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openssl/crypto/ec/ecp_nistp224.c
Bodo Möller 396cb5657b More C language police work.
2010-08-27 13:17:59 +00:00

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/* crypto/ec/ecp_nistp224.c */
/*
* Written by Emilia Kasper (Google) for the OpenSSL project.
*/
/* ====================================================================
* Copyright (c) 2000-2010 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.OpenSSL.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* licensing@OpenSSL.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.OpenSSL.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
* OF THE POSSIBILITY OF SUCH DAMAGE.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com).
*
*/
/*
* 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
*/
#ifdef EC_NISTP224_64_GCC_128
#include <stdint.h>
#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;
/******************************************************************************/
/* INTERNAL REPRESENTATION OF FIELD ELEMENTS
*
* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
* where each slice a_i is a 64-bit word, i.e., a field element is an fslice
* array a with 4 elements, where a[i] = a_i.
* Outputs from multiplications are represented as unreduced polynomials
* 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
* where each b_i is a 128-bit word. We ensure that inputs to each field
* 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 a_i < 2^57. We only reduce to the unique minimal
* representation at the end of the computation.
*
*/
typedef uint64_t fslice;
/* Field element represented as a byte arrary.
* 28*8 = 224 bits is also the group order size for the elliptic curve. */
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
* b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for
* (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity,
* gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc.
* Points are given in Jacobian projective coordinates: words 0-3 represent the
* X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the
* Y-coordinate and words 8-11 represent the Z-coordinate. */
static const fslice gmul[16][3][4] = {
{{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
{0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
{0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
{0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
{0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
{0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
{0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
{0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
{0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
{0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
{0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
{0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
{0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
{0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
{0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
{{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
{0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
{0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}
};
/* Precomputation for the group generator. */
typedef struct {
fslice g_pre_comp[16][3][4];
int references;
} NISTP224_PRE_COMP;
const EC_METHOD *EC_GFp_nistp224_method(void)
{
static const EC_METHOD ret = {
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,
ec_GFp_simple_set_compressed_coordinates,
ec_GFp_simple_point2oct,
ec_GFp_simple_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(fslice out[4], 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+21))) & 0x00ffffffffffffff;
}
static void felem_to_bin28(u8 out[28], const fslice in[4])
{
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(fslice out[4], 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 fslice in[4])
{
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.
*
*/
/* Sum two field elements: out += in */
static void felem_sum64(fslice out[4], const fslice in[4])
{
out[0] += in[0];
out[1] += in[1];
out[2] += in[2];
out[3] += in[3];
}
/* Subtract field elements: out -= in */
/* Assumes in[i] < 2^57 */
static void felem_diff64(fslice out[4], const fslice in[4])
{
static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2);
static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2);
