2 * Copyright 2011-2018 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
29 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
30 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
31 * work which got its smarts from Daniel J. Bernstein's work on the same.
34 #include <openssl/opensslconf.h>
35 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
36 NON_EMPTY_TRANSLATION_UNIT
41 # include <openssl/err.h>
44 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
45 /* even with gcc, the typedef won't work for 32-bit platforms */
46 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
48 typedef __int128_t int128_t;
50 # error "Your compiler doesn't appear to support 128-bit integer types"
58 * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
59 * can serialise an element of this field into 32 bytes. We call this an
63 typedef u8 felem_bytearray[32];
66 * These are the parameters of P256, taken from FIPS 186-3, page 86. These
67 * values are big-endian.
69 static const felem_bytearray nistp256_curve_params[5] = {
70 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
71 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
72 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
73 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
74 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
75 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
76 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
77 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */
78 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
79 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
80 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
81 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
82 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
83 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
84 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
85 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
86 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
87 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
88 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
89 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
93 * The representation of field elements.
94 * ------------------------------------
96 * We represent field elements with either four 128-bit values, eight 128-bit
97 * values, or four 64-bit values. The field element represented is:
98 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
100 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
102 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
103 * apart, but are 128-bits wide, the most significant bits of each limb overlap
104 * with the least significant bits of the next.
106 * A field element with four limbs is an 'felem'. One with eight limbs is a
109 * A field element with four, 64-bit values is called a 'smallfelem'. Small
110 * values are used as intermediate values before multiplication.
115 typedef uint128_t limb;
116 typedef limb felem[NLIMBS];
117 typedef limb longfelem[NLIMBS * 2];
118 typedef u64 smallfelem[NLIMBS];
120 /* This is the value of the prime as four 64-bit words, little-endian. */
121 static const u64 kPrime[4] =
122 { 0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul };
123 static const u64 bottom63bits = 0x7ffffffffffffffful;
126 * bin32_to_felem takes a little-endian byte array and converts it into felem
127 * form. This assumes that the CPU is little-endian.
129 static void bin32_to_felem(felem out, const u8 in[32])
131 out[0] = *((u64 *)&in[0]);
132 out[1] = *((u64 *)&in[8]);
133 out[2] = *((u64 *)&in[16]);
134 out[3] = *((u64 *)&in[24]);
138 * smallfelem_to_bin32 takes a smallfelem and serialises into a little
139 * endian, 32 byte array. This assumes that the CPU is little-endian.
141 static void smallfelem_to_bin32(u8 out[32], const smallfelem in)
143 *((u64 *)&out[0]) = in[0];
144 *((u64 *)&out[8]) = in[1];
145 *((u64 *)&out[16]) = in[2];
146 *((u64 *)&out[24]) = in[3];
149 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
150 static void flip_endian(u8 *out, const u8 *in, unsigned len)
153 for (i = 0; i < len; ++i)
154 out[i] = in[len - 1 - i];
157 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
158 static int BN_to_felem(felem out, const BIGNUM *bn)
160 felem_bytearray b_in;
161 felem_bytearray b_out;
164 /* BN_bn2bin eats leading zeroes */
165 memset(b_out, 0, sizeof(b_out));
166 num_bytes = BN_num_bytes(bn);
167 if (num_bytes > sizeof(b_out)) {
168 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
171 if (BN_is_negative(bn)) {
172 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
175 num_bytes = BN_bn2bin(bn, b_in);
176 flip_endian(b_out, b_in, num_bytes);
177 bin32_to_felem(out, b_out);
181 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
182 static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in)
184 felem_bytearray b_in, b_out;
185 smallfelem_to_bin32(b_in, in);
186 flip_endian(b_out, b_in, sizeof(b_out));
187 return BN_bin2bn(b_out, sizeof(b_out), out);
195 static void smallfelem_one(smallfelem out)
203 static void smallfelem_assign(smallfelem out, const smallfelem in)
211 static void felem_assign(felem out, const felem in)
219 /* felem_sum sets out = out + in. */
220 static void felem_sum(felem out, const felem in)
228 /* felem_small_sum sets out = out + in. */
229 static void felem_small_sum(felem out, const smallfelem in)
237 /* felem_scalar sets out = out * scalar */
238 static void felem_scalar(felem out, const u64 scalar)
246 /* longfelem_scalar sets out = out * scalar */
247 static void longfelem_scalar(longfelem out, const u64 scalar)
259 # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
260 # define two105 (((limb)1) << 105)
261 # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
263 /* zero105 is 0 mod p */
264 static const felem zero105 =
265 { two105m41m9, two105, two105m41p9, two105m41p9 };
268 * smallfelem_neg sets |out| to |-small|
270 * out[i] < out[i] + 2^105
272 static void smallfelem_neg(felem out, const smallfelem small)
274 /* In order to prevent underflow, we subtract from 0 mod p. */
275 out[0] = zero105[0] - small[0];
276 out[1] = zero105[1] - small[1];
277 out[2] = zero105[2] - small[2];
278 out[3] = zero105[3] - small[3];
282 * felem_diff subtracts |in| from |out|
286 * out[i] < out[i] + 2^105
288 static void felem_diff(felem out, const felem in)
291 * In order to prevent underflow, we add 0 mod p before subtracting.
293 out[0] += zero105[0];
294 out[1] += zero105[1];
295 out[2] += zero105[2];
296 out[3] += zero105[3];
304 # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
305 # define two107 (((limb)1) << 107)
306 # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
308 /* zero107 is 0 mod p */
309 static const felem zero107 =
310 { two107m43m11, two107, two107m43p11, two107m43p11 };
313 * An alternative felem_diff for larger inputs |in|
314 * felem_diff_zero107 subtracts |in| from |out|
318 * out[i] < out[i] + 2^107
320 static void felem_diff_zero107(felem out, const felem in)
323 * In order to prevent underflow, we add 0 mod p before subtracting.
325 out[0] += zero107[0];
326 out[1] += zero107[1];
327 out[2] += zero107[2];
328 out[3] += zero107[3];
337 * longfelem_diff subtracts |in| from |out|
341 * out[i] < out[i] + 2^70 + 2^40
343 static void longfelem_diff(longfelem out, const longfelem in)
345 static const limb two70m8p6 =
346 (((limb) 1) << 70) - (((limb) 1) << 8) + (((limb) 1) << 6);
347 static const limb two70p40 = (((limb) 1) << 70) + (((limb) 1) << 40);
348 static const limb two70 = (((limb) 1) << 70);
349 static const limb two70m40m38p6 =
350 (((limb) 1) << 70) - (((limb) 1) << 40) - (((limb) 1) << 38) +
352 static const limb two70m6 = (((limb) 1) << 70) - (((limb) 1) << 6);
354 /* add 0 mod p to avoid underflow */
358 out[3] += two70m40m38p6;
364 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
375 # define two64m0 (((limb)1) << 64) - 1
376 # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
377 # define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
378 # define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
380 /* zero110 is 0 mod p */
381 static const felem zero110 = { two64m0, two110p32m0, two64m46, two64m32 };
384 * felem_shrink converts an felem into a smallfelem. The result isn't quite
385 * minimal as the value may be greater than p.
392 static void felem_shrink(smallfelem out, const felem in)
397 static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */
400 tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64));
403 tmp[2] = zero110[2] + (u64)in[2];
404 tmp[0] = zero110[0] + in[0];
405 tmp[1] = zero110[1] + in[1];
406 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
409 * We perform two partial reductions where we eliminate the high-word of
410 * tmp[3]. We don't update the other words till the end.
