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/* crypto/ec/ecp_nistp224.c */
/*
* Written by Emilia Kasper (Google) for the OpenSSL project.
*/
/* Copyright 2011 Google Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License");
*
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/*
* A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
*
* Inspired by Daniel J. Bernstein's public domain nistp224 implementation
* and Adam Langley's public domain 64-bit C implementation of curve25519
*/
#include <openssl/opensslconf.h>
#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
# ifndef OPENSSL_SYS_VMS
# include <stdint.h>
# else
# include <inttypes.h>
# endif
# include <string.h>
# include <openssl/err.h>
# include "ec_lcl.h"
# if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
/* even with gcc, the typedef won't work for 32-bit platforms */
typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
* platforms */
# else
# error "Need GCC 3.1 or later to define type uint128_t"
# endif
typedef uint8_t u8;
typedef uint64_t u64;
typedef int64_t s64;
/******************************************************************************/
/*-
* INTERNAL REPRESENTATION OF FIELD ELEMENTS
*
* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
* using 64-bit coefficients called 'limbs',
* and sometimes (for multiplication results) as
* b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
* using 128-bit coefficients called 'widelimbs'.
* A 4-limb representation is an 'felem';
* a 7-widelimb representation is a 'widefelem'.
* Even within felems, bits of adjacent limbs overlap, and we don't always
* reduce the representations: we ensure that inputs to each felem
* multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
* and fit into a 128-bit word without overflow. The coefficients are then
* again partially reduced to obtain an felem satisfying a_i < 2^57.
* We only reduce to the unique minimal representation at the end of the
* computation.
*/
typedef uint64_t limb;
typedef uint128_t widelimb;
typedef limb felem[4];
typedef widelimb widefelem[7];
/*
* Field element represented as a byte arrary. 28*8 = 224 bits is also the
* group order size for the elliptic curve, and we also use this type for
* scalars for point multiplication.
*/
typedef u8 felem_bytearray[28];
static const felem_bytearray nistp224_curve_params[5] = {
{0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
{0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF,
0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
{0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA,
0x27, 0x0B, 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
{0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22,
0x34, 0x32, 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
{0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}
};
/*-
* Precomputed multiples of the standard generator
* Points are given in coordinates (X, Y, Z) where Z normally is 1
* (0 for the point at infinity).
* For each field element, slice a_0 is word 0, etc.
*
* The table has 2 * 16 elements, starting with the following:
* index | bits | point
* ------+---------+------------------------------
* 0 | 0 0 0 0 | 0G
* 1 | 0 0 0 1 | 1G
* 2 | 0 0 1 0 | 2^56G
* 3 | 0 0 1 1 | (2^56 + 1)G
* 4 | 0 1 0 0 | 2^112G
* 5 | 0 1 0 1 | (2^112 + 1)G
* 6 | 0 1 1 0 | (2^112 + 2^56)G
* 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
* 8 | 1 0 0 0 | 2^168G
* 9 | 1 0 0 1 | (2^168 + 1)G
* 10 | 1 0 1 0 | (2^168 + 2^56)G
* 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
* 12 | 1 1 0 0 | (2^168 + 2^112)G
* 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
* 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
* 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
* followed by a copy of this with each element multiplied by 2^28.
*
* The reason for this is so that we can clock bits into four different
* locations when doing simple scalar multiplies against the base point,
* and then another four locations using the second 16 elements.
*/
static const felem gmul[2][16][3] = { {{{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}},
{{0x3280d6115c1d21, 0xc1d356c2112234,
0x7f321390b94a03, 0xb70e0cbd6bb4bf},
{0xd5819985007e34, 0x75a05a07476444,
0xfb4c22dfe6cd43, 0xbd376388b5f723},
{1, 0, 0, 0}},
{{0xfd9675666ebbe9, 0xbca7664d40ce5e,
0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
{0x29e0b892dc9c43, 0xece8608436e662,
0xdc858f185310d0, 0x9812dd4eb8d321},
{1, 0, 0, 0}},
{{0x6d3e678d5d8eb8, 0x559eed1cb362f1,
0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
{0xf19f90ed50266d, 0xabf2b4bf65f9df,
0x313865468fafec, 0x5cb379ba910a17},
{1, 0, 0, 0}},
{{0x0641966cab26e3, 0x91fb2991fab0a0,
0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
{0x7510407766af5d, 0x84d929610d5450,
0x81d77aae82f706, 0x6916f6d4338c5b},
{1, 0, 0, 0}},
{{0xea95ac3b1f15c6, 0x086000905e82d4,
0xdd323ae4d1c8b1, 0x932b56be7685a3},
{0x9ef93dea25dbbf, 0x41665960f390f0,
0xfdec76dbe2a8a7, 0x523e80f019062a},
{1, 0, 0, 0}},
{{0x822fdd26732c73, 0xa01c83531b5d0f,
0x363f37347c1ba4, 0xc391b45c84725c},
{0xbbd5e1b2d6ad24, 0xddfbcde19dfaec,
0xc393da7e222a7f, 0x1efb7890ede244},
{1, 0, 0, 0}},
{{0x4c9e90ca217da1, 0xd11beca79159bb,
0xff8d33c2c98b7c, 0x2610b39409f849},
{0x44d1352ac64da0, 0xcdbb7b2c46b4fb,
0x966c079b753c89, 0xfe67e4e820b112},
{1, 0, 0, 0}},
{{0xe28cae2df5312d, 0xc71b61d16f5c6e,
0x79b7619a3e7c4c, 0x05c73240899b47},
{0x9f7f6382c73e3a, 0x18615165c56bda,
0x641fab2116fd56, 0x72855882b08394},
{1, 0, 0, 0}},
{{0x0469182f161c09, 0x74a98ca8d00fb5,
0xb89da93489a3e0, 0x41c98768fb0c1d},
{0xe5ea05fb32da81, 0x3dce9ffbca6855,
0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
{1, 0, 0, 0}},
{{0xdab22b2333e87f, 0x4430137a5dd2f6,
0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
{0x764a7df0c8fda5, 0x185ba5c3fa2044,
0x9281d688bcbe50, 0xc40331df893881},
{1, 0, 0, 0}},
{{0xb89530796f0f60, 0xade92bd26909a3,
0x1a0c83fb4884da, 0x1765bf22a5a984},