static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) -
(((uint64_t) 1) << 42) - (((uint64_t) 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: out128 -= in128 */
/* Assumes in[i] < 2^119 */
static void felem_diff128(uint128_t out[7], const uint128_t in[4])
{
static const uint128_t two120 = ((uint128_t) 1) << 120;
static const uint128_t two120m64 = (((uint128_t) 1) << 120) -
(((uint128_t) 1) << 64);
static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) -
(((uint128_t) 1) << 104) - (((uint128_t) 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(uint128_t out[7], const fslice in[4])
{
static const uint128_t two64p8 = (((uint128_t) 1) << 64) +
(((uint128_t) 1) << 8);
static const uint128_t two64m8 = (((uint128_t) 1) << 64) -
(((uint128_t) 1) << 8);
static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) -
(((uint128_t) 1) << 48) - (((uint128_t) 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: out64 = out64 * scalar
* The scalars we actually use are small, so results fit without overflow */
static void felem_scalar64(fslice out[4], const fslice scalar)
{
out[0] *= scalar;
out[1] *= scalar;
out[2] *= scalar;
out[3] *= scalar;
}
/* Multiply an unreduced field element by a scalar: out128 = out128 * scalar
* The scalars we actually use are small, so results fit without overflow */
static void felem_scalar128(uint128_t out[7], const uint128_t 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(uint128_t out[7], const fslice in[4])
{
out[0] = ((uint128_t) in[0]) * in[0];
out[1] = ((uint128_t) in[0]) * in[1] * 2;
out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1];
out[3] = ((uint128_t) in[0]) * in[3] * 2 +
((uint128_t) in[1]) * in[2] * 2;
out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2];
out[5] = ((uint128_t) in[2]) * in[3] * 2;
out[6] = ((uint128_t) in[3]) * in[3];
}
/* Multiply two field elements: out = in1 * in2 */
static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4])
{
out[0] = ((uint128_t) in1[0]) * in2[0];
out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0];
out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] +
((uint128_t) in1[2]) * in2[0];
out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] +
((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0];
out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] +
((uint128_t) in1[3]) * in2[1];
out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2];
out[6] = ((uint128_t) in1[3]) * in2[3];
}
/* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126,
* ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */
static void felem_reduce(fslice out[4], const uint128_t in[7])
{
static const uint128_t two127p15 = (((uint128_t) 1) << 127) +
(((uint128_t) 1) << 15);
static const uint128_t two127m71 = (((uint128_t) 1) << 127) -
(((uint128_t) 1) << 71);
static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) -
(((uint128_t) 1) << 71) - (((uint128_t) 1) << 55);
uint128_t 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];
output[4] = 0;
/* 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 */
/* Eliminate output[4] */
output[2] += output[4] >> 16;
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;
out[1] = output[1] & 0x00ffffffffffffff;
output[3] += output[2] >> 56;
out[2] = output[2] & 0x00ffffffffffffff;
/* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
* out[3] < 2^57 (due to final carry) */
out[3] = output[3];
}
/* Reduce to unique minimal representation */
static void felem_contract(fslice out[4], const fslice in[4])
{
static const int64_t two56 = ((uint64_t) 1) << 56;
/* 0 <= in < 2^225 */
/* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
int64_t tmp[4], a;
tmp[0] = (int64_t) in[0] - (in[3] >> 56);
tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000);
tmp[2] = (int64_t) in[2];
tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff;
/* eliminate negative coefficients */
a = tmp[0] >> 63;
tmp[0] += two56 & a;
tmp[1] -= 1 & a;
a = tmp[1] >> 63;
tmp[1] += two56 & a;
tmp[2] -= 1 & a;
a = tmp[2] >> 63;
tmp[2] += two56 & a;
tmp[3] -= 1 & a;
a = tmp[3] >> 63;
tmp[3] += two56 & a;
tmp[0] += 1 & a;
tmp[1] -= (1 & a) << 40;
/* carry 1 -> 2 -> 3 */
tmp[2] += tmp[1] >> 56;
tmp[1] &= 0x00ffffffffffffff;
tmp[3] += tmp[2] >> 56;
tmp[2] &= 0x00ffffffffffffff;
/* 0 <= in < 2^224 + 2^96 - 1 */
/* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