412 a = tmp[3] >> 64; /* a < 2^46 */
413 tmp[3] = (u64)tmp[3];
415 tmp[3] += ((limb) a) << 32;
419 a = tmp[3] >> 64; /* a < 2^15 */
420 b += a; /* b < 2^46 + 2^15 < 2^47 */
421 tmp[3] = (u64)tmp[3];
423 tmp[3] += ((limb) a) << 32;
424 /* tmp[3] < 2^64 + 2^47 */
427 * This adjusts the other two words to complete the two partial
431 tmp[1] -= (((limb) b) << 32);
434 * In order to make space in tmp[3] for the carry from 2 -> 3, we
435 * conditionally subtract kPrime if tmp[3] is large enough.
437 high = (u64)(tmp[3] >> 64);
438 /* As tmp[3] < 2^65, high is either 1 or 0 */
442 * all ones if the high word of tmp[3] is 1
443 * all zeros if the high word of tmp[3] if 0
446 mask = 0 - (low >> 63);
449 * all ones if the MSB of low is 1
450 * all zeros if the MSB of low if 0
454 /* if low was greater than kPrime3Test then the MSB is zero */
456 low = 0 - (low >> 63);
459 * all ones if low was > kPrime3Test
460 * all zeros if low was <= kPrime3Test
462 mask = (mask & low) | high;
463 tmp[0] -= mask & kPrime[0];
464 tmp[1] -= mask & kPrime[1];
465 /* kPrime[2] is zero, so omitted */
466 tmp[3] -= mask & kPrime[3];
467 /* tmp[3] < 2**64 - 2**32 + 1 */
469 tmp[1] += ((u64)(tmp[0] >> 64));
470 tmp[0] = (u64)tmp[0];
471 tmp[2] += ((u64)(tmp[1] >> 64));
472 tmp[1] = (u64)tmp[1];
473 tmp[3] += ((u64)(tmp[2] >> 64));
474 tmp[2] = (u64)tmp[2];
483 /* smallfelem_expand converts a smallfelem to an felem */
484 static void smallfelem_expand(felem out, const smallfelem in)
493 * smallfelem_square sets |out| = |small|^2
497 * out[i] < 7 * 2^64 < 2^67
499 static void smallfelem_square(longfelem out, const smallfelem small)
504 a = ((uint128_t) small[0]) * small[0];
510 a = ((uint128_t) small[0]) * small[1];
517 a = ((uint128_t) small[0]) * small[2];
524 a = ((uint128_t) small[0]) * small[3];
530 a = ((uint128_t) small[1]) * small[2];
537 a = ((uint128_t) small[1]) * small[1];
543 a = ((uint128_t) small[1]) * small[3];
550 a = ((uint128_t) small[2]) * small[3];
558 a = ((uint128_t) small[2]) * small[2];
564 a = ((uint128_t) small[3]) * small[3];
572 * felem_square sets |out| = |in|^2
576 * out[i] < 7 * 2^64 < 2^67
578 static void felem_square(longfelem out, const felem in)
581 felem_shrink(small, in);
582 smallfelem_square(out, small);
586 * smallfelem_mul sets |out| = |small1| * |small2|
591 * out[i] < 7 * 2^64 < 2^67
593 static void smallfelem_mul(longfelem out, const smallfelem small1,
594 const smallfelem small2)
599 a = ((uint128_t) small1[0]) * small2[0];
605 a = ((uint128_t) small1[0]) * small2[1];
611 a = ((uint128_t) small1[1]) * small2[0];
617 a = ((uint128_t) small1[0]) * small2[2];
623 a = ((uint128_t) small1[1]) * small2[1];
629 a = ((uint128_t) small1[2]) * small2[0];
635 a = ((uint128_t) small1[0]) * small2[3];
641 a = ((uint128_t) small1[1]) * small2[2];
647 a = ((uint128_t) small1[2]) * small2[1];
653 a = ((uint128_t) small1[3]) * small2[0];
659 a = ((uint128_t) small1[1]) * small2[3];
665 a = ((uint128_t) small1[2]) * small2[2];
671 a = ((uint128_t) small1[3]) * small2[1];
677 a = ((uint128_t) small1[2]) * small2[3];
683 a = ((uint128_t) small1[3]) * small2[2];
689 a = ((uint128_t) small1[3]) * small2[3];
697 * felem_mul sets |out| = |in1| * |in2|
702 * out[i] < 7 * 2^64 < 2^67
704 static void felem_mul(longfelem out, const felem in1, const felem in2)
706 smallfelem small1, small2;
707 felem_shrink(small1, in1);
708 felem_shrink(small2, in2);
709 smallfelem_mul(out, small1, small2);
713 * felem_small_mul sets |out| = |small1| * |in2|
718 * out[i] < 7 * 2^64 < 2^67
720 static void felem_small_mul(longfelem out, const smallfelem small1,
724 felem_shrink(small2, in2);
725 smallfelem_mul(out, small1, small2);
728 # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
729 # define two100 (((limb)1) << 100)
730 # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
731 /* zero100 is 0 mod p */
732 static const felem zero100 =
733 { two100m36m4, two100, two100m36p4, two100m36p4 };
736 * Internal function for the different flavours of felem_reduce.
737 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
739 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
740 * out[1] >= in[7] + 2^32*in[4]
741 * out[2] >= in[5] + 2^32*in[5]
742 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
744 * out[0] <= out[0] + in[4] + 2^32*in[5]
745 * out[1] <= out[1] + in[5] + 2^33*in[6]
746 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
747 * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
749 static void felem_reduce_(felem out, const longfelem in)
752 /* combine common terms from below */
753 c = in[4] + (in[5] << 32);
761 /* the remaining terms */
762 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
763 out[1] -= (in[4] << 32);
764 out[3] += (in[4] << 32);
766 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
767 out[2] -= (in[5] << 32);
769 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
771 out[0] -= (in[6] << 32);
772 out[1] += (in[6] << 33);
773 out[2] += (in[6] * 2);
774 out[3] -= (in[6] << 32);
776 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
778 out[0] -= (in[7] << 32);
779 out[2] += (in[7] << 33);
780 out[3] += (in[7] * 3);
784 * felem_reduce converts a longfelem into an felem.
785 * To be called directly after felem_square or felem_mul.
787 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
788 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
792 static void felem_reduce(felem out, const longfelem in)
794 out[0] = zero100[0] + in[0];
795 out[1] = zero100[1] + in[1];
796 out[2] = zero100[2] + in[2];
797 out[3] = zero100[3] + in[3];
799 felem_reduce_(out, in);
802 * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
803 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
804 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
805 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
807 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
808 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
809 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
810 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
815 * felem_reduce_zero105 converts a larger longfelem into an felem.
821 static void felem_reduce_zero105(felem out, const longfelem in)
823 out[0] = zero105[0] + in[0];
824 out[1] = zero105[1] + in[1];
825 out[2] = zero105[2] + in[2];
826 out[3] = zero105[3] + in[3];
828 felem_reduce_(out, in);
831 * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
832 * out[1] > 2^105 - 2^71 - 2^103 > 0
833 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
834 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
836 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
837 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
838 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
839 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
844 * subtract_u64 sets *result = *result - v and *carry to one if the
845 * subtraction underflowed.
