/* Copyright (c) 2015, Google Inc.

 *

 * Permission to use, copy, modify, and/or 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 <openssl/base.h>

#include <openssl/bn.h>

#include <openssl/ec.h>

#include <openssl/err.h>

#include <openssl/mem.h>

#include <assert.h>

#include <string.h>

#include "internal.h"

#include "../delocate.h"

#include "../../internal.h"

#if defined(BORINGSSL_HAS_UINT128) && !defined(OPENSSL_SMALL)

// 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-p224_limb

// representation is an 'p224_felem'; a 7-p224_widelimb representation is a

// 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we

// don't always reduce the representations: we ensure that inputs to each

// p224_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 p224_felem satisfying a_i <

// 2^57. We only reduce to the unique minimal representation at the end of the

// computation.

typedef uint64_t p224_limb;

typedef uint128_t p224_widelimb;

typedef p224_limb p224_felem[4];

typedef p224_widelimb p224_widefelem[7];

// 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 p224_felem g_p224_pre_comp[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}}}};

// Helper functions to convert field elements to/from internal representation

static void p224_generic_to_felem(p224_felem out, const EC_FELEM *in) {

  // |p224_felem|'s minimal representation uses four 56-bit words. |EC_FELEM|

  // uses four 64-bit words. (The top-most word only has 32 bits.)

  out[0] = in->words[0] & 0x00ffffffffffffff;

  out[1] = ((in->words[0] >> 56) | (in->words[1] << 8)) & 0x00ffffffffffffff;

  out[2] = ((in->words[1] >> 48) | (in->words[2] << 16)) & 0x00ffffffffffffff;

  out[3] = ((in->words[2] >> 40) | (in->words[3] << 24)) & 0x00ffffffffffffff;

}

// Requires 0 <= in < 2*p (always call p224_felem_reduce first)

static void p224_felem_to_generic(EC_FELEM *out, const p224_felem in) {

  // Reduce to unique minimal representation.

  static const int64_t two56 = ((p224_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 <= tmp < p

  p224_felem tmp2;

  tmp2[0] = tmp[0];

  tmp2[1] = tmp[1];

  tmp2[2] = tmp[2];

  tmp2[3] = tmp[3];

  // |p224_felem|'s minimal representation uses four 56-bit words. |EC_FELEM|

  // uses four 64-bit words. (The top-most word only has 32 bits.)

  out->words[0] = tmp2[0] | (tmp2[1] << 56);

  out->words[1] = (tmp2[1] >> 8) | (tmp2[2] << 48);

  out->words[2] = (tmp2[2] >> 16) | (tmp2[3] << 40);

  out->words[3] = tmp2[3] >> 24;

}

// 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 p224_felem_assign(p224_felem out, const p224_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 p224_felem_sum(p224_felem out, const p224_felem in) {

  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 p224_felem_diff(p224_felem out, const p224_felem in) {

  static const p224_limb two58p2 =

      (((p224_limb)1) << 58) + (((p224_limb)1) << 2);

  static const p224_limb two58m2 =

      (((p224_limb)1) << 58) - (((p224_limb)1) << 2);

  static const p224_limb two58m42m2 =

      (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_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 p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) {

  static const p224_widelimb two120 = ((p224_widelimb)1) << 120;

  static const p224_widelimb two120m64 =

      (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64);

  static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) -

                                             (((p224_widelimb)1) << 104) -

                                             (((p224_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 p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) {

  static const p224_widelimb two64p8 =

      (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8);

  static const p224_widelimb two64m8 =

      (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8);

  static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) -

                                          (((p224_widelimb)1) << 48) -

                                          (((p224_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 p224_felem_scalar(p224_felem out, const p224_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 p224_widefelem_scalar(p224_widefelem out,

                                  const p224_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 p224_felem_square(p224_widefelem out, const p224_felem in) {

  p224_limb tmp0, tmp1, tmp2;

  tmp0 = 2 * in[0];

  tmp1 = 2 * in[1];

  tmp2 = 2 * in[2];

  out[0] = ((p224_widelimb)in[0]) * in[0];

  out[1] = ((p224_widelimb)in[0]) * tmp1;

  out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1];

  out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2;

  out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2];

  out[5] = ((p224_widelimb)in[3]) * tmp2;

  out[6] = ((p224_widelimb)in[3]) * in[3];