{0x772a9ee75db09e, 0x23bc6c67cec16f,
0x4c1edba8b14e2f, 0xe2a215d9611369},
{1, 0, 0, 0}},
{{0x571e509fb5efb3, 0xade88696410552,
0xc8ae85fada74fe, 0x6c7e4be83bbde3},
{0xff9f51160f4652, 0xb47ce2495a6539,
0xa2946c53b582f4, 0x286d2db3ee9a60},
{1, 0, 0, 0}},
{{0x40bbd5081a44af, 0x0995183b13926c,
0xbcefba6f47f6d0, 0x215619e9cc0057},
{0x8bc94d3b0df45e, 0xf11c54a3694f6f,
0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
{1, 0, 0, 0}},
{{0xb17048ab3e1c7b, 0xac38f36ff8a1d8,
0x1c29819435d2c6, 0xc813132f4c07e9},
{0x2891425503b11f, 0x08781030579fea,
0xf5426ba5cc9674, 0x1e28ebf18562bc},
{1, 0, 0, 0}},
{{0x9f31997cc864eb, 0x06cd91d28b5e4c,
0xff17036691a973, 0xf1aef351497c58},
{0xdd1f2d600564ff, 0xdead073b1402db,
0x74a684435bd693, 0xeea7471f962558},
{1, 0, 0, 0}}},
{{{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}},
{{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
{0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
{1, 0, 0, 0}},
{{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
{0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
{1, 0, 0, 0}},
{{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
{0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
{1, 0, 0, 0}},
{{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
{0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
{1, 0, 0, 0}},
{{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
{0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
{1, 0, 0, 0}},
{{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
{0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
{1, 0, 0, 0}},
{{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
{0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
{1, 0, 0, 0}},
{{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
{0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
{1, 0, 0, 0}},
{{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
{0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
{1, 0, 0, 0}},
{{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
{0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
{1, 0, 0, 0}},
{{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
{0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
{1, 0, 0, 0}},
{{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
{0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
{1, 0, 0, 0}},
{{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
{0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
{1, 0, 0, 0}},
{{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
{0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
{1, 0, 0, 0}},
{{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
{0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
{1, 0, 0, 0}}}
};
/* Precomputation for the group generator. */
typedef struct {
felem g_pre_comp[2][16][3];
int references;
} NISTP224_PRE_COMP;
const EC_METHOD *EC_GFp_nistp224_method(void)
{
static const EC_METHOD ret = {
EC_FLAGS_DEFAULT_OCT,
NID_X9_62_prime_field,
ec_GFp_nistp224_group_init,
ec_GFp_simple_group_finish,
ec_GFp_simple_group_clear_finish,
ec_GFp_nist_group_copy,
ec_GFp_nistp224_group_set_curve,
ec_GFp_simple_group_get_curve,
ec_GFp_simple_group_get_degree,
ec_GFp_simple_group_check_discriminant,
ec_GFp_simple_point_init,
ec_GFp_simple_point_finish,
ec_GFp_simple_point_clear_finish,
ec_GFp_simple_point_copy,
ec_GFp_simple_point_set_to_infinity,
ec_GFp_simple_set_Jprojective_coordinates_GFp,
ec_GFp_simple_get_Jprojective_coordinates_GFp,
ec_GFp_simple_point_set_affine_coordinates,
ec_GFp_nistp224_point_get_affine_coordinates,
0 /* point_set_compressed_coordinates */ ,
0 /* point2oct */ ,
0 /* oct2point */ ,
ec_GFp_simple_add,
ec_GFp_simple_dbl,
ec_GFp_simple_invert,
ec_GFp_simple_is_at_infinity,
ec_GFp_simple_is_on_curve,
ec_GFp_simple_cmp,
ec_GFp_simple_make_affine,
ec_GFp_simple_points_make_affine,
ec_GFp_nistp224_points_mul,
ec_GFp_nistp224_precompute_mult,
ec_GFp_nistp224_have_precompute_mult,
ec_GFp_nist_field_mul,
ec_GFp_nist_field_sqr,
0 /* field_div */ ,
0 /* field_encode */ ,
0 /* field_decode */ ,
0 /* field_set_to_one */
};
return &ret;
}
/*
* Helper functions to convert field elements to/from internal representation
*/
static void bin28_to_felem(felem out, const u8 in[28])
{
out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
out[3] = (*((const uint64_t *)(in+20))) >> 8;
}
static void felem_to_bin28(u8 out[28], const felem in)
{
unsigned i;
for (i = 0; i < 7; ++i) {
out[i] = in[0] >> (8 * i);
out[i + 7] = in[1] >> (8 * i);
out[i + 14] = in[2] >> (8 * i);
out[i + 21] = in[3] >> (8 * i);
}
}
/* To preserve endianness when using BN_bn2bin and BN_bin2bn */
static void flip_endian(u8 *out, const u8 *in, unsigned len)
{
unsigned i;
for (i = 0; i < len; ++i)
out[i] = in[len - 1 - i];
}
/* From OpenSSL BIGNUM to internal representation */
static int BN_to_felem(felem out, const BIGNUM *bn)
{
felem_bytearray b_in;
felem_bytearray b_out;
unsigned num_bytes;
/* BN_bn2bin eats leading zeroes */
memset(b_out, 0, sizeof b_out);
num_bytes = BN_num_bytes(bn);
if (num_bytes > sizeof b_out) {
ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
return 0;
}
if (BN_is_negative(bn)) {
ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
return 0;
}
num_bytes = BN_bn2bin(bn, b_in);
flip_endian(b_out, b_in, num_bytes);
bin28_to_felem(out, b_out);
return 1;
}
/* From internal representation to OpenSSL BIGNUM */
static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
{
felem_bytearray b_in, b_out;
felem_to_bin28(b_in, in);
flip_endian(b_out, b_in, sizeof b_out);
return BN_bin2bn(b_out, sizeof b_out, out);
}
/******************************************************************************/
/*-
* FIELD OPERATIONS
*
* Field operations, using the internal representation of field elements.
* NB! These operations are specific to our point multiplication and cannot be
* expected to be correct in general - e.g., multiplication with a large scalar
* will cause an overflow.