847 static void subtract_u64(u64 *result, u64 *carry, u64 v)
849 uint128_t r = *result;
851 *carry = (r >> 64) & 1;
856 * felem_contract converts |in| to its unique, minimal representation. On
857 * entry: in[i] < 2^109
859 static void felem_contract(smallfelem out, const felem in)
862 u64 all_equal_so_far = 0, result = 0, carry;
864 felem_shrink(out, in);
865 /* small is minimal except that the value might be > p */
869 * We are doing a constant time test if out >= kPrime. We need to compare
870 * each u64, from most-significant to least significant. For each one, if
871 * all words so far have been equal (m is all ones) then a non-equal
872 * result is the answer. Otherwise we continue.
874 for (i = 3; i < 4; i--) {
876 uint128_t a = ((uint128_t) kPrime[i]) - out[i];
878 * if out[i] > kPrime[i] then a will underflow and the high 64-bits
881 result |= all_equal_so_far & ((u64)(a >> 64));
884 * if kPrime[i] == out[i] then |equal| will be all zeros and the
885 * decrement will make it all ones.
887 equal = kPrime[i] ^ out[i];
889 equal &= equal << 32;
890 equal &= equal << 16;
895 equal = 0 - (equal >> 63);
897 all_equal_so_far &= equal;
901 * if all_equal_so_far is still all ones then the two values are equal
902 * and so out >= kPrime is true.
904 result |= all_equal_so_far;
906 /* if out >= kPrime then we subtract kPrime. */
907 subtract_u64(&out[0], &carry, result & kPrime[0]);
908 subtract_u64(&out[1], &carry, carry);
909 subtract_u64(&out[2], &carry, carry);
910 subtract_u64(&out[3], &carry, carry);
912 subtract_u64(&out[1], &carry, result & kPrime[1]);
913 subtract_u64(&out[2], &carry, carry);
914 subtract_u64(&out[3], &carry, carry);
916 subtract_u64(&out[2], &carry, result & kPrime[2]);
917 subtract_u64(&out[3], &carry, carry);
919 subtract_u64(&out[3], &carry, result & kPrime[3]);
922 static void smallfelem_square_contract(smallfelem out, const smallfelem in)
927 smallfelem_square(longtmp, in);
928 felem_reduce(tmp, longtmp);
929 felem_contract(out, tmp);
932 static void smallfelem_mul_contract(smallfelem out, const smallfelem in1,
933 const smallfelem in2)
938 smallfelem_mul(longtmp, in1, in2);
939 felem_reduce(tmp, longtmp);
940 felem_contract(out, tmp);
944 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
949 static limb smallfelem_is_zero(const smallfelem small)
954 u64 is_zero = small[0] | small[1] | small[2] | small[3];
956 is_zero &= is_zero << 32;
957 is_zero &= is_zero << 16;
958 is_zero &= is_zero << 8;
959 is_zero &= is_zero << 4;
960 is_zero &= is_zero << 2;
961 is_zero &= is_zero << 1;
962 is_zero = 0 - (is_zero >> 63);
964 is_p = (small[0] ^ kPrime[0]) |
965 (small[1] ^ kPrime[1]) |
966 (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]);
974 is_p = 0 - (is_p >> 63);
979 result |= ((limb) is_zero) << 64;
983 static int smallfelem_is_zero_int(const void *small)
985 return (int)(smallfelem_is_zero(small) & ((limb) 1));
989 * felem_inv calculates |out| = |in|^{-1}
991 * Based on Fermat's Little Theorem:
993 * a^{p-1} = 1 (mod p)
994 * a^{p-2} = a^{-1} (mod p)
996 static void felem_inv(felem out, const felem in)
999 /* each e_I will hold |in|^{2^I - 1} */
1000 felem e2, e4, e8, e16, e32, e64;
1004 felem_square(tmp, in);
1005 felem_reduce(ftmp, tmp); /* 2^1 */
1006 felem_mul(tmp, in, ftmp);
1007 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
1008 felem_assign(e2, ftmp);
1009 felem_square(tmp, ftmp);
1010 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
1011 felem_square(tmp, ftmp);
1012 felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */
1013 felem_mul(tmp, ftmp, e2);
1014 felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */
1015 felem_assign(e4, ftmp);
1016 felem_square(tmp, ftmp);
1017 felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */
1018 felem_square(tmp, ftmp);
1019 felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */
1020 felem_square(tmp, ftmp);
1021 felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */
1022 felem_square(tmp, ftmp);
1023 felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */
1024 felem_mul(tmp, ftmp, e4);
1025 felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */
1026 felem_assign(e8, ftmp);
1027 for (i = 0; i < 8; i++) {
1028 felem_square(tmp, ftmp);
1029 felem_reduce(ftmp, tmp);
1031 felem_mul(tmp, ftmp, e8);
1032 felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */
1033 felem_assign(e16, ftmp);
1034 for (i = 0; i < 16; i++) {
1035 felem_square(tmp, ftmp);
1036 felem_reduce(ftmp, tmp);
1038 felem_mul(tmp, ftmp, e16);
1039 felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */
1040 felem_assign(e32, ftmp);
1041 for (i = 0; i < 32; i++) {
1042 felem_square(tmp, ftmp);
1043 felem_reduce(ftmp, tmp);
1045 felem_assign(e64, ftmp);
1046 felem_mul(tmp, ftmp, in);
1047 felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */
1048 for (i = 0; i < 192; i++) {
1049 felem_square(tmp, ftmp);
1050 felem_reduce(ftmp, tmp);
1051 } /* 2^256 - 2^224 + 2^192 */
1053 felem_mul(tmp, e64, e32);
1054 felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */
1055 for (i = 0; i < 16; i++) {
1056 felem_square(tmp, ftmp2);
1057 felem_reduce(ftmp2, tmp);
1059 felem_mul(tmp, ftmp2, e16);
1060 felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */
1061 for (i = 0; i < 8; i++) {
1062 felem_square(tmp, ftmp2);
1063 felem_reduce(ftmp2, tmp);
1065 felem_mul(tmp, ftmp2, e8);
1066 felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */
1067 for (i = 0; i < 4; i++) {
1068 felem_square(tmp, ftmp2);
1069 felem_reduce(ftmp2, tmp);
1071 felem_mul(tmp, ftmp2, e4);
1072 felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */
1073 felem_square(tmp, ftmp2);
1074 felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */
1075 felem_square(tmp, ftmp2);
1076 felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */
1077 felem_mul(tmp, ftmp2, e2);
1078 felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */
1079 felem_square(tmp, ftmp2);
1080 felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */
1081 felem_square(tmp, ftmp2);
1082 felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */
1083 felem_mul(tmp, ftmp2, in);
1084 felem_reduce(ftmp2, tmp); /* 2^96 - 3 */
1086 felem_mul(tmp, ftmp2, ftmp);
1087 felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1090 static void smallfelem_inv_contract(smallfelem out, const smallfelem in)
1094 smallfelem_expand(tmp, in);
1095 felem_inv(tmp, tmp);
1096 felem_contract(out, tmp);
1103 * Building on top of the field operations we have the operations on the
1104 * elliptic curve group itself. Points on the curve are represented in Jacobian
1109 * point_double calculates 2*(x_in, y_in, z_in)
1111 * The method is taken from:
1112 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1114 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1115 * while x_out == y_in is not (maybe this works, but it's not tested).
