------------------------
1435 * 2 | 0 0 1 0 | 2^64G
1436 * 3 | 0 0 1 1 | (2^64 + 1)G
1437 * 4 | 0 1 0 0 | 2^128G
1438 * 5 | 0 1 0 1 | (2^128 + 1)G
1439 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1440 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1441 * 8 | 1 0 0 0 | 2^192G
1442 * 9 | 1 0 0 1 | (2^192 + 1)G
1443 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1444 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1445 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1446 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1447 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1448 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1449 * followed by a copy of this with each element multiplied by 2^32.
1451 * The reason for this is so that we can clock bits into four different
1452 * locations when doing simple scalar multiplies against the base point,
1453 * and then another four locations using the second 16 elements.
1455 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1457 /* gmul is the table of precomputed base points */
1458 static const smallfelem gmul
[2][16][3] = {
1462 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1463 0x6b17d1f2e12c4247},
1464 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1465 0x4fe342e2fe1a7f9b},
1467 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1468 0x0fa822bc2811aaa5},
1469 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1470 0xbff44ae8f5dba80d},
1472 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1473 0x300a4bbc89d6726f},
1474 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1475 0x72aac7e0d09b4644},
1477 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1478 0x447d739beedb5e67},
1479 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1480 0x2d4825ab834131ee},
1482 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1483 0xef9519328a9c72ff},
1484 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1485 0x611e9fc37dbb2c9b},
1487 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1488 0x550663797b51f5d8},
1489 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1490 0x157164848aecb851},
1492 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1493 0xeb5d7745b21141ea},
1494 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1495 0xeafd72ebdbecc17b},
1497 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1498 0xa6d39677a7849276},
1499 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1500 0x674f84749b0b8816},
1502 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1503 0x4e769e7672c9ddad},
1504 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1505 0x42b99082de830663},
1507 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1508 0x78878ef61c6ce04d},
1509 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1510 0xb6cb3f5d7b72c321},
1512 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1513 0x0c88bc4d716b1287},
1514 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1515 0xdd5ddea3f3901dc6},
1517 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1518 0x68f344af6b317466},
1519 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1520 0x31b9c405f8540a20},
1522 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1523 0x4052bf4b6f461db9},
1524 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1525 0xfecf4d5190b0fc61},
1527 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1528 0x1eddbae2c802e41a},
1529 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1530 0x43104d86560ebcfc},
1532 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1533 0xb48e26b484f7a21c},
1534 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1535 0xfac015404d4d3dab},
1540 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1541 0x7fe36b40af22af89},
1542 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1543 0xe697d45825b63624},
1545 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1546 0x4a5b506612a677a6},
1547 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1548 0xeb13461ceac089f1},
1550 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1551 0x0781b8291c6a220a},
1552 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1553 0x690cde8df0151593},
1555 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1556 0x8a535f566ec73617},
1557 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1558 0x0455c08468b08bd7},
1560 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1561 0x06bada7ab77f8276},
1562 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1563 0x5b476dfd0e6cb18a},
1565 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1566 0x3e29864e8a2ec908},
1567 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1568 0x239b90ea3dc31e7e},
1570 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1571 0x820f4dd949f72ff7},
1572 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1573 0x140406ec783a05ec},
1575 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1576 0x68f6b8542783dfee},
1577 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1578 0xcbe1feba92e40ce6},
1580 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1581 0xd0b2f94d2f420109},
1582 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1583 0x971459828b0719e5},
1585 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1586 0x961610004a866aba},
1587 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1588 0x7acb9fadcee75e44},
1590 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1591 0x24eb9acca333bf5b},
1592 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1593 0x69f891c5acd079cc},
1595 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1596 0xe51f547c5972a107},
1597 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1598 0x1c309a2b25bb1387},
1600 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1601 0x20b87b8aa2c4e503},
1602 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1603 0xf5c6fa49919776be},
1605 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1606 0x1ed7d1b9332010b9},
1607 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1608 0x3a2b03f03217257a},
1610 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1611 0x15fee545c78dd9f6},
1612 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1613 0x4ab5b6b2b8753f81},
1618 * select_point selects the |idx|th point from a precomputation table and
1621 static void select_point(const u64 idx
, unsigned int size
,
1622 const smallfelem pre_comp
[16][3], smallfelem out
[3])
1625 u64
*outlimbs
= &out
[0][0];
1627 memset(out
, 0, sizeof(*out
) * 3);
1629 for (i
= 0; i
< size
; i
++) {
1630 const u64
*inlimbs
= (u64
*)&pre_comp
[i
][0][0];
1637 for (j
= 0; j
< NLIMBS
* 3; j
++)
1638 outlimbs
[j
] |= inlimbs
[j
] & mask
;
1642 /* get_bit returns the |i|th bit in |in| */
1643 static char get_bit(const felem_bytearray in
, int i
)
1645 if ((i
< 0) || (i
>= 256))
1647 return (in
[i
>> 3] >> (i
& 7)) & 1;
1651 * Interleaved point multiplication using precomputed point multiples: The
1652 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1653 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1654 * generator, using certain (large) precomputed multiples in g_pre_comp.
