2 * Copyright 2011-2020 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * ECDSA low level APIs are deprecated for public use, but still ok for
30 #include "internal/deprecated.h"
33 * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
35 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
36 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
37 * work which got its smarts from Daniel J. Bernstein's work on the same.
40 #include <openssl/opensslconf.h>
44 #include <openssl/err.h>
47 #if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
48 /* even with gcc, the typedef won't work for 32-bit platforms */
49 typedef __uint128_t uint128_t
; /* nonstandard; implemented by gcc on 64-bit
51 typedef __int128_t int128_t
;
53 # error "Your compiler doesn't appear to support 128-bit integer types"
61 * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
62 * can serialize an element of this field into 32 bytes. We call this an
66 typedef u8 felem_bytearray
[32];
69 * These are the parameters of P256, taken from FIPS 186-3, page 86. These
70 * values are big-endian.
72 static const felem_bytearray nistp256_curve_params
[5] = {
73 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
74 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
75 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
76 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
77 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
78 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
79 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
80 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc},
81 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, /* b */
82 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
83 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
84 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
85 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
86 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
87 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
88 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
89 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
90 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
91 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
92 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
96 * The representation of field elements.
97 * ------------------------------------
99 * We represent field elements with either four 128-bit values, eight 128-bit
100 * values, or four 64-bit values. The field element represented is:
101 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
103 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
105 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
106 * apart, but are 128-bits wide, the most significant bits of each limb overlap
107 * with the least significant bits of the next.
109 * A field element with four limbs is an 'felem'. One with eight limbs is a
112 * A field element with four, 64-bit values is called a 'smallfelem'. Small
113 * values are used as intermediate values before multiplication.
118 typedef uint128_t limb
;
119 typedef limb felem
[NLIMBS
];
120 typedef limb longfelem
[NLIMBS
* 2];
121 typedef u64 smallfelem
[NLIMBS
];
123 /* This is the value of the prime as four 64-bit words, little-endian. */
124 static const u64 kPrime
[4] =
125 { 0xfffffffffffffffful
, 0xffffffff, 0, 0xffffffff00000001ul
};
126 static const u64 bottom63bits
= 0x7ffffffffffffffful
;
129 * bin32_to_felem takes a little-endian byte array and converts it into felem
130 * form. This assumes that the CPU is little-endian.
132 static void bin32_to_felem(felem out
, const u8 in
[32])
134 out
[0] = *((u64
*)&in
[0]);
135 out
[1] = *((u64
*)&in
[8]);
136 out
[2] = *((u64
*)&in
[16]);
137 out
[3] = *((u64
*)&in
[24]);
141 * smallfelem_to_bin32 takes a smallfelem and serializes into a little
142 * endian, 32 byte array. This assumes that the CPU is little-endian.
144 static void smallfelem_to_bin32(u8 out
[32], const smallfelem in
)
146 *((u64
*)&out
[0]) = in
[0];
147 *((u64
*)&out
[8]) = in
[1];
148 *((u64
*)&out
[16]) = in
[2];
149 *((u64
*)&out
[24]) = in
[3];
152 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
153 static int BN_to_felem(felem out
, const BIGNUM
*bn
)
155 felem_bytearray b_out
;
158 if (BN_is_negative(bn
)) {
159 ERR_raise(ERR_LIB_EC
, EC_R_BIGNUM_OUT_OF_RANGE
);
162 num_bytes
= BN_bn2lebinpad(bn
, b_out
, sizeof(b_out
));
164 ERR_raise(ERR_LIB_EC
, EC_R_BIGNUM_OUT_OF_RANGE
);
167 bin32_to_felem(out
, b_out
);
171 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
172 static BIGNUM
*smallfelem_to_BN(BIGNUM
*out
, const smallfelem in
)
174 felem_bytearray b_out
;
175 smallfelem_to_bin32(b_out
, in
);
176 return BN_lebin2bn(b_out
, sizeof(b_out
), out
);
184 static void smallfelem_one(smallfelem out
)
192 static void smallfelem_assign(smallfelem out
, const smallfelem in
)
200 static void felem_assign(felem out
, const felem in
)
208 /* felem_sum sets out = out + in. */
209 static void felem_sum(felem out
, const felem in
)
217 /* felem_small_sum sets out = out + in. */
218 static void felem_small_sum(felem out
, const smallfelem in
)
226 /* felem_scalar sets out = out * scalar */
227 static void felem_scalar(felem out
, const u64 scalar
)
235 /* longfelem_scalar sets out = out * scalar */
236 static void longfelem_scalar(longfelem out
, const u64 scalar
)
248 #define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
249 #define two105 (((limb)1) << 105)
250 #define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
252 /* zero105 is 0 mod p */
253 static const felem zero105
=
254 { two105m41m9
, two105
, two105m41p9
, two105m41p9
};
257 * smallfelem_neg sets |out| to |-small|
259 * out[i] < out[i] + 2^105
261 static void smallfelem_neg(felem out
, const smallfelem small
)
263 /* In order to prevent underflow, we subtract from 0 mod p. */
264 out
[0] = zero105
[0] - small
[0];
265 out
[1] = zero105
[1] - small
[1];
266 out
[2] = zero105
[2] - small
[2];
267 out
[3] = zero105
[3] - small
[3];
271 * felem_diff subtracts |in| from |out|
275 * out[i] < out[i] + 2^105
277 static void felem_diff(felem out
, const felem in
)
280 * In order to prevent underflow, we add 0 mod p before subtracting.
282 out
[0] += zero105
[0];
283 out
[1] += zero105
[1];
284 out
[2] += zero105
[2];
285 out
[3] += zero105
[3];
293 #define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
294 #define two107 (((limb)1) << 107)
295 #define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
297 /* zero107 is 0 mod p */
298 static const felem zero107
=
299 { two107m43m11
, two107
, two107m43p11
, two107m43p11
};
302 * An alternative felem_diff for larger inputs |in|
303 * felem_diff_zero107 subtracts |in| from |out|
307 * out[i] < out[i] + 2^107
309 static void felem_diff_zero107(felem out
, const felem in
)
312 * In order to prevent underflow, we add 0 mod p before subtracting.
314 out
[0] += zero107
[0];
315 out
[1] += zero107
[1];
316 out
[2] += zero107
[2];
317 out
[3] += zero107
[3];
326 * longfelem_diff subtracts |in| from |out|
330 * out[i] < out[i] + 2^70 + 2^40
332 static void longfelem_diff(longfelem out
, const longfelem in
)
334 static const limb two70m8p6
=
335 (((limb
) 1) << 70) - (((limb
) 1) << 8) + (((limb
) 1) << 6);
336 static const limb two70p40
= (((limb
) 1) << 70) + (((limb
) 1) << 40);
337 static const limb two70
= (((limb
) 1) << 70);
338 static const limb two70m40m38p6
=
339 (((limb
) 1) << 70) - (((limb
) 1) << 40) - (((limb
) 1) << 38) +
341 static const limb two70m6
= (((limb
) 1) << 70) - (((limb
) 1) << 6);
343 /* add 0 mod p to avoid underflow */
347 out
[3] += two70m40m38p6
;
353 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
364 #define two64m0 (((limb)1) << 64) - 1
365 #define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
366 #define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
367 #define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
369 /* zero110 is 0 mod p */
370 static const felem zero110
= { two64m0
, two110p32m0
, two64m46
, two64m32
};
373 * felem_shrink converts an felem into a smallfelem. The result isn't quite
374 * minimal as the value may be greater than p.
381 static void felem_shrink(smallfelem out
, const felem in
)
386 static const u64 kPrime3Test
= 0x7fffffff00000001ul
; /* 2^63 - 2^32 + 1 */
389 tmp
[3] = zero110
[3] + in
[3] + ((u64
)(in
[2] >> 64));
392 tmp
[2] = zero110
[2] + (u64
)in
[2];
393 tmp
[0] = zero110
[0] + in
[0];
394 tmp
[1] = zero110
[1] + in
[1];
395 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
398 * We perform two partial reductions where we eliminate the high-word of
399 * tmp[3]. We don't update the other words till the end.
401 a
= tmp
[3] >> 64; /* a < 2^46 */
402 tmp
[3] = (u64
)tmp
[3];
404 tmp
[3] += ((limb
) a
) << 32;
408 a
= tmp
[3] >> 64; /* a < 2^15 */
409 b
+= a
; /* b < 2^46 + 2^15 < 2^47 */
410 tmp
[3] = (u64
)tmp
[3];
412 tmp
[3] += ((limb
) a
) << 32;
413 /* tmp[3] < 2^64 + 2^47 */
416 * This adjusts the other two words to complete the two partial
420 tmp
[1] -= (((limb
) b
) << 32);
423 * In order to make space in tmp[3] for the carry from 2 -> 3, we
424 * conditionally subtract kPrime if tmp[3] is large enough.