}

// Multiply two field elements: out = in1 * in2

static void p224_felem_mul(p224_widefelem out, const p224_felem in1,

                           const p224_felem in2) {

  out[0] = ((p224_widelimb)in1[0]) * in2[0];

  out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0];

  out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] +

           ((p224_widelimb)in1[2]) * in2[0];

  out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] +

           ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0];

  out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] +

           ((p224_widelimb)in1[3]) * in2[1];

  out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2];

  out[6] = ((p224_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 p224_felem_reduce(p224_felem out, const p224_widefelem in) {

  static const p224_widelimb two127p15 =

      (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15);

  static const p224_widelimb two127m71 =

      (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71);

  static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) -

                                            (((p224_widelimb)1) << 71) -

                                            (((p224_widelimb)1) << 55);

  p224_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];

}

// Get negative value: out = -in

// Requires in[i] < 2^63,

// ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16

static void p224_felem_neg(p224_felem out, const p224_felem in) {

  p224_widefelem tmp = {0};

  p224_felem_diff_128_64(tmp, in);

  p224_felem_reduce(out, tmp);

}

// 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 p224_limb p224_felem_is_zero(const p224_felem in) {

  p224_limb zero = in[0] | in[1] | in[2] | in[3];

  zero = (((int64_t)(zero)-1) >> 63) & 1;

  p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) |

                     (in[2] ^ 0x00ffffffffffffff) |

                     (in[3] ^ 0x00ffffffffffffff);

  two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1;

  p224_limb 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 p224_felem_inv(p224_felem out, const p224_felem in) {

  p224_felem ftmp, ftmp2, ftmp3, ftmp4;

  p224_widefelem tmp;

  p224_felem_square(tmp, in);

  p224_felem_reduce(ftmp, tmp);  // 2

  p224_felem_mul(tmp, in, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^2 - 1

  p224_felem_square(tmp, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^3 - 2

  p224_felem_mul(tmp, in, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^3 - 1

  p224_felem_square(tmp, ftmp);

  p224_felem_reduce(ftmp2, tmp);  // 2^4 - 2

  p224_felem_square(tmp, ftmp2);

  p224_felem_reduce(ftmp2, tmp);  // 2^5 - 4

  p224_felem_square(tmp, ftmp2);

  p224_felem_reduce(ftmp2, tmp);  // 2^6 - 8

  p224_felem_mul(tmp, ftmp2, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^6 - 1

  p224_felem_square(tmp, ftmp);

  p224_felem_reduce(ftmp2, tmp);  // 2^7 - 2

  for (size_t i = 0; i < 5; ++i) {  // 2^12 - 2^6

    p224_felem_square(tmp, ftmp2);

    p224_felem_reduce(ftmp2, tmp);

  }

  p224_felem_mul(tmp, ftmp2, ftmp);

  p224_felem_reduce(ftmp2, tmp);  // 2^12 - 1

  p224_felem_square(tmp, ftmp2);

  p224_felem_reduce(ftmp3, tmp);  // 2^13 - 2

  for (size_t i = 0; i < 11; ++i) {  // 2^24 - 2^12

    p224_felem_square(tmp, ftmp3);

    p224_felem_reduce(ftmp3, tmp);

  }

  p224_felem_mul(tmp, ftmp3, ftmp2);

  p224_felem_reduce(ftmp2, tmp);  // 2^24 - 1

  p224_felem_square(tmp, ftmp2);

  p224_felem_reduce(ftmp3, tmp);  // 2^25 - 2

  for (size_t i = 0; i < 23; ++i) {  // 2^48 - 2^24

    p224_felem_square(tmp, ftmp3);

    p224_felem_reduce(ftmp3, tmp);

  }

  p224_felem_mul(tmp, ftmp3, ftmp2);

  p224_felem_reduce(ftmp3, tmp);  // 2^48 - 1

  p224_felem_square(tmp, ftmp3);

  p224_felem_reduce(ftmp4, tmp);  // 2^49 - 2

  for (size_t i = 0; i < 47; ++i) {  // 2^96 - 2^48

    p224_felem_square(tmp, ftmp4);

    p224_felem_reduce(ftmp4, tmp);

  }

  p224_felem_mul(tmp, ftmp3, ftmp4);

  p224_felem_reduce(ftmp3, tmp);  // 2^96 - 1

  p224_felem_square(tmp, ftmp3);

  p224_felem_reduce(ftmp4, tmp);  // 2^97 - 2

  for (size_t i = 0; i < 23; ++i) {  // 2^120 - 2^24

    p224_felem_square(tmp, ftmp4);

    p224_felem_reduce(ftmp4, tmp);

  }

  p224_felem_mul(tmp, ftmp2, ftmp4);

  p224_felem_reduce(ftmp2, tmp);  // 2^120 - 1

  for (size_t i = 0; i < 6; ++i) {  // 2^126 - 2^6

    p224_felem_square(tmp, ftmp2);

    p224_felem_reduce(ftmp2, tmp);