1655 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1657 static void batch_mul(felem x_out
, felem y_out
, felem z_out
,
1658 const felem_bytearray scalars
[],
1659 const unsigned num_points
, const u8
*g_scalar
,
1660 const int mixed
, const smallfelem pre_comp
[][17][3],
1661 const smallfelem g_pre_comp
[2][16][3])
1664 unsigned num
, gen_mul
= (g_scalar
!= NULL
);
1670 /* set nq to the point at infinity */
1671 memset(nq
, 0, sizeof(nq
));
1674 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1675 * of the generator (two in each of the last 32 rounds) and additions of
1676 * other points multiples (every 5th round).
1678 skip
= 1; /* save two point operations in the first
1680 for (i
= (num_points
? 255 : 31); i
>= 0; --i
) {
1683 point_double(nq
[0], nq
[1], nq
[2], nq
[0], nq
[1], nq
[2]);
1685 /* add multiples of the generator */
1686 if (gen_mul
&& (i
<= 31)) {
1687 /* first, look 32 bits upwards */
1688 bits
= get_bit(g_scalar
, i
+ 224) << 3;
1689 bits
|= get_bit(g_scalar
, i
+ 160) << 2;
1690 bits
|= get_bit(g_scalar
, i
+ 96) << 1;
1691 bits
|= get_bit(g_scalar
, i
+ 32);
1692 /* select the point to add, in constant time */
1693 select_point(bits
, 16, g_pre_comp
[1], tmp
);
1696 /* Arg 1 below is for "mixed" */
1697 point_add(nq
[0], nq
[1], nq
[2],
1698 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1700 smallfelem_expand(nq
[0], tmp
[0]);
1701 smallfelem_expand(nq
[1], tmp
[1]);
1702 smallfelem_expand(nq
[2], tmp
[2]);
1706 /* second, look at the current position */
1707 bits
= get_bit(g_scalar
, i
+ 192) << 3;
1708 bits
|= get_bit(g_scalar
, i
+ 128) << 2;
1709 bits
|= get_bit(g_scalar
, i
+ 64) << 1;
1710 bits
|= get_bit(g_scalar
, i
);
1711 /* select the point to add, in constant time */
1712 select_point(bits
, 16, g_pre_comp
[0], tmp
);
1713 /* Arg 1 below is for "mixed" */
1714 point_add(nq
[0], nq
[1], nq
[2],
1715 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1718 /* do other additions every 5 doublings */
1719 if (num_points
&& (i
% 5 == 0)) {
1720 /* loop over all scalars */
1721 for (num
= 0; num
< num_points
; ++num
) {
1722 bits
= get_bit(scalars
[num
], i
+ 4) << 5;
1723 bits
|= get_bit(scalars
[num
], i
+ 3) << 4;
1724 bits
|= get_bit(scalars
[num
], i
+ 2) << 3;
1725 bits
|= get_bit(scalars
[num
], i
+ 1) << 2;
1726 bits
|= get_bit(scalars
[num
], i
) << 1;
1727 bits
|= get_bit(scalars
[num
], i
- 1);
1728 ec_GFp_nistp_recode_scalar_bits(&sign
, &digit
, bits
);
1731 * select the point to add or subtract, in constant time
1733 select_point(digit
, 17, pre_comp
[num
], tmp
);
1734 smallfelem_neg(ftmp
, tmp
[1]); /* (X, -Y, Z) is the negative
1736 copy_small_conditional(ftmp
, tmp
[1], (((limb
) sign
) - 1));
1737 felem_contract(tmp
[1], ftmp
);
1740 point_add(nq
[0], nq
[1], nq
[2],
1741 nq