426 high
= (u64
)(tmp
[3] >> 64);
427 /* As tmp[3] < 2^65, high is either 1 or 0 */
431 * all ones if the high word of tmp[3] is 1
432 * all zeros if the high word of tmp[3] if 0
435 mask
= 0 - (low
>> 63);
438 * all ones if the MSB of low is 1
439 * all zeros if the MSB of low if 0
443 /* if low was greater than kPrime3Test then the MSB is zero */
445 low
= 0 - (low
>> 63);
448 * all ones if low was > kPrime3Test
449 * all zeros if low was <= kPrime3Test
451 mask
= (mask
& low
) | high
;
452 tmp
[0] -= mask
& kPrime
[0];
453 tmp
[1] -= mask
& kPrime
[1];
454 /* kPrime[2] is zero, so omitted */
455 tmp
[3] -= mask
& kPrime
[3];
456 /* tmp[3] < 2**64 - 2**32 + 1 */
458 tmp
[1] += ((u64
)(tmp
[0] >> 64));
459 tmp
[0] = (u64
)tmp
[0];
460 tmp
[2] += ((u64
)(tmp
[1] >> 64));
461 tmp
[1] = (u64
)tmp
[1];
462 tmp
[3] += ((u64
)(tmp
[2] >> 64));
463 tmp
[2] = (u64
)tmp
[2];
472 /* smallfelem_expand converts a smallfelem to an felem */
473 static void smallfelem_expand(felem out
, const smallfelem in
)
482 * smallfelem_square sets |out| = |small|^2
486 * out[i] < 7 * 2^64 < 2^67
488 static void smallfelem_square(longfelem out
, const smallfelem small
)
493 a
= ((uint128_t
) small
[0]) * small
[0];
499 a
= ((uint128_t
) small
[0]) * small
[1];
506 a
= ((uint128_t
) small
[0]) * small
[2];
513 a
= ((uint128_t
) small
[0]) * small
[3];
519 a
= ((uint128_t
) small
[1]) * small
[2];
526 a
= ((uint128_t
) small
[1]) * small
[1];
532 a
= ((uint128_t
) small
[1]) * small
[3];
539 a
= ((uint128_t
) small
[2]) * small
[3];
547 a
= ((uint128_t
) small
[2]) * small
[2];
553 a
= ((uint128_t
) small
[3]) * small
[3];
561 * felem_square sets |out| = |in|^2
565 * out[i] < 7 * 2^64 < 2^67
567 static void felem_square(longfelem out
, const felem in
)
570 felem_shrink(small
, in
);
571 smallfelem_square(out
, small
);
575 * smallfelem_mul sets |out| = |small1| * |small2|
580 * out[i] < 7 * 2^64 < 2^67
582 static void smallfelem_mul(longfelem out
, const smallfelem small1
,
583 const smallfelem small2
)
588 a
= ((uint128_t
) small1
[0]) * small2
[0];
594 a
= ((uint128_t
) small1
[0]) * small2
[1];
600 a
= ((uint128_t
) small1
[1]) * small2
[0];
606 a
= ((uint128_t
) small1
[0]) * small2
[2];
612 a
= ((uint128_t
) small1
[1]) * small2
[1];
618 a
= ((uint128_t
) small1
[2]) * small2
[0];
624 a
= ((uint128_t
) small1
[0]) * small2
[3];
630 a
= ((uint128_t
) small1
[1]) * small2
[2];
636 a
= ((uint128_t
) small1
[2]) * small2
[1];
642 a
= ((uint128_t
) small1
[3]) * small2
[0];
648 a
= ((uint128_t
) small1
[1]) * small2
[3];
654 a
= ((uint128_t
) small1
[2]) * small2
[2];
660 a
= ((uint128_t
) small1
[3]) * small2
[1];
666 a
= ((uint128_t
) small1
[2]) * small2
[3];
672 a
= ((uint128_t
) small1
[3]) * small2
[2];
678 a
= ((uint128_t
) small1
[3]) * small2
[3];
686 * felem_mul sets |out| = |in1| * |in2|
691 * out[i] < 7 * 2^64 < 2^67
693 static void felem_mul(longfelem out
, const felem in1
, const felem in2
)
695 smallfelem small1
, small2
;
696 felem_shrink(small1
, in1
);
697 felem_shrink(small2
, in2
);
698 smallfelem_mul(out
, small1
, small2
);
702 * felem_small_mul sets |out| = |small1| * |in2|
707 * out[i] < 7 * 2^64 < 2^67
709 static void felem_small_mul(longfelem out
, const smallfelem small1
,
713 felem_shrink(small2
, in2
);
714 smallfelem_mul(out
, small1
, small2
);
717 #define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
718 #define two100 (((limb)1) << 100)
719 #define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
720 /* zero100 is 0 mod p */
721 static const felem zero100
=
722 { two100m36m4
, two100
, two100m36p4
, two100m36p4
};
725 * Internal function for the different flavours of felem_reduce.
726 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
728 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
729 * out[1] >= in[7] + 2^32*in[4]
730 * out[2] >= in[5] + 2^32*in[5]
731 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
733 * out[0] <= out[0] + in[4] + 2^32*in[5]
734 * out[1] <= out[1] + in[5] + 2^33*in[6]
735 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
736 * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
738 static void felem_reduce_(felem out
, const longfelem in
)
741 /* combine common terms from below */
742 c
= in
[4] + (in
[5] << 32);
750 /* the remaining terms */
751 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
752 out
[1] -= (in
[4] << 32);
753 out
[3] += (in
[4] << 32);
755 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
756 out
[2] -= (in
[5] << 32);
758 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
760 out
[0] -= (in
[6] << 32);
761 out
[1] += (in
[6] << 33);
762 out
[2] += (in
[6] * 2);
763 out
[3] -= (in
[6] << 32);
765 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
767 out
[0] -= (in
[7] << 32);
768 out
[2] += (in
[7] << 33);
769 out
[3] += (in
[7] * 3);
773 * felem_reduce converts a longfelem into an felem.
774 * To be called directly after felem_square or felem_mul.
776 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
777 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
781 static void felem_reduce(felem out
, const longfelem in
)
783 out
[0] = zero100
[0] + in
[0];
784 out
[1] = zero100
[1] + in
[1];
785 out
[2] = zero100
[2] + in
[2];
786 out
[3] = zero100
[3] + in
[3];
788 felem_reduce_(out
, in
);
791 * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
792 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
793 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
794 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
796 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
797 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
798 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
799 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
804 * felem_reduce_zero105 converts a larger longfelem into an felem.
810 static void felem_reduce_zero105(felem out
, const longfelem in
)
812 out
[0] = zero105
[0] + in
[0];
813 out
[1] = zero105
[1] + in
[1];
814 out
[2] = zero105
[2] + in
[2];
815 out
[3] = zero105
[3] + in
[3];
817 felem_reduce_(out
, in
);
820 * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
821 * out[1] > 2^105 - 2^71 - 2^103 > 0
822 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
823 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
825 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
826 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
827 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
828 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
833 * subtract_u64 sets *result = *result - v and *carry to one if the
834 * subtraction underflowed.
836 static void subtract_u64(u64
*result
, u64
*carry
, u64 v
)
838 uint128_t r
= *result
;
840 *carry
= (r
>> 64) & 1;
845 * felem_contract converts |in| to its unique, minimal representation. On
846 * entry: in[i] < 2^109
848 static void felem_contract(smallfelem out
, const felem in
)
851 u64 all_equal_so_far
= 0, result
= 0, carry
;
853 felem_shrink(out
, in
);
854 /* small is minimal except that the value might be > p */
858 * We are doing a constant time test if out >= kPrime. We need to compare
859 * each u64, from most-significant to least significant. For each one, if
860 * all words so far have been equal (m is all ones) then a non-equal
861 * result is the answer. Otherwise we continue.
863 for (i
= 3; i
< 4; i
--) {
865 uint128_t a
= ((uint128_t
) kPrime
[i
]) - out
[i
];
867 * if out[i] > kPrime[i] then a will underflow and the high 64-bits
870 result
|= all_equal_so_far
& ((u64
)(a
>> 64));
873 * if kPrime[i] == out[i] then |equal| will be all zeros and the
874 * decrement will make it all ones.
876 equal
= kPrime
[i
] ^ out
[i
];
878 equal
&= equal
<< 32;
879 equal
&= equal
<< 16;
884 equal
= 0 - (equal
>> 63);
886 all_equal_so_far
&= equal
;
890 * if all_equal_so_far is still all ones then the two values are equal
891 * and so out >= kPrime is true.
