2 * Copyright 2011-2016 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (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(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
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 "Need GCC 3.1 or later to define type uint128_t"
59 * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
60 * can serialise an element of this field into 32 bytes. We call this an
64 typedef u8 felem_bytearray
[32];
67 * These are the parameters of P256, taken from FIPS 186-3, page 86. These
68 * values are big-endian.
70 static const felem_bytearray nistp256_curve_params
[5] = {
71 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
72 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
73 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
74 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
75 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
76 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
77 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
78 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */
79 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
80 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
81 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
82 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
83 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
84 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
85 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
86 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
87 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
88 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
89 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
90 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
94 * The representation of field elements.
95 * ------------------------------------
97 * We represent field elements with either four 128-bit values, eight 128-bit
98 * values, or four 64-bit values. The field element represented is:
99 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
101 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
103 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
104 * apart, but are 128-bits wide, the most significant bits of each limb overlap
105 * with the least significant bits of the next.
107 * A field element with four limbs is an 'felem'. One with eight limbs is a
110 * A field element with four, 64-bit values is called a 'smallfelem'. Small
111 * values are used as intermediate values before multiplication.
116 typedef uint128_t limb
;
117 typedef limb felem
[NLIMBS
];
118 typedef limb longfelem
[NLIMBS
* 2];
119 typedef u64 smallfelem
[NLIMBS
];
121 /* This is the value of the prime as four 64-bit words, little-endian. */
122 static const u64 kPrime
[4] =
123 { 0xfffffffffffffffful
, 0xffffffff, 0, 0xffffffff00000001ul
};
124 static const u64 bottom63bits
= 0x7ffffffffffffffful
;
127 * bin32_to_felem takes a little-endian byte array and converts it into felem
128 * form. This assumes that the CPU is little-endian.
130 static void bin32_to_felem(felem out
, const u8 in
[32])
132 out
[0] = *((u64
*)&in
[0]);
133 out
[1] = *((u64
*)&in
[8]);
134 out
[2] = *((u64
*)&in
[16]);
135 out
[3] = *((u64
*)&in
[24]);
139 * smallfelem_to_bin32 takes a smallfelem and serialises into a little
140 * endian, 32 byte array. This assumes that the CPU is little-endian.
142 static void smallfelem_to_bin32(u8 out
[32], const smallfelem in
)
144 *((u64
*)&out
[0]) = in
[0];
145 *((u64
*)&out
[8]) = in
[1];
146 *((u64
*)&out
[16]) = in
[2];
147 *((u64
*)&out
[24]) = in
[3];
150 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
151 static void flip_endian(u8
*out
, const u8
*in
, unsigned len
)
154 for (i
= 0; i
< len
; ++i
)
155 out
[i
] = in
[len
- 1 - i
];
158 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
159 static int BN_to_felem(felem out
, const BIGNUM
*bn
)
161 felem_bytearray b_in
;
162 felem_bytearray b_out
;
165 /* BN_bn2bin eats leading zeroes */
166 memset(b_out
, 0, sizeof(b_out
));
167 num_bytes
= BN_num_bytes(bn
);
168 if (num_bytes
> sizeof b_out
) {
169 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
172 if (BN_is_negative(bn
)) {
173 ECerr(EC_F_BN_TO_FELEM
, EC_R_BIGNUM_OUT_OF_RANGE
);
176 num_bytes
= BN_bn2bin(bn
, b_in
);
177 flip_endian(b_out
, b_in
, num_bytes
);
178 bin32_to_felem(out
, b_out
);
182 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
183 static BIGNUM
*smallfelem_to_BN(BIGNUM
*out
, const smallfelem in
)
185 felem_bytearray b_in
, b_out
;
186 smallfelem_to_bin32(b_in
, in
);
187 flip_endian(b_out
, b_in
, sizeof b_out
);
188 return BN_bin2bn(b_out
, sizeof b_out
, out
);
196 static void smallfelem_one(smallfelem out
)
204 static void smallfelem_assign(smallfelem out
, const smallfelem in
)
212 static void felem_assign(felem out
, const felem in
)
220 /* felem_sum sets out = out + in. */
221 static void felem_sum(felem out
, const felem in
)
229 /* felem_small_sum sets out = out + in. */
230 static void felem_small_sum(felem out
, const smallfelem in
)
238 /* felem_scalar sets out = out * scalar */
239 static void felem_scalar(felem out
, const u64 scalar
)
247 /* longfelem_scalar sets out = out * scalar */
248 static void longfelem_scalar(longfelem out
, const u64 scalar
)
260 # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
261 # define two105 (((limb)1) << 105)
262 # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
264 /* zero105 is 0 mod p */
265 static const felem zero105
=
266 { two105m41m9
, two105
, two105m41p9
, two105m41p9
};
269 * smallfelem_neg sets |out| to |-small|
271 * out[i] < out[i] + 2^105
273 static void smallfelem_neg(felem out
, const smallfelem small
)
275 /* In order to prevent underflow, we subtract from 0 mod p. */
276 out
[0] = zero105
[0] - small
[0];
277 out
[1] = zero105
[1] - small
[1];
278 out
[2] = zero105
[2] - small
[2];
279 out
[3] = zero105
[3] - small
[3];
283 * felem_diff subtracts |in| from |out|
287 * out[i] < out[i] + 2^105
289 static void felem_diff(felem out
, const felem in
)
292 * In order to prevent underflow, we add 0 mod p before subtracting.
294 out
[0] += zero105
[0];
295 out
[1] += zero105
[1];
296 out
[2] += zero105
[2];
297 out
[3] += zero105
[3];
305 # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
306 # define two107 (((limb)1) << 107)
307 # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
309 /* zero107 is 0 mod p */
310 static const felem zero107
=
311 { two107m43m11
, two107
, two107m43p11
, two107m43p11
};
314 * An alternative felem_diff for larger inputs |in|
315 * felem_diff_zero107 subtracts |in| from |out|
319 * out[i] < out[i] + 2^107
321 static void felem_diff_zero107(felem out
, const felem in
)
324 * In order to prevent underflow, we add 0 mod p before subtracting.
326 out
[0] += zero107
[0];
327 out
[1] += zero107
[1];
328 out
[2] += zero107
[2];
329 out
[3] += zero107
[3];
338 * longfelem_diff subtracts |in| from |out|
342 * out[i] < out[i] + 2^70 + 2^40
344 static void longfelem_diff(longfelem out
, const longfelem in
)
346 static const limb two70m8p6
=
347 (((limb
) 1) << 70) - (((limb
) 1) << 8) + (((limb
) 1) << 6);
348 static const limb two70p40
= (((limb
) 1) << 70) + (((limb
) 1) << 40);
349 static const limb two70
= (((limb
) 1) << 70);
350 static const limb two70m40m38p6
=
351 (((limb
) 1) << 70) - (((limb
) 1) << 40) - (((limb
) 1) << 38) +
353 static const limb two70m6
= (((limb
) 1) << 70) - (((limb
) 1) << 6);
355 /* add 0 mod p to avoid underflow */
359 out
[3] += two70m40m38p6
;
365 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
376 # define two64m0 (((limb)1) << 64) - 1
377 # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
378 # define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
379 # define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
381 /* zero110 is 0 mod p */
382 static const felem zero110
= { two64m0
, two110p32m0
, two64m46
, two64m32
};
385 * felem_shrink converts an felem into a smallfelem. The result isn't quite
386 * minimal as the value may be greater than p.
393 static void felem_shrink(smallfelem out
, const felem in
)
398 static const u64 kPrime3Test
= 0x7fffffff00000001ul
; /* 2^63 - 2^32 + 1 */
401 tmp
[3] = zero110
[3] + in
[3] + ((u64
)(in
[2] >> 64));
404 tmp
[2] = zero110
[2] + (u64
)in
[2];
405 tmp
[0] = zero110
[0] + in
[0];
406 tmp
[1] = zero110
[1] + in
[1];
407 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
410 * We perform two partial reductions where we eliminate the high-word of
411 * tmp[3]. We don't update the other words till the end.