1118 point_double(felem x_out, felem y_out, felem z_out,
1119 const felem x_in, const felem y_in, const felem z_in)
1121 longfelem tmp, tmp2;
1122 felem delta, gamma, beta, alpha, ftmp, ftmp2;
1123 smallfelem small1, small2;
1125 felem_assign(ftmp, x_in);
1126 /* ftmp[i] < 2^106 */
1127 felem_assign(ftmp2, x_in);
1128 /* ftmp2[i] < 2^106 */
1131 felem_square(tmp, z_in);
1132 felem_reduce(delta, tmp);
1133 /* delta[i] < 2^101 */
1136 felem_square(tmp, y_in);
1137 felem_reduce(gamma, tmp);
1138 /* gamma[i] < 2^101 */
1139 felem_shrink(small1, gamma);
1141 /* beta = x*gamma */
1142 felem_small_mul(tmp, small1, x_in);
1143 felem_reduce(beta, tmp);
1144 /* beta[i] < 2^101 */
1146 /* alpha = 3*(x-delta)*(x+delta) */
1147 felem_diff(ftmp, delta);
1148 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1149 felem_sum(ftmp2, delta);
1150 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1151 felem_scalar(ftmp2, 3);
1152 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1153 felem_mul(tmp, ftmp, ftmp2);
1154 felem_reduce(alpha, tmp);
1155 /* alpha[i] < 2^101 */
1156 felem_shrink(small2, alpha);
1158 /* x' = alpha^2 - 8*beta */
1159 smallfelem_square(tmp, small2);
1160 felem_reduce(x_out, tmp);
1161 felem_assign(ftmp, beta);
1162 felem_scalar(ftmp, 8);
1163 /* ftmp[i] < 8 * 2^101 = 2^104 */
1164 felem_diff(x_out, ftmp);
1165 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1167 /* z' = (y + z)^2 - gamma - delta */
1168 felem_sum(delta, gamma);
1169 /* delta[i] < 2^101 + 2^101 = 2^102 */
1170 felem_assign(ftmp, y_in);
1171 felem_sum(ftmp, z_in);
1172 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1173 felem_square(tmp, ftmp);
1174 felem_reduce(z_out, tmp);
1175 felem_diff(z_out, delta);
1176 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1178 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1179 felem_scalar(beta, 4);
1180 /* beta[i] < 4 * 2^101 = 2^103 */
1181 felem_diff_zero107(beta, x_out);
1182 /* beta[i] < 2^107 + 2^103 < 2^108 */
1183 felem_small_mul(tmp, small2, beta);
1184 /* tmp[i] < 7 * 2^64 < 2^67 */
1185 smallfelem_square(tmp2, small1);
1186 /* tmp2[i] < 7 * 2^64 */
1187 longfelem_scalar(tmp2, 8);
1188 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1189 longfelem_diff(tmp, tmp2);
1190 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1191 felem_reduce_zero105(y_out, tmp);
1192 /* y_out[i] < 2^106 */
1196 * point_double_small is the same as point_double, except that it operates on
1200 point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out,
1201 const smallfelem x_in, const smallfelem y_in,
1202 const smallfelem z_in)
1204 felem felem_x_out, felem_y_out, felem_z_out;
1205 felem felem_x_in, felem_y_in, felem_z_in;
1207 smallfelem_expand(felem_x_in, x_in);
1208 smallfelem_expand(felem_y_in, y_in);
1209 smallfelem_expand(felem_z_in, z_in);
1210 point_double(felem_x_out, felem_y_out, felem_z_out,
1211 felem_x_in, felem_y_in, felem_z_in);
1212 felem_shrink(x_out, felem_x_out);
1213 felem_shrink(y_out, felem_y_out);
1214 felem_shrink(z_out, felem_z_out);
1217 /* copy_conditional copies in to out iff mask is all ones. */
1218 static void copy_conditional(felem out, const felem in, limb mask)
1221 for (i = 0; i < NLIMBS; ++i) {
1222 const limb tmp = mask & (in[i] ^ out[i]);
1227 /* copy_small_conditional copies in to out iff mask is all ones. */
1228 static void copy_small_conditional(felem out, const smallfelem in, limb mask)
1231 const u64 mask64 = mask;
1232 for (i = 0; i < NLIMBS; ++i) {
1233 out[i] = ((limb) (in[i] & mask64)) | (out[i] & ~mask);
1238 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
1240 * The method is taken from:
1241 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1242 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1244 * This function includes a branch for checking whether the two input points
1245 * are equal, (while not equal to the point at infinity). This case never
1246 * happens during single point multiplication, so there is no timing leak for
1247 * ECDH or ECDSA signing.
1249 static void point_add(felem x3, felem y3, felem z3,
1250 const felem x1, const felem y1, const felem z1,
1251 const int mixed, const smallfelem x2,
1252 const smallfelem y2, const smallfelem z2)
1254 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1255 longfelem tmp, tmp2;
1256 smallfelem small1, small2, small3, small4, small5;
1257 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1259 felem_shrink(small3, z1);
1261 z1_is_zero = smallfelem_is_zero(small3);
1262 z2_is_zero = smallfelem_is_zero(z2);
1264 /* ftmp = z1z1 = z1**2 */
1265 smallfelem_square(tmp, small3);
1266 felem_reduce(ftmp, tmp);
1267 /* ftmp[i] < 2^101 */
1268 felem_shrink(small1, ftmp);
1271 /* ftmp2 = z2z2 = z2**2 */
1272 smallfelem_square(tmp, z2);
1273 felem_reduce(ftmp2, tmp);
1274 /* ftmp2[i] < 2^101 */
1275 felem_shrink(small2, ftmp2);
1277 felem_shrink(small5, x1);
1279 /* u1 = ftmp3 = x1*z2z2 */
1280 smallfelem_mul(tmp, small5, small2);
1281 felem_reduce(ftmp3, tmp);
1282 /* ftmp3[i] < 2^101 */
1284 /* ftmp5 = z1 + z2 */
1285 felem_assign(ftmp5, z1);
1286 felem_small_sum(ftmp5, z2);
1287 /* ftmp5[i] < 2^107 */
1289 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1290 felem_square(tmp, ftmp5);
1291 felem_reduce(ftmp5, tmp);
1292 /* ftmp2 = z2z2 + z1z1 */
1293 felem_sum(ftmp2, ftmp);
1294 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1295 felem_diff(ftmp5, ftmp2);
1296 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1298 /* ftmp2 = z2 * z2z2 */
1299 smallfelem_mul(tmp, small2, z2);
1300 felem_reduce(ftmp2, tmp);
1302 /* s1 = ftmp2 = y1 * z2**3 */
1303 felem_mul(tmp, y1, ftmp2);
1304 felem_reduce(ftmp6, tmp);
1305 /* ftmp6[i] < 2^101 */
1308 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
1311 /* u1 = ftmp3 = x1*z2z2 */
1312 felem_assign(ftmp3, x1);
1313 /* ftmp3[i] < 2^106 */
1316 felem_assign(ftmp5, z1);
1317 felem_scalar(ftmp5, 2);
1318 /* ftmp5[i] < 2*2^106 = 2^107 */
1320 /* s1 = ftmp2 = y1 * z2**3 */
1321 felem_assign(ftmp6, y1);
1322 /* ftmp6[i] < 2^106 */
1326 smallfelem_mul(tmp, x2, small1);
1327 felem_reduce(ftmp4, tmp);
1329 /* h = ftmp4 = u2 - u1 */
1330 felem_diff_zero107(ftmp4, ftmp3);
1331 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1332 felem_shrink(small4, ftmp4);
1334 x_equal = smallfelem_is_zero(small4);
1336 /* z_out = ftmp5 * h */
1337 felem_small_mul(tmp, small4, ftmp5);
1338 felem_reduce(z_out, tmp);
1339 /* z_out[i] < 2^101 */
1341 /* ftmp = z1 * z1z1 */
1342 smallfelem_mul(tmp, small1, small3);
1343 felem_reduce(ftmp, tmp);
1345 /* s2 = tmp = y2 * z1**3 */
1346 felem_small_mul(tmp, y2, ftmp);
1347 felem_reduce(ftmp5, tmp);
1349 /* r = ftmp5 = (s2 - s1)*2 */
1350 felem_diff_zero107(ftmp5, ftmp6);
1351 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1352 felem_scalar(ftmp5, 2);
1353 /* ftmp5[i] < 2^109 */
1354 felem_shrink(small1, ftmp5);
1355 y_equal = smallfelem_is_zero(small1);
1357 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1358 point_double(x3, y3, z3, x1, y1, z1);
1362 /* I = ftmp = (2h)**2 */
1363 felem_assign(ftmp, ftmp4);
1364 felem_scalar(ftmp, 2);
1365 /* ftmp[i] < 2*2^108 = 2^109 */
1366 felem_square(tmp, ftmp);
1367 felem_reduce(ftmp, tmp);
1369 /* J = ftmp2 = h * I */
1370 felem_mul(tmp, ftmp4, ftmp);
1371 felem_reduce(ftmp2, tmp);
1373 /* V = ftmp4 = U1 * I */
1374 felem_mul(tmp, ftmp3, ftmp);
1375 felem_reduce(ftmp4, tmp);
1377 /* x_out = r**2 - J - 2V */
1378 smallfelem_square(tmp, small1);
1379 felem_reduce(x_out, tmp);
1380 felem_assign(ftmp3, ftmp4);
1381 felem_scalar(ftmp4, 2);
1382 felem_sum(ftmp4, ftmp2);
1383 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1384 felem_diff(x_out, ftmp4);
1385 /* x_out[i] < 2^105 + 2^101 */
1387 /* y_out = r(V-x_out) - 2 * s1 * J */
1388 felem_diff_zero107(ftmp3, x_out);
1389 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1390 felem_small_mul(tmp, small1, ftmp3);
1391 felem_mul(tmp2, ftmp6, ftmp2);
1392 longfelem_scalar(tmp2, 2);
1393 /* tmp2[i] < 2*2^67 = 2^68 */
1394 longfelem_diff(tmp, tmp2);
1395 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1396 felem_reduce_zero105(y_out, tmp);
1397 /* y_out[i] < 2^106 */
1399 copy_small_conditional(x_out, x2, z1_is_zero);
1400 copy_conditional(x_out, x1, z2_is_zero);
1401 copy_small_conditional(y_out, y2, z1_is_zero);
1402 copy_conditional(y_out, y1, z2_is_zero);