  }

  p224_felem_mul(tmp, ftmp2, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^126 - 1

  p224_felem_square(tmp, ftmp);

  p224_felem_reduce(ftmp, tmp);  // 2^127 - 2

  p224_felem_mul(tmp, ftmp, in);

  p224_felem_reduce(ftmp, tmp);  // 2^127 - 1

  for (size_t i = 0; i < 97; ++i) {  // 2^224 - 2^97

    p224_felem_square(tmp, ftmp);

    p224_felem_reduce(ftmp, tmp);

  }

  p224_felem_mul(tmp, ftmp, ftmp3);

  p224_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 p224_copy_conditional(p224_felem out, const p224_felem in,

                                  p224_limb icopy) {

  // icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one

  const p224_limb copy = -icopy;

  for (size_t i = 0; i < 4; ++i) {

    const p224_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 p224_point_double(p224_felem x_out, p224_felem y_out,

                              p224_felem z_out, const p224_felem x_in,

                              const p224_felem y_in, const p224_felem z_in) {

  p224_widefelem tmp, tmp2;

  p224_felem delta, gamma, beta, alpha, ftmp, ftmp2;

  p224_felem_assign(ftmp, x_in);

  p224_felem_assign(ftmp2, x_in);

  // delta = z^2

  p224_felem_square(tmp, z_in);

  p224_felem_reduce(delta, tmp);

  // gamma = y^2

  p224_felem_square(tmp, y_in);

  p224_felem_reduce(gamma, tmp);

  // beta = x*gamma

  p224_felem_mul(tmp, x_in, gamma);

  p224_felem_reduce(beta, tmp);

  // alpha = 3*(x-delta)*(x+delta)

  p224_felem_diff(ftmp, delta);

  // ftmp[i] < 2^57 + 2^58 + 2 < 2^59

  p224_felem_sum(ftmp2, delta);

  // ftmp2[i] < 2^57 + 2^57 = 2^58

  p224_felem_scalar(ftmp2, 3);

  // ftmp2[i] < 3 * 2^58 < 2^60

  p224_felem_mul(tmp, ftmp, ftmp2);

  // tmp[i] < 2^60 * 2^59 * 4 = 2^121

  p224_felem_reduce(alpha, tmp);

  // x' = alpha^2 - 8*beta

  p224_felem_square(tmp, alpha);

  // tmp[i] < 4 * 2^57 * 2^57 = 2^116

  p224_felem_assign(ftmp, beta);

  p224_felem_scalar(ftmp, 8);

  // ftmp[i] < 8 * 2^57 = 2^60

  p224_felem_diff_128_64(tmp, ftmp);

  // tmp[i] < 2^116 + 2^64 + 8 < 2^117

  p224_felem_reduce(x_out, tmp);

  // z' = (y + z)^2 - gamma - delta

  p224_felem_sum(delta, gamma);

  // delta[i] < 2^57 + 2^57 = 2^58

  p224_felem_assign(ftmp, y_in);

  p224_felem_sum(ftmp, z_in);

  // ftmp[i] < 2^57 + 2^57 = 2^58

  p224_felem_square(tmp, ftmp);

  // tmp[i] < 4 * 2^58 * 2^58 = 2^118

  p224_felem_diff_128_64(tmp, delta);

  // tmp[i] < 2^118 + 2^64 + 8 < 2^119

  p224_felem_reduce(z_out, tmp);

  // y' = alpha*(4*beta - x') - 8*gamma^2

  p224_felem_scalar(beta, 4);

  // beta[i] < 4 * 2^57 = 2^59

  p224_felem_diff(beta, x_out);

  // beta[i] < 2^59 + 2^58 + 2 < 2^60

  p224_felem_mul(tmp, alpha, beta);

  // tmp[i] < 4 * 2^57 * 2^60 = 2^119

  p224_felem_square(tmp2, gamma);

  // tmp2[i] < 4 * 2^57 * 2^57 = 2^116

  p224_widefelem_scalar(tmp2, 8);

  // tmp2[i] < 8 * 2^116 = 2^119

  p224_widefelem_diff(tmp, tmp2);

  // tmp[i] < 2^119 + 2^120 < 2^121

  p224_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 p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3,

                           const p224_felem x1, const p224_felem y1,

                           const p224_felem z1, const int mixed,

                           const p224_felem x2, const p224_felem y2,

                           const p224_felem z2) {

  p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;

  p224_widefelem tmp, tmp2;

  p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal;

  if (!mixed) {

    // ftmp2 = z2^2

    p224_felem_square(tmp, z2);

    p224_felem_reduce(ftmp2, tmp);