413 a
= tmp
[3] >> 64; /* a < 2^46 */
414 tmp
[3] = (u64
)tmp
[3];
416 tmp
[3] += ((limb
) a
) << 32;
420 a
= tmp
[3] >> 64; /* a < 2^15 */
421 b
+= a
; /* b < 2^46 + 2^15 < 2^47 */
422 tmp
[3] = (u64
)tmp
[3];
424 tmp
[3] += ((limb
) a
) << 32;
425 /* tmp[3] < 2^64 + 2^47 */
428 * This adjusts the other two words to complete the two partial
432 tmp
[1] -= (((limb
) b
) << 32);
435 * In order to make space in tmp[3] for the carry from 2 -> 3, we
436 * conditionally subtract kPrime if tmp[3] is large enough.
439 /* As tmp[3] < 2^65, high is either 1 or 0 */
444 * all ones if the high word of tmp[3] is 1
445 * all zeros if the high word of tmp[3] if 0 */
450 * all ones if the MSB of low is 1
451 * all zeros if the MSB of low if 0 */
454 /* if low was greater than kPrime3Test then the MSB is zero */
459 * all ones if low was > kPrime3Test
460 * all zeros if low was <= kPrime3Test */
461 mask
= (mask
& low
) | high
;
462 tmp
[0] -= mask
& kPrime
[0];
463 tmp
[1] -= mask
& kPrime
[1];
464 /* kPrime[2] is zero, so omitted */
465 tmp
[3] -= mask
& kPrime
[3];
466 /* tmp[3] < 2**64 - 2**32 + 1 */
468 tmp
[1] += ((u64
)(tmp
[0] >> 64));
469 tmp
[0] = (u64
)tmp
[0];
470 tmp
[2] += ((u64
)(tmp
[1] >> 64));
471 tmp
[1] = (u64
)tmp
[1];
472 tmp
[3] += ((u64
)(tmp
[2] >> 64));
473 tmp
[2] = (u64
)tmp
[2];
482 /* smallfelem_expand converts a smallfelem to an felem */
483 static void smallfelem_expand(felem out
, const smallfelem in
)
492 * smallfelem_square sets |out| = |small|^2
496 * out[i] < 7 * 2^64 < 2^67
498 static void smallfelem_square(longfelem out
, const smallfelem small
)
503 a
= ((uint128_t
) small
[0]) * small
[0];
509 a
= ((uint128_t
) small
[0]) * small
[1];
516 a
= ((uint128_t
) small
[0]) * small
[2];
523 a
= ((uint128_t
) small
[0]) * small
[3];
529 a
= ((uint128_t
) small
[1]) * small
[2];
536 a
= ((uint128_t
) small
[1]) * small
[1];
542 a
= ((uint128_t
) small
[1]) * small
[3];
549 a
= ((uint128_t
) small
[2]) * small
[3];
557 a
= ((uint128_t
) small
[2]) * small
[2];
563 a
= ((uint128_t
) small
[3]) * small
[3];
571 * felem_square sets |out| = |in|^2
575 * out[i] < 7 * 2^64 < 2^67
577 static void felem_square(longfelem out
, const felem in
)
580 felem_shrink(small
, in
);
581 smallfelem_square(out
, small
);
585 * smallfelem_mul sets |out| = |small1| * |small2|
590 * out[i] < 7 * 2^64 < 2^67
592 static void smallfelem_mul(longfelem out
, const smallfelem small1
,
593 const smallfelem small2
)
598 a
= ((uint128_t
) small1
[0]) * small2
[0];
604 a
= ((uint128_t
) small1
[0]) * small2
[1];
610 a
= ((uint128_t
) small1
[1]) * small2
[0];
616 a
= ((uint128_t
) small1
[0]) * small2
[2];
622 a
= ((uint128_t
) small1
[1]) * small2
[1];
628 a
= ((uint128_t
) small1
[2]) * small2
[0];
634 a
= ((uint128_t
) small1
[0]) * small2
[3];
640 a
= ((uint128_t
) small1
[1]) * small2
[2];
646 a
= ((uint128_t
) small1
[2]) * small2
[1];
652 a
= ((uint128_t
) small1
[3]) * small2
[0];
658 a
= ((uint128_t
) small1
[1]) * small2
[3];
664 a
= ((uint128_t
) small1
[2]) * small2
[2];
670 a
= ((uint128_t
) small1
[3]) * small2
[1];
676 a
= ((uint128_t
) small1
[2]) * small2
[3];
682 a
= ((uint128_t
) small1
[3]) * small2
[2];
688 a
= ((uint128_t
) small1
[3]) * small2
[3];
696 * felem_mul sets |out| = |in1| * |in2|
701 * out[i] < 7 * 2^64 < 2^67
703 static void felem_mul(longfelem out
, const felem in1
, const felem in2
)
705 smallfelem small1
, small2
;
706 felem_shrink(small1
, in1
);
707 felem_shrink(small2
, in2
);
708 smallfelem_mul(out
, small1
, small2
);
712 * felem_small_mul sets |out| = |small1| * |in2|
717 * out[i] < 7 * 2^64 < 2^67
719 static void felem_small_mul(longfelem out
, const smallfelem small1
,
723 felem_shrink(small2
, in2
);
724 smallfelem_mul(out
, small1
, small2
);
727 # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
728 # define two100 (((limb)1) << 100)
729 # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
730 /* zero100 is 0 mod p */
731 static const felem zero100
=
732 { two100m36m4
, two100
, two100m36p4
, two100m36p4
};
735 * Internal function for the different flavours of felem_reduce.
736 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
738 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
739 * out[1] >= in[7] + 2^32*in[4]
740 * out[2] >= in[5] + 2^32*in[5]
741 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
743 * out[0] <= out[0] + in[4] + 2^32*in[5]
744 * out[1] <= out[1] + in[5] + 2^33*in[6]
745 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
746 * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
748 static void felem_reduce_(felem out
, const longfelem in
)
751 /* combine common terms from below */
752 c
= in
[4] + (in
[5] << 32);
760 /* the remaining terms */
761 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
762 out
[1] -= (in
[4] << 32);
763 out
[3] += (in
[4] << 32);
765 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
766 out
[2] -= (in
[5] << 32);
768 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
770 out
[0] -= (in
[6] << 32);
771 out
[1] += (in
[6] << 33);
772 out
[2] += (in
[6] * 2);
773 out
[3] -= (in
[6] << 32);
775 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
777 out
[0] -= (in
[7] << 32);
778 out
[2] += (in
[7] << 33);
779 out
[3] += (in
[7] * 3);
783 * felem_reduce converts a longfelem into an felem.
784 * To be called directly after felem_square or felem_mul.
786 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
787 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
791 static void felem_reduce(felem out
, const longfelem in
)
793 out
[0] = zero100
[0] + in
[0];
794 out
[1] = zero100
[1] + in
[1];
795 out
[2] = zero100
[2] + in
[2];
796 out
[3] = zero100
[3] + in
[3];
798 felem_reduce_(out
, in
);
801 * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
802 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
803 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
804 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
806 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
807 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
808 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
809 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
814 * felem_reduce_zero105 converts a larger longfelem into an felem.
820 static void felem_reduce_zero105(felem out
, const longfelem in
)
822 out
[0] = zero105
[0] + in
[0];
823 out
[1] = zero105
[1] + in
[1];
824 out
[2] = zero105
[2] + in
[2];
825 out
[3] = zero105
[3] + in
[3];
827 felem_reduce_(out
, in
);
830 * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
831 * out[1] > 2^105 - 2^71 - 2^103 > 0
832 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
833 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
835 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
836 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
837 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
838 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
843 * subtract_u64 sets *result = *result - v and *carry to one if the
844 * subtraction underflowed.
846 static void subtract_u64(u64
*result
, u64
*carry
, u64 v
)
848 uint128_t r
= *result
;
850 *carry
= (r
>> 64) & 1;
855 * felem_contract converts |in| to its unique, minimal representation. On
856 * entry: in[i] < 2^109
858 static void felem_contract(smallfelem out
, const felem in
)
861 u64 all_equal_so_far
= 0, result
= 0, carry
;
863 felem_shrink(out
, in
);
864 /* small is minimal except that the value might be > p */
868 * We are doing a constant time test if out >= kPrime. We need to compare
869 * each u64, from most-significant to least significant. For each one, if
870 * all words so far have been equal (m is all ones) then a non-equal
871 * result is the answer. Otherwise we continue.
873 for (i
= 3; i
< 4; i
--) {
875 uint128_t a
= ((uint128_t
) kPrime
[i
]) - out
[i
];
877 * if out[i] > kPrime[i] then a will underflow and the high 64-bits
880 result
|= all_equal_so_far
& ((u64
)(a
>> 64));
883 * if kPrime[i] == out[i] then |equal| will be all zeros and the
884 * decrement will make it all ones.