*
*/
static void felem_one(felem out)
{
out[0] = 1;
out[1] = 0;
out[2] = 0;
out[3] = 0;
}
static void felem_assign(felem out, const felem in)
{
out[0] = in[0];
out[1] = in[1];
out[2] = in[2];
out[3] = in[3];
}
/* Sum two field elements: out += in */
static void felem_sum(felem out, const felem in)
{
out[0] += in[0];
out[1] += in[1];
out[2] += in[2];
out[3] += in[3];
}
/* Get negative value: out = -in */
/* Assumes in[i] < 2^57 */
static void felem_neg(felem out, const felem in)
{
static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
static const limb two58m42m2 = (((limb) 1) << 58) -
(((limb) 1) << 42) - (((limb) 1) << 2);
/* Set to 0 mod 2^224-2^96+1 to ensure out > in */
out[0] = two58p2 - in[0];
out[1] = two58m42m2 - in[1];
out[2] = two58m2 - in[2];
out[3] = two58m2 - in[3];
}
/* Subtract field elements: out -= in */
/* Assumes in[i] < 2^57 */
static void felem_diff(felem out, const felem in)
{
static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
static const limb two58m42m2 = (((limb) 1) << 58) -
(((limb) 1) << 42) - (((limb) 1) << 2);
/* Add 0 mod 2^224-2^96+1 to ensure out > in */
out[0] += two58p2;
out[1] += two58m42m2;
out[2] += two58m2;
out[3] += two58m2;
out[0] -= in[0];
out[1] -= in[1];
out[2] -= in[2];
out[3] -= in[3];
}
/* Subtract in unreduced 128-bit mode: out -= in */
/* Assumes in[i] < 2^119 */
static void widefelem_diff(widefelem out, const widefelem in)
{
static const widelimb two120 = ((widelimb) 1) << 120;
static const widelimb two120m64 = (((widelimb) 1) << 120) -
(((widelimb) 1) << 64);
static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
(((widelimb) 1) << 104) - (((widelimb) 1) << 64);
/* Add 0 mod 2^224-2^96+1 to ensure out > in */
out[0] += two120;
out[1] += two120m64;
out[2] += two120m64;
out[3] += two120;
out[4] += two120m104m64;
out[5] += two120m64;
out[6] += two120m64;
out[0] -= in[0];
out[1] -= in[1];
out[2] -= in[2];
out[3] -= in[3];
out[4] -= in[4];
out[5] -= in[5];
out[6] -= in[6];
}
/* Subtract in mixed mode: out128 -= in64 */
/* in[i] < 2^63 */
static void felem_diff_128_64(widefelem out, const felem in)
{
static const widelimb two64p8 = (((widelimb) 1) << 64) +
(((widelimb) 1) << 8);
static const widelimb two64m8 = (((widelimb) 1) << 64) -
(((widelimb) 1) << 8);
static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
(((widelimb) 1) << 48) - (((widelimb) 1) << 8);
/* Add 0 mod 2^224-2^96+1 to ensure out > in */
out[0] += two64p8;
out[1] += two64m48m8;
out[2] += two64m8;
out[3] += two64m8;
out[0] -= in[0];
out[1] -= in[1];
out[2] -= in[2];
out[3] -= in[3];
}
/*
* Multiply a field element by a scalar: out = out * scalar The scalars we
* actually use are small, so results fit without overflow
*/
static void felem_scalar(felem out, const limb scalar)
{
out[0] *= scalar;
out[1] *= scalar;
out[2] *= scalar;
out[3] *= scalar;
}
/*
* Multiply an unreduced field element by a scalar: out = out * scalar The
* scalars we actually use are small, so results fit without overflow
*/
static void widefelem_scalar(widefelem out, const widelimb scalar)
{
out[0] *= scalar;
out[1] *= scalar;
out[2] *= scalar;
out[3] *= scalar;
out[4] *= scalar;
out[5] *= scalar;
out[6] *= scalar;
}
/* Square a field element: out = in^2 */
static void felem_square(widefelem out, const felem in)
{
limb tmp0, tmp1, tmp2;
tmp0 = 2 * in[0];
tmp1 = 2 * in[1];
tmp2 = 2 * in[2];
out[0] = ((widelimb) in[0]) * in[0];
out[1] = ((widelimb) in[0]) * tmp1;
out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
out[3] = ((widelimb) in[3]) * tmp0 + ((widelimb) in[1]) * tmp2;
out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
out[5] = ((widelimb) in[3]) * tmp2;
out[6] = ((widelimb) in[3]) * in[3];
}
/* Multiply two field elements: out = in1 * in2 */
static void felem_mul(widefelem out, const felem in1, const felem in2)
{
out[0] = ((widelimb) in1[0]) * in2[0];
out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
((widelimb) in1[2]) * in2[0];
out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
((widelimb) in1[3]) * in2[1];
out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
out[6] = ((widelimb) in1[3]) * in2[3];
}
/*-
* Reduce seven 128-bit coefficients to four 64-bit coefficients.