    // ftmp4 = z2^3

    p224_felem_mul(tmp, ftmp2, z2);

    p224_felem_reduce(ftmp4, tmp);

    // ftmp4 = z2^3*y1

    p224_felem_mul(tmp2, ftmp4, y1);

    p224_felem_reduce(ftmp4, tmp2);

    // ftmp2 = z2^2*x1

    p224_felem_mul(tmp2, ftmp2, x1);

    p224_felem_reduce(ftmp2, tmp2);

  } else {

    // We'll assume z2 = 1 (special case z2 = 0 is handled later)

    // ftmp4 = z2^3*y1

    p224_felem_assign(ftmp4, y1);

    // ftmp2 = z2^2*x1

    p224_felem_assign(ftmp2, x1);

  }

  // ftmp = z1^2

  p224_felem_square(tmp, z1);

  p224_felem_reduce(ftmp, tmp);

  // ftmp3 = z1^3

  p224_felem_mul(tmp, ftmp, z1);

  p224_felem_reduce(ftmp3, tmp);

  // tmp = z1^3*y2

  p224_felem_mul(tmp, ftmp3, y2);

  // tmp[i] < 4 * 2^57 * 2^57 = 2^116

  // ftmp3 = z1^3*y2 - z2^3*y1

  p224_felem_diff_128_64(tmp, ftmp4);

  // tmp[i] < 2^116 + 2^64 + 8 < 2^117

  p224_felem_reduce(ftmp3, tmp);

  // tmp = z1^2*x2

  p224_felem_mul(tmp, ftmp, x2);

  // tmp[i] < 4 * 2^57 * 2^57 = 2^116

  // ftmp = z1^2*x2 - z2^2*x1

  p224_felem_diff_128_64(tmp, ftmp2);

  // tmp[i] < 2^116 + 2^64 + 8 < 2^117

  p224_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 = p224_felem_is_zero(ftmp);

  y_equal = p224_felem_is_zero(ftmp3);

  z1_is_zero = p224_felem_is_zero(z1);

  z2_is_zero = p224_felem_is_zero(z2);

  // In affine coordinates, (X_1, Y_1) == (X_2, Y_2)

  p224_limb is_nontrivial_double =

      x_equal & y_equal & (1 - z1_is_zero) & (1 - z2_is_zero);

  if (constant_time_declassify_w(is_nontrivial_double)) {

    p224_point_double(x3, y3, z3, x1, y1, z1);

    return;

  }

  // ftmp5 = z1*z2

  if (!mixed) {

    p224_felem_mul(tmp, z1, z2);

    p224_felem_reduce(ftmp5, tmp);

  } else {

    // special case z2 = 0 is handled later

    p224_felem_assign(ftmp5, z1);

  }

  // z_out = (z1^2*x2 - z2^2*x1)*(z1*z2)

  p224_felem_mul(tmp, ftmp, ftmp5);

  p224_felem_reduce(z_out, tmp);

  // ftmp = (z1^2*x2 - z2^2*x1)^2

  p224_felem_assign(ftmp5, ftmp);

  p224_felem_square(tmp, ftmp);

  p224_felem_reduce(ftmp, tmp);

  // ftmp5 = (z1^2*x2 - z2^2*x1)^3

  p224_felem_mul(tmp, ftmp, ftmp5);

  p224_felem_reduce(ftmp5, tmp);

  // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2

  p224_felem_mul(tmp, ftmp2, ftmp);

  p224_felem_reduce(ftmp2, tmp);

  // tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3

  p224_felem_mul(tmp, ftmp4, ftmp5);

  // tmp[i] < 4 * 2^57 * 2^57 = 2^116

  // tmp2 = (z1^3*y2 - z2^3*y1)^2

  p224_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

  p224_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

  p224_felem_assign(ftmp5, ftmp2);

  p224_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 */

  p224_felem_diff_128_64(tmp2, ftmp5);

  // tmp2[i] < 2^117 + 2^64 + 8 < 2^118

  p224_felem_reduce(x_out, tmp2);

  // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out

  p224_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)

  p224_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 */

  p224_widefelem_diff(tmp2, tmp);

  // tmp2[i] < 2^118 + 2^120 < 2^121

  p224_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

  p224_copy_conditional(x_out, x2, z1_is_zero);

  p224_copy_conditional(x_out, x1, z2_is_zero);

  p224_copy_conditional(y_out, y2, z1_is_zero);

  p224_copy_conditional(y_out, y1, z2_is_zero);

  p224_copy_conditional(z_out, z2, z1_is_zero);

  p224_copy_conditional(z_out, z1, z2_is_zero);

  p224_felem_assign(x3, x_out);

  p224_felem_assign(y3, y_out);

  p224_felem_assign(z3, z_out);