886 equal
= kPrime
[i
] ^ out
[i
];
888 equal
&= equal
<< 32;
889 equal
&= equal
<< 16;
894 equal
= ((s64
) equal
) >> 63;
896 all_equal_so_far
&= equal
;
900 * if all_equal_so_far is still all ones then the two values are equal
901 * and so out >= kPrime is true.
903 result
|= all_equal_so_far
;
905 /* if out >= kPrime then we subtract kPrime. */
906 subtract_u64(&out
[0], &carry
, result
& kPrime
[0]);
907 subtract_u64(&out
[1], &carry
, carry
);
908 subtract_u64(&out
[2], &carry
, carry
);
909 subtract_u64(&out
[3], &carry
, carry
);
911 subtract_u64(&out
[1], &carry
, result
& kPrime
[1]);
912 subtract_u64(&out
[2], &carry
, carry
);
913 subtract_u64(&out
[3], &carry
, carry
);
915 subtract_u64(&out
[2], &carry
, result
& kPrime
[2]);
916 subtract_u64(&out
[3], &carry
, carry
);
918 subtract_u64(&out
[3], &carry
, result
& kPrime
[3]);
921 static void smallfelem_square_contract(smallfelem out
, const smallfelem in
)
926 smallfelem_square(longtmp
, in
);
927 felem_reduce(tmp
, longtmp
);
928 felem_contract(out
, tmp
);
931 static void smallfelem_mul_contract(smallfelem out
, const smallfelem in1
,
932 const smallfelem in2
)
937 smallfelem_mul(longtmp
, in1
, in2
);
938 felem_reduce(tmp
, longtmp
);
939 felem_contract(out
, tmp
);
943 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
948 static limb
smallfelem_is_zero(const smallfelem small
)
953 u64 is_zero
= small
[0] | small
[1] | small
[2] | small
[3];
955 is_zero
&= is_zero
<< 32;
956 is_zero
&= is_zero
<< 16;
957 is_zero
&= is_zero
<< 8;
958 is_zero
&= is_zero
<< 4;
959 is_zero
&= is_zero
<< 2;
960 is_zero
&= is_zero
<< 1;
961 is_zero
= ((s64
) is_zero
) >> 63;
963 is_p
= (small
[0] ^ kPrime
[0]) |
964 (small
[1] ^ kPrime
[1]) |
965 (small
[2] ^ kPrime
[2]) | (small
[3] ^ kPrime
[3]);
973 is_p
= ((s64
) is_p
) >> 63;
978 result
|= ((limb
) is_zero
) << 64;
982 static int smallfelem_is_zero_int(const smallfelem small
)
984 return (int)(smallfelem_is_zero(small
) & ((limb
) 1));
988 * felem_inv calculates |out| = |in|^{-1}
990 * Based on Fermat's Little Theorem:
992 * a^{p-1} = 1 (mod p)
993 * a^{p-2} = a^{-1} (mod p)
995 static void felem_inv(felem out
, const felem in
)
998 /* each e_I will hold |in|^{2^I - 1} */
999 felem e2
, e4
, e8
, e16
, e32
, e64
;
1003 felem_square(tmp
, in
);
1004 felem_reduce(ftmp
, tmp
); /* 2^1 */
1005 felem_mul(tmp
, in
, ftmp
);
1006 felem_reduce(ftmp
, tmp
); /* 2^2 - 2^0 */
1007 felem_assign(e2
, ftmp
);
1008 felem_square(tmp
, ftmp
);
1009 felem_reduce(ftmp
, tmp
); /* 2^3 - 2^1 */
1010 felem_square(tmp
, ftmp
);
1011 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^2 */
1012 felem_mul(tmp
, ftmp
, e2
);
1013 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^0 */
1014 felem_assign(e4
, ftmp
);
1015 felem_square(tmp
, ftmp
);
1016 felem_reduce(ftmp
, tmp
); /* 2^5 - 2^1 */
1017 felem_square(tmp
, ftmp
);
1018 felem_reduce(ftmp
, tmp
); /* 2^6 - 2^2 */
1019 felem_square(tmp
, ftmp
);
1020 felem_reduce(ftmp
, tmp
); /* 2^7 - 2^3 */
1021 felem_square(tmp
, ftmp
);
1022 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^4 */
1023 felem_mul(tmp
, ftmp
, e4
);
1024 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^0 */
1025 felem_assign(e8
, ftmp
);
1026 for (i
= 0; i
< 8; i
++) {
1027 felem_square(tmp
, ftmp
);
1028 felem_reduce(ftmp
, tmp
);
1030 felem_mul(tmp
, ftmp
, e8
);
1031 felem_reduce(ftmp
, tmp
); /* 2^16 - 2^0 */
1032 felem_assign(e16
, ftmp
);
1033 for (i
= 0; i
< 16; i
++) {
1034 felem_square(tmp
, ftmp
);
1035 felem_reduce(ftmp
, tmp
);
1037 felem_mul(tmp
, ftmp
, e16
);
1038 felem_reduce(ftmp
, tmp
); /* 2^32 - 2^0 */
1039 felem_assign(e32
, ftmp
);
1040 for (i
= 0; i
< 32; i
++) {
1041 felem_square(tmp
, ftmp
);
1042 felem_reduce(ftmp
, tmp
);
1044 felem_assign(e64
, ftmp
);
1045 felem_mul(tmp
, ftmp
, in
);
1046 felem_reduce(ftmp
, tmp
); /* 2^64 - 2^32 + 2^0 */
1047 for (i
= 0; i
< 192; i
++) {
1048 felem_square(tmp
, ftmp
);
1049 felem_reduce(ftmp
, tmp
);
1050 } /* 2^256 - 2^224 + 2^192 */
1052 felem_mul(tmp
, e64
, e32
);
1053 felem_reduce(ftmp2
, tmp
); /* 2^64 - 2^0 */
1054 for (i
= 0; i
< 16; i
++) {
1055 felem_square(tmp
, ftmp2
);
1056 felem_reduce(ftmp2
, tmp
);
1058 felem_mul(tmp
, ftmp2
, e16
);
1059 felem_reduce(ftmp2
, tmp
); /* 2^80 - 2^0 */
1060 for (i
= 0; i
< 8; i
++) {
1061 felem_square(tmp
, ftmp2
);
1062 felem_reduce(ftmp2
, tmp
);
1064 felem_mul(tmp
, ftmp2
, e8
);
1065 felem_reduce(ftmp2
, tmp
); /* 2^88 - 2^0 */
1066 for (i
= 0; i
< 4; i
++) {
1067 felem_square(tmp
, ftmp2
);
1068 felem_reduce(ftmp2
, tmp
);
1070 felem_mul(tmp
, ftmp2
, e4
);
1071 felem_reduce(ftmp2
, tmp
); /* 2^92 - 2^0 */
1072 felem_square(tmp
, ftmp2
);
1073 felem_reduce(ftmp2
, tmp
); /* 2^93 - 2^1 */
1074 felem_square(tmp
, ftmp2
);
1075 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^2 */
1076 felem_mul(tmp
, ftmp2
, e2
);
1077 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^0 */
1078 felem_square(tmp
, ftmp2
);
1079 felem_reduce(ftmp2
, tmp
); /* 2^95 - 2^1 */
1080 felem_square(tmp
, ftmp2
);
1081 felem_reduce(ftmp2
, tmp
); /* 2^96 - 2^2 */
1082 felem_mul(tmp
, ftmp2
, in
);
1083 felem_reduce(ftmp2
, tmp
); /* 2^96 - 3 */
1085 felem_mul(tmp
, ftmp2
, ftmp
);
1086 felem_reduce(out
, tmp
); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1089 static void smallfelem_inv_contract(smallfelem out
, const smallfelem in
)
1093 smallfelem_expand(tmp
, in
);
1094 felem_inv(tmp
, tmp
);
1095 felem_contract(out
, tmp
);
1102 * Building on top of the field operations we have the operations on the
1103 * elliptic curve group itself. Points on the curve are represented in Jacobian
1108 * point_double calculates 2*(x_in, y_in, z_in)
1110 * The method is taken from:
1111 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1113 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1114 * while x_out == y_in is not (maybe this works, but it's not tested).