893 result
|= all_equal_so_far
;
895 /* if out >= kPrime then we subtract kPrime. */
896 subtract_u64(&out
[0], &carry
, result
& kPrime
[0]);
897 subtract_u64(&out
[1], &carry
, carry
);
898 subtract_u64(&out
[2], &carry
, carry
);
899 subtract_u64(&out
[3], &carry
, carry
);
901 subtract_u64(&out
[1], &carry
, result
& kPrime
[1]);
902 subtract_u64(&out
[2], &carry
, carry
);
903 subtract_u64(&out
[3], &carry
, carry
);
905 subtract_u64(&out
[2], &carry
, result
& kPrime
[2]);
906 subtract_u64(&out
[3], &carry
, carry
);
908 subtract_u64(&out
[3], &carry
, result
& kPrime
[3]);
911 static void smallfelem_square_contract(smallfelem out
, const smallfelem in
)
916 smallfelem_square(longtmp
, in
);
917 felem_reduce(tmp
, longtmp
);
918 felem_contract(out
, tmp
);
921 static void smallfelem_mul_contract(smallfelem out
, const smallfelem in1
,
922 const smallfelem in2
)
927 smallfelem_mul(longtmp
, in1
, in2
);
928 felem_reduce(tmp
, longtmp
);
929 felem_contract(out
, tmp
);
933 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
938 static limb
smallfelem_is_zero(const smallfelem small
)
943 u64 is_zero
= small
[0] | small
[1] | small
[2] | small
[3];
945 is_zero
&= is_zero
<< 32;
946 is_zero
&= is_zero
<< 16;
947 is_zero
&= is_zero
<< 8;
948 is_zero
&= is_zero
<< 4;
949 is_zero
&= is_zero
<< 2;
950 is_zero
&= is_zero
<< 1;
951 is_zero
= 0 - (is_zero
>> 63);
953 is_p
= (small
[0] ^ kPrime
[0]) |
954 (small
[1] ^ kPrime
[1]) |
955 (small
[2] ^ kPrime
[2]) | (small
[3] ^ kPrime
[3]);
963 is_p
= 0 - (is_p
>> 63);
968 result
|= ((limb
) is_zero
) << 64;
972 static int smallfelem_is_zero_int(const void *small
)
974 return (int)(smallfelem_is_zero(small
) & ((limb
) 1));
978 * felem_inv calculates |out| = |in|^{-1}
980 * Based on Fermat's Little Theorem:
982 * a^{p-1} = 1 (mod p)
983 * a^{p-2} = a^{-1} (mod p)
985 static void felem_inv(felem out
, const felem in
)
988 /* each e_I will hold |in|^{2^I - 1} */
989 felem e2
, e4
, e8
, e16
, e32
, e64
;
993 felem_square(tmp
, in
);
994 felem_reduce(ftmp
, tmp
); /* 2^1 */
995 felem_mul(tmp
, in
, ftmp
);
996 felem_reduce(ftmp
, tmp
); /* 2^2 - 2^0 */
997 felem_assign(e2
, ftmp
);
998 felem_square(tmp
, ftmp
);
999 felem_reduce(ftmp
, tmp
); /* 2^3 - 2^1 */
1000 felem_square(tmp
, ftmp
);
1001 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^2 */
1002 felem_mul(tmp
, ftmp
, e2
);
1003 felem_reduce(ftmp
, tmp
); /* 2^4 - 2^0 */
1004 felem_assign(e4
, ftmp
);
1005 felem_square(tmp
, ftmp
);
1006 felem_reduce(ftmp
, tmp
); /* 2^5 - 2^1 */
1007 felem_square(tmp
, ftmp
);
1008 felem_reduce(ftmp
, tmp
); /* 2^6 - 2^2 */
1009 felem_square(tmp
, ftmp
);
1010 felem_reduce(ftmp
, tmp
); /* 2^7 - 2^3 */
1011 felem_square(tmp
, ftmp
);
1012 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^4 */
1013 felem_mul(tmp
, ftmp
, e4
);
1014 felem_reduce(ftmp
, tmp
); /* 2^8 - 2^0 */
1015 felem_assign(e8
, ftmp
);
1016 for (i
= 0; i
< 8; i
++) {
1017 felem_square(tmp
, ftmp
);
1018 felem_reduce(ftmp
, tmp
);
1020 felem_mul(tmp
, ftmp
, e8
);
1021 felem_reduce(ftmp
, tmp
); /* 2^16 - 2^0 */
1022 felem_assign(e16
, ftmp
);
1023 for (i
= 0; i
< 16; i
++) {
1024 felem_square(tmp
, ftmp
);
1025 felem_reduce(ftmp
, tmp
);
1027 felem_mul(tmp
, ftmp
, e16
);
1028 felem_reduce(ftmp
, tmp
); /* 2^32 - 2^0 */
1029 felem_assign(e32
, ftmp
);
1030 for (i
= 0; i
< 32; i
++) {
1031 felem_square(tmp
, ftmp
);
1032 felem_reduce(ftmp
, tmp
);
1034 felem_assign(e64
, ftmp
);
1035 felem_mul(tmp
, ftmp
, in
);
1036 felem_reduce(ftmp
, tmp
); /* 2^64 - 2^32 + 2^0 */
1037 for (i
= 0; i
< 192; i
++) {
1038 felem_square(tmp
, ftmp
);
1039 felem_reduce(ftmp
, tmp
);
1040 } /* 2^256 - 2^224 + 2^192 */
1042 felem_mul(tmp
, e64
, e32
);
1043 felem_reduce(ftmp2
, tmp
); /* 2^64 - 2^0 */
1044 for (i
= 0; i
< 16; i
++) {
1045 felem_square(tmp
, ftmp2
);
1046 felem_reduce(ftmp2
, tmp
);
1048 felem_mul(tmp
, ftmp2
, e16
);
1049 felem_reduce(ftmp2
, tmp
); /* 2^80 - 2^0 */
1050 for (i
= 0; i
< 8; i
++) {
1051 felem_square(tmp
, ftmp2
);
1052 felem_reduce(ftmp2
, tmp
);
1054 felem_mul(tmp
, ftmp2
, e8
);
1055 felem_reduce(ftmp2
, tmp
); /* 2^88 - 2^0 */
1056 for (i
= 0; i
< 4; i
++) {
1057 felem_square(tmp
, ftmp2
);
1058 felem_reduce(ftmp2
, tmp
);
1060 felem_mul(tmp
, ftmp2
, e4
);
1061 felem_reduce(ftmp2
, tmp
); /* 2^92 - 2^0 */
1062 felem_square(tmp
, ftmp2
);
1063 felem_reduce(ftmp2
, tmp
); /* 2^93 - 2^1 */
1064 felem_square(tmp
, ftmp2
);
1065 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^2 */
1066 felem_mul(tmp
, ftmp2
, e2
);
1067 felem_reduce(ftmp2
, tmp
); /* 2^94 - 2^0 */
1068 felem_square(tmp
, ftmp2
);
1069 felem_reduce(ftmp2
, tmp
); /* 2^95 - 2^1 */
1070 felem_square(tmp
, ftmp2
);
1071 felem_reduce(ftmp2
, tmp
); /* 2^96 - 2^2 */
1072 felem_mul(tmp
, ftmp2
, in
);
1073 felem_reduce(ftmp2
, tmp
); /* 2^96 - 3 */
1075 felem_mul(tmp
, ftmp2
, ftmp
);
1076 felem_reduce(out
, tmp
); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1079 static void smallfelem_inv_contract(smallfelem out
, const smallfelem in
)
1083 smallfelem_expand(tmp
, in
);
1084 felem_inv(tmp
, tmp
);
1085 felem_contract(out
, tmp
);
1092 * Building on top of the field operations we have the operations on the
1093 * elliptic curve group itself. Points on the curve are represented in Jacobian
1098 * point_double calculates 2*(x_in, y_in, z_in)
1100 * The method is taken from:
1101 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1103 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1104 * while x_out == y_in is not (maybe this works, but it's not tested).