tmp[0] -= (tmp[3] >> 56);
tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000);
tmp[3] &= 0x00ffffffffffffff;
/* eliminate negative coefficients */
a = tmp[0] >> 63;
tmp[0] += two56 & a;
tmp[1] -= 1 & a;
a = tmp[1] >> 63;
tmp[1] += two56 & a;
tmp[2] -= 1 & a;
a = tmp[2] >> 63;
tmp[2] += two56 & a;
tmp[3] -= 1 & a;
a = tmp[3] >> 63;
tmp[3] += two56 & a;
tmp[0] += 1 & a;
tmp[1] -= (1 & a) << 40;
/* carry 1 -> 2 -> 3 */
tmp[2] += tmp[1] >> 56;
tmp[1] &= 0x00ffffffffffffff;
tmp[3] += tmp[2] >> 56;
tmp[2] &= 0x00ffffffffffffff;
/* Now 0 <= in < 2^224 */
/* if in > 2^224 - 2^96, reduce */
/* a = 0 iff in > 2^224 - 2^96, i.e.,
* the high 128 bits are all 1 and the lower part is non-zero */
a = (tmp[3] + 1) | (tmp[2] + 1) |
((tmp[1] | 0x000000ffffffffff) + 1) |
((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63));
/* turn a into an all-one mask (if a = 0) or an all-zero mask */
a = ((a & 0x00ffffffffffffff) - 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;
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 fslice felem_is_zero(const fslice in[4])
{
fslice 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);
}
/* Invert a field element */
/* Computation chain copied from djb's code */
static void felem_inv(fslice out[4], const fslice in[4])
{
fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4];
uint128_t tmp[7];
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(fslice *out, const fslice *in, unsigned len, fslice icopy)
{
unsigned i;
/* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
const fslice copy = -icopy;
for (i = 0; i < len; ++i)
{
const fslice tmp = copy & (in[i] ^ out[i]);
out[i] ^= tmp;
}
}
/* Copy in constant time:
* if isel == 1, copy in2 to out,
* if isel == 0, copy in1 to out. */
static void select_conditional(fslice *out, const fslice *in1, const fslice *in2,
unsigned len, fslice isel)
{
unsigned i;
/* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */
const fslice sel = -isel;
for (i = 0; i < len; ++i)
{
const fslice tmp = sel & (in1[i] ^ in2[i]);
out[i] = in1[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(fslice x_out[4], fslice y_out[4], fslice z_out[4],
const fslice x_in[4], const fslice y_in[4], const fslice z_in[4])
{
uint128_t tmp[7], tmp2[7];
fslice delta[4];
fslice gamma[4];
fslice beta[4];
fslice alpha[4];
fslice ftmp[4], ftmp2[4];
memcpy(ftmp, x_in, 4 * sizeof(fslice));
memcpy(ftmp2, x_in, 4 * sizeof(fslice));
/* 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_diff64(ftmp, delta);
/* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
felem_sum64(ftmp2, delta);
/* ftmp2[i] < 2^57 + 2^57 = 2^58 */
felem_scalar64(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 */
memcpy(ftmp, beta, 4 * sizeof(fslice));
felem_scalar64(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_sum64(delta, gamma);
/* delta[i] < 2^57 + 2^57 = 2^58 */
memcpy(ftmp, y_in, 4 * sizeof(fslice));
felem_sum64(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_scalar64(beta, 4);
/* beta[i] < 4 * 2^57 = 2^59 */
felem_diff64(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 */
felem_scalar128(tmp2, 8);
/* tmp2[i] < 8 * 2^116 = 2^119 */
felem_diff128(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 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(fslice x3[4], fslice y3[4], fslice z3[4],
const fslice x1[4], const fslice y1[4], const fslice z1[4],
const fslice x2[4], const fslice y2[4], const fslice z2[4])
{
fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4];
uint128_t tmp[7], tmp2[7];
fslice z1_is_zero, z2_is_zero, x_equal, y_equal;
/* ftmp = z1^2 */
felem_square(tmp, z1);
felem_reduce(ftmp, tmp);
/* ftmp2 = z2^2 */
felem_square(tmp, z2);
felem_reduce(ftmp2, tmp);
/* ftmp3 = z1^3 */
felem_mul(tmp, ftmp, z1);
felem_reduce(ftmp3, tmp);
/* ftmp4 = z2^3 */
felem_mul(tmp, ftmp2, z2);
felem_reduce(ftmp4, tmp);
/* ftmp3 = z1^3*y2 */
felem_mul(tmp, ftmp3, y2);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
/* ftmp4 = z2^3*y1 */
felem_mul(tmp2, ftmp4, y1);
felem_reduce(ftmp4, tmp2);
/* 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);
/* ftmp = z1^2*x2 */
felem_mul(tmp, ftmp, x2);
/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
/* ftmp2 =z2^2*x1 */
felem_mul(tmp2, ftmp2, x1);
felem_reduce(ftmp2, tmp2);
/* ftmp = z1^2*x2 - z2^2*x1 */
felem_diff128(tmp, tmp2);
/* 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 */