1117 point_double(felem x_out
, felem y_out
, felem z_out
,
1118 const felem x_in
, const felem y_in
, const felem z_in
)
1120 longfelem tmp
, tmp2
;
1121 felem delta
, gamma
, beta
, alpha
, ftmp
, ftmp2
;
1122 smallfelem small1
, small2
;
1124 felem_assign(ftmp
, x_in
);
1125 /* ftmp[i] < 2^106 */
1126 felem_assign(ftmp2
, x_in
);
1127 /* ftmp2[i] < 2^106 */
1130 felem_square(tmp
, z_in
);
1131 felem_reduce(delta
, tmp
);
1132 /* delta[i] < 2^101 */
1135 felem_square(tmp
, y_in
);
1136 felem_reduce(gamma
, tmp
);
1137 /* gamma[i] < 2^101 */
1138 felem_shrink(small1
, gamma
);
1140 /* beta = x*gamma */
1141 felem_small_mul(tmp
, small1
, x_in
);
1142 felem_reduce(beta
, tmp
);
1143 /* beta[i] < 2^101 */
1145 /* alpha = 3*(x-delta)*(x+delta) */
1146 felem_diff(ftmp
, delta
);
1147 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1148 felem_sum(ftmp2
, delta
);
1149 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1150 felem_scalar(ftmp2
, 3);
1151 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1152 felem_mul(tmp
, ftmp
, ftmp2
);
1153 felem_reduce(alpha
, tmp
);
1154 /* alpha[i] < 2^101 */
1155 felem_shrink(small2
, alpha
);
1157 /* x' = alpha^2 - 8*beta */
1158 smallfelem_square(tmp
, small2
);
1159 felem_reduce(x_out
, tmp
);
1160 felem_assign(ftmp
, beta
);
1161 felem_scalar(ftmp
, 8);
1162 /* ftmp[i] < 8 * 2^101 = 2^104 */
1163 felem_diff(x_out
, ftmp
);
1164 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1166 /* z' = (y + z)^2 - gamma - delta */
1167 felem_sum(delta
, gamma
);
1168 /* delta[i] < 2^101 + 2^101 = 2^102 */
1169 felem_assign(ftmp
, y_in
);
1170 felem_sum(ftmp
, z_in
);
1171 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1172 felem_square(tmp
, ftmp
);
1173 felem_reduce(z_out
, tmp
);
1174 felem_diff(z_out
, delta
);
1175 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1177 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1178 felem_scalar(beta
, 4);
1179 /* beta[i] < 4 * 2^101 = 2^103 */
1180 felem_diff_zero107(beta
, x_out
);
1181 /* beta[i] < 2^107 + 2^103 < 2^108 */
1182 felem_small_mul(tmp
, small2
, beta
);
1183 /* tmp[i] < 7 * 2^64 < 2^67 */
1184 smallfelem_square(tmp2
, small1
);
1185 /* tmp2[i] < 7 * 2^64 */
1186 longfelem_scalar(tmp2
, 8);
1187 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1188 longfelem_diff(tmp
, tmp2
);
1189 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1190 felem_reduce_zero105(y_out
, tmp
);
1191 /* y_out[i] < 2^106 */
1195 * point_double_small is the same as point_double, except that it operates on
1199 point_double_small(smallfelem x_out
, smallfelem y_out
, smallfelem z_out
,
1200 const smallfelem x_in
, const smallfelem y_in
,
1201 const smallfelem z_in
)
1203 felem felem_x_out
, felem_y_out
, felem_z_out
;
1204 felem felem_x_in
, felem_y_in
, felem_z_in
;
1206 smallfelem_expand(felem_x_in
, x_in
);
1207 smallfelem_expand(felem_y_in
, y_in
);
1208 smallfelem_expand(felem_z_in
, z_in
);
1209 point_double(felem_x_out
, felem_y_out
, felem_z_out
,
1210 felem_x_in
, felem_y_in
, felem_z_in
);
1211 felem_shrink(x_out
, felem_x_out
);
1212 felem_shrink(y_out
, felem_y_out
);
1213 felem_shrink(z_out
, felem_z_out
);
1216 /* copy_conditional copies in to out iff mask is all ones. */
1217 static void copy_conditional(felem out
, const felem in
, limb mask
)
1220 for (i
= 0; i
< NLIMBS
; ++i
) {
1221 const limb tmp
= mask
& (in
[i
] ^ out
[i
]);
1226 /* copy_small_conditional copies in to out iff mask is all ones. */
1227 static void copy_small_conditional(felem out
, const smallfelem in
, limb mask
)
1230 const u64 mask64
= mask
;
1231 for (i
= 0; i
< NLIMBS
; ++i
) {
1232 out
[i
] = ((limb
) (in
[i
] & mask64
)) | (out
[i
] & ~mask
);
1237 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1239 * The method is taken from:
1240 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1241 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1243 * This function includes a branch for checking whether the two input points
1244 * are equal, (while not equal to the point at infinity). This case never
1245 * happens during single point multiplication, so there is no timing leak for
1246 * ECDH or ECDSA signing.
1248 static void point_add(felem x3
, felem y3
, felem z3
,
1249 const felem x1
, const felem y1
, const felem z1
,
1250 const int mixed
, const smallfelem x2
,
1251 const smallfelem y2
, const smallfelem z2
)
1253 felem ftmp
, ftmp2
, ftmp3
, ftmp4
, ftmp5
, ftmp6
, x_out
, y_out
, z_out
;
1254 longfelem tmp
, tmp2
;
1255 smallfelem small1
, small2
, small3
, small4
, small5
;
1256 limb x_equal
, y_equal
, z1_is_zero
, z2_is_zero
;
1258 felem_shrink(small3
, z1
);
1260 z1_is_zero
= smallfelem_is_zero(small3
);
1261 z2_is_zero
= smallfelem_is_zero(z2
);
1263 /* ftmp = z1z1 = z1**2 */
1264 smallfelem_square(tmp
, small3
);
1265 felem_reduce(ftmp
, tmp
);
1266 /* ftmp[i] < 2^101 */
1267 felem_shrink(small1
, ftmp
);
1270 /* ftmp2 = z2z2 = z2**2 */
1271 smallfelem_square(tmp
, z2
);
1272 felem_reduce(ftmp2
, tmp
);
1273 /* ftmp2[i] < 2^101 */
1274 felem_shrink(small2
, ftmp2
);
1276 felem_shrink(small5
, x1
);
1278 /* u1 = ftmp3 = x1*z2z2 */
1279 smallfelem_mul(tmp
, small5
, small2
);
1280 felem_reduce(ftmp3
, tmp
);
1281 /* ftmp3[i] < 2^101 */
1283 /* ftmp5 = z1 + z2 */
1284 felem_assign(ftmp5
, z1
);
1285 felem_small_sum(ftmp5
, z2
);
1286 /* ftmp5[i] < 2^107 */
1288 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1289 felem_square(tmp
, ftmp5
);
1290 felem_reduce(ftmp5
, tmp
);
1291 /* ftmp2 = z2z2 + z1z1 */
1292 felem_sum(ftmp2
, ftmp
);
1293 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1294 felem_diff(ftmp5
, ftmp2
);
1295 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1297 /* ftmp2 = z2 * z2z2 */
1298 smallfelem_mul(tmp
, small2
, z2
);
1299 felem_reduce(ftmp2
, tmp
);
1301 /* s1 = ftmp2 = y1 * z2**3 */
1302 felem_mul(tmp
, y1
, ftmp2
);
1303 felem_reduce(ftmp6
, tmp
);
1304 /* ftmp6[i] < 2^101 */
1307 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1310 /* u1 = ftmp3 = x1*z2z2 */
1311 felem_assign(ftmp3
, x1
);
1312 /* ftmp3[i] < 2^106 */
1315 felem_assign(ftmp5
, z1
);
1316 felem_scalar(ftmp5
, 2);
1317 /* ftmp5[i] < 2*2^106 = 2^107 */
1319 /* s1 = ftmp2 = y1 * z2**3 */
1320 felem_assign(ftmp6
, y1
);
1321 /* ftmp6[i] < 2^106 */
1325 smallfelem_mul(tmp
, x2
, small1
);
1326 felem_reduce(ftmp4
, tmp
);
1328 /* h = ftmp4 = u2 - u1 */
1329 felem_diff_zero107(ftmp4
, ftmp3
);
1330 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1331 felem_shrink(small4
, ftmp4
);
1333 x_equal
= smallfelem_is_zero(small4
);
1335 /* z_out = ftmp5 * h */
1336 felem_small_mul(tmp
, small4
, ftmp5
);
1337 felem_reduce(z_out
, tmp
);
1338 /* z_out[i] < 2^101 */
1340 /* ftmp = z1 * z1z1 */
1341 smallfelem_mul(tmp
, small1
, small3
);
1342 felem_reduce(ftmp
, tmp
);
1344 /* s2 = tmp = y2 * z1**3 */
1345 felem_small_mul(tmp
, y2
, ftmp
);
1346 felem_reduce(ftmp5
, tmp
);
1348 /* r = ftmp5 = (s2 - s1)*2 */
1349 felem_diff_zero107(ftmp5
, ftmp6
);
1350 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1351 felem_scalar(ftmp5
, 2);
1352 /* ftmp5[i] < 2^109 */
1353 felem_shrink(small1
, ftmp5
);
1354 y_equal
= smallfelem_is_zero(small1
);
1356 if (x_equal
&& y_equal
&& !z1_is_zero
&& !z2_is_zero
) {
1357 point_double(x3
, y3
, z3
, x1
, y1
, z1
);
1361 /* I = ftmp = (2h)**2 */
1362 felem_assign(ftmp
, ftmp4
);
1363 felem_scalar(ftmp
, 2);
1364 /* ftmp[i] < 2*2^108 = 2^109 */
1365 felem_square(tmp
, ftmp
);
1366 felem_reduce(ftmp
, tmp
);
1368 /* J = ftmp2 = h * I */
1369 felem_mul(tmp
, ftmp4
, ftmp
);
1370 felem_reduce(ftmp2
, tmp
);
1372 /* V = ftmp4 = U1 * I */
1373 felem_mul(tmp
, ftmp3
, ftmp
);
1374 felem_reduce(ftmp4
, tmp
);
1376 /* x_out = r**2 - J - 2V */
1377 smallfelem_square(tmp
, small1
);
1378 felem_reduce(x_out
, tmp
);
1379 felem_assign(ftmp3
, ftmp4
);
1380 felem_scalar(ftmp4
, 2);
1381 felem_sum(ftmp4
, ftmp2
);
1382 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1383 felem_diff(x_out
, ftmp4
);
1384 /* x_out[i] < 2^105 + 2^101 */
1386 /* y_out = r(V-x_out) - 2 * s1 * J */
1387 felem_diff_zero107(ftmp3
, x_out
);
1388 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1389 felem_small_mul(tmp
, small1
, ftmp3
);
1390 felem_mul(tmp2
, ftmp6
, ftmp2
);
1391 longfelem_scalar(tmp2
, 2);
1392 /* tmp2[i] < 2*2^67 = 2^68 */
1393 longfelem_diff(tmp
, tmp2
);
1394 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1395 felem_reduce_zero105(y_out
, tmp
);
1396 /* y_out[i] < 2^106 */
1398 copy_small_conditional(x_out
, x2
, z1_is_zero
);
1399 copy_conditional(x_out