1403 copy_small_conditional(z_out, z2, z1_is_zero);
1404 copy_conditional(z_out, z1, z2_is_zero);
1405 felem_assign(x3, x_out);
1406 felem_assign(y3, y_out);
1407 felem_assign(z3, z_out);
1411 * point_add_small is the same as point_add, except that it operates on
1414 static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3,
1415 smallfelem x1, smallfelem y1, smallfelem z1,
1416 smallfelem x2, smallfelem y2, smallfelem z2)
1418 felem felem_x3, felem_y3, felem_z3;
1419 felem felem_x1, felem_y1, felem_z1;
1420 smallfelem_expand(felem_x1, x1);
1421 smallfelem_expand(felem_y1, y1);
1422 smallfelem_expand(felem_z1, z1);
1423 point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0,
1425 felem_shrink(x3, felem_x3);
1426 felem_shrink(y3, felem_y3);
1427 felem_shrink(z3, felem_z3);
1431 * Base point pre computation
1432 * --------------------------
1434 * Two different sorts of precomputed tables are used in the following code.
1435 * Each contain various points on the curve, where each point is three field
1436 * elements (x, y, z).
1438 * For the base point table, z is usually 1 (0 for the point at infinity).
1439 * This table has 2 * 16 elements, starting with the following:
1440 * index | bits | point
1441 * ------+---------+------------------------------
1444 * 2 | 0 0 1 0 | 2^64G
1445 * 3 | 0 0 1 1 | (2^64 + 1)G
1446 * 4 | 0 1 0 0 | 2^128G
1447 * 5 | 0 1 0 1 | (2^128 + 1)G
1448 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1449 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1450 * 8 | 1 0 0 0 | 2^192G
1451 * 9 | 1 0 0 1 | (2^192 + 1)G
1452 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1453 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1454 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1455 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1456 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1457 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1458 * followed by a copy of this with each element multiplied by 2^32.
1460 * The reason for this is so that we can clock bits into four different
1461 * locations when doing simple scalar multiplies against the base point,
1462 * and then another four locations using the second 16 elements.
1464 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1466 /* gmul is the table of precomputed base points */
1467 static const smallfelem gmul[2][16][3] = {
1471 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1472 0x6b17d1f2e12c4247},
1473 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1474 0x4fe342e2fe1a7f9b},
1476 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1477 0x0fa822bc2811aaa5},
1478 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1479 0xbff44ae8f5dba80d},
1481 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1482 0x300a4bbc89d6726f},
1483 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1484 0x72aac7e0d09b4644},
1486 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1487 0x447d739beedb5e67},
1488 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1489 0x2d4825ab834131ee},
1491 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1492 0xef9519328a9c72ff},
1493 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1494 0x611e9fc37dbb2c9b},
1496 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1497 0x550663797b51f5d8},
1498 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1499 0x157164848aecb851},
1501 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1502 0xeb5d7745b21141ea},
1503 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1504 0xeafd72ebdbecc17b},
1506 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1507 0xa6d39677a7849276},
1508 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1509 0x674f84749b0b8816},
1511 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1512 0x4e769e7672c9ddad},
1513 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1514 0x42b99082de830663},
1516 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1517 0x78878ef61c6ce04d},
1518 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1519 0xb6cb3f5d7b72c321},
1521 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1522 0x0c88bc4d716b1287},
1523 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1524 0xdd5ddea3f3901dc6},
1526 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1527 0x68f344af6b317466},
1528 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1529 0x31b9c405f8540a20},
1531 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1532 0x4052bf4b6f461db9},
1533 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1534 0xfecf4d5190b0fc61},
1536 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1537 0x1eddbae2c802e41a},
1538 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1539 0x43104d86560ebcfc},
1541 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1542 0xb48e26b484f7a21c},
1543 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1544 0xfac015404d4d3dab},
1549 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1550 0x7fe36b40af22af89},
1551 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1552 0xe697d45825b63624},
1554 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1555 0x4a5b506612a677a6},
1556 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1557 0xeb13461ceac089f1},
1559 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1560 0x0781b8291c6a220a},
1561 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1562 0x690cde8df0151593},
1564 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1565 0x8a535f566ec73617},
1566 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1567 0x0455c08468b08bd7},
1569 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1570 0x06bada7ab77f8276},
1571 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1572 0x5b476dfd0e6cb18a},
1574 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1575 0x3e29864e8a2ec908},
1576 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1577 0x239b90ea3dc31e7e},
1579 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1580 0x820f4dd949f72ff7},
1581 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1582 0x140406ec783a05ec},
1584 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1585 0x68f6b8542783dfee},
1586 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1587 0xcbe1feba92e40ce6},
1589 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1590 0xd0b2f94d2f420109},
1591 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1592 0x971459828b0719e5},
1594 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1595 0x961610004a866aba},
1596 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1597 0x7acb9fadcee75e44},
1599 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1600 0x24eb9acca333bf5b},
1601 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1602 0x69f891c5acd079cc},
1604 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1605 0xe51f547c5972a107},
1606 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1607 0x1c309a2b25bb1387},
1609 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1610 0x20b87b8aa2c4e503},
1611 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1612 0xf5c6fa49919776be},
1614 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1615 0x1ed7d1b9332010b9},
1616 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1617 0x3a2b03f03217257a},
1619 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1620 0x15fee545c78dd9f6},
1621 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1622 0x4ab5b6b2b8753f81},
1627 * select_point selects the |idx|th point from a precomputation table and
1630 static void select_point(const u64 idx, unsigned int size,
1631 const smallfelem pre_comp[16][3], smallfelem out[3])
1634 u64 *outlimbs = &out[0][0];
1636 memset(out, 0, sizeof(*out) * 3);
1638 for (i = 0; i < size; i++) {
1639 const u64 *inlimbs = (u64 *)&pre_comp[i][0][0];
1646 for (j = 0; j < NLIMBS * 3; j++)
1647 outlimbs[j] |= inlimbs[j] & mask;
1651 /* get_bit returns the |i|th bit in |in| */
1652 static char get_bit(const felem_bytearray in, int i)
1654 if ((i < 0) || (i >= 256))
1656 return (in[i >> 3] >> (i & 7)) & 1;
1660 * Interleaved point multiplication using precomputed point multiples: The
1661 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1662 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1663 * generator, using certain (large) precomputed multiples in g_pre_comp.