* Requires in[i] < 2^126,
* ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
static void felem_reduce(felem out, const widefelem in)
{
static const widelimb two127p15 = (((widelimb) 1) << 127) +
(((widelimb) 1) << 15);
static const widelimb two127m71 = (((widelimb) 1) << 127) -
(((widelimb) 1) << 71);
static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
(((widelimb) 1) << 71) - (((widelimb) 1) << 55);
widelimb output[5];
/* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
output[0] = in[0] + two127p15;
output[1] = in[1] + two127m71m55;
output[2] = in[2] + two127m71;
output[3] = in[3];
output[4] = in[4];
/* Eliminate in[4], in[5], in[6] */
output[4] += in[6] >> 16;
output[3] += (in[6] & 0xffff) << 40;
output[2] -= in[6];
output[3] += in[5] >> 16;
output[2] += (in[5] & 0xffff) << 40;
output[1] -= in[5];
output[2] += output[4] >> 16;
output[1] += (output[4] & 0xffff) << 40;
output[0] -= output[4];
/* Carry 2 -> 3 -> 4 */
output[3] += output[2] >> 56;
output[2] &= 0x00ffffffffffffff;
output[4] = output[3] >> 56;
output[3] &= 0x00ffffffffffffff;
/* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
/* Eliminate output[4] */
output[2] += output[4] >> 16;
/* output[2] < 2^56 + 2^56 = 2^57 */
output[1] += (output[4] & 0xffff) << 40;
output[0] -= output[4];
/* Carry 0 -> 1 -> 2 -> 3 */
output[1] += output[0] >> 56;
out[0] = output[0] & 0x00ffffffffffffff;
output[2] += output[1] >> 56;
/* output[2] < 2^57 + 2^72 */
out[1] = output[1] & 0x00ffffffffffffff;
output[3] += output[2] >> 56;
/* output[3] <= 2^56 + 2^16 */
out[2] = output[2] & 0x00ffffffffffffff;
/*-
* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
* out[3] <= 2^56 + 2^16 (due to final carry),
* so out < 2*p
*/
out[3] = output[3];
}
static void felem_square_reduce(felem out, const felem in)
{
widefelem tmp;
felem_square(tmp, in);
felem_reduce(out, tmp);
}
static void felem_mul_reduce(felem out, const felem in1, const felem in2)
{
widefelem tmp;
felem_mul(tmp, in1, in2);
felem_reduce(out, tmp);
}
/*
* Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
* call felem_reduce first)
*/
static void felem_contract(felem out, const felem in)
{
static const int64_t two56 = ((limb) 1) << 56;
/* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
/* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
int64_t tmp[4], a;
tmp[0] = in[0];
tmp[1] = in[1];
tmp[2] = in[2];
tmp[3] = in[3];
/* Case 1: a = 1 iff in >= 2^224 */
a = (in[3] >> 56);
tmp[0] -= a;
tmp[1] += a << 40;
tmp[3] &= 0x00ffffffffffffff;
/*
* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
* and the lower part is non-zero
*/
a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
(((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
a &= 0x00ffffffffffffff;
/* turn a into an all-one mask (if a = 0) or an all-zero mask */
a = (a - 1) >> 63;
/* subtract 2^224 - 2^96 + 1 if a is all-one */
tmp[3] &= a ^ 0xffffffffffffffff;
tmp[2] &= a ^ 0xffffffffffffffff;
tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
tmp[0] -= 1 & a;
/*
* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
* non-zero, so we only need one step
*/
a = tmp[0] >> 63;
tmp[0] += two56 & a;
tmp[1] -= 1 & a;
/* carry 1 -> 2 -> 3 */
tmp[2] += tmp[1] >> 56;
tmp[1] &= 0x00ffffffffffffff;
tmp[3] += tmp[2] >> 56;
tmp[2] &= 0x00ffffffffffffff;
/* Now 0 <= out < p */
out[0] = tmp[0];
out[1] = tmp[1];
out[2] = tmp[2];
out[3] = tmp[3];
}
/*
* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
* elements are reduced to in < 2^225, so we only need to check three cases:
* 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
*/
static limb felem_is_zero(const felem in)
{
limb zero, two224m96p1, two225m97p2;
zero = in[0] | in[1] | in[2] | in[3];
zero = (((int64_t) (zero) - 1) >> 63) & 1;
two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
| (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
| (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
return (zero | two224m96p1 | two225m97p2);
}
static limb felem_is_zero_int(const felem in)
{
return (int)(felem_is_zero(in) & ((limb) 1));
}
/* Invert a field element */
/* Computation chain copied from djb's code */
static void felem_inv(felem out, const felem in)
{
felem ftmp, ftmp2, ftmp3, ftmp4;
widefelem tmp;
unsigned i;
felem_square(tmp, in);
felem_reduce(ftmp, tmp); /* 2 */
felem_mul(tmp, in, ftmp);
felem_reduce(ftmp, tmp); /* 2^2 - 1 */
felem_square(tmp, ftmp);
felem_reduce(ftmp, tmp); /* 2^3 - 2 */
felem_mul(tmp, in, ftmp);
felem_reduce(ftmp, tmp); /* 2^3 - 1 */
felem_square(tmp, ftmp);
felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
felem_mul(tmp, ftmp2, ftmp);
felem_reduce(ftmp, tmp); /* 2^6 - 1 */
felem_square(tmp, ftmp);
felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp2, tmp);
}
felem_mul(tmp, ftmp2, ftmp);
felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
felem_square(tmp, ftmp3);
felem_reduce(ftmp3, tmp);
}
felem_mul(tmp, ftmp3, ftmp2);
felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
felem_square(tmp, ftmp3);
felem_reduce(ftmp3, tmp);
}
felem_mul(tmp, ftmp3, ftmp2);
felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
felem_square(tmp, ftmp3);
felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
felem_square(tmp, ftmp4);
felem_reduce(ftmp4, tmp);
}
felem_mul(tmp, ftmp3, ftmp4);
felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
felem_square(tmp, ftmp3);
felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
felem_square(tmp, ftmp4);
felem_reduce(ftmp4, tmp);
}
felem_mul(tmp, ftmp2, ftmp4);
felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
felem_square(tmp, ftmp2);
felem_reduce(ftmp2, tmp);
}
felem_mul(tmp, ftmp2, ftmp);
felem_reduce(ftmp, tmp); /* 2^126 - 1 */
felem_square(tmp, ftmp);
felem_reduce(ftmp, tmp); /* 2^127 - 2 */
felem_mul(tmp, ftmp, in);
felem_reduce(ftmp, tmp); /* 2^127 - 1 */
for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
felem_square(tmp, ftmp);
felem_reduce(ftmp, tmp);
}
felem_mul(tmp, ftmp, ftmp3);
felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
}
/*
* Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
* out to itself.
*/
static void copy_conditional(felem out, const felem in, limb icopy)
{
unsigned i;
/*
* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
*/
const limb copy = -icopy;
for (i = 0; i < 4; ++i) {
const limb tmp = copy & (in[i] ^ out[i]);
out[i] ^= tmp;
}
}
/******************************************************************************/
/*-
* ELLIPTIC CURVE POINT OPERATIONS
*
* Points are represented in Jacobian projective coordinates:
* (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
* or to the point at infinity if Z == 0.
*
*/
/*-
* Double an elliptic curve point:
* (X', Y', Z') = 2 * (X, Y, Z), where
* X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
* Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
* Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
* Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
* while x_out == y_in is not (maybe this works, but it's not tested).