[0], nq
[1], nq
[2],
1742 mixed
, tmp
[0], tmp
[1], tmp
[2]);
1744 smallfelem_expand(nq
[0], tmp
[0]);
1745 smallfelem_expand(nq
[1], tmp
[1]);
1746 smallfelem_expand(nq
[2], tmp
[2]);
1752 felem_assign(x_out
, nq
[0]);
1753 felem_assign(y_out
, nq
[1]);
1754 felem_assign(z_out
, nq
[2]);
1757 /* Precomputation for the group generator. */
1758 struct nistp256_pre_comp_st
{
1759 smallfelem g_pre_comp
[2][16][3];
1763 const EC_METHOD
*EC_GFp_nistp256_method(void)
1765 static const EC_METHOD ret
= {
1766 EC_FLAGS_DEFAULT_OCT
,
1767 NID_X9_62_prime_field
,
1768 ec_GFp_nistp256_group_init
,
1769 ec_GFp_simple_group_finish
,
1770 ec_GFp_simple_group_clear_finish
,
1771 ec_GFp_nist_group_copy
,
1772 ec_GFp_nistp256_group_set_curve
,
1773 ec_GFp_simple_group_get_curve
,
1774 ec_GFp_simple_group_get_degree
,
1775 ec_GFp_simple_group_check_discriminant
,
1776 ec_GFp_simple_point_init
,
1777 ec_GFp_simple_point_finish
,
1778 ec_GFp_simple_point_clear_finish
,
1779 ec_GFp_simple_point_copy
,
1780 ec_GFp_simple_point_set_to_infinity
,
1781 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
1782 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
1783 ec_GFp_simple_point_set_affine_coordinates
,
1784 ec_GFp_nistp256_point_get_affine_coordinates
,
1785 0 /* point_set_compressed_coordinates */ ,
1790 ec_GFp_simple_invert
,
1791 ec_GFp_simple_is_at_infinity
,
1792 ec_GFp_simple_is_on_curve
,
1794 ec_GFp_simple_make_affine
,
1795 ec_GFp_simple_points_make_affine
,
1796 ec_GFp_nistp256_points_mul
,
1797 ec_GFp_nistp256_precompute_mult
,
1798 ec_GFp_nistp256_have_precompute_mult
,
1799 ec_GFp_nist_field_mul
,
1800 ec_GFp_nist_field_sqr
,
1802 0 /* field_encode */ ,
1803 0 /* field_decode */ ,
1804 0 /* field_set_to_one */
1810 /******************************************************************************/
1812 * FUNCTIONS TO MANAGE PRECOMPUTATION
1815 static NISTP256_PRE_COMP
*nistp256_pre_comp_new()
1817 NISTP256_PRE_COMP
*ret
= NULL
;
1818 ret
= OPENSSL_malloc(sizeof(*ret
));
1820 ECerr(EC_F_NISTP256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1823 memset(ret
->g_pre_comp
, 0, sizeof(ret
->g_pre_comp
));
1824 ret
->references
= 1;
1828 NISTP256_PRE_COMP
*EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP
*p
)
1831 CRYPTO_add(&p
->references
, 1, CRYPTO_LOCK_EC_PRE_COMP
);
1835 void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP
*pre
)
1838 || CRYPTO_add(&pre
->references
, -1, CRYPTO_LOCK_EC_PRE_COMP
) > 0)
1843 /******************************************************************************/
1845 * OPENSSL EC_METHOD FUNCTIONS
1848 int ec_GFp_nistp256_group_init(EC_GROUP
*group
)
1851 ret
= ec_GFp_simple_group_init(group
);
1852 group
->a_is_minus3
= 1;
1856 int ec_GFp_nistp256_group_set_curve(EC_GROUP
*group
, const BIGNUM
*p
,