1664 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1666 static void batch_mul(felem x_out, felem y_out, felem z_out,
1667 const felem_bytearray scalars[],
1668 const unsigned num_points, const u8 *g_scalar,
1669 const int mixed, const smallfelem pre_comp[][17][3],
1670 const smallfelem g_pre_comp[2][16][3])
1673 unsigned num, gen_mul = (g_scalar != NULL);
1679 /* set nq to the point at infinity */
1680 memset(nq, 0, sizeof(nq));
1683 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1684 * of the generator (two in each of the last 32 rounds) and additions of
1685 * other points multiples (every 5th round).
1687 skip = 1; /* save two point operations in the first
1689 for (i = (num_points ? 255 : 31); i >= 0; --i) {
1692 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1694 /* add multiples of the generator */
1695 if (gen_mul && (i <= 31)) {
1696 /* first, look 32 bits upwards */
1697 bits = get_bit(g_scalar, i + 224) << 3;
1698 bits |= get_bit(g_scalar, i + 160) << 2;
1699 bits |= get_bit(g_scalar, i + 96) << 1;
1700 bits |= get_bit(g_scalar, i + 32);
1701 /* select the point to add, in constant time */
1702 select_point(bits, 16, g_pre_comp[1], tmp);
1705 /* Arg 1 below is for "mixed" */
1706 point_add(nq[0], nq[1], nq[2],
1707 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1709 smallfelem_expand(nq[0], tmp[0]);
1710 smallfelem_expand(nq[1], tmp[1]);
1711 smallfelem_expand(nq[2], tmp[2]);
1715 /* second, look at the current position */
1716 bits = get_bit(g_scalar, i + 192) << 3;
1717 bits |= get_bit(g_scalar, i + 128) << 2;
1718 bits |= get_bit(g_scalar, i + 64) << 1;
1719 bits |= get_bit(g_scalar, i);
1720 /* select the point to add, in constant time */
1721 select_point(bits, 16, g_pre_comp[0], tmp);
1722 /* Arg 1 below is for "mixed" */
1723 point_add(nq[0], nq[1], nq[2],
1724 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1727 /* do other additions every 5 doublings */
1728 if (num_points && (i % 5 == 0)) {
1729 /* loop over all scalars */
1730 for (num = 0; num < num_points; ++num) {
1731 bits = get_bit(scalars[num], i + 4) << 5;
1732 bits |= get_bit(scalars[num], i + 3) << 4;
1733 bits |= get_bit(scalars[num], i + 2) << 3;
1734 bits |= get_bit(scalars[num], i + 1) << 2;
1735 bits |= get_bit(scalars[num], i) << 1;
1736 bits |= get_bit(scalars[num], i - 1);
1737 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1740 * select the point to add or subtract, in constant time
1742 select_point(digit, 17, pre_comp[num], tmp);
1743 smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative
1745 copy_small_conditional(ftmp, tmp[1], (((limb) sign) - 1));
1746 felem_contract(tmp[1], ftmp);
1749 point_add(nq[0], nq[1], nq[2],
1750 nq[0], nq[1], nq[2],
1751 mixed, tmp[0], tmp[1], tmp[2]);
1753 smallfelem_expand(nq[0], tmp[0]);
1754 smallfelem_expand(nq[1], tmp[1]);
1755 smallfelem_expand(nq[2], tmp[2]);
1761 felem_assign(x_out, nq[0]);
1762 felem_assign(y_out, nq[1]);
1763 felem_assign(z_out, nq[2]);
1766 /* Precomputation for the group generator. */
1767 struct nistp256_pre_comp_st {
1768 smallfelem g_pre_comp[2][16][3];
1769 CRYPTO_REF_COUNT references;
1770 CRYPTO_RWLOCK *lock;
1773 const EC_METHOD *EC_GFp_nistp256_method(void)
1775 static const EC_METHOD ret = {
1776 EC_FLAGS_DEFAULT_OCT,
1777 NID_X9_62_prime_field,
1778 ec_GFp_nistp256_group_init,
1779 ec_GFp_simple_group_finish,
1780 ec_GFp_simple_group_clear_finish,
1781 ec_GFp_nist_group_copy,
1782 ec_GFp_nistp256_group_set_curve,
1783 ec_GFp_simple_group_get_curve,
1784 ec_GFp_simple_group_get_degree,
1785 ec_group_simple_order_bits,
1786 ec_GFp_simple_group_check_discriminant,
1787 ec_GFp_simple_point_init,
1788 ec_GFp_simple_point_finish,
1789 ec_GFp_simple_point_clear_finish,
1790 ec_GFp_simple_point_copy,
1791 ec_GFp_simple_point_set_to_infinity,
1792 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1793 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1794 ec_GFp_simple_point_set_affine_coordinates,
1795 ec_GFp_nistp256_point_get_affine_coordinates,
1796 0 /* point_set_compressed_coordinates */ ,
1801 ec_GFp_simple_invert,
1802 ec_GFp_simple_is_at_infinity,
1803 ec_GFp_simple_is_on_curve,
1805 ec_GFp_simple_make_affine,
1806 ec_GFp_simple_points_make_affine,
1807 ec_GFp_nistp256_points_mul,
1808 ec_GFp_nistp256_precompute_mult,
1809 ec_GFp_nistp256_have_precompute_mult,
1810 ec_GFp_nist_field_mul,
1811 ec_GFp_nist_field_sqr,
1813 ec_GFp_simple_field_inv,
1814 0 /* field_encode */ ,
1815 0 /* field_decode */ ,
1816 0, /* field_set_to_one */
1817 ec_key_simple_priv2oct,
1818 ec_key_simple_oct2priv,
1819 0, /* set private */
1820 ec_key_simple_generate_key,
1821 ec_key_simple_check_key,
1822 ec_key_simple_generate_public_key,
1825 ecdh_simple_compute_key,
1826 0, /* field_inverse_mod_ord */
1827 0, /* blind_coordinates */
1829 0, /* ladder_step */
1836 /******************************************************************************/
1838 * FUNCTIONS TO MANAGE PRECOMPUTATION
1841 static NISTP256_PRE_COMP *nistp256_pre_comp_new(void)
1843 NISTP256_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1846 ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1850 ret->references = 1;
1852 ret->lock = CRYPTO_THREAD_lock_new();
1853 if (ret->lock == NULL) {
1854 ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1861 NISTP256_PRE_COMP *EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP *p)
1865 CRYPTO_UP_REF(&p->references, &i, p->lock);
1869 void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP *pre)
1876 CRYPTO_DOWN_REF(&pre->references, &i, pre->lock);
1877 REF_PRINT_COUNT("EC_nistp256", x);
1880 REF_ASSERT_ISNT(i < 0);
1882 CRYPTO_THREAD_lock_free(pre->lock);
1886 /******************************************************************************/
1888 * OPENSSL EC_METHOD FUNCTIONS
1891 int ec_GFp_nistp256_group_init(EC_GROUP *group)
1894 ret = ec_GFp_simple_group_init(group);
1895 group->a_is_minus3 = 1;
1899 int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1900 const BIGNUM *a, const BIGNUM *b,
1904 BIGNUM *curve_p, *curve_a, *curve_b;
1906 BN_CTX *new_ctx = NULL;
1909 new_ctx = BN_CTX_new();
1915 curve_p = BN_CTX_get(ctx);
1916 curve_a = BN_CTX_get(ctx);