*/
static void
point_double(felem x_out, felem y_out, felem z_out,
const felem x_in, const felem y_in, const felem z_in)
{
widefelem tmp, tmp2;
felem delta, gamma, beta, alpha, ftmp, ftmp2;
felem_assign(ftmp, x_in);
felem_assign(ftmp2, x_in);
/* delta = z^2 */
felem_square(tmp, z_in);
felem_reduce(delta, tmp);
/* gamma = y^2 */
felem_square(tmp, y_in);
felem_reduce(gamma, tmp);
/* beta = x*gamma */
felem_mul(tmp, x_in, gamma);
felem_reduce(beta, tmp);
/* alpha = 3*(x-delta)*(x+delta) */
felem_diff(ftmp, delta);
/* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
felem_sum(ftmp2, delta);
/* ftmp2[i] < 2^57 + 2^57 = 2^58 */
felem_scalar(ftmp2, 3);
/* ftmp2[i] < 3 * 2^58 < 2^60 */
felem_mul(tmp, ftmp, ftmp2);
/* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
felem_reduce(alpha, tmp);
/* x' = alpha^2 - 8*beta */
felem_square(tmp, alpha);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
felem_assign(ftmp, beta);
felem_scalar(ftmp, 8);
/* ftmp[i] < 8 * 2^57 = 2^60 */
felem_diff_128_64(tmp, ftmp);
/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
felem_reduce(x_out, tmp);
/* z' = (y + z)^2 - gamma - delta */
felem_sum(delta, gamma);
/* delta[i] < 2^57 + 2^57 = 2^58 */
felem_assign(ftmp, y_in);
felem_sum(ftmp, z_in);
/* ftmp[i] < 2^57 + 2^57 = 2^58 */
felem_square(tmp, ftmp);
/* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
felem_diff_128_64(tmp, delta);
/* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
felem_reduce(z_out, tmp);
/* y' = alpha*(4*beta - x') - 8*gamma^2 */
felem_scalar(beta, 4);
/* beta[i] < 4 * 2^57 = 2^59 */
felem_diff(beta, x_out);
/* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
felem_mul(tmp, alpha, beta);
/* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
felem_square(tmp2, gamma);
/* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
widefelem_scalar(tmp2, 8);
/* tmp2[i] < 8 * 2^116 = 2^119 */
widefelem_diff(tmp, tmp2);
/* tmp[i] < 2^119 + 2^120 < 2^121 */
felem_reduce(y_out, tmp);
}
/*-
* Add two elliptic curve points:
* (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
* X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
* 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
* Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 - X_3) -
* Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
* Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
*
* This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
*/
/*
* This function is not entirely constant-time: it includes a branch for
* checking whether the two input points are equal, (while not equal to the
* point at infinity). This case never happens during single point
* multiplication, so there is no timing leak for ECDH or ECDSA signing.
*/
static void point_add(felem x3, felem y3, felem z3,
const felem x1, const felem y1, const felem z1,
const int mixed, const felem x2, const felem y2,
const felem z2)
{
felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
widefelem tmp, tmp2;
limb z1_is_zero, z2_is_zero, x_equal, y_equal;
if (!mixed) {
/* ftmp2 = z2^2 */
felem_square(tmp, z2);
felem_reduce(ftmp2, tmp);
/* ftmp4 = z2^3 */
felem_mul(tmp, ftmp2, z2);
felem_reduce(ftmp4, tmp);
/* ftmp4 = z2^3*y1 */
felem_mul(tmp2, ftmp4, y1);
felem_reduce(ftmp4, tmp2);
/* ftmp2 = z2^2*x1 */
felem_mul(tmp2, ftmp2, x1);
felem_reduce(ftmp2, tmp2);
} else {
/*
* We'll assume z2 = 1 (special case z2 = 0 is handled later)
*/
/* ftmp4 = z2^3*y1 */
felem_assign(ftmp4, y1);
/* ftmp2 = z2^2*x1 */
felem_assign(ftmp2, x1);
}
/* ftmp = z1^2 */
felem_square(tmp, z1);
felem_reduce(ftmp, tmp);
/* ftmp3 = z1^3 */
felem_mul(tmp, ftmp, z1);
felem_reduce(ftmp3, tmp);
/* tmp = z1^3*y2 */
felem_mul(tmp, ftmp3, y2);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
/* ftmp3 = z1^3*y2 - z2^3*y1 */
felem_diff_128_64(tmp, ftmp4);
/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
felem_reduce(ftmp3, tmp);
/* tmp = z1^2*x2 */
felem_mul(tmp, ftmp, x2);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
/* ftmp = z1^2*x2 - z2^2*x1 */
felem_diff_128_64(tmp, ftmp2);
/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
felem_reduce(ftmp, tmp);
/*
* the formulae are incorrect if the points are equal so we check for
* this and do doubling if this happens
*/
x_equal = felem_is_zero(ftmp);
y_equal = felem_is_zero(ftmp3);
z1_is_zero = felem_is_zero(z1);
z2_is_zero = felem_is_zero(z2);
/* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
point_double(x3, y3, z3, x1, y1, z1);
return;
}
/* ftmp5 = z1*z2 */
if (!mixed) {
felem_mul(tmp, z1, z2);
felem_reduce(ftmp5, tmp);
} else {
/* special case z2 = 0 is handled later */
felem_assign(ftmp5, z1);
}
/* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
felem_mul(tmp, ftmp, ftmp5);
felem_reduce(z_out, tmp);
/* ftmp = (z1^2*x2 - z2^2*x1)^2 */
felem_assign(ftmp5, ftmp);
felem_square(tmp, ftmp);
felem_reduce(ftmp, tmp);
/* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
felem_mul(tmp, ftmp, ftmp5);
felem_reduce(ftmp5, tmp);
/* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
felem_mul(tmp, ftmp2, ftmp);
felem_reduce(ftmp2, tmp);
/* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
felem_mul(tmp, ftmp4, ftmp5);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
/* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
felem_square(tmp2, ftmp3);
/* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
/* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
felem_diff_128_64(tmp2, ftmp5);
/* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
/* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
felem_assign(ftmp5, ftmp2);
felem_scalar(ftmp5, 2);
/* ftmp5[i] < 2 * 2^57 = 2^58 */
/*-
* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
* 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
*/
felem_diff_128_64(tmp2, ftmp5);
/* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
felem_reduce(x_out, tmp2);
/* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
felem_diff(ftmp2, x_out);
/* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
/*
* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
*/
felem_mul(tmp2, ftmp3, ftmp2);
/* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
/*-
* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
* z2^3*y1*(z1^2*x2 - z2^2*x1)^3
*/
widefelem_diff(tmp2, tmp);
/* tmp2[i] < 2^118 + 2^120 < 2^121 */
felem_reduce(y_out, tmp2);
/*
* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
* the point at infinity, so we need to check for this separately
*/
/*
* if point 1 is at infinity, copy point 2 to output, and vice versa
*/
copy_conditional(x_out, x2, z1_is_zero);
copy_conditional(x_out, x1, z2_is_zero);
copy_conditional(y_out, y2, z1_is_zero);
copy_conditional(y_out, y1, z2_is_zero);
copy_conditional(z_out, z2, z1_is_zero);
copy_conditional(z_out, z1, z2_is_zero);
felem_assign(x3, x_out);
felem_assign(y3, y_out);
felem_assign(z3, z_out);
}
/*
* select_point selects the |idx|th point from a precomputation table and
* copies it to out.