1857 const BIGNUM
*a
, const BIGNUM
*b
,
1861 BN_CTX
*new_ctx
= NULL
;
1862 BIGNUM
*curve_p
, *curve_a
, *curve_b
;
1865 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
1868 if (((curve_p
= BN_CTX_get(ctx
)) == NULL
) ||
1869 ((curve_a
= BN_CTX_get(ctx
)) == NULL
) ||
1870 ((curve_b
= BN_CTX_get(ctx
)) == NULL
))
1872 BN_bin2bn(nistp256_curve_params
[0], sizeof(felem_bytearray
), curve_p
);
1873 BN_bin2bn(nistp256_curve_params
[1], sizeof(felem_bytearray
), curve_a
);
1874 BN_bin2bn(nistp256_curve_params
[2], sizeof(felem_bytearray
), curve_b
);
1875 if ((BN_cmp(curve_p
, p
)) || (BN_cmp(curve_a
, a
)) || (BN_cmp(curve_b
, b
))) {
1876 ECerr(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE
,
1877 EC_R_WRONG_CURVE_PARAMETERS
);
1880 group
->field_mod_func
= BN_nist_mod_256
;
1881 ret
= ec_GFp_simple_group_set_curve(group
, p
, a
, b
, ctx
);
1884 BN_CTX_free(new_ctx
);
1889 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1892 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP
*group
,
1893 const EC_POINT
*point
,
1894 BIGNUM
*x
, BIGNUM
*y
,
1897 felem z1
, z2
, x_in
, y_in
;
1898 smallfelem x_out
, y_out
;
1901 if (EC_POINT_is_at_infinity(group
, point
)) {
1902 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1903 EC_R_POINT_AT_INFINITY
);
1906 if ((!BN_to_felem(x_in
, point
->X
)) || (!BN_to_felem(y_in
, point
->Y
)) ||
1907 (!BN_to_felem(z1
, point
->Z
)))
1910 felem_square(tmp
, z2
);
1911 felem_reduce(z1
, tmp
);
1912 felem_mul(tmp
, x_in
, z1
);
1913 felem_reduce(x_in
, tmp
);
1914 felem_contract(x_out
, x_in
);
1916 if (!smallfelem_to_BN(x
, x_out
)) {
1917 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1922 felem_mul(tmp
, z1
, z2
);
1923 felem_reduce(z1
, tmp
);
1924 felem_mul(tmp
, y_in
, z1
);
1925 felem_reduce(y_in
, tmp
);
1926 felem_contract(y_out
, y_in
);
1928 if (!smallfelem_to_BN(y
, y_out
)) {
1929 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1937 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
1938 static void make_points_affine(size_t num
, smallfelem points
[][3],
1939 smallfelem tmp_smallfelems
[])
1942 * Runs in constant time, unless an input is the point at infinity (which
1943 * normally shouldn't happen).
1945 ec_GFp_nistp_points_make_affine_internal(num
,
1949 (void (*)(void *))smallfelem_one
,
1950 (int (*)(const void *))
1951 smallfelem_is_zero_int
,
1952 (void (*)(void *, const void *))
1954 (void (*)(void *, const void *))
1955 smallfelem_square_contract
,
1957 (void *, const void *,
1959 smallfelem_mul_contract
,
1960 (void (*)(void *, const void *))
1961 smallfelem_inv_contract
,
1962 /* nothing to contract */
1963 (void (*)(void *, const void *))
1968 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1969 * values Result is stored in r (r can equal one of the inputs).