1107 point_double(felem x_out
, felem y_out
, felem z_out
,
1108 const felem x_in
, const felem y_in
, const felem z_in
)
1110 longfelem tmp
, tmp2
;
1111 felem delta
, gamma
, beta
, alpha
, ftmp
, ftmp2
;
1112 smallfelem small1
, small2
;
1114 felem_assign(ftmp
, x_in
);
1115 /* ftmp[i] < 2^106 */
1116 felem_assign(ftmp2
, x_in
);
1117 /* ftmp2[i] < 2^106 */
1120 felem_square(tmp
, z_in
);
1121 felem_reduce(delta
, tmp
);
1122 /* delta[i] < 2^101 */
1125 felem_square(tmp
, y_in
);
1126 felem_reduce(gamma
, tmp
);
1127 /* gamma[i] < 2^101 */
1128 felem_shrink(small1
, gamma
);
1130 /* beta = x*gamma */
1131 felem_small_mul(tmp
, small1
, x_in
);
1132 felem_reduce(beta
, tmp
);
1133 /* beta[i] < 2^101 */
1135 /* alpha = 3*(x-delta)*(x+delta) */
1136 felem_diff(ftmp
, delta
);
1137 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1138 felem_sum(ftmp2
, delta
);
1139 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1140 felem_scalar(ftmp2
, 3);
1141 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1142 felem_mul(tmp
, ftmp
, ftmp2
);
1143 felem_reduce(alpha
, tmp
);
1144 /* alpha[i] < 2^101 */
1145 felem_shrink(small2
, alpha
);
1147 /* x' = alpha^2 - 8*beta */
1148 smallfelem_square(tmp
, small2
);
1149 felem_reduce(x_out
, tmp
);
1150 felem_assign(ftmp
, beta
);
1151 felem_scalar(ftmp
, 8);
1152 /* ftmp[i] < 8 * 2^101 = 2^104 */
1153 felem_diff(x_out
, ftmp
);
1154 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1156 /* z' = (y + z)^2 - gamma - delta */
1157 felem_sum(delta
, gamma
);
1158 /* delta[i] < 2^101 + 2^101 = 2^102 */
1159 felem_assign(ftmp
, y_in
);
1160 felem_sum(ftmp
, z_in
);
1161 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1162 felem_square(tmp
, ftmp
);
1163 felem_reduce(z_out
, tmp
);
1164 felem_diff(z_out
, delta
);
1165 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1167 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1168 felem_scalar(beta
, 4);
1169 /* beta[i] < 4 * 2^101 = 2^103 */
1170 felem_diff_zero107(beta
, x_out
);
1171 /* beta[i] < 2^107 + 2^103 < 2^108 */
1172 felem_small_mul(tmp
, small2
, beta
);
1173 /* tmp[i] < 7 * 2^64 < 2^67 */
1174 smallfelem_square(tmp2
, small1
);
1175 /* tmp2[i] < 7 * 2^64 */
1176 longfelem_scalar(tmp2
, 8);
1177 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1178 longfelem_diff(tmp
, tmp2
);
1179 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1180 felem_reduce_zero105(y_out
, tmp
);
1181 /* y_out[i] < 2^106 */
1185 * point_double_small is the same as point_double, except that it operates on
1189 point_double_small(smallfelem x_out
, smallfelem y_out
, smallfelem z_out
,
1190 const smallfelem x_in
, const smallfelem y_in
,
1191 const smallfelem z_in
)
1193 felem felem_x_out
, felem_y_out
, felem_z_out
;
1194 felem felem_x_in
, felem_y_in
, felem_z_in
;
1196 smallfelem_expand(felem_x_in
, x_in
);
1197 smallfelem_expand(felem_y_in
, y_in
);
1198 smallfelem_expand(felem_z_in
, z_in
);
1199 point_double(felem_x_out
, felem_y_out
, felem_z_out
,
1200 felem_x_in
, felem_y_in
, felem_z_in
);
1201 felem_shrink(x_out
, felem_x_out
);
1202 felem_shrink(y_out
, felem_y_out
);
1203 felem_shrink(z_out
, felem_z_out
);
1206 /* copy_conditional copies in to out iff mask is all ones. */
1207 static void copy_conditional(felem out
, const felem in
, limb mask
)
1210 for (i
= 0; i
< NLIMBS
; ++i
) {
1211 const limb tmp
= mask
& (in
[i
] ^ out
[i
]);
1216 /* copy_small_conditional copies in to out iff mask is all ones. */
1217 static void copy_small_conditional(felem out
, const smallfelem in
, limb mask
)
1220 const u64 mask64
= mask
;
1221 for (i
= 0; i
< NLIMBS
; ++i
) {
1222 out
[i
] = ((limb
) (in
[i
] & mask64
)) | (out
[i
] & ~mask
);
1227 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1229 * The method is taken from:
1230 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1231 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1233 * This function includes a branch for checking whether the two input points
1234 * are equal, (while not equal to the point at infinity). This case never
1235 * happens during single point multiplication, so there is no timing leak for
1236 * ECDH or ECDSA signing.
1238 static void point_add(felem x3
, felem y3
, felem z3
,
1239 const felem x1
, const felem y1
, const felem z1
,
1240 const int mixed
, const smallfelem x2
,
1241 const smallfelem y2
, const smallfelem z2
)
1243 felem ftmp
, ftmp2
, ftmp3
, ftmp4
, ftmp5
, ftmp6
, x_out
, y_out
, z_out
;
1244 longfelem tmp
, tmp2
;
1245 smallfelem small1
, small2
, small3
, small4
, small5
;
1246 limb x_equal
, y_equal
, z1_is_zero
, z2_is_zero
;
1249 felem_shrink(small3
, z1
);
1251 z1_is_zero
= smallfelem_is_zero(small3
);
1252 z2_is_zero
= smallfelem_is_zero(z2
);
1254 /* ftmp = z1z1 = z1**2 */
1255 smallfelem_square(tmp
, small3
);
1256 felem_reduce(ftmp
, tmp
);
1257 /* ftmp[i] < 2^101 */
1258 felem_shrink(small1
, ftmp
);
1261 /* ftmp2 = z2z2 = z2**2 */
1262 smallfelem_square(tmp
, z2
);
1263 felem_reduce(ftmp2
, tmp
);
1264 /* ftmp2[i] < 2^101 */
1265 felem_shrink(small2
, ftmp2
);
1267 felem_shrink(small5
, x1
);
1269 /* u1 = ftmp3 = x1*z2z2 */
1270 smallfelem_mul(tmp
, small5
, small2
);
1271 felem_reduce(ftmp3
, tmp
);
1272 /* ftmp3[i] < 2^101 */
1274 /* ftmp5 = z1 + z2 */
1275 felem_assign(ftmp5
, z1
);
1276 felem_small_sum(ftmp5
, z2
);
1277 /* ftmp5[i] < 2^107 */
1279 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1280 felem_square(tmp
, ftmp5
);
1281 felem_reduce(ftmp5
, tmp
);
1282 /* ftmp2 = z2z2 + z1z1 */
1283 felem_sum(ftmp2
, ftmp
);
1284 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1285 felem_diff(ftmp5
, ftmp2
);
1286 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1288 /* ftmp2 = z2 * z2z2 */
1289 smallfelem_mul(tmp
, small2
, z2
);
1290 felem_reduce(ftmp2
, tmp
);
1292 /* s1 = ftmp2 = y1 * z2**3 */
1293 felem_mul(tmp
, y1
, ftmp2
);
1294 felem_reduce(ftmp6
, tmp
);
1295 /* ftmp6[i] < 2^101 */
1298 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1301 /* u1 = ftmp3 = x1*z2z2 */
1302 felem_assign(ftmp3
, x1
);
1303 /* ftmp3[i] < 2^106 */
1306 felem_assign(ftmp5
, z1
);
1307 felem_scalar(ftmp5
, 2);
1308 /* ftmp5[i] < 2*2^106 = 2^107 */
1310 /* s1 = ftmp2 = y1 * z2**3 */
1311 felem_assign(ftmp6
, y1
);
1312 /* ftmp6[i] < 2^106 */
1316 smallfelem_mul(tmp
, x2
, small1
);
1317 felem_reduce(ftmp4
, tmp
);
1319 /* h = ftmp4 = u2 - u1 */
1320 felem_diff_zero107(ftmp4
, ftmp3
);
1321 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1322 felem_shrink(small4
, ftmp4
);
1324 x_equal
= smallfelem_is_zero(small4
);
1326 /* z_out = ftmp5 * h */
1327 felem_small_mul(tmp
, small4
, ftmp5
);
1328 felem_reduce(z_out
, tmp
);
1329 /* z_out[i] < 2^101 */
1331 /* ftmp = z1 * z1z1 */
1332 smallfelem_mul(tmp
, small1
, small3
);
1333 felem_reduce(ftmp
, tmp
);
1335 /* s2 = tmp = y2 * z1**3 */
1336 felem_small_mul(tmp
, y2
, ftmp
);
1337 felem_reduce(ftmp5
, tmp
);
1339 /* r = ftmp5 = (s2 - s1)*2 */
1340 felem_diff_zero107(ftmp5
, ftmp6
);
1341 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1342 felem_scalar(ftmp5
, 2);
1343 /* ftmp5[i] < 2^109 */
1344 felem_shrink(small1
, ftmp5
);
1345 y_equal
= smallfelem_is_zero(small1
);
1348 * The formulae are incorrect if the points are equal, in affine coordinates
1349 * (X_1, Y_1) == (X_2, Y_2), so we check for this and do doubling if this
1352 * We use bitwise operations to avoid potential side-channels introduced by
1353 * the short-circuiting behaviour of boolean operators.
1355 * The special case of either point being the point at infinity (z1 and/or
1356 * z2 are zero), is handled separately later on in this function, so we
1357 * avoid jumping to point_double here in those special cases.
1359 points_equal
= (x_equal
& y_equal
& (~z1_is_zero
) & (~z2_is_zero
));
1363 * This is obviously not constant-time but, as mentioned before, this
1364 * case never happens during single point multiplication, so there is no
1365 * timing leak for ECDH or ECDSA signing.