* The pre_comp array argument should be size of |size| argument
*/
static void select_point(const u64 idx, unsigned int size,
const felem pre_comp[][3], felem out[3])
{
unsigned i, j;
limb *outlimbs = &out[0][0];
memset(outlimbs, 0, 3 * sizeof(felem));
for (i = 0; i < size; i++) {
const limb *inlimbs = &pre_comp[i][0][0];
u64 mask = i ^ idx;
mask |= mask >> 4;
mask |= mask >> 2;
mask |= mask >> 1;
mask &= 1;
mask--;
for (j = 0; j < 4 * 3; j++)
outlimbs[j] |= inlimbs[j] & mask;
}
}
/* get_bit returns the |i|th bit in |in| */
static char get_bit(const felem_bytearray in, unsigned i)
{
if (i >= 224)
return 0;
return (in[i >> 3] >> (i & 7)) & 1;
}
/*
* Interleaved point multiplication using precomputed point multiples: The
* small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
* in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
* generator, using certain (large) precomputed multiples in g_pre_comp.
* Output point (X, Y, Z) is stored in x_out, y_out, z_out
*/
static void batch_mul(felem x_out, felem y_out, felem z_out,
const felem_bytearray scalars[],
const unsigned num_points, const u8 *g_scalar,
const int mixed, const felem pre_comp[][17][3],
const felem g_pre_comp[2][16][3])
{
int i, skip;
unsigned num;
unsigned gen_mul = (g_scalar != NULL);
felem nq[3], tmp[4];
u64 bits;
u8 sign, digit;
/* set nq to the point at infinity */
memset(nq, 0, 3 * sizeof(felem));
/*
* Loop over all scalars msb-to-lsb, interleaving additions of multiples
* of the generator (two in each of the last 28 rounds) and additions of
* other points multiples (every 5th round).
*/
skip = 1; /* save two point operations in the first
* round */
for (i = (num_points ? 220 : 27); i >= 0; --i) {
/* double */
if (!skip)
point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
/* add multiples of the generator */
if (gen_mul && (i <= 27)) {
/* first, look 28 bits upwards */
bits = get_bit(g_scalar, i + 196) << 3;
bits |= get_bit(g_scalar, i + 140) << 2;
bits |= get_bit(g_scalar, i + 84) << 1;
bits |= get_bit(g_scalar, i + 28);
/* select the point to add, in constant time */
select_point(bits, 16, g_pre_comp[1], tmp);
if (!skip) {
/* value 1 below is argument for "mixed" */
point_add(nq[0], nq[1], nq[2],
nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
} else {
memcpy(nq, tmp, 3 * sizeof(felem));
skip = 0;
}
/* second, look at the current position */
bits = get_bit(g_scalar, i + 168) << 3;
bits |= get_bit(g_scalar, i + 112) << 2;
bits |= get_bit(g_scalar, i + 56) << 1;
bits |= get_bit(g_scalar, i);
/* select the point to add, in constant time */
select_point(bits, 16, g_pre_comp[0], tmp);
point_add(nq[0], nq[1], nq[2],
nq[0], nq[1], nq[2],
1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
}
/* do other additions every 5 doublings */
if (num_points && (i % 5 == 0)) {
/* loop over all scalars */
for (num = 0; num < num_points; ++num) {
bits = get_bit(scalars[num], i + 4) << 5;
bits |= get_bit(scalars[num], i + 3) << 4;
bits |= get_bit(scalars[num], i + 2) << 3;
bits |= get_bit(scalars[num], i + 1) << 2;
bits |= get_bit(scalars[num], i) << 1;
bits |= get_bit(scalars[num], i - 1);
ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
/* select the point to add or subtract */
select_point(digit, 17, pre_comp[num], tmp);
felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
* point */
copy_conditional(tmp[1], tmp[3], sign);
if (!skip) {
point_add(nq[0], nq[1], nq[2],
nq[0], nq[1], nq[2],
mixed, tmp[0], tmp[1], tmp[2]);
} else {
memcpy(nq, tmp, 3 * sizeof(felem));
skip = 0;
}
}
}
}
felem_assign(x_out, nq[0]);
felem_assign(y_out, nq[1]);
felem_assign(z_out, nq[2]);
}
/******************************************************************************/
/*
* FUNCTIONS TO MANAGE PRECOMPUTATION
*/
static NISTP224_PRE_COMP *nistp224_pre_comp_new()
{
NISTP224_PRE_COMP *ret = NULL;
ret = (NISTP224_PRE_COMP *) OPENSSL_malloc(sizeof *ret);
if (!ret) {
ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
return ret;
}
memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
ret->references = 1;
return ret;
}
static void *nistp224_pre_comp_dup(void *src_)
{
NISTP224_PRE_COMP *src = src_;
/* no need to actually copy, these objects never change! */
CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
return src_;
}
static void nistp224_pre_comp_free(void *pre_)
{
int i;
NISTP224_PRE_COMP *pre = pre_;
if (!pre)
return;
i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
if (i > 0)
return;
OPENSSL_free(pre);
}
static void nistp224_pre_comp_clear_free(void *pre_)
{
int i;
NISTP224_PRE_COMP *pre = pre_;
if (!pre)
return;
i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
if (i > 0)
return;
OPENSSL_cleanse(pre, sizeof *pre);
OPENSSL_free(pre);
}
/******************************************************************************/
/*
* OPENSSL EC_METHOD FUNCTIONS
*/
int ec_GFp_nistp224_group_init(EC_GROUP *group)
{
int ret;
ret = ec_GFp_simple_group_init(group);
group->a_is_minus3 = 1;
return ret;
}
int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
const BIGNUM *a, const BIGNUM *b,
BN_CTX *ctx)
{
int ret = 0;
BN_CTX *new_ctx = NULL;
BIGNUM *curve_p, *curve_a, *curve_b;
if (ctx == NULL)
if ((ctx = new_ctx = BN_CTX_new()) == NULL)
return 0;
BN_CTX_start(ctx);
if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
((curve_a = BN_CTX_get(ctx)) == NULL) ||
((curve_b = BN_CTX_get(ctx)) == NULL))
goto err;
BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