, x1
, z2_is_zero
);
1400 copy_small_conditional(y_out
, y2
, z1_is_zero
);
1401 copy_conditional(y_out
, y1
, z2_is_zero
);
1402 copy_small_conditional(z_out
, z2
, z1_is_zero
);
1403 copy_conditional(z_out
, z1
, z2_is_zero
);
1404 felem_assign(x3
, x_out
);
1405 felem_assign(y3
, y_out
);
1406 felem_assign(z3
, z_out
);
1410 * point_add_small is the same as point_add, except that it operates on
1413 static void point_add_small(smallfelem x3
, smallfelem y3
, smallfelem z3
,
1414 smallfelem x1
, smallfelem y1
, smallfelem z1
,
1415 smallfelem x2
, smallfelem y2
, smallfelem z2
)
1417 felem felem_x3
, felem_y3
, felem_z3
;
1418 felem felem_x1
, felem_y1
, felem_z1
;
1419 smallfelem_expand(felem_x1
, x1
);
1420 smallfelem_expand(felem_y1
, y1
);
1421 smallfelem_expand(felem_z1
, z1
);
1422 point_add(felem_x3
, felem_y3
, felem_z3
, felem_x1
, felem_y1
, felem_z1
, 0,
1424 felem_shrink(x3
, felem_x3
);
1425 felem_shrink(y3
, felem_y3
);
1426 felem_shrink(z3
, felem_z3
);
1430 * Base point pre computation
1431 * --------------------------
1433 * Two different sorts of precomputed tables are used in the following code.
1434 * Each contain various points on the curve, where each point is three field
1435 * elements (x, y, z).
1437 * For the base point table, z is usually 1 (0 for the point at infinity).
1438 * This table has 2 * 16 elements, starting with the following:
1439 * index | bits | point
1440 * ------+---------+------------------------------
1443 * 2 | 0 0 1 0 | 2^64G
1444 * 3 | 0 0 1 1 | (2^64 + 1)G
1445 * 4 | 0 1 0 0 | 2^128G
1446 * 5 | 0 1 0 1 | (2^128 + 1)G
1447 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1448 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1449 * 8 | 1 0 0 0 | 2^192G
1450 * 9 | 1 0 0 1 | (2^192 + 1)G
1451 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1452 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1453 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1454 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1455 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1456 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1457 * followed by a copy of this with each element multiplied by 2^32.
1459 * The reason for this is so that we can clock bits into four different
1460 * locations when doing simple scalar multiplies against the base point,
1461 * and then another four locations using the second 16 elements.
1463 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1465 /* gmul is the table of precomputed base points */
1466 static const smallfelem gmul
[2][16][3] = {
1470 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1471 0x6b17d1f2e12c4247},
1472 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1473 0x4fe342e2fe1a7f9b},
1475 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1476 0x0fa822bc2811aaa5},
1477 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1478 0xbff44ae8f5dba80d},
1480 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1481 0x300a4bbc89d6726f},
1482 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1483 0x72aac7e0d09b4644},
1485 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1486 0x447d739beedb5e67},
1487 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1488 0x2d4825ab834131ee},
1490 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1491 0xef9519328a9c72ff},
1492 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1493 0x611e9fc37dbb2c9b},
1495 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1496 0x550663797b51f5d8},
1497 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1498 0x157164848aecb851},
1500 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1501 0xeb5d7745b21141ea},
1502 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1503 0xeafd72ebdbecc17b},
1505 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1506 0xa6d39677a7849276},
1507 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1508 0x674f84749b0b8816},
1510 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1511 0x4e769e7672c9ddad},
1512 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1513 0x42b99082de830663},
1515 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1516 0x78878ef61c6ce04d},
1517 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1518 0xb6cb3f5d7b72c321},
1520 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1521 0x0c88bc4d716b1287},
1522 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1523 0xdd5ddea3f3901dc6},
1525 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1526 0x68f344af6b317466},
1527 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1528 0x31b9c405f8540a20},
1530 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1531 0x4052bf4b6f461db9},
1532 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1533 0xfecf4d5190b0fc61},
1535 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1536 0x1eddbae2c802e41a},
1537 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1538 0x43104d86560ebcfc},
1540 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1541 0xb48e26b484f7a21c},
1542 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1543 0xfac015404d4d3dab},
1548 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1549 0x7fe36b40af22af89},
1550 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1551 0xe697d45825b63624},
1553 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1554 0x4a5b506612a677a6},
1555 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1556 0xeb13461ceac089f1},
1558 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1559 0x0781b8291c6a220a},
1560 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1561 0x690cde8df0151593},
1563 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1564 0x8a535f566ec73617},
1565 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1566 0x0455c08468b08bd7},
1568 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1569 0x06bada7ab77f8276},
1570 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1571 0x5b476dfd0e6cb18a},
1573 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1574 0x3e29864e8a2ec908},
1575 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1576 0x239b90ea3dc31e7e},
1578 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1579 0x820f4dd949f72ff7},
1580 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1581 0x140406ec783a05ec},
1583 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1584 0x68f6b8542783dfee},
1585 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1586 0xcbe1feba92e40ce6},
1588 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1589 0xd0b2f94d2f420109},
1590 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1591 0x971459828b0719e5},
1593 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1594 0x961610004a866aba},
1595 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1596 0x7acb9fadcee75e44},
1598 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1599 0x24eb9acca333bf5b},
1600 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1601 0x69f891c5acd079cc},
1603 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1604 0xe51f547c5972a107},
1605 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1606 0x1c309a2b25bb1387},
1608 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1609 0x20b87b8aa2c4e503},
1610 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1611 0xf5c6fa49919776be},
1613 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1614 0x1ed7d1b9332010b9},
1615 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1616 0x3a2b03f03217257a},
1618 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1619 0x15fee545c78dd9f6},
1620 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1621 0x4ab5b6b2b8753f81},
1626 * select_point selects the |idx|th point from a precomputation table and
1629 static void select_point(const u64 idx
, unsigned int size
,
1630 const smallfelem pre_comp
[16][3], smallfelem out
[3])
1633 u64
*outlimbs
= &out
[0][0];
1635 memset(out
, 0, sizeof(*out
) * 3);
1637 for (i
= 0; i
< size
; i
++) {
1638 const u64
*inlimbs
= (u64
*)&pre_comp
[i
][0][0];
1645 for (j
= 0; j
< NLIMBS
* 3; j
++)
1646 outlimbs
[j
] |= inlimbs
[j
] & mask
;
1650 /* get_bit returns the |i|th bit in |in| */
1651 static char get_bit(const felem_bytearray in
, int i
)
1653 if ((i
< 0) || (i
>= 256))
1655 return (in
[i
>> 3] >> (i
& 7)) & 1;
1659 * Interleaved point multiplication using precomputed point multiples: The
1660 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1661 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1662 * generator, using certain (large) precomputed multiples in g_pre_comp.