1367 point_double(x3
, y3
, z3
, x1
, y1
, z1
);
1371 /* I = ftmp = (2h)**2 */
1372 felem_assign(ftmp
, ftmp4
);
1373 felem_scalar(ftmp
, 2);
1374 /* ftmp[i] < 2*2^108 = 2^109 */
1375 felem_square(tmp
, ftmp
);
1376 felem_reduce(ftmp
, tmp
);
1378 /* J = ftmp2 = h * I */
1379 felem_mul(tmp
, ftmp4
, ftmp
);
1380 felem_reduce(ftmp2
, tmp
);
1382 /* V = ftmp4 = U1 * I */
1383 felem_mul(tmp
, ftmp3
, ftmp
);
1384 felem_reduce(ftmp4
, tmp
);
1386 /* x_out = r**2 - J - 2V */
1387 smallfelem_square(tmp
, small1
);
1388 felem_reduce(x_out
, tmp
);
1389 felem_assign(ftmp3
, ftmp4
);
1390 felem_scalar(ftmp4
, 2);
1391 felem_sum(ftmp4
, ftmp2
);
1392 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1393 felem_diff(x_out
, ftmp4
);
1394 /* x_out[i] < 2^105 + 2^101 */
1396 /* y_out = r(V-x_out) - 2 * s1 * J */
1397 felem_diff_zero107(ftmp3
, x_out
);
1398 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1399 felem_small_mul(tmp
, small1
, ftmp3
);
1400 felem_mul(tmp2
, ftmp6
, ftmp2
);
1401 longfelem_scalar(tmp2
, 2);
1402 /* tmp2[i] < 2*2^67 = 2^68 */
1403 longfelem_diff(tmp
, tmp2
);
1404 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1405 felem_reduce_zero105(y_out
, tmp
);
1406 /* y_out[i] < 2^106 */
1408 copy_small_conditional(x_out
, x2
, z1_is_zero
);
1409 copy_conditional(x_out
, x1
, z2_is_zero
);
1410 copy_small_conditional(y_out
, y2
, z1_is_zero
);
1411 copy_conditional(y_out
, y1
, z2_is_zero
);
1412 copy_small_conditional(z_out
, z2
, z1_is_zero
);
1413 copy_conditional(z_out
, z1
, z2_is_zero
);
1414 felem_assign(x3
, x_out
);
1415 felem_assign(y3
, y_out
);
1416 felem_assign(z3
, z_out
);
1420 * point_add_small is the same as point_add, except that it operates on
1423 static void point_add_small(smallfelem x3
, smallfelem y3
, smallfelem z3
,
1424 smallfelem x1
, smallfelem y1
, smallfelem z1
,
1425 smallfelem x2
, smallfelem y2
, smallfelem z2
)
1427 felem felem_x3
, felem_y3
, felem_z3
;
1428 felem felem_x1
, felem_y1
, felem_z1
;
1429 smallfelem_expand(felem_x1
, x1
);
1430 smallfelem_expand(felem_y1
, y1
);
1431 smallfelem_expand(felem_z1
, z1
);
1432 point_add(felem_x3
, felem_y3
, felem_z3
, felem_x1
, felem_y1
, felem_z1
, 0,
1434 felem_shrink(x3
, felem_x3
);
1435 felem_shrink(y3
, felem_y3
);
1436 felem_shrink(z3
, felem_z3
);
1440 * Base point pre computation
1441 * --------------------------
1443 * Two different sorts of precomputed tables are used in the following code.
1444 * Each contain various points on the curve, where each point is three field
1445 * elements (x, y, z).
1447 * For the base point table, z is usually 1 (0 for the point at infinity).
1448 * This table has 2 * 16 elements, starting with the following:
1449 * index | bits | point
1450 * ------+---------+------------------------------
1453 * 2 | 0 0 1 0 | 2^64G
1454 * 3 | 0 0 1 1 | (2^64 + 1)G
1455 * 4 | 0 1 0 0 | 2^128G
1456 * 5 | 0 1 0 1 | (2^128 + 1)G
1457 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1458 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1459 * 8 | 1 0 0 0 | 2^192G
1460 * 9 | 1 0 0 1 | (2^192 + 1)G
1461 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1462 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1463 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1464 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1465 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1466 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1467 * followed by a copy of this with each element multiplied by 2^32.
1469 * The reason for this is so that we can clock bits into four different
1470 * locations when doing simple scalar multiplies against the base point,
1471 * and then another four locations using the second 16 elements.
1473 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1475 /* gmul is the table of precomputed base points */
1476 static const smallfelem gmul
[2][16][3] = {
1480 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1481 0x6b17d1f2e12c4247},
1482 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1483 0x4fe342e2fe1a7f9b},
1485 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1486 0x0fa822bc2811aaa5},
1487 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1488 0xbff44ae8f5dba80d},
1490 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1491 0x300a4bbc89d6726f},
1492 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1493 0x72aac7e0d09b4644},
1495 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1496 0x447d739beedb5e67},
1497 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1498 0x2d4825ab834131ee},
1500 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1501 0xef9519328a9c72ff},
1502 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1503 0x611e9fc37dbb2c9b},
1505 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1506 0x550663797b51f5d8},
1507 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1508 0x157164848aecb851},
1510 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1511 0xeb5d7745b21141ea},
1512 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1513 0xeafd72ebdbecc17b},
1515 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1516 0xa6d39677a7849276},
1517 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1518 0x674f84749b0b8816},
1520 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1521 0x4e769e7672c9ddad},
1522 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1523 0x42b99082de830663},
1525 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1526 0x78878ef61c6ce04d},
1527 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1528 0xb6cb3f5d7b72c321},
1530 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1531 0x0c88bc4d716b1287},
1532 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1533 0xdd5ddea3f3901dc6},
1535 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1536 0x68f344af6b317466},
1537 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1538 0x31b9c405f8540a20},
1540 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1541 0x4052bf4b6f461db9},
1542 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1543 0xfecf4d5190b0fc61},
1545 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1546 0x1eddbae2c802e41a},
1547 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1548 0x43104d86560ebcfc},
1550 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1551 0xb48e26b484f7a21c},
1552 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1553 0xfac015404d4d3dab},
1558 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1559 0x7fe36b40af22af89},
1560 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1561 0xe697d45825b63624},
1563 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1564 0x4a5b506612a677a6},
1565 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1566 0xeb13461ceac089f1},
1568 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1569 0x0781b8291c6a220a},
1570 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1571 0x690cde8df0151593},
1573 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1574 0x8a535f566ec73617},
1575 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1576 0x0455c08468b08bd7},
1578 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1579 0x06bada7ab77f8276},
1580 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1581 0x5b476dfd0e6cb18a},
1583 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1584 0x3e29864e8a2ec908},
1585 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1586 0x239b90ea3dc31e7e},
1588 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1589 0x820f4dd949f72ff7},
1590 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1591 0x140406ec783a05ec},
1593 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1594 0x68f6b8542783dfee},
1595 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1596 0xcbe1feba92e40ce6},
1598 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1599 0xd0b2f94d2f420109},
1600 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1601 0x971459828b0719e5},
1603 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1604 0x961610004a866aba},
1605 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1606 0x7acb9fadcee75e44},
1608 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1609 0x24eb9acca333bf5b},
1610 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1611 0x69f891c5acd079cc},
1613 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1614 0xe51f547c5972a107},
1615 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1616 0x1c309a2b25bb1387},
1618 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1619 0x20b87b8aa2c4e503},
1620 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1621 0xf5c6fa49919776be},
1623 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1624 0x1ed7d1b9332010b9},
1625 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1626 0x3a2b03f03217257a},
1628 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1629 0x15fee545c78dd9f6},
1630 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1631 0x4ab5b6b2b8753f81},
1636 * select_point selects the |idx|th point from a precomputation table and
1639 static void select_point(const u64 idx
, unsigned int size
,
1640 const smallfelem pre_comp
[16][3], smallfelem out
[3])
1643 u64
*outlimbs
= &out
[0][0];
1645 memset(out
, 0, sizeof(*out
) * 3);
1647 for (i
= 0; i
< size
; i
++) {
1648 const u64
*inlimbs
= (u64
*)&pre_comp
[i
][0][0];
1655 for (j
= 0; j
< NLIMBS
* 3; j
++)
1656 outlimbs
[j
] |= inlimbs
[j
] & mask
;
1660 /* get_bit returns the |i|th bit in |in| */
1661 static char get_bit(const felem_bytearray in
, int i
)
1663 if ((i
< 0) || (i
>= 256))
1665 return (in
[i
>> 3] >> (i
& 7)) & 1;
1669 * Interleaved point multiplication using precomputed point multiples: The
1670 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1671 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1672 * generator, using certain (large) precomputed multiples in g_pre_comp.