EC_R_WRONG_CURVE_PARAMETERS);
goto err;
}
group->field_mod_func = BN_nist_mod_224;
ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
err:
BN_CTX_end(ctx);
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
return ret;
}
/*
* Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
* (X/Z^2, Y/Z^3)
*/
int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
const EC_POINT *point,
BIGNUM *x, BIGNUM *y,
BN_CTX *ctx)
{
felem z1, z2, x_in, y_in, x_out, y_out;
widefelem tmp;
if (EC_POINT_is_at_infinity(group, point)) {
ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
EC_R_POINT_AT_INFINITY);
return 0;
}
if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
(!BN_to_felem(z1, &point->Z)))
return 0;
felem_inv(z2, z1);
felem_square(tmp, z2);
felem_reduce(z1, tmp);
felem_mul(tmp, x_in, z1);
felem_reduce(x_in, tmp);
felem_contract(x_out, x_in);
if (x != NULL) {
if (!felem_to_BN(x, x_out)) {
ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
ERR_R_BN_LIB);
return 0;
}
}
felem_mul(tmp, z1, z2);
felem_reduce(z1, tmp);
felem_mul(tmp, y_in, z1);
felem_reduce(y_in, tmp);
felem_contract(y_out, y_in);
if (y != NULL) {
if (!felem_to_BN(y, y_out)) {
ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
ERR_R_BN_LIB);
return 0;
}
}
return 1;
}
static void make_points_affine(size_t num, felem points[ /* num */ ][3],
felem tmp_felems[ /* num+1 */ ])
{
/*
* Runs in constant time, unless an input is the point at infinity (which
* normally shouldn't happen).
*/
ec_GFp_nistp_points_make_affine_internal(num,
points,
sizeof(felem),
tmp_felems,
(void (*)(void *))felem_one,
(int (*)(const void *))
felem_is_zero_int,
(void (*)(void *, const void *))
felem_assign,
(void (*)(void *, const void *))
felem_square_reduce, (void (*)
(void *,
const void
*,
const void
*))
felem_mul_reduce,
(void (*)(void *, const void *))
felem_inv,
(void (*)(void *, const void *))
felem_contract);
}
/*
* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
* values Result is stored in r (r can equal one of the inputs).
*/
int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
const BIGNUM *scalar, size_t num,
const EC_POINT *points[],
const BIGNUM *scalars[], BN_CTX *ctx)
{
int ret = 0;
int j;
unsigned i;
int mixed = 0;
BN_CTX *new_ctx = NULL;
BIGNUM *x, *y, *z, *tmp_scalar;
felem_bytearray g_secret;
felem_bytearray *secrets = NULL;
felem(*pre_comp)[17][3] = NULL;
felem *tmp_felems = NULL;
felem_bytearray tmp;
unsigned num_bytes;
int have_pre_comp = 0;
size_t num_points = num;
felem x_in, y_in, z_in, x_out, y_out, z_out;
NISTP224_PRE_COMP *pre = NULL;
const felem(*g_pre_comp)[16][3] = NULL;
EC_POINT *generator = NULL;
const EC_POINT *p = NULL;
const BIGNUM *p_scalar = NULL;
if (ctx == NULL)
if ((ctx = new_ctx = BN_CTX_new()) == NULL)
return 0;
BN_CTX_start(ctx);
if (((x = BN_CTX_get(ctx)) == NULL) ||
((y = BN_CTX_get(ctx)) == NULL) ||
((z = BN_CTX_get(ctx)) == NULL) ||
((tmp_scalar = BN_CTX_get(ctx)) == NULL))
goto err;
if (scalar != NULL) {
pre = EC_EX_DATA_get_data(group->extra_data,
nistp224_pre_comp_dup,
nistp224_pre_comp_free,
nistp224_pre_comp_clear_free);
if (pre)
/* we have precomputation, try to use it */
g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
else
/* try to use the standard precomputation */
g_pre_comp = &gmul[0];
generator = EC_POINT_new(group);
if (generator == NULL)
goto err;
/* get the generator from precomputation */
if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
!felem_to_BN(y, g_pre_comp[0][1][1]) ||
!felem_to_BN(z, g_pre_comp[0][1][2])) {
ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
goto err;
}
if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
generator, x, y, z,
ctx))
goto err;
if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
/* precomputation matches generator */
have_pre_comp = 1;
else
/*
* we don't have valid precomputation: treat the generator as a
* random point
*/
num_points = num_points + 1;
}
if (num_points > 0) {
if (num_points >= 3) {
/*
* unless we precompute multiples for just one or two points,
* converting those into affine form is time well spent
*/
mixed = 1;
}
secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
if (mixed)
tmp_felems =
OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
if ((secrets == NULL) || (pre_comp == NULL)
|| (mixed && (tmp_felems == NULL))) {
ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
goto err;
}
/*
* we treat NULL scalars as 0, and NULL points as points at infinity,
* i.e., they contribute nothing to the linear combination
*/
memset(secrets, 0, num_points * sizeof(felem_bytearray));
memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
for (i = 0; i < num_points; ++i) {
if (i == num)
/* the generator */
{
p = EC_GROUP_get0_generator(group);
p_scalar = scalar;
} else
/* the i^th point */
{
p = points[i];
p_scalar = scalars[i];
}
if ((p_scalar != NULL) && (p != NULL)) {
/* reduce scalar to 0 <= scalar < 2^224 */
if ((BN_num_bits(p_scalar) > 224)
|| (BN_is_negative(p_scalar))) {
/*
* this is an unusual input, and we don't guarantee
* constant-timeness
*/
if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
goto err;
}
num_bytes = BN_bn2bin(tmp_scalar, tmp);
} else
num_bytes = BN_bn2bin(p_scalar, tmp);
flip_endian(secrets[i], tmp, num_bytes);
/* precompute multiples */
if ((!BN_to_felem(x_out, &p->X)) ||
(!BN_to_felem(y_out, &p->Y)) ||
(!BN_to_felem(z_out, &p->Z)))
goto err;