1663 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1665 static void batch_mul(felem x_out
, felem y_out
, felem z_out
,
1666 const felem_bytearray scalars
[],
1667 const unsigned num_points
, const u8
*g_scalar
,
1668 const int mixed
, const smallfelem pre_comp
[][17][3],
1669 const smallfelem g_pre_comp
[2][16][3])
1672 unsigned num
, gen_mul
= (g_scalar
!= NULL
);
1678 /* set nq to the point at infinity */
1679 memset(nq
, 0, sizeof(nq
));
1682 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1683 * of the generator (two in each of the last 32 rounds) and additions of
1684 * other points multiples (every 5th round).
1686 skip
= 1; /* save two point operations in the first
1688 for (i
= (num_points
? 255 : 31); i
>= 0; --i
) {
1691 point_double(nq
[0], nq
[1], nq
[2], nq
[0], nq
[1], nq
[2]);
1693 /* add multiples of the generator */
1694 if (gen_mul
&& (i
<= 31)) {
1695 /* first, look 32 bits upwards */
1696 bits
= get_bit(g_scalar
, i
+ 224) << 3;
1697 bits
|= get_bit(g_scalar
, i
+ 160) << 2;
1698 bits
|= get_bit(g_scalar
, i
+ 96) << 1;
1699 bits
|= get_bit(g_scalar
, i
+ 32);
1700 /* select the point to add, in constant time */
1701 select_point(bits
, 16, g_pre_comp
[1], tmp
);
1704 /* Arg 1 below is for "mixed" */
1705 point_add(nq
[0], nq
[1], nq
[2],
1706 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1708 smallfelem_expand(nq
[0], tmp
[0]);
1709 smallfelem_expand(nq
[1], tmp
[1]);
1710 smallfelem_expand(nq
[2], tmp
[2]);
1714 /* second, look at the current position */
1715 bits
= get_bit(g_scalar
, i
+ 192) << 3;
1716 bits
|= get_bit(g_scalar
, i
+ 128) << 2;
1717 bits
|= get_bit(g_scalar
, i
+ 64) << 1;
1718 bits
|= get_bit(g_scalar
, i
);
1719 /* select the point to add, in constant time */
1720 select_point(bits
, 16, g_pre_comp
[0], tmp
);
1721 /* Arg 1 below is for "mixed" */
1722 point_add(nq
[0], nq
[1], nq
[2],
1723 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1726 /* do other additions every 5 doublings */
1727 if (num_points
&& (i
% 5 == 0)) {
1728 /* loop over all scalars */
1729 for (num
= 0; num
< num_points
; ++num
) {
1730 bits
= get_bit(scalars
[num
], i
+ 4) << 5;
1731 bits
|= get_bit(scalars
[num
], i
+ 3) << 4;
1732 bits
|= get_bit(scalars
[num
], i
+ 2) << 3;
1733 bits
|= get_bit(scalars
[num
], i
+ 1) << 2;
1734 bits
|= get_bit(scalars
[num
], i
) << 1;
1735 bits
|= get_bit(scalars
[num
], i
- 1);
1736 ec_GFp_nistp_recode_scalar_bits(&sign
, &digit
, bits
);
1739 * select the point to add or subtract, in constant time
1741 select_point(digit
, 17, pre_comp
[num
], tmp
);
1742 smallfelem_neg(ftmp
, tmp
[1]); /* (X, -Y, Z) is the negative
1744 copy_small_conditional(ftmp
, tmp
[1], (((limb
) sign
) - 1));
1745 felem_contract(tmp
[1], ftmp
);
1748 point_add(nq
[0], nq
[1], nq
[2],
1749 nq
[0], nq
[1], nq
[2],
1750 mixed
, tmp
[0], tmp
[1], tmp
[2]);
1752 smallfelem_expand(nq
[0], tmp
[0]);
1753 smallfelem_expand(nq
[1], tmp
[1]);
1754 smallfelem_expand(nq
[2], tmp
[2]);
1760 felem_assign(x_out
, nq
[0]);
1761 felem_assign(y_out
, nq
[1]);
1762 felem_assign(z_out
, nq
[2]);
1765 /* Precomputation for the group generator. */
1766 struct nistp256_pre_comp_st
{
1767 smallfelem g_pre_comp
[2][16][3];
1768 CRYPTO_REF_COUNT references
;
1769 CRYPTO_RWLOCK
*lock
;
1772 const EC_METHOD
*EC_GFp_nistp256_method(void)
1774 static const EC_METHOD ret
= {
1775 EC_FLAGS_DEFAULT_OCT
,
1776 NID_X9_62_prime_field
,
1777 ec_GFp_nistp256_group_init
,
1778 ec_GFp_simple_group_finish
,
1779 ec_GFp_simple_group_clear_finish
,
1780 ec_GFp_nist_group_copy
,
1781 ec_GFp_nistp256_group_set_curve
,
1782 ec_GFp_simple_group_get_curve
,
1783 ec_GFp_simple_group_get_degree
,
1784 ec_group_simple_order_bits
,
1785 ec_GFp_simple_group_check_discriminant
,
1786 ec_GFp_simple_point_init
,
1787 ec_GFp_simple_point_finish
,
1788 ec_GFp_simple_point_clear_finish
,
1789 ec_GFp_simple_point_copy
,
1790 ec_GFp_simple_point_set_to_infinity
,
1791 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
1792 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
1793 ec_GFp_simple_point_set_affine_coordinates
,
1794 ec_GFp_nistp256_point_get_affine_coordinates
,
1795 0 /* point_set_compressed_coordinates */ ,
1800 ec_GFp_simple_invert
,
1801 ec_GFp_simple_is_at_infinity
,
1802 ec_GFp_simple_is_on_curve
,
1804 ec_GFp_simple_make_affine
,
1805 ec_GFp_simple_points_make_affine
,
1806 ec_GFp_nistp256_points_mul
,
1807 ec_GFp_nistp256_precompute_mult
,
1808 ec_GFp_nistp256_have_precompute_mult
,
1809 ec_GFp_nist_field_mul
,
1810 ec_GFp_nist_field_sqr
,
1812 0 /* field_encode */ ,
1813 0 /* field_decode */ ,
1814 0, /* field_set_to_one */
1815 ec_key_simple_priv2oct
,
1816 ec_key_simple_oct2priv
,
1817 0, /* set private */
1818 ec_key_simple_generate_key
,
1819 ec_key_simple_check_key
,
1820 ec_key_simple_generate_public_key
,
1823 ecdh_simple_compute_key
1829 /******************************************************************************/
1831 * FUNCTIONS TO MANAGE PRECOMPUTATION
1834 static NISTP256_PRE_COMP
*nistp256_pre_comp_new()
1836 NISTP256_PRE_COMP
*ret
= OPENSSL_zalloc(sizeof(*ret
));
1839 ECerr(EC_F_NISTP256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1843 ret
->references
= 1;
1845 ret
->lock
= CRYPTO_THREAD_lock_new();
1846 if (ret
->lock
== NULL
) {
1847 ECerr(EC_F_NISTP256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1854 NISTP256_PRE_COMP
*EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP
*p
)
1858 CRYPTO_UP_REF(&p
->references
, &i
, p
->lock
);
1862 void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP
*pre
)
1869 CRYPTO_DOWN_REF(&pre
->references
, &i
, pre
->lock
);
1870 REF_PRINT_COUNT("EC_nistp256", x
);
1873 REF_ASSERT_ISNT(i
< 0);
1875 CRYPTO_THREAD_lock_free(pre
->lock
);
1879 /******************************************************************************/
1881 * OPENSSL EC_METHOD FUNCTIONS
1884 int ec_GFp_nistp256_group_init(EC_GROUP
*group
)
1887 ret
= ec_GFp_simple_group_init(group
);
1888 group
->a_is_minus3
= 1;
1892 int ec_GFp_nistp256_group_set_curve(EC_GROUP
*group
, const BIGNUM
*p
,
1893 const BIGNUM
*a
, const BIGNUM
*b
,
1897 BN_CTX
*new_ctx
= NULL
;
1898 BIGNUM
*curve_p
, *curve_a