1673 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1675 static void batch_mul(felem x_out
, felem y_out
, felem z_out
,
1676 const felem_bytearray scalars
[],
1677 const unsigned num_points
, const u8
*g_scalar
,
1678 const int mixed
, const smallfelem pre_comp
[][17][3],
1679 const smallfelem g_pre_comp
[2][16][3])
1682 unsigned num
, gen_mul
= (g_scalar
!= NULL
);
1688 /* set nq to the point at infinity */
1689 memset(nq
, 0, sizeof(nq
));
1692 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1693 * of the generator (two in each of the last 32 rounds) and additions of
1694 * other points multiples (every 5th round).
1696 skip
= 1; /* save two point operations in the first
1698 for (i
= (num_points
? 255 : 31); i
>= 0; --i
) {
1701 point_double(nq
[0], nq
[1], nq
[2], nq
[0], nq
[1], nq
[2]);
1703 /* add multiples of the generator */
1704 if (gen_mul
&& (i
<= 31)) {
1705 /* first, look 32 bits upwards */
1706 bits
= get_bit(g_scalar
, i
+ 224) << 3;
1707 bits
|= get_bit(g_scalar
, i
+ 160) << 2;
1708 bits
|= get_bit(g_scalar
, i
+ 96) << 1;
1709 bits
|= get_bit(g_scalar
, i
+ 32);
1710 /* select the point to add, in constant time */
1711 select_point(bits
, 16, g_pre_comp
[1], tmp
);
1714 /* Arg 1 below is for "mixed" */
1715 point_add(nq
[0], nq
[1], nq
[2],
1716 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1718 smallfelem_expand(nq
[0], tmp
[0]);
1719 smallfelem_expand(nq
[1], tmp
[1]);
1720 smallfelem_expand(nq
[2], tmp
[2]);
1724 /* second, look at the current position */
1725 bits
= get_bit(g_scalar
, i
+ 192) << 3;
1726 bits
|= get_bit(g_scalar
, i
+ 128) << 2;
1727 bits
|= get_bit(g_scalar
, i
+ 64) << 1;
1728 bits
|= get_bit(g_scalar
, i
);
1729 /* select the point to add, in constant time */
1730 select_point(bits
, 16, g_pre_comp
[0], tmp
);
1731 /* Arg 1 below is for "mixed" */
1732 point_add(nq
[0], nq
[1], nq
[2],
1733 nq
[0], nq
[1], nq
[2], 1, tmp
[0], tmp
[1], tmp
[2]);
1736 /* do other additions every 5 doublings */
1737 if (num_points
&& (i
% 5 == 0)) {
1738 /* loop over all scalars */
1739 for (num
= 0; num
< num_points
; ++num
) {
1740 bits
= get_bit(scalars
[num
], i
+ 4) << 5;
1741 bits
|= get_bit(scalars
[num
], i
+ 3) << 4;
1742 bits
|= get_bit(scalars
[num
], i
+ 2) << 3;
1743 bits
|= get_bit(scalars
[num
], i
+ 1) << 2;
1744 bits
|= get_bit(scalars
[num
], i
) << 1;
1745 bits
|= get_bit(scalars
[num
], i
- 1);
1746 ec_GFp_nistp_recode_scalar_bits(&sign
, &digit
, bits
);
1749 * select the point to add or subtract, in constant time
1751 select_point(digit
, 17, pre_comp
[num
], tmp
);
1752 smallfelem_neg(ftmp
, tmp
[1]); /* (X, -Y, Z) is the negative
1754 copy_small_conditional(ftmp
, tmp
[1], (((limb
) sign
) - 1));
1755 felem_contract(tmp
[1], ftmp
);
1758 point_add(nq
[0], nq
[1], nq
[2],
1759 nq
[0], nq
[1], nq
[2],
1760 mixed
, tmp
[0], tmp
[1], tmp
[2]);
1762 smallfelem_expand(nq
[0], tmp
[0]);
1763 smallfelem_expand(nq
[1], tmp
[1]);
1764 smallfelem_expand(nq
[2], tmp
[2]);
1770 felem_assign(x_out
, nq
[0]);
1771 felem_assign(y_out
, nq
[1]);
1772 felem_assign(z_out
, nq
[2]);
1775 /* Precomputation for the group generator. */
1776 struct nistp256_pre_comp_st
{
1777 smallfelem g_pre_comp
[2][16][3];
1778 CRYPTO_REF_COUNT references
;
1779 CRYPTO_RWLOCK
*lock
;
1782 const EC_METHOD
*EC_GFp_nistp256_method(void)
1784 static const EC_METHOD ret
= {
1785 EC_FLAGS_DEFAULT_OCT
,
1786 NID_X9_62_prime_field
,
1787 ec_GFp_nistp256_group_init
,
1788 ec_GFp_simple_group_finish
,
1789 ec_GFp_simple_group_clear_finish
,
1790 ec_GFp_nist_group_copy
,
1791 ec_GFp_nistp256_group_set_curve
,
1792 ec_GFp_simple_group_get_curve
,
1793 ec_GFp_simple_group_get_degree
,
1794 ec_group_simple_order_bits
,
1795 ec_GFp_simple_group_check_discriminant
,
1796 ec_GFp_simple_point_init
,
1797 ec_GFp_simple_point_finish
,
1798 ec_GFp_simple_point_clear_finish
,
1799 ec_GFp_simple_point_copy
,
1800 ec_GFp_simple_point_set_to_infinity
,
1801 ec_GFp_simple_point_set_affine_coordinates
,
1802 ec_GFp_nistp256_point_get_affine_coordinates
,
1803 0 /* point_set_compressed_coordinates */ ,
1808 ec_GFp_simple_invert
,
1809 ec_GFp_simple_is_at_infinity
,
1810 ec_GFp_simple_is_on_curve
,
1812 ec_GFp_simple_make_affine
,
1813 ec_GFp_simple_points_make_affine
,
1814 ec_GFp_nistp256_points_mul
,
1815 ec_GFp_nistp256_precompute_mult
,
1816 ec_GFp_nistp256_have_precompute_mult
,
1817 ec_GFp_nist_field_mul
,
1818 ec_GFp_nist_field_sqr
,
1820 ec_GFp_simple_field_inv
,
1821 0 /* field_encode */ ,
1822 0 /* field_decode */ ,
1823 0, /* field_set_to_one */
1824 ec_key_simple_priv2oct
,
1825 ec_key_simple_oct2priv
,
1826 0, /* set private */
1827 ec_key_simple_generate_key
,
1828 ec_key_simple_check_key
,
1829 ec_key_simple_generate_public_key
,
1832 ecdh_simple_compute_key
,
1833 ecdsa_simple_sign_setup
,
1834 ecdsa_simple_sign_sig
,
1835 ecdsa_simple_verify_sig
,
1836 0, /* field_inverse_mod_ord */
1837 0, /* blind_coordinates */
1839 0, /* ladder_step */
1846 /******************************************************************************/
1848 * FUNCTIONS TO MANAGE PRECOMPUTATION
1851 static NISTP256_PRE_COMP
*nistp256_pre_comp_new(void)
1853 NISTP256_PRE_COMP
*ret
= OPENSSL_zalloc(sizeof(*ret
));
1856 ERR_raise(ERR_LIB_EC
, ERR_R_MALLOC_FAILURE
);
1860 ret
->references
= 1;
1862 ret
->lock
= CRYPTO_THREAD_lock_new();
1863 if (ret
->lock
== NULL
) {
1864 ERR_raise(ERR_LIB_EC
, ERR_R_MALLOC_FAILURE
);
1871 NISTP256_PRE_COMP
*EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP
*p
)
1875 CRYPTO_UP_REF(&p
->references
, &i
, p
->lock
);
1879 void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP
*pre
)
1886 CRYPTO_DOWN_REF(&pre
->references
, &i
, pre
->lock
);
1887 REF_PRINT_COUNT("EC_nistp256", x
);
1890 REF_ASSERT_ISNT(i
< 0);
1892 CRYPTO_THREAD_lock_free(pre
->lock
);
1896 /******************************************************************************/
1898 * OPENSSL EC_METHOD FUNCTIONS
1901 int ec_GFp_nistp256_group_init(EC_GROUP
*group
)
1904 ret
= ec_GFp_simple_group_init(group
);
1905 group
->a_is_minus3
= 1;
1909 int ec_GFp_nistp256_group_set_curve(EC_GROUP
*group
, const BIGNUM
*p
,
1910 const BIGNUM
*a
, const BIGNUM
*b
,
1914 BIGNUM
*curve_p
, *curve_a
, *curve_b
;
1916 BN_CTX
*new_ctx
= NULL
;
1919 ctx
= new_ctx
= BN_CTX_new();
1925 curve_p
= BN_CTX_get(ctx
);
1926 curve_a
= BN_CTX_get(ctx
);
1927 curve_b
= BN_CTX_get(ctx
);
1928 if (curve_b
== NULL
)
1930 BN_bin2bn(nistp256_curve_params
[0], sizeof(felem_bytearray
), curve_p
);
1931 BN_bin2bn(nistp256_curve_params
[1], sizeof(felem_bytearray
), curve_a
);
1932 BN_bin2bn(nistp256_curve_params
[2], sizeof(felem_bytearray
), curve_b
);
1933 if ((BN_cmp(curve_p
, p
)) || (BN_cmp(curve_a
, a
)) || (BN_cmp(curve_b
, b
))) {
1934 ERR_raise(ERR_LIB_EC
, EC_R_WRONG_CURVE_PARAMETERS
);
1937 group
->field_mod_func
= BN_nist_mod_256
;
1938 ret
= ec_GFp_simple_group_set_curve(group
, p
, a
, b
, ctx
);
1942 BN_CTX_free(new_ctx
);
1948 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1951 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP
*group
,
1952 const EC_POINT
*point
,
1953 BIGNUM
*x
, BIGNUM
*y
,
1956 felem z1
, z2
, x_in
, y_in
;
1957 smallfelem x_out
, y_out
;
1960 if (EC_POINT_is_at_infinity(group
, point
)) {
1961 ERR_raise(ERR_LIB_EC
, EC_R_POINT_AT_INFINITY
);
1964 if ((!BN_to_felem(x_in
, point
->X
)) || (!BN_to_felem(y_in
, point
->Y
)) ||
1965 (!BN_to_felem(z1
, point
->Z
)))
1968 felem_square(tmp
, z2
);
1969 felem_reduce(z1
, tmp
);
1970 felem_mul(tmp
, x_in
, z1
);
1971 felem_reduce(x_in
, tmp
);
1972 felem_contract(x_out
, x_in
);
1974 if (!smallfelem_to_BN(x
, x_out
)) {
1975 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
1979 felem_mul(tmp
, z1
, z2
);
1980 felem_reduce(z1
, tmp
);
1981 felem_mul(tmp
, y_in
, z1
);
1982 felem_reduce(y_in
, tmp
);
1983 felem_contract(y_out
, y_in
);
1985 if (!smallfelem_to_BN(y
, y_out
)) {
1986 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
1993 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
1994 static void make_points_affine(size_t num
, smallfelem points
[][3],
1995 smallfelem tmp_smallfelems
[])
1998 * Runs in constant time, unless an input is the point at infinity (which
1999 * normally shouldn't happen).