1917 curve_b = BN_CTX_get(ctx);
1918 if (curve_b == NULL)
1920 BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p);
1921 BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a);
1922 BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b);
1923 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1924 ECerr(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE,
1925 EC_R_WRONG_CURVE_PARAMETERS);
1928 group->field_mod_func = BN_nist_mod_256;
1929 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1933 BN_CTX_free(new_ctx);
1939 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1942 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group,
1943 const EC_POINT *point,
1944 BIGNUM *x, BIGNUM *y,
1947 felem z1, z2, x_in, y_in;
1948 smallfelem x_out, y_out;
1951 if (EC_POINT_is_at_infinity(group, point)) {
1952 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
1953 EC_R_POINT_AT_INFINITY);
1956 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1957 (!BN_to_felem(z1, point->Z)))
1960 felem_square(tmp, z2);
1961 felem_reduce(z1, tmp);
1962 felem_mul(tmp, x_in, z1);
1963 felem_reduce(x_in, tmp);
1964 felem_contract(x_out, x_in);
1966 if (!smallfelem_to_BN(x, x_out)) {
1967 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
1972 felem_mul(tmp, z1, z2);
1973 felem_reduce(z1, tmp);
1974 felem_mul(tmp, y_in, z1);
1975 felem_reduce(y_in, tmp);
1976 felem_contract(y_out, y_in);
1978 if (!smallfelem_to_BN(y, y_out)) {
1979 ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
1987 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
1988 static void make_points_affine(size_t num, smallfelem points[][3],
1989 smallfelem tmp_smallfelems[])
1992 * Runs in constant time, unless an input is the point at infinity (which
1993 * normally shouldn't happen).
1995 ec_GFp_nistp_points_make_affine_internal(num,
1999 (void (*)(void *))smallfelem_one,
2000 smallfelem_is_zero_int,
2001 (void (*)(void *, const void *))
2003 (void (*)(void *, const void *))
2004 smallfelem_square_contract,
2006 (void *, const void *,
2008 smallfelem_mul_contract,
2009 (void (*)(void *, const void *))
2010 smallfelem_inv_contract,
2011 /* nothing to contract */
2012 (void (*)(void *, const void *))
2017 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
2018 * values Result is stored in r (r can equal one of the inputs).
2020 int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r,
2021 const BIGNUM *scalar, size_t num,
2022 const EC_POINT *points[],
2023 const BIGNUM *scalars[], BN_CTX *ctx)
2028 BIGNUM *x, *y, *z, *tmp_scalar;
2029 felem_bytearray g_secret;
2030 felem_bytearray *secrets = NULL;
2031 smallfelem (*pre_comp)[17][3] = NULL;
2032 smallfelem *tmp_smallfelems = NULL;
2033 felem_bytearray tmp;
2034 unsigned i, num_bytes;
2035 int have_pre_comp = 0;
2036 size_t num_points = num;
2037 smallfelem x_in, y_in, z_in;
2038 felem x_out, y_out, z_out;
2039 NISTP256_PRE_COMP *pre = NULL;
2040 const smallfelem(*g_pre_comp)[16][3] = NULL;
2041 EC_POINT *generator = NULL;
2042 const EC_POINT *p = NULL;
2043 const BIGNUM *p_scalar = NULL;
2046 x = BN_CTX_get(ctx);
2047 y = BN_CTX_get(ctx);
2048 z = BN_CTX_get(ctx);
2049 tmp_scalar = BN_CTX_get(ctx);
2050 if (tmp_scalar == NULL)
2053 if (scalar != NULL) {
2054 pre = group->pre_comp.nistp256;
2056 /* we have precomputation, try to use it */
2057 g_pre_comp = (const smallfelem(*)[16][3])pre->g_pre_comp;
2059 /* try to use the standard precomputation */
2060 g_pre_comp = &gmul[0];
2061 generator = EC_POINT_new(group);
2062 if (generator == NULL)
2064 /* get the generator from precomputation */
2065 if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) ||
2066 !smallfelem_to_BN(y, g_pre_comp[0][1][1]) ||
2067 !smallfelem_to_BN(z, g_pre_comp[0][1][2])) {
2068 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
2071 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
2075 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
2076 /* precomputation matches generator */
2080 * we don't have valid precomputation: treat the generator as a
2085 if (num_points > 0) {
2086 if (num_points >= 3) {
2088 * unless we precompute multiples for just one or two points,
2089 * converting those into affine form is time well spent
2093 secrets = OPENSSL_malloc(sizeof(*secrets) * num_points);
2094 pre_comp = OPENSSL_malloc(sizeof(*pre_comp) * num_points);
2097 OPENSSL_malloc(sizeof(*tmp_smallfelems) * (num_points * 17 + 1));
2098 if ((secrets == NULL) || (pre_comp == NULL)
2099 || (mixed && (tmp_smallfelems == NULL))) {
2100 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
2105 * we treat NULL scalars as 0, and NULL points as points at infinity,
2106 * i.e., they contribute nothing to the linear combination
2108 memset(secrets, 0, sizeof(*secrets) * num_points);
2109 memset(pre_comp, 0, sizeof(*pre_comp) * num_points);
2110 for (i = 0; i < num_points; ++i) {
2113 * we didn't have a valid precomputation, so we pick the
2117 p = EC_GROUP_get0_generator(group);
2120 /* the i^th point */
2123 p_scalar = scalars[i];
2125 if ((p_scalar != NULL) && (p != NULL)) {
2126 /* reduce scalar to 0 <= scalar < 2^256 */
2127 if ((BN_num_bits(p_scalar) > 256)
2128 || (BN_is_negative(p_scalar))) {
2130 * this is an unusual input, and we don't guarantee
2133 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
2134 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
2137 num_bytes = BN_bn2bin(tmp_scalar, tmp);
2139 num_bytes = BN_bn2bin(p_scalar, tmp);
2140 flip_endian(secrets[i], tmp, num_bytes);
2141 /* precompute multiples */
2142 if ((!BN_to_felem(x_out, p->X)) ||
2143 (!BN_to_felem(y_out, p->Y)) ||
2144 (!BN_to_felem(z_out, p->Z)))
2146 felem_shrink(pre_comp[i][1][0], x_out);
2147 felem_shrink(pre_comp[i][1][1], y_out);
2148 felem_shrink(pre_comp[i][1][2], z_out);
2149 for (j = 2; j <= 16; ++j) {
2151 point_add_small(pre_comp[i][j][0], pre_comp[i][j][1],
2152 pre_comp[i][j][2], pre_comp[i][1][0],
2153 pre_comp[i][1][1], pre_comp[i][1][2],
2154 pre_comp[i][j - 1][0],
2155 pre_comp[i][j - 1][1],
2156 pre_comp[i][j - 1][2]);
2158 point_double_small(pre_comp[i][j][0],
2161 pre_comp[i][j / 2][0],
2162 pre_comp[i][j / 2][1],
2163 pre_comp[i][j / 2][2]);