1971 int ec_GFp_nistp256_points_mul(const EC_GROUP
*group
, EC_POINT
*r
,
1972 const BIGNUM
*scalar
, size_t num
,
1973 const EC_POINT
*points
[],
1974 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
1979 BN_CTX
*new_ctx
= NULL
;
1980 BIGNUM
*x
, *y
, *z
, *tmp_scalar
;
1981 felem_bytearray g_secret
;
1982 felem_bytearray
*secrets
= NULL
;
1983 smallfelem (*pre_comp
)[17][3] = NULL
;
1984 smallfelem
*tmp_smallfelems
= NULL
;
1985 felem_bytearray tmp
;
1986 unsigned i
, num_bytes
;
1987 int have_pre_comp
= 0;
1988 size_t num_points
= num
;
1989 smallfelem x_in
, y_in
, z_in
;
1990 felem x_out
, y_out
, z_out
;
1991 NISTP256_PRE_COMP
*pre
= NULL
;
1992 const smallfelem(*g_pre_comp
)[16][3] = NULL
;
1993 EC_POINT
*generator
= NULL
;
1994 const EC_POINT
*p
= NULL
;
1995 const BIGNUM
*p_scalar
= NULL
;
1998 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
2001 if (((x
= BN_CTX_get(ctx
)) == NULL
) ||
2002 ((y
= BN_CTX_get(ctx
)) == NULL
) ||
2003 ((z
= BN_CTX_get(ctx
)) == NULL
) ||
2004 ((tmp_scalar
= BN_CTX_get(ctx
)) == NULL
))
2007 if (scalar
!= NULL
) {
2008 pre
= group
->pre_comp
.nistp256
;
2010 /* we have precomputation, try to use it */
2011 g_pre_comp
= (const smallfelem(*)[16][3])pre
->g_pre_comp
;
2013 /* try to use the standard precomputation */
2014 g_pre_comp
= &gmul
[0];
2015 generator
= EC_POINT_new(group
);
2016 if (generator
== NULL
)
2018 /* get the generator from precomputation */
2019 if (!smallfelem_to_BN(x
, g_pre_comp
[0][1][0]) ||
2020 !smallfelem_to_BN(y
, g_pre_comp
[0][1][1]) ||
2021 !smallfelem_to_BN(z
, g_pre_comp
[0][1][2])) {
2022 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2025 if (!EC_POINT_set_Jprojective_coordinates_GFp(group
,
2029 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
2030 /* precomputation matches generator */
2034 * we don't have valid precomputation: treat the generator as a
2039 if (num_points
> 0) {
2040 if (num_points
>= 3) {
2042 * unless we precompute multiples for just one or two points,
2043 * converting those into affine form is time well spent
2047 secrets
= OPENSSL_malloc(sizeof(*secrets
) * num_points
);
2048 pre_comp
= OPENSSL_malloc(sizeof(*pre_comp
) * num_points
);
2051 OPENSSL_malloc(sizeof(*tmp_smallfelems
) * (num_points
* 17 + 1));
2052 if ((secrets
== NULL
) || (pre_comp
== NULL
)
2053 || (mixed
&& (tmp_smallfelems
== NULL
))) {
2054 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
2059 * we treat NULL scalars as 0, and NULL points as points at infinity,
2060 * i.e., they contribute nothing to the linear combination
2062 memset(secrets
, 0, sizeof(*secrets
) * num_points
);
2063 memset(pre_comp
, 0, sizeof(*pre_comp
) * num_points
);
2064 for (i
= 0; i
< num_points
; ++i
) {
2067 * we didn't have a valid precomputation, so we pick the
2071 p
= EC_GROUP_get0_generator(group
);
2074 /* the i^th point */
2077 p_scalar
= scalars
[i
];
2079 if ((p_scalar
!= NULL
) && (p
!= NULL
)) {
2080 /* reduce scalar to 0 <= scalar < 2^256 */
2081 if ((BN_num_bits(p_scalar
) > 256)
2082 || (BN_is_negative(p_scalar
))) {
2084 * this is an unusual input, and we don't guarantee
2087 if (!BN_nnmod(tmp_scalar
, p_scalar
, group
->order
, ctx
)) {
2088 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2091 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
2093 num_bytes
= BN_bn2bin(p_scalar
, tmp
);
2094 flip_endian(secrets
[i
], tmp
, num_bytes
);
2095 /* precompute multiples */
2096 if ((!BN_to_felem(x_out
, p
->X
)) ||
2097 (!BN_to_felem(y_out
, p
->Y
)) ||
2098 (!BN_to_felem(z_out
, p
->Z
)))
2100 felem_shrink(pre_comp
[i
][1][0], x_out