, *curve_b
;
1901 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
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
);
1921 BN_CTX_free(new_ctx
);
1926 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1929 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP
*group
,
1930 const EC_POINT
*point
,
1931 BIGNUM
*x
, BIGNUM
*y
,
1934 felem z1
, z2
, x_in
, y_in
;
1935 smallfelem x_out
, y_out
;
1938 if (EC_POINT_is_at_infinity(group
, point
)) {
1939 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1940 EC_R_POINT_AT_INFINITY
);
1943 if ((!BN_to_felem(x_in
, point
->X
)) || (!BN_to_felem(y_in
, point
->Y
)) ||
1944 (!BN_to_felem(z1
, point
->Z
)))
1947 felem_square(tmp
, z2
);
1948 felem_reduce(z1
, tmp
);
1949 felem_mul(tmp
, x_in
, z1
);
1950 felem_reduce(x_in
, tmp
);
1951 felem_contract(x_out
, x_in
);
1953 if (!smallfelem_to_BN(x
, x_out
)) {
1954 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1959 felem_mul(tmp
, z1
, z2
);
1960 felem_reduce(z1
, tmp
);
1961 felem_mul(tmp
, y_in
, z1
);
1962 felem_reduce(y_in
, tmp
);
1963 felem_contract(y_out
, y_in
);
1965 if (!smallfelem_to_BN(y
, y_out
)) {
1966 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES
,
1974 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
1975 static void make_points_affine(size_t num
, smallfelem points
[][3],
1976 smallfelem tmp_smallfelems
[])
1979 * Runs in constant time, unless an input is the point at infinity (which
1980 * normally shouldn't happen).
1982 ec_GFp_nistp_points_make_affine_internal(num
,
1986 (void (*)(void *))smallfelem_one
,
1987 (int (*)(const void *))
1988 smallfelem_is_zero_int
,
1989 (void (*)(void *, const void *))
1991 (void (*)(void *, const void *))
1992 smallfelem_square_contract
,
1994 (void *, const void *,
1996 smallfelem_mul_contract
,
1997 (void (*)(void *, const void *))
1998 smallfelem_inv_contract
,
1999 /* nothing to contract */
2000 (void (*)(void *, const void *))
2005 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
2006 * values Result is stored in r (r can equal one of the inputs).
2008 int ec_GFp_nistp256_points_mul(const EC_GROUP
*group
, EC_POINT
*r
,
2009 const BIGNUM
*scalar
, size_t num
,
2010 const EC_POINT
*points
[],
2011 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
2016 BN_CTX
*new_ctx
= NULL
;
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
;
2022 felem_bytearray tmp
;
2023 unsigned i
, num_bytes
;
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 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
2038 x
= BN_CTX_get(ctx
);
2039 y
= BN_CTX_get(ctx
);
2040 z
= BN_CTX_get(ctx
);
2041 tmp_scalar
= BN_CTX_get(ctx
);
2042 if (tmp_scalar
== NULL
)
2045 if (scalar
!= NULL
) {
2046 pre
= group
->pre_comp
.nistp256
;
2048 /* we have precomputation, try to use it */
2049 g_pre_comp
= (const smallfelem(*)[16][3])pre
->g_pre_comp
;
2051 /* try to use the standard precomputation */
2052 g_pre_comp
= &gmul
[0];
2053 generator
= EC_POINT_new(group
);
2054 if (generator
== NULL
)
2056 /* get the generator from precomputation */
2057 if (!smallfelem_to_BN(x
, g_pre_comp
[0][1][0]) ||
2058 !smallfelem_to_BN(y
, g_pre_comp
[0][1][1]) ||
2059 !smallfelem_to_BN(z
, g_pre_comp
[0][1][2])) {
2060 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2063 if (!EC_POINT_set_Jprojective_coordinates_GFp(group
,
2067 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
2068 /* precomputation matches generator */
2072 * we don't have valid precomputation: treat the generator as a
2077 if (num_points
> 0) {
2078 if (num_points
>= 3) {
2080 * unless we precompute multiples for just one or two points,
2081 * converting those into affine form is time well spent
2085 secrets
= OPENSSL_malloc(sizeof(*secrets
) * num_points
);
2086 pre_comp
= OPENSSL_malloc(sizeof(*pre_comp
) * num_points
);
2089 OPENSSL_malloc(sizeof(*tmp_smallfelems
) * (num_points
* 17 + 1));
2090 if ((secrets
== NULL
) || (pre_comp
== NULL
)
2091 || (mixed
&& (tmp_smallfelems
== NULL
))) {
2092 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
2097 * we treat NULL scalars as 0, and NULL points as points at infinity,
2098 * i.e., they contribute nothing to the linear combination
2100 memset(secrets
, 0, sizeof(*secrets
) * num_points
);
2101 memset(pre_comp
, 0, sizeof(*pre_comp
) * num_points
);
2102 for (i
= 0; i
< num_points
; ++i
) {
2105 * we didn't have a valid precomputation, so we pick the
2109 p
= EC_GROUP_get0_generator(group
);
2112 /* the i^th point */
2115 p_scalar
= scalars
[i
];
2117 if ((p_scalar
!= NULL
) && (p
!= NULL
)) {
2118 /* reduce scalar to 0 <= scalar < 2^256 */
2119 if ((BN_num_bits(p_scalar
) > 256)
2120 || (BN_is_negative(p_scalar
))) {
2122 * this is an unusual input, and we don't guarantee
2125 if (!BN_nnmod(tmp_scalar
, p_scalar
, group
->order
, ctx
)) {
2126 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2129 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
2131 num_bytes
= BN_bn2bin(p_scalar
, tmp
);
2132 flip_endian(secrets
[i
], tmp
, num_bytes
);
2133 /* precompute multiples */
2134 if ((!BN_to_felem(x_out
, p
->X
)) ||
2135 (!BN_to_felem(y_out
, p
->Y
)) ||
2136 (!BN_to_felem(z_out
, p
->Z
)))
2138 felem_shrink(pre_comp
[i
][1][0], x_out
);
2139 felem_shrink(pre_comp
[i
][1][1], y_out
);
2140 felem_shrink(pre_comp
[i
][1][2], z_out
);
2141 for (j
= 2; j
<= 16; ++j
) {
2143 point_add_small(pre_comp
[i
][j
][0], pre_comp
[i
][j
][1],
2144 pre_comp
[i
][j
][2], pre_comp
[i
][1][0],
2145 pre_comp
[i
][1][1], pre_comp
[i
][1][2],
2146 pre_comp
[i
][j
- 1][0],
2147 pre_comp
[i
][j
- 1][1],
2148 pre_comp
[i
][j
- 1][2]);
2150 point_double_small(pre_comp
[i
][j
][0],
2153 pre_comp
[i
][j
/ 2][0],
2154 pre_comp
[i
][j
/ 2][1],
2155 pre_comp
[i
][j
/ 2][2]);
2161 make_points_affine(num_points
* 17, pre_comp
[0], tmp_smallfelems
);
2164 /* the scalar for the generator */
2165 if ((scalar
!= NULL
) && (have_pre_comp
)) {
2166 memset(g_secret
, 0, sizeof(g_secret
));
2167 /* reduce scalar to 0 <= scalar < 2^256 */
2168 if ((BN_num_bits(scalar
) > 256) || (BN_is_negative(scalar
))) {
2170 * this is an unusual input, and we don't guarantee
2173 if (!BN_nnmod(tmp_scalar
, scalar
, group