2169 make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems);
2172 /* the scalar for the generator */
2173 if ((scalar != NULL) && (have_pre_comp)) {
2174 memset(g_secret, 0, sizeof(g_secret));
2175 /* reduce scalar to 0 <= scalar < 2^256 */
2176 if ((BN_num_bits(scalar) > 256) || (BN_is_negative(scalar))) {
2178 * this is an unusual input, and we don't guarantee
2181 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
2182 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
2185 num_bytes = BN_bn2bin(tmp_scalar, tmp);
2187 num_bytes = BN_bn2bin(scalar, tmp);
2188 flip_endian(g_secret, tmp, num_bytes);
2189 /* do the multiplication with generator precomputation */
2190 batch_mul(x_out, y_out, z_out,
2191 (const felem_bytearray(*))secrets, num_points,
2193 mixed, (const smallfelem(*)[17][3])pre_comp, g_pre_comp);
2195 /* do the multiplication without generator precomputation */
2196 batch_mul(x_out, y_out, z_out,
2197 (const felem_bytearray(*))secrets, num_points,
2198 NULL, mixed, (const smallfelem(*)[17][3])pre_comp, NULL);
2199 /* reduce the output to its unique minimal representation */
2200 felem_contract(x_in, x_out);
2201 felem_contract(y_in, y_out);
2202 felem_contract(z_in, z_out);
2203 if ((!smallfelem_to_BN(x, x_in)) || (!smallfelem_to_BN(y, y_in)) ||
2204 (!smallfelem_to_BN(z, z_in))) {
2205 ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
2208 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2212 EC_POINT_free(generator);
2213 OPENSSL_free(secrets);
2214 OPENSSL_free(pre_comp);
2215 OPENSSL_free(tmp_smallfelems);
2219 int ec_GFp_nistp256_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
2222 NISTP256_PRE_COMP *pre = NULL;
2225 EC_POINT *generator = NULL;
2226 smallfelem tmp_smallfelems[32];
2227 felem x_tmp, y_tmp, z_tmp;
2229 BN_CTX *new_ctx = NULL;
2232 /* throw away old precomputation */
2233 EC_pre_comp_free(group);
2237 new_ctx = BN_CTX_new();
2243 x = BN_CTX_get(ctx);
2244 y = BN_CTX_get(ctx);
2247 /* get the generator */
2248 if (group->generator == NULL)
2250 generator = EC_POINT_new(group);
2251 if (generator == NULL)
2253 BN_bin2bn(nistp256_curve_params[3], sizeof(felem_bytearray), x);
2254 BN_bin2bn(nistp256_curve_params[4], sizeof(felem_bytearray), y);
2255 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
2257 if ((pre = nistp256_pre_comp_new()) == NULL)
2260 * if the generator is the standard one, use built-in precomputation
2262 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2263 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2266 if ((!BN_to_felem(x_tmp, group->generator->X)) ||
2267 (!BN_to_felem(y_tmp, group->generator->Y)) ||
2268 (!BN_to_felem(z_tmp, group->generator->Z)))
2270 felem_shrink(pre->g_pre_comp[0][1][0], x_tmp);
2271 felem_shrink(pre->g_pre_comp[0][1][1], y_tmp);
2272 felem_shrink(pre->g_pre_comp[0][1][2], z_tmp);
2274 * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
2275 * 2^160*G, 2^224*G for the second one
2277 for (i = 1; i <= 8; i <<= 1) {
2278 point_double_small(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
2279 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
2280 pre->g_pre_comp[0][i][1],
2281 pre->g_pre_comp[0][i][2]);
2282 for (j = 0; j < 31; ++j) {
2283 point_double_small(pre->g_pre_comp[1][i][0],
2284 pre->g_pre_comp[1][i][1],
2285 pre->g_pre_comp[1][i][2],
2286 pre->g_pre_comp[1][i][0],
2287 pre->g_pre_comp[1][i][1],
2288 pre->g_pre_comp[1][i][2]);
2292 point_double_small(pre->g_pre_comp[0][2 * i][0],
2293 pre->g_pre_comp[0][2 * i][1],
2294 pre->g_pre_comp[0][2 * i][2],
2295 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
2296 pre->g_pre_comp[1][i][2]);
2297 for (j = 0; j < 31; ++j) {
2298 point_double_small(pre->g_pre_comp[0][2 * i][0],
2299 pre->g_pre_comp[0][2 * i][1],
2300 pre->g_pre_comp[0][2 * i][2],
2301 pre->g_pre_comp[0][2 * i][0],
2302 pre->g_pre_comp[0][2 * i][1],
2303 pre->g_pre_comp[0][2 * i][2]);
2306 for (i = 0; i < 2; i++) {
2307 /* g_pre_comp[i][0] is the point at infinity */
2308 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
2309 /* the remaining multiples */
2310 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2311 point_add_small(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
2312 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
2313 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
2314 pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
2315 pre->g_pre_comp[i][2][2]);
2316 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2317 point_add_small(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
2318 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
2319 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
2320 pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
2321 pre->g_pre_comp[i][2][2]);
2322 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2323 point_add_small(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
2324 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
2325 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
2326 pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
2327 pre->g_pre_comp[i][4][2]);
2329 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
2331 point_add_small(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
2332 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
2333 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
2334 pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
2335 pre->g_pre_comp[i][2][2]);
2336 for (j = 1; j < 8; ++j) {
2337 /* odd multiples: add G resp. 2^32*G */
2338 point_add_small(pre->g_pre_comp[i][2 * j + 1][0],
2339 pre->g_pre_comp[i][2 * j + 1][1],
2340 pre->g_pre_comp[i][2 * j + 1][2],
2341 pre->g_pre_comp[i][2 * j][0],
2342 pre->g_pre_comp[i][2 * j][1],
2343 pre->g_pre_comp[i][2 * j][2],
2344 pre->g_pre_comp[i][1][0],
2345 pre->g_pre_comp[i][1][1],
2346 pre->g_pre_comp[i][1][2]);
2349 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_smallfelems);
2352 SETPRECOMP(group, nistp256, pre);
2358 EC_POINT_free(generator);
2360 BN_CTX_free(new_ctx);
2362 EC_nistp256_pre_comp_free(pre);
2366 int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP *group)
2368 return HAVEPRECOMP(group, nistp256);