);
2101 felem_shrink(pre_comp
[i
][1][1], y_out
);
2102 felem_shrink(pre_comp
[i
][1][2], z_out
);
2103 for (j
= 2; j
<= 16; ++j
) {
2105 point_add_small(pre_comp
[i
][j
][0], pre_comp
[i
][j
][1],
2106 pre_comp
[i
][j
][2], pre_comp
[i
][1][0],
2107 pre_comp
[i
][1][1], pre_comp
[i
][1][2],
2108 pre_comp
[i
][j
- 1][0],
2109 pre_comp
[i
][j
- 1][1],
2110 pre_comp
[i
][j
- 1][2]);
2112 point_double_small(pre_comp
[i
][j
][0],
2115 pre_comp
[i
][j
/ 2][0],
2116 pre_comp
[i
][j
/ 2][1],
2117 pre_comp
[i
][j
/ 2][2]);
2123 make_points_affine(num_points
* 17, pre_comp
[0], tmp_smallfelems
);
2126 /* the scalar for the generator */
2127 if ((scalar
!= NULL
) && (have_pre_comp
)) {
2128 memset(g_secret
, 0, sizeof(g_secret
));
2129 /* reduce scalar to 0 <= scalar < 2^256 */
2130 if ((BN_num_bits(scalar
) > 256) || (BN_is_negative(scalar
))) {
2132 * this is an unusual input, and we don't guarantee
2135 if (!BN_nnmod(tmp_scalar
, scalar
, group
->order
, ctx
)) {
2136 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2139 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
2141 num_bytes
= BN_bn2bin(scalar
, tmp
);
2142 flip_endian(g_secret
, tmp
, num_bytes
);
2143 /* do the multiplication with generator precomputation */
2144 batch_mul(x_out
, y_out
, z_out
,
2145 (const felem_bytearray(*))secrets
, num_points
,
2147 mixed
, (const smallfelem(*)[17][3])pre_comp
, g_pre_comp
);
2149 /* do the multiplication without generator precomputation */
2150 batch_mul(x_out
, y_out
, z_out
,
2151 (const felem_bytearray(*))secrets
, num_points
,
2152 NULL
, mixed
, (const smallfelem(*)[17][3])pre_comp
, NULL
);
2153 /* reduce the output to its unique minimal representation */
2154 felem_contract(x_in
, x_out
);
2155 felem_contract(y_in
, y_out
);
2156 felem_contract(z_in
, z_out
);
2157 if ((!smallfelem_to_BN(x
, x_in
)) || (!smallfelem_to_BN(y
, y_in
)) ||
2158 (!smallfelem_to_BN(z
, z_in
))) {
2159 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2162 ret
= EC_POINT_set_Jprojective_coordinates_GFp(group
, r
, x
, y
, z
, ctx
);
2166 EC_POINT_free(generator
);
2167 BN_CTX_free(new_ctx
);
2168 OPENSSL_free(secrets
);
2169 OPENSSL_free(pre_comp
);
2170 OPENSSL_free(tmp_smallfelems
);
2174 int ec_GFp_nistp256_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
2177 NISTP256_PRE_COMP
*pre
= NULL
;
2179 BN_CTX
*new_ctx
= NULL
;
2181 EC_POINT
*generator
= NULL
;
2182 smallfelem tmp_smallfelems
[32];
2183 felem x_tmp
, y_tmp
, z_tmp
;
2185 /* throw away old precomputation */
2186 EC_pre_comp_free(group
);
2188 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
2191 if (((x
= BN_CTX_get(ctx
)) == NULL
) || ((y
= BN_CTX_get(ctx
)) == NULL
))
2193 /* get the generator */
2194 if (group
->generator
== NULL
)
2196 generator
= EC_POINT_new(group
);
2197 if (generator
== NULL
)
2199 BN_bin2bn(nistp256_curve_params
[3], sizeof(felem_bytearray
), x
);
2200 BN_bin2bn(nistp256_curve_params
[4], sizeof(felem_bytearray
), y
);
2201 if (!EC_POINT_set_affine_coordinates_GFp(group
, generator
, x
, y
, ctx
))
2203 if ((pre
= nistp256_pre_comp_new()) == NULL
)
2206 * if the generator is the standard one, use built-in precomputation
2208 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
)) {
2209 memcpy(pre
->g_pre_comp
, gmul
, sizeof(pre
->g_pre_comp
));
2213 if ((!BN_to_felem(x_tmp
, group
->generator
->X
)) ||
2214 (!BN_to_felem(y_tmp
, group
->generator
->Y
)) ||
2215 (!BN_to_felem(z_tmp
, group
->generator
->Z
)))
2217 felem_shrink(pre
->g_pre_comp
[0][1][0], x_tmp
);
2218 felem_shrink(pre
->g_pre_comp
[0][1][1], y_tmp
);
2219 felem_shrink(pre