->order
, ctx
)) {
2174 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2177 num_bytes
= BN_bn2bin(tmp_scalar
, tmp
);
2179 num_bytes
= BN_bn2bin(scalar
, tmp
);
2180 flip_endian(g_secret
, tmp
, num_bytes
);
2181 /* do the multiplication with generator precomputation */
2182 batch_mul(x_out
, y_out
, z_out
,
2183 (const felem_bytearray(*))secrets
, num_points
,
2185 mixed
, (const smallfelem(*)[17][3])pre_comp
, g_pre_comp
);
2187 /* do the multiplication without generator precomputation */
2188 batch_mul(x_out
, y_out
, z_out
,
2189 (const felem_bytearray(*))secrets
, num_points
,
2190 NULL
, mixed
, (const smallfelem(*)[17][3])pre_comp
, NULL
);
2191 /* reduce the output to its unique minimal representation */
2192 felem_contract(x_in
, x_out
);
2193 felem_contract(y_in
, y_out
);
2194 felem_contract(z_in
, z_out
);
2195 if ((!smallfelem_to_BN(x
, x_in
)) || (!smallfelem_to_BN(y
, y_in
)) ||
2196 (!smallfelem_to_BN(z
, z_in
))) {
2197 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL
, ERR_R_BN_LIB
);
2200 ret
= EC_POINT_set_Jprojective_coordinates_GFp(group
, r
, x
, y
, z
, ctx
);
2204 EC_POINT_free(generator
);
2205 BN_CTX_free(new_ctx
);
2206 OPENSSL_free(secrets
);
2207 OPENSSL_free(pre_comp
);
2208 OPENSSL_free(tmp_smallfelems
);
2212 int ec_GFp_nistp256_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
2215 NISTP256_PRE_COMP
*pre
= NULL
;
2217 BN_CTX
*new_ctx
= NULL
;
2219 EC_POINT
*generator
= NULL
;
2220 smallfelem tmp_smallfelems
[32];
2221 felem x_tmp
, y_tmp
, z_tmp
;
2223 /* throw away old precomputation */
2224 EC_pre_comp_free(group
);
2226 if ((ctx
= new_ctx
= BN_CTX_new()) == NULL
)
2229 x
= BN_CTX_get(ctx
);
2230 y
= BN_CTX_get(ctx
);
2233 /* get the generator */
2234 if (group
->generator
== NULL
)
2236 generator
= EC_POINT_new(group
);
2237 if (generator
== NULL
)
2239 BN_bin2bn(nistp256_curve_params
[3], sizeof(felem_bytearray
), x
);
2240 BN_bin2bn(nistp256_curve_params
[4], sizeof(felem_bytearray
), y
);
2241 if (!EC_POINT_set_affine_coordinates_GFp(group
, generator
, x
, y
, ctx
))
2243 if ((pre
= nistp256_pre_comp_new()) == NULL
)
2246 * if the generator is the standard one, use built-in precomputation
2248 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
)) {
2249 memcpy(pre
->g_pre_comp
, gmul
, sizeof(pre
->g_pre_comp
));
2252 if ((!BN_to_felem(x_tmp
, group
->generator
->X
)) ||
2253 (!BN_to_felem(y_tmp
, group
->generator
->Y
)) ||
2254 (!BN_to_felem(z_tmp
, group
->generator
->Z
)))
2256 felem_shrink(pre
->g_pre_comp
[0][1][0], x_tmp
);
2257 felem_shrink(pre
->g_pre_comp
[0][1][1], y_tmp
);
2258 felem_shrink(pre
->g_pre_comp
[0][1][2], z_tmp
);
2260 * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
2261 * 2^160*G, 2^224*G for the second one
2263 for (i
= 1; i
<= 8; i
<<= 1) {
2264 point_double_small(pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2265 pre
->g_pre_comp
[1][i
][2], pre
->g_pre_comp
[0][i
][0],
2266 pre
->g_pre_comp
[0][i
][1],
2267 pre
->g_pre_comp
[0][i
][2]);
2268 for (j
= 0; j
< 31; ++j
) {
2269 point_double_small(pre
->g_pre_comp
[1][i
][0],
2270 pre
->g_pre_comp
[1][i
][1],
2271 pre
->g_pre_comp
[1][i
][2],
2272 pre
->g_pre_comp
[1][i
][0],
2273 pre
->g_pre_comp
[1][i
][1],
2274 pre
->g_pre_comp
[1][i
][2]);
2278 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2279 pre
->g_pre_comp
[0][2 * i
][1],
2280 pre
->g_pre_comp
[0][2 * i
][2],
2281 pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2282 pre
->g_pre_comp
[1][i
][2]);
2283 for (j
= 0; j
< 31; ++j
) {
2284 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2285 pre
->g_pre_comp
[0][2 * i
][1],
2286 pre
->g_pre_comp
[0][2 * i
][2],
2287 pre
->g_pre_comp
[0][2 * i
][0],
2288 pre
->g_pre_comp
[0][2 * i
][1],
2289 pre
->g_pre_comp
[0][2 * i
][2]);
2292 for (i
= 0; i
< 2; i
++) {
2293 /* g_pre_comp[i][0] is the point at infinity */
2294 memset(pre
->g_pre_comp
[i
][0], 0, sizeof(pre
->g_pre_comp
[i
][0]));
2295 /* the remaining multiples */
2296 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2297 point_add_small(pre
->g_pre_comp
[i
][6][0], pre
->g_pre_comp
[i
][6][1],
2298 pre
->g_pre_comp
[i
][6][2], pre
->g_pre_comp
[i
][4][0],
2299 pre
->g_pre_comp
[i
][4][1], pre
->g_pre_comp
[i
][4][2],
2300 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2301 pre
->g_pre_comp
[i
][2][2]);
2302 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2303 point_add_small(pre
->g_pre_comp
[i
][10][0], pre
->g_pre_comp
[i
][10][1],
2304 pre
->g_pre_comp
[i
][10][2], pre
->g_pre_comp
[i
][8][0],
2305 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2306 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2307 pre
->g_pre_comp
[i
][2][2]);
2308 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2309 point_add_small(pre
->g_pre_comp
[i
][12][0], pre
->g_pre_comp
[i
][12][1],
2310 pre
->g_pre_comp
[i
][12][2], pre
->g_pre_comp
[i
][8][0],
2311 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2312 pre
->g_pre_comp
[i
][4][0], pre
->g_pre_comp
[i
][4][1],
2313 pre
->g_pre_comp
[i
][4][2]);
2315 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
2317 point_add_small(pre
->g_pre_comp
[i
][14][0], pre
->g_pre_comp
[i
][14][1],
2318 pre
->g_pre_comp
[i
][14][2], pre
->g_pre_comp
[i
][12][0],
2319 pre
->g_pre_comp
[i
][12][1], pre
->g_pre_comp
[i
][12][2],
2320 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2321 pre
->g_pre_comp
[i
][2][2]);
2322 for (j
= 1; j
< 8; ++j
) {
2323 /* odd multiples: add G resp. 2^32*G */
2324 point_add_small(pre
->g_pre_comp
[i
][2 * j
+ 1][0],
2325 pre
->g_pre_comp
[i
][2 * j
+ 1][1],
2326 pre
->g_pre_comp
[i
][2 * j
+ 1][2],
2327 pre
->g_pre_comp
[i
][2 * j
][0],
2328 pre
->g_pre_comp
[i
][2 * j
][1],
2329 pre
->g_pre_comp
[i
][2 * j
][2],
2330 pre
->g_pre_comp
[i
][1][0],
2331 pre
->g_pre_comp
[i
][1][1],
2332 pre
->g_pre_comp
[i
][1][2]);
2335 make_points_affine(31, &(pre
->g_pre_comp
[0][1]), tmp_smallfelems
);
2338 SETPRECOMP(group
, nistp256
, pre
);
2344 EC_POINT_free(generator
);
2345 BN_CTX_free(new_ctx
);
2346 EC_nistp256_pre_comp_free(pre
);
2350 int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP
*group
)
2352 return HAVEPRECOMP(group
, nistp256
);