2001 ec_GFp_nistp_points_make_affine_internal(num
,
2005 (void (*)(void *))smallfelem_one
,
2006 smallfelem_is_zero_int
,
2007 (void (*)(void *, const void *))
2009 (void (*)(void *, const void *))
2010 smallfelem_square_contract
,
2012 (void *, const void *,
2014 smallfelem_mul_contract
,
2015 (void (*)(void *, const void *))
2016 smallfelem_inv_contract
,
2017 /* nothing to contract */
2018 (void (*)(void *, const void *))
2023 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
2024 * values Result is stored in r (r can equal one of the inputs).
2026 int ec_GFp_nistp256_points_mul(const EC_GROUP
*group
, EC_POINT
*r
,
2027 const BIGNUM
*scalar
, size_t num
,
2028 const EC_POINT
*points
[],
2029 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
2034 BIGNUM
*x
, *y
, *z
, *tmp_scalar
;
2035 felem_bytearray g_secret
;
2036 felem_bytearray
*secrets
= NULL
;
2037 smallfelem (*pre_comp
)[17][3] = NULL
;
2038 smallfelem
*tmp_smallfelems
= NULL
;
2041 int have_pre_comp
= 0;
2042 size_t num_points
= num
;
2043 smallfelem x_in
, y_in
, z_in
;
2044 felem x_out
, y_out
, z_out
;
2045 NISTP256_PRE_COMP
*pre
= NULL
;
2046 const smallfelem(*g_pre_comp
)[16][3] = NULL
;
2047 EC_POINT
*generator
= NULL
;
2048 const EC_POINT
*p
= NULL
;
2049 const BIGNUM
*p_scalar
= NULL
;
2052 x
= BN_CTX_get(ctx
);
2053 y
= BN_CTX_get(ctx
);
2054 z
= BN_CTX_get(ctx
);
2055 tmp_scalar
= BN_CTX_get(ctx
);
2056 if (tmp_scalar
== NULL
)
2059 if (scalar
!= NULL
) {
2060 pre
= group
->pre_comp
.nistp256
;
2062 /* we have precomputation, try to use it */
2063 g_pre_comp
= (const smallfelem(*)[16][3])pre
->g_pre_comp
;
2065 /* try to use the standard precomputation */
2066 g_pre_comp
= &gmul
[0];
2067 generator
= EC_POINT_new(group
);
2068 if (generator
== NULL
)
2070 /* get the generator from precomputation */
2071 if (!smallfelem_to_BN(x
, g_pre_comp
[0][1][0]) ||
2072 !smallfelem_to_BN(y
, g_pre_comp
[0][1][1]) ||
2073 !smallfelem_to_BN(z
, g_pre_comp
[0][1][2])) {
2074 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
2077 if (!ec_GFp_simple_set_Jprojective_coordinates_GFp(group
, generator
, x
,
2080 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
))
2081 /* precomputation matches generator */
2085 * we don't have valid precomputation: treat the generator as a
2090 if (num_points
> 0) {
2091 if (num_points
>= 3) {
2093 * unless we precompute multiples for just one or two points,
2094 * converting those into affine form is time well spent
2098 secrets
= OPENSSL_malloc(sizeof(*secrets
) * num_points
);
2099 pre_comp
= OPENSSL_malloc(sizeof(*pre_comp
) * num_points
);
2102 OPENSSL_malloc(sizeof(*tmp_smallfelems
) * (num_points
* 17 + 1));
2103 if ((secrets
== NULL
) || (pre_comp
== NULL
)
2104 || (mixed
&& (tmp_smallfelems
== NULL
))) {
2105 ERR_raise(ERR_LIB_EC
, ERR_R_MALLOC_FAILURE
);
2110 * we treat NULL scalars as 0, and NULL points as points at infinity,
2111 * i.e., they contribute nothing to the linear combination
2113 memset(secrets
, 0, sizeof(*secrets
) * num_points
);
2114 memset(pre_comp
, 0, sizeof(*pre_comp
) * num_points
);
2115 for (i
= 0; i
< num_points
; ++i
) {
2118 * we didn't have a valid precomputation, so we pick the
2121 p
= EC_GROUP_get0_generator(group
);
2124 /* the i^th point */
2126 p_scalar
= scalars
[i
];
2128 if ((p_scalar
!= NULL
) && (p
!= NULL
)) {
2129 /* reduce scalar to 0 <= scalar < 2^256 */
2130 if ((BN_num_bits(p_scalar
) > 256)
2131 || (BN_is_negative(p_scalar
))) {
2133 * this is an unusual input, and we don't guarantee
2136 if (!BN_nnmod(tmp_scalar
, p_scalar
, group
->order
, ctx
)) {
2137 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
2140 num_bytes
= BN_bn2lebinpad(tmp_scalar
,
2141 secrets
[i
], sizeof(secrets
[i
]));
2143 num_bytes
= BN_bn2lebinpad(p_scalar
,
2144 secrets
[i
], sizeof(secrets
[i
]));
2146 if (num_bytes
< 0) {
2147 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
2150 /* precompute multiples */
2151 if ((!BN_to_felem(x_out
, p
->X
)) ||
2152 (!BN_to_felem(y_out
, p
->Y
)) ||
2153 (!BN_to_felem(z_out
, p
->Z
)))
2155 felem_shrink(pre_comp
[i
][1][0], x_out
);
2156 felem_shrink(pre_comp
[i
][1][1], y_out
);
2157 felem_shrink(pre_comp
[i
][1][2], z_out
);
2158 for (j
= 2; j
<= 16; ++j
) {
2160 point_add_small(pre_comp
[i
][j
][0], pre_comp
[i
][j
][1],
2161 pre_comp
[i
][j
][2], pre_comp
[i
][1][0],
2162 pre_comp
[i
][1][1], pre_comp
[i
][1][2],
2163 pre_comp
[i
][j
- 1][0],
2164 pre_comp
[i
][j
- 1][1],
2165 pre_comp
[i
][j
- 1][2]);
2167 point_double_small(pre_comp
[i
][j
][0],
2170 pre_comp
[i
][j
/ 2][0],
2171 pre_comp
[i
][j
/ 2][1],
2172 pre_comp
[i
][j
/ 2][2]);
2178 make_points_affine(num_points
* 17, pre_comp
[0], tmp_smallfelems
);
2181 /* the scalar for the generator */
2182 if ((scalar
!= NULL
) && (have_pre_comp
)) {
2183 memset(g_secret
, 0, sizeof(g_secret
));
2184 /* reduce scalar to 0 <= scalar < 2^256 */
2185 if ((BN_num_bits(scalar
) > 256) || (BN_is_negative(scalar
))) {
2187 * this is an unusual input, and we don't guarantee
2190 if (!BN_nnmod(tmp_scalar
, scalar
, group
->order
, ctx
)) {
2191 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
2194 num_bytes
= BN_bn2lebinpad(tmp_scalar
, g_secret
, sizeof(g_secret
));
2196 num_bytes
= BN_bn2lebinpad(scalar
, g_secret
, sizeof(g_secret
));
2198 /* do the multiplication with generator precomputation */
2199 batch_mul(x_out
, y_out
, z_out
,
2200 (const felem_bytearray(*))secrets
, num_points
,
2202 mixed
, (const smallfelem(*)[17][3])pre_comp
, g_pre_comp
);
2204 /* do the multiplication without generator precomputation */
2205 batch_mul(x_out
, y_out