felem_mul(tmp, z1, z2);
felem_reduce(ftmp5, tmp);
/* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */
felem_mul(tmp, ftmp, ftmp5);
felem_reduce(z3, tmp);
/* ftmp = (z1^2*x2 - z2^2*x1)^2 */
memcpy(ftmp5, ftmp, 4 * sizeof(fslice));
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);
/* ftmp4 = 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 */
memcpy(ftmp5, ftmp2, 4 * sizeof(fslice));
felem_scalar64(ftmp5, 2);
/* ftmp5[i] < 2 * 2^57 = 2^58 */
/* x3 = (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(x3, tmp2);
/* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */
felem_diff64(ftmp2, x3);
/* 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 - x3) */
felem_mul(tmp2, ftmp3, ftmp2);
/* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
/* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) -
z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
felem_diff128(tmp2, tmp);
/* tmp2[i] < 2^118 + 2^120 < 2^121 */
felem_reduce(y3, tmp2);
/* the result (x3, y3, z3) 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(x3, x2, 4, z1_is_zero);
copy_conditional(x3, x1, 4, z2_is_zero);
copy_conditional(y3, y2, 4, z1_is_zero);
copy_conditional(y3, y1, 4, z2_is_zero);
copy_conditional(z3, z2, 4, z1_is_zero);
copy_conditional(z3, z1, 4, z2_is_zero);
}
/* Select a point from an array of 16 precomputed point multiples,
* in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point
* pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */
static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4],
fslice out[12])
{
fslice tmp[5][12];
select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]);
select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]);
select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]);
select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]);
select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]);
select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]);
select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]);
select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]);
select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]);
select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]);
select_conditional(out, tmp[1], tmp[4], 12, bits[0]);
}
/* Interleaved point multiplication using precomputed point multiples:
* The small point multiples 0*P, 1*P, ..., 15*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(fslice x_out[4], fslice y_out[4], fslice z_out[4],
const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4])
{
unsigned i, j, num;
unsigned gen_mul = (g_scalar != NULL);
fslice nq[12], nqt[12], tmp[12];
fslice bits[4];
u8 byte;
/* set nq to the point at infinity */
memset(nq, 0, 12 * sizeof(fslice));
/* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble,
* double 4 times, then add the precomputed point multiples.
* If we are also adding multiples of the generator, then interleave
* these additions with the last 56 doublings. */
for (i = (num_points ? 28 : 7); i > 0; --i)
{
for (j = 0; j < 8; ++j)
{
/* double once */
point_double(nq, nq+4, nq+8, nq, nq+4, nq+8);
/* add multiples of the generator */
if ((gen_mul) && (i <= 7))
{
bits[3] = (g_scalar[i+20] >> (7-j)) & 1;
bits[2] = (g_scalar[i+13] >> (7-j)) & 1;
bits[1] = (g_scalar[i+6] >> (7-j)) & 1;
bits[0] = (g_scalar[i-1] >> (7-j)) & 1;
/* select the point to add, in constant time */
select_point(bits, g_pre_comp, tmp);
memcpy(nqt, nq, 12 * sizeof(fslice));
point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8,
tmp, tmp+4, tmp+8);
}
/* do an addition after every 4 doublings */
if (j % 4 == 3)
{
/* loop over all scalars */
for (num = 0; num < num_points; ++num)
{
byte = scalars[num][i-1];
bits[3] = (byte >> (10-j)) & 1;
bits[2] = (byte >> (9-j)) & 1;
bits[1] = (byte >> (8-j)) & 1;
bits[0] = (byte >> (7-j)) & 1;
/* select the point to add */
select_point(bits,
pre_comp[num], tmp);
memcpy(nqt, nq, 12 * sizeof(fslice));
point_add(nq, nq+4, nq+8, nqt, nqt+4,
nqt+8, tmp, tmp+4, tmp+8);
}
}
}
}
memcpy(x_out, nq, 4 * sizeof(fslice));
memcpy(y_out, nq+4, 4 * sizeof(fslice));
memcpy(z_out, nq+8, 4 * sizeof(fslice));
}
/******************************************************************************/
/* FUNCTIONS TO MANAGE PRECOMPUTATION
*/
static NISTP224_PRE_COMP *nistp224_pre_comp_new()
{
NISTP224_PRE_COMP *ret = NULL;
ret = (NISTP224_PRE_COMP *)OPENSSL_malloc(sizeof(NISTP224_PRE_COMP));