->g_pre_comp
[0][1][2], z_tmp
);
2221 * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
2222 * 2^160*G, 2^224*G for the second one
2224 for (i
= 1; i
<= 8; i
<<= 1) {
2225 point_double_small(pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2226 pre
->g_pre_comp
[1][i
][2], pre
->g_pre_comp
[0][i
][0],
2227 pre
->g_pre_comp
[0][i
][1],
2228 pre
->g_pre_comp
[0][i
][2]);
2229 for (j
= 0; j
< 31; ++j
) {
2230 point_double_small(pre
->g_pre_comp
[1][i
][0],
2231 pre
->g_pre_comp
[1][i
][1],
2232 pre
->g_pre_comp
[1][i
][2],
2233 pre
->g_pre_comp
[1][i
][0],
2234 pre
->g_pre_comp
[1][i
][1],
2235 pre
->g_pre_comp
[1][i
][2]);
2239 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2240 pre
->g_pre_comp
[0][2 * i
][1],
2241 pre
->g_pre_comp
[0][2 * i
][2],
2242 pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2243 pre
->g_pre_comp
[1][i
][2]);
2244 for (j
= 0; j
< 31; ++j
) {
2245 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2246 pre
->g_pre_comp
[0][2 * i
][1],
2247 pre
->g_pre_comp
[0][2 * i
][2],
2248 pre
->g_pre_comp
[0][2 * i
][0],
2249 pre
->g_pre_comp
[0][2 * i
][1],
2250 pre
->g_pre_comp
[0][2 * i
][2]);
2253 for (i
= 0; i
< 2; i
++) {
2254 /* g_pre_comp[i][0] is the point at infinity */
2255 memset(pre
->g_pre_comp
[i
][0], 0, sizeof(pre
->g_pre_comp
[i
][0]));
2256 /* the remaining multiples */
2257 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2258 point_add_small(pre
->g_pre_comp
[i
][6][0], pre
->g_pre_comp
[i
][6][1],
2259 pre
->g_pre_comp
[i
][6][2], pre
->g_pre_comp
[i
][4][0],
2260 pre
->g_pre_comp
[i
][4][1], pre
->g_pre_comp
[i
][4][2],
2261 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2262 pre
->g_pre_comp
[i
][2][2]);
2263 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2264 point_add_small(pre
->g_pre_comp
[i
][10][0], pre
->g_pre_comp
[i
][10][1],
2265 pre
->g_pre_comp
[i
][10][2], pre
->g_pre_comp
[i
][8][0],
2266 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2267 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2268 pre
->g_pre_comp
[i
][2][2]);
2269 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2270 point_add_small(pre
->g_pre_comp
[i
][12][0], pre
->g_pre_comp
[i
][12][1],
2271 pre
->g_pre_comp
[i
][12][2], pre
->g_pre_comp
[i
][8][0],
2272 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2273 pre
->g_pre_comp
[i
][4][0], pre
->g_pre_comp
[i
][4][1],
2274 pre
->g_pre_comp
[i
][4][2]);
2276 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
2278 point_add_small(pre
->g_pre_comp
[i
][14][0], pre
->g_pre_comp
[i
][14][1],
2279 pre
->g_pre_comp
[i
][14][2], pre
->g_pre_comp
[i
][12][0],
2280 pre
->g_pre_comp
[i
][12][1], pre
->g_pre_comp
[i
][12][2],
2281 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2282 pre
->g_pre_comp
[i
][2][2]);
2283 for (j
= 1; j
< 8; ++j
) {
2284 /* odd multiples: add G resp. 2^32*G */
2285 point_add_small(pre
->g_pre_comp
[i
][2 * j
+ 1][0],
2286 pre
->g_pre_comp
[i
][2 * j
+ 1][1],
2287 pre
->g_pre_comp
[i
][2 * j
+ 1][2],
2288 pre
->g_pre_comp
[i
][2 * j
][0],
2289 pre
->g_pre_comp
[i
][2 * j
][1],
2290 pre
->g_pre_comp
[i
][2 * j
][2],
2291 pre
->g_pre_comp
[i
][1][0],
2292 pre
->g_pre_comp
[i
][1][1],
2293 pre
->g_pre_comp
[i
][1][2]);
2296 make_points_affine(31, &(pre
->g_pre_comp
[0][1]), tmp_smallfelems
);
2298 SETPRECOMP(group
, nistp256
, pre
);
2304 EC_POINT_free(generator
);
2305 BN_CTX_free(new_ctx
);
2306 EC_nistp256_pre_comp_free(pre
);
2310 int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP
*group
)
2312 return HAVEPRECOMP(group
, nistp256
);
2315 static void *dummy
= &dummy
;