dmake: do not set MAKEFLAGS=k

[unleashed/tickless.git] / lib / libcrypto / ec / ecp_nistp224.c

/* $OpenBSD: ecp_nistp224.c,v 1.19 2017/05/02 03:59:44 deraadt Exp $ */
/*
 * Written by Emilia Kasper (Google) for the OpenSSL project.
 */
/*
 * Copyright (c) 2011 Google Inc.
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/*
 * 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 <stdint.h>
#include <string.h>

#include <openssl/opensslconf.h>

#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128

#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 = {
                .flags = EC_FLAGS_DEFAULT_OCT,
                .field_type = NID_X9_62_prime_field,
                .group_init = ec_GFp_nistp224_group_init,
                .group_finish = ec_GFp_simple_group_finish,
                .group_clear_finish = ec_GFp_simple_group_clear_finish,
                .group_copy = ec_GFp_nist_group_copy,
                .group_set_curve = ec_GFp_nistp224_group_set_curve,
                .group_get_curve = ec_GFp_simple_group_get_curve,
                .group_get_degree = ec_GFp_simple_group_get_degree,
                .group_check_discriminant =
                    ec_GFp_simple_group_check_discriminant,
                .point_init = ec_GFp_simple_point_init,
                .point_finish = ec_GFp_simple_point_finish,
                .point_clear_finish = ec_GFp_simple_point_clear_finish,
                .point_copy = ec_GFp_simple_point_copy,
                .point_set_to_infinity = ec_GFp_simple_point_set_to_infinity,
                .point_set_Jprojective_coordinates_GFp =
                    ec_GFp_simple_set_Jprojective_coordinates_GFp,
                .point_get_Jprojective_coordinates_GFp =
                    ec_GFp_simple_get_Jprojective_coordinates_GFp,
                .point_set_affine_coordinates =
                    ec_GFp_simple_point_set_affine_coordinates,
                .point_get_affine_coordinates =
                    ec_GFp_nistp224_point_get_affine_coordinates,
                .add = ec_GFp_simple_add,
                .dbl = ec_GFp_simple_dbl,
                .invert = ec_GFp_simple_invert,
                .is_at_infinity = ec_GFp_simple_is_at_infinity,
                .is_on_curve = ec_GFp_simple_is_on_curve,
                .point_cmp = ec_GFp_simple_cmp,
                .make_affine = ec_GFp_simple_make_affine,
                .points_make_affine = ec_GFp_simple_points_make_affine,
                .mul = ec_GFp_nistp224_points_mul,
                .precompute_mult = ec_GFp_nistp224_precompute_mult,
                .have_precompute_mult = ec_GFp_nistp224_have_precompute_mult,
                .field_mul = ec_GFp_nist_field_mul,
                .field_sqr = ec_GFp_nist_field_sqr
        };

        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 + 21))) & 0x00ffffffffffffff;
}

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) {
                ECerror(EC_R_BIGNUM_OUT_OF_RANGE);
                return 0;
        }
        if (BN_is_negative(bn)) {
                ECerror(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. */
static void 
select_point(const u64 idx, unsigned int size, const felem pre_comp[ /* size */ ][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) {
                                point_add(nq[0], nq[1], nq[2],
                                    nq[0], nq[1], nq[2],
                                    1 /* mixed */ , 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 = malloc(sizeof *ret);
        if (!ret) {
                ECerror(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;

        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;

        freezero(pre, sizeof *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))) {
                ECerror(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);
        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) > 0) {
                ECerror(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)) {
                        ECerror(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)) {
                        ECerror(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])) {
                        ECerror(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 = calloc(num_points, sizeof(felem_bytearray));
                pre_comp = calloc(num_points, 17 * 3 * sizeof(felem));
                if (mixed) {
                        /* XXX should do more int overflow checking */
                        tmp_felems = reallocarray(NULL,
                            (num_points * 17 + 1), sizeof(felem));
                }
                if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL))) {
                        ECerror(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
                 */
                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)) {
                                                ECerror(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)) {
                                ECerror(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))) {
                ECerror(ERR_R_BN_LIB);
                goto err;
        }
        ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);

err:
        BN_CTX_end(ctx);
        EC_POINT_free(generator);
        BN_CTX_free(new_ctx);
        free(secrets);
        free(pre_comp);
        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);
        EC_POINT_free(generator);
        BN_CTX_free(new_ctx);
        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;
}

#endif