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)
{
fslice z1[4], z2[4], x_in[4], y_in[4], x_out[4], y_out[4];
uint128_t tmp[7];
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;
}
/* 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 i, j;
BN_CTX *new_ctx = NULL;
BIGNUM *x, *y, *z, *tmp_scalar;
felem_bytearray g_secret;
felem_bytearray *secrets = NULL;
fslice (*pre_comp)[16][3][4] = NULL;
felem_bytearray tmp;
unsigned num_bytes;
int have_pre_comp = 0;
size_t num_points = num;
fslice x_in[4], y_in[4], z_in[4], x_out[4], y_out[4], z_out[4];
NISTP224_PRE_COMP *pre = NULL;
fslice (*g_pre_comp)[3][4] = 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 = pre->g_pre_comp;
else
/* try to use the standard precomputation */
g_pre_comp = (fslice (*)[3][4]) gmul;
generator = EC_POINT_new(group);
if (generator == NULL)
goto err;
/* get the generator from precomputation */
if (!felem_to_BN(x, g_pre_comp[1][0]) ||
!felem_to_BN(y, g_pre_comp[1][1]) ||
!felem_to_BN(z, g_pre_comp[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;
}
secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
pre_comp = OPENSSL_malloc(num_points * 16 * 3 * 4 * sizeof(fslice));
if ((num_points) && ((secrets == NULL) || (pre_comp == 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 * 16 * 3 * 4 * sizeof(fslice));
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))
{
num_bytes = BN_num_bytes(p_scalar);
/* reduce scalar to 0 <= scalar < 2^224 */
if ((num_bytes > sizeof(felem_bytearray)) || (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
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;
memcpy(pre_comp[i][1][0], x_out, 4 * sizeof(fslice));
memcpy(pre_comp[i][1][1], y_out, 4 * sizeof(fslice));
memcpy(pre_comp[i][1][2], z_out, 4 * sizeof(fslice));
for (j = 1; j < 8; ++j)
{
point_double(pre_comp[i][2*j][0],
pre_comp[i][2*j][1],
pre_comp[i][2*j][2],
pre_comp[i][j][0],
pre_comp[i][j][1],
pre_comp[i][j][2]);
point_add(pre_comp[i][2*j+1][0],
pre_comp[i][2*j+1][1],
pre_comp[i][2*j+1][2],
pre_comp[i][1][0],
pre_comp[i][1][1],
pre_comp[i][1][2],
pre_comp[i][2*j][0],
pre_comp[i][2*j][1],
pre_comp[i][2*j][2]);
}
}
}
/* the scalar for the generator */
if ((scalar != NULL) && (have_pre_comp))
{
memset(g_secret, 0, sizeof g_secret);
num_bytes = BN_num_bytes(scalar);
/* reduce scalar to 0 <= scalar < 2^224 */
if ((num_bytes > sizeof(felem_bytearray)) || (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
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, (const fslice (*)[16][3][4]) pre_comp,
(const fslice (*)[3][4]) 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, (const fslice (*)[16][3][4]) 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);
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;
/* 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[1][0], &group->generator->X)) ||
(!BN_to_felem(pre->g_pre_comp[1][1], &group->generator->Y)) ||
(!BN_to_felem(pre->g_pre_comp[1][2], &group->generator->Z)))
goto err;
/* compute 2^56*G, 2^112*G, 2^168*G */
for (i = 1; i < 5; ++i)
{
point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1],
pre->g_pre_comp[2*i][2], pre->g_pre_comp[i][0],
pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
for (j = 0; j < 55; ++j)
{
point_double(pre->g_pre_comp[2*i][0],
pre->g_pre_comp[2*i][1],
pre->g_pre_comp[2*i][2],
pre->g_pre_comp[2*i][0],
pre->g_pre_comp[2*i][1],
pre->g_pre_comp[2*i][2]);
}
}
/* g_pre_comp[0] is the point at infinity */
memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
/* the remaining multiples */
/* 2^56*G + 2^112*G */
point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
pre->g_pre_comp[2][2]);
/* 2^56*G + 2^168*G */
point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
pre->g_pre_comp[2][2]);
/* 2^112*G + 2^168*G */
point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
pre->g_pre_comp[4][2]);
/* 2^56*G + 2^112*G + 2^168*G */
point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
pre->g_pre_comp[2][2]);
for (i = 1; i < 8; ++i)
{
/* odd multiples: add G */
point_add(pre->g_pre_comp[2*i+1][0], pre->g_pre_comp[2*i+1][1],
pre->g_pre_comp[2*i+1][2], pre->g_pre_comp[2*i][0],
pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2],
pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
pre->g_pre_comp[1][2]);
}
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
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