, z_out
,
2206 (const felem_bytearray(*))secrets
, num_points
,
2207 NULL
, mixed
, (const smallfelem(*)[17][3])pre_comp
, NULL
);
2209 /* reduce the output to its unique minimal representation */
2210 felem_contract(x_in
, x_out
);
2211 felem_contract(y_in
, y_out
);
2212 felem_contract(z_in
, z_out
);
2213 if ((!smallfelem_to_BN(x
, x_in
)) || (!smallfelem_to_BN(y
, y_in
)) ||
2214 (!smallfelem_to_BN(z
, z_in
))) {
2215 ERR_raise(ERR_LIB_EC
, ERR_R_BN_LIB
);
2218 ret
= ec_GFp_simple_set_Jprojective_coordinates_GFp(group
, r
, x
, y
, z
, ctx
);
2222 EC_POINT_free(generator
);
2223 OPENSSL_free(secrets
);
2224 OPENSSL_free(pre_comp
);
2225 OPENSSL_free(tmp_smallfelems
);
2229 int ec_GFp_nistp256_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
2232 NISTP256_PRE_COMP
*pre
= NULL
;
2235 EC_POINT
*generator
= NULL
;
2236 smallfelem tmp_smallfelems
[32];
2237 felem x_tmp
, y_tmp
, z_tmp
;
2239 BN_CTX
*new_ctx
= NULL
;
2242 /* throw away old precomputation */
2243 EC_pre_comp_free(group
);
2247 ctx
= new_ctx
= BN_CTX_new();
2253 x
= BN_CTX_get(ctx
);
2254 y
= BN_CTX_get(ctx
);
2257 /* get the generator */
2258 if (group
->generator
== NULL
)
2260 generator
= EC_POINT_new(group
);
2261 if (generator
== NULL
)
2263 BN_bin2bn(nistp256_curve_params
[3], sizeof(felem_bytearray
), x
);
2264 BN_bin2bn(nistp256_curve_params
[4], sizeof(felem_bytearray
), y
);
2265 if (!EC_POINT_set_affine_coordinates(group
, generator
, x
, y
, ctx
))
2267 if ((pre
= nistp256_pre_comp_new()) == NULL
)
2270 * if the generator is the standard one, use built-in precomputation
2272 if (0 == EC_POINT_cmp(group
, generator
, group
->generator
, ctx
)) {
2273 memcpy(pre
->g_pre_comp
, gmul
, sizeof(pre
->g_pre_comp
));
2276 if ((!BN_to_felem(x_tmp
, group
->generator
->X
)) ||
2277 (!BN_to_felem(y_tmp
, group
->generator
->Y
)) ||
2278 (!BN_to_felem(z_tmp
, group
->generator
->Z
)))
2280 felem_shrink(pre
->g_pre_comp
[0][1][0], x_tmp
);
2281 felem_shrink(pre
->g_pre_comp
[0][1][1], y_tmp
);
2282 felem_shrink(pre
->g_pre_comp
[0][1][2], z_tmp
);
2284 * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
2285 * 2^160*G, 2^224*G for the second one
2287 for (i
= 1; i
<= 8; i
<<= 1) {
2288 point_double_small(pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2289 pre
->g_pre_comp
[1][i
][2], pre
->g_pre_comp
[0][i
][0],
2290 pre
->g_pre_comp
[0][i
][1],
2291 pre
->g_pre_comp
[0][i
][2]);
2292 for (j
= 0; j
< 31; ++j
) {
2293 point_double_small(pre
->g_pre_comp
[1][i
][0],
2294 pre
->g_pre_comp
[1][i
][1],
2295 pre
->g_pre_comp
[1][i
][2],
2296 pre
->g_pre_comp
[1][i
][0],
2297 pre
->g_pre_comp
[1][i
][1],
2298 pre
->g_pre_comp
[1][i
][2]);
2302 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2303 pre
->g_pre_comp
[0][2 * i
][1],
2304 pre
->g_pre_comp
[0][2 * i
][2],
2305 pre
->g_pre_comp
[1][i
][0], pre
->g_pre_comp
[1][i
][1],
2306 pre
->g_pre_comp
[1][i
][2]);
2307 for (j
= 0; j
< 31; ++j
) {
2308 point_double_small(pre
->g_pre_comp
[0][2 * i
][0],
2309 pre
->g_pre_comp
[0][2 * i
][1],
2310 pre
->g_pre_comp
[0][2 * i
][2],
2311 pre
->g_pre_comp
[0][2 * i
][0],
2312 pre
->g_pre_comp
[0][2 * i
][1],
2313 pre
->g_pre_comp
[0][2 * i
][2]);
2316 for (i
= 0; i
< 2; i
++) {
2317 /* g_pre_comp[i][0] is the point at infinity */
2318 memset(pre
->g_pre_comp
[i
][0], 0, sizeof(pre
->g_pre_comp
[i
][0]));
2319 /* the remaining multiples */
2320 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2321 point_add_small(pre
->g_pre_comp
[i
][6][0], pre
->g_pre_comp
[i
][6][1],
2322 pre
->g_pre_comp
[i
][6][2], pre
->g_pre_comp
[i
][4][0],
2323 pre
->g_pre_comp
[i
][4][1], pre
->g_pre_comp
[i
][4][2],
2324 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2325 pre
->g_pre_comp
[i
][2][2]);
2326 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2327 point_add_small(pre
->g_pre_comp
[i
][10][0], pre
->g_pre_comp
[i
][10][1],
2328 pre
->g_pre_comp
[i
][10][2], pre
->g_pre_comp
[i
][8][0],
2329 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2330 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2331 pre
->g_pre_comp
[i
][2][2]);
2332 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2333 point_add_small(pre
->g_pre_comp
[i
][12][0], pre
->g_pre_comp
[i
][12][1],
2334 pre
->g_pre_comp
[i
][12][2], pre
->g_pre_comp
[i
][8][0],
2335 pre
->g_pre_comp
[i
][8][1], pre
->g_pre_comp
[i
][8][2],
2336 pre
->g_pre_comp
[i
][4][0], pre
->g_pre_comp
[i
][4][1],
2337 pre
->g_pre_comp
[i
][4][2]);
2339 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
2341 point_add_small(pre
->g_pre_comp
[i
][14][0], pre
->g_pre_comp
[i
][14][1],
2342 pre
->g_pre_comp
[i
][14][2], pre
->g_pre_comp
[i
][12][0],
2343 pre
->g_pre_comp
[i
][12][1], pre
->g_pre_comp
[i
][12][2],
2344 pre
->g_pre_comp
[i
][2][0], pre
->g_pre_comp
[i
][2][1],
2345 pre
->g_pre_comp
[i
][2][2]);
2346 for (j
= 1; j
< 8; ++j
) {
2347 /* odd multiples: add G resp. 2^32*G */
2348 point_add_small(pre
->g_pre_comp
[i
][2 * j
+ 1][0],
2349 pre
->g_pre_comp
[i
][2 * j
+ 1][1],
2350 pre
->g_pre_comp
[i
][2 * j
+ 1][2],
2351 pre
->g_pre_comp
[i
][2 * j
][0],
2352 pre
->g_pre_comp
[i
][2 * j
][1],
2353 pre
->g_pre_comp
[i
][2 * j
][2],
2354 pre
->g_pre_comp
[i
][1][0],
2355 pre
->g_pre_comp
[i
][1][1],
2356 pre
->g_pre_comp
[i
][1][2]);
2359 make_points_affine(31, &(pre
->g_pre_comp
[0][1]), tmp_smallfelems
);
2362 SETPRECOMP(group
, nistp256
, pre
);
2368 EC_POINT_free(generator
);
2370 BN_CTX_free(new_ctx
);
2372 EC_nistp256_pre_comp_free(pre
);
2376 int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP
*group
)
2378 return HAVEPRECOMP(group
, nistp256
);