felem_assign(pre_comp[i][1][0], x_out);
felem_assign(pre_comp[i][1][1], y_out);
felem_assign(pre_comp[i][1][2], z_out);
for (j = 2; j <= 16; ++j) {
if (j & 1) {
point_add(pre_comp[i][j][0], pre_comp[i][j][1],
pre_comp[i][j][2], pre_comp[i][1][0],
pre_comp[i][1][1], pre_comp[i][1][2], 0,
pre_comp[i][j - 1][0],
pre_comp[i][j - 1][1],
pre_comp[i][j - 1][2]);
} else {
point_double(pre_comp[i][j][0], pre_comp[i][j][1],
pre_comp[i][j][2], pre_comp[i][j / 2][0],
pre_comp[i][j / 2][1],
pre_comp[i][j / 2][2]);
}
}
}
}
if (mixed)
make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
}
/* the scalar for the generator */
if ((scalar != NULL) && (have_pre_comp)) {
memset(g_secret, 0, sizeof g_secret);
/* reduce scalar to 0 <= scalar < 2^224 */
if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
/*
* this is an unusual input, and we don't guarantee
* constant-timeness
*/
if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) {
ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
goto err;
}
num_bytes = BN_bn2bin(tmp_scalar, tmp);
} else
num_bytes = BN_bn2bin(scalar, tmp);
flip_endian(g_secret, tmp, num_bytes);
/* do the multiplication with generator precomputation */
batch_mul(x_out, y_out, z_out,
(const felem_bytearray(*))secrets, num_points,
g_secret,
mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
} else
/* do the multiplication without generator precomputation */
batch_mul(x_out, y_out, z_out,
(const felem_bytearray(*))secrets, num_points,
NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
/* reduce the output to its unique minimal representation */
felem_contract(x_in, x_out);
felem_contract(y_in, y_out);
felem_contract(z_in, z_out);
if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
(!felem_to_BN(z, z_in))) {
ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
goto err;
}
ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
err:
BN_CTX_end(ctx);
if (generator != NULL)
EC_POINT_free(generator);
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
if (secrets != NULL)
OPENSSL_free(secrets);
if (pre_comp != NULL)
OPENSSL_free(pre_comp);
if (tmp_felems != NULL)
OPENSSL_free(tmp_felems);
return ret;
}
int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
{
int ret = 0;
NISTP224_PRE_COMP *pre = NULL;
int i, j;
BN_CTX *new_ctx = NULL;
BIGNUM *x, *y;
EC_POINT *generator = NULL;
felem tmp_felems[32];
/* throw away old precomputation */
EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
nistp224_pre_comp_free,
nistp224_pre_comp_clear_free);
if (ctx == NULL)
if ((ctx = new_ctx = BN_CTX_new()) == NULL)
return 0;
BN_CTX_start(ctx);
if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
goto err;
/* get the generator */
if (group->generator == NULL)
goto err;
generator = EC_POINT_new(group);
if (generator == NULL)
goto err;
BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
goto err;
if ((pre = nistp224_pre_comp_new()) == NULL)
goto err;
/*
* if the generator is the standard one, use built-in precomputation
*/
if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
ret = 1;
goto err;
}
if ((!BN_to_felem(pre->g_pre_comp[0][1][0], &group->generator->X)) ||
(!BN_to_felem(pre->g_pre_comp[0][1][1], &group->generator->Y)) ||
(!BN_to_felem(pre->g_pre_comp[0][1][2], &group->generator->Z)))
goto err;
/*
* compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
* 2^140*G, 2^196*G for the second one
*/
for (i = 1; i <= 8; i <<= 1) {
point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
for (j = 0; j < 27; ++j) {
point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
}
if (i == 8)
break;
point_double(pre->g_pre_comp[0][2 * i][0],
pre->g_pre_comp[0][2 * i][1],
pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
for (j = 0; j < 27; ++j) {
point_double(pre->g_pre_comp[0][2 * i][0],
pre->g_pre_comp[0][2 * i][1],
pre->g_pre_comp[0][2 * i][2],
pre->g_pre_comp[0][2 * i][0],
pre->g_pre_comp[0][2 * i][1],
pre->g_pre_comp[0][2 * i][2]);
}
}
for (i = 0; i < 2; i++) {
/* g_pre_comp[i][0] is the point at infinity */
memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
/* the remaining multiples */
/* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
pre->g_pre_comp[i][2][2]);
/* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
pre->g_pre_comp[i][2][2]);
/* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
pre->g_pre_comp[i][4][2]);
/*
* 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
*/
point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
pre->g_pre_comp[i][2][2]);
for (j = 1; j < 8; ++j) {
/* odd multiples: add G resp. 2^28*G */
point_add(pre->g_pre_comp[i][2 * j + 1][0],
pre->g_pre_comp[i][2 * j + 1][1],
pre->g_pre_comp[i][2 * j + 1][2],
pre->g_pre_comp[i][2 * j][0],
pre->g_pre_comp[i][2 * j][1],
pre->g_pre_comp[i][2 * j][2], 0,
pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
pre->g_pre_comp[i][1][2]);
}
}
make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
nistp224_pre_comp_free,
nistp224_pre_comp_clear_free))
goto err;
ret = 1;
pre = NULL;
err:
BN_CTX_end(ctx);
if (generator != NULL)
EC_POINT_free(generator);
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
if (pre)
nistp224_pre_comp_free(pre);
return ret;
}
int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
{
if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
nistp224_pre_comp_free,
nistp224_pre_comp_clear_free)
!= NULL)
return 1;
else
return 0;
}
#else
static void *dummy = &dummy;
#endif