}

// p224_select_point selects the |idx|th point from a precomputation table and

// copies it to out.

static void p224_select_point(const uint64_t idx, size_t size,

                              const p224_felem pre_comp[/*size*/][3],

                              p224_felem out[3]) {

  p224_limb *outlimbs = &out[0][0];

  OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem));

  for (size_t i = 0; i < size; i++) {

    const p224_limb *inlimbs = &pre_comp[i][0][0];

    static_assert(sizeof(uint64_t) <= sizeof(crypto_word_t),

                  "crypto_word_t too small");

    static_assert(sizeof(size_t) <= sizeof(crypto_word_t),

                  "crypto_word_t too small");

    // Without a value barrier, Clang adds a branch here.

    uint64_t mask = value_barrier_w(constant_time_eq_w(i, idx));

    for (size_t j = 0; j < 4 * 3; j++) {

      outlimbs[j] |= inlimbs[j] & mask;

    }

  }

}

// p224_get_bit returns the |i|th bit in |in|.

static crypto_word_t p224_get_bit(const EC_SCALAR *in, size_t i) {

  if (i >= 224) {

    return 0;

  }

  static_assert(sizeof(in->words[0]) == 8, "BN_ULONG is not 64-bit");

  return (in->words[i >> 6] >> (i & 63)) & 1;

}

// Takes the Jacobian coordinates (X, Y, Z) of a point and returns

// (X', Y') = (X/Z^2, Y/Z^3)

static int ec_GFp_nistp224_point_get_affine_coordinates(

    const EC_GROUP *group, const EC_JACOBIAN *point, EC_FELEM *x,

    EC_FELEM *y) {

  if (constant_time_declassify_int(

          ec_GFp_simple_is_at_infinity(group, point))) {

    OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);

    return 0;

  }

  p224_felem z1, z2;

  p224_widefelem tmp;

  p224_generic_to_felem(z1, &point->Z);

  p224_felem_inv(z2, z1);

  p224_felem_square(tmp, z2);

  p224_felem_reduce(z1, tmp);

  if (x != NULL) {

    p224_felem x_in, x_out;

    p224_generic_to_felem(x_in, &point->X);

    p224_felem_mul(tmp, x_in, z1);

    p224_felem_reduce(x_out, tmp);

    p224_felem_to_generic(x, x_out);

  }

  if (y != NULL) {

    p224_felem y_in, y_out;

    p224_generic_to_felem(y_in, &point->Y);

    p224_felem_mul(tmp, z1, z2);

    p224_felem_reduce(z1, tmp);

    p224_felem_mul(tmp, y_in, z1);

    p224_felem_reduce(y_out, tmp);

    p224_felem_to_generic(y, y_out);

  }

  return 1;

}

static void ec_GFp_nistp224_add(const EC_GROUP *group, EC_JACOBIAN *r,

                                const EC_JACOBIAN *a, const EC_JACOBIAN *b) {

  p224_felem x1, y1, z1, x2, y2, z2;

  p224_generic_to_felem(x1, &a->X);

  p224_generic_to_felem(y1, &a->Y);

  p224_generic_to_felem(z1, &a->Z);

  p224_generic_to_felem(x2, &b->X);

  p224_generic_to_felem(y2, &b->Y);

  p224_generic_to_felem(z2, &b->Z);

  p224_point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2, z2);

  // The outputs are already reduced, but still need to be contracted.

  p224_felem_to_generic(&r->X, x1);

  p224_felem_to_generic(&r->Y, y1);

  p224_felem_to_generic(&r->Z, z1);

}

static void ec_GFp_nistp224_dbl(const EC_GROUP *group, EC_JACOBIAN *r,

                                const EC_JACOBIAN *a) {

  p224_felem x, y, z;

  p224_generic_to_felem(x, &a->X);

  p224_generic_to_felem(y, &a->Y);

  p224_generic_to_felem(z, &a->Z);

  p224_point_double(x, y, z, x, y, z);

  // The outputs are already reduced, but still need to be contracted.

  p224_felem_to_generic(&r->X, x);

  p224_felem_to_generic(&r->Y, y);

  p224_felem_to_generic(&r->Z, z);

}

static void ec_GFp_nistp224_make_precomp(p224_felem out[17][3],

                                         const EC_JACOBIAN *p) {

  OPENSSL_memset(out[0], 0, sizeof(p224_felem) * 3);

  p224_generic_to_felem(out[1][0], &p->X);

  p224_generic_to_felem(out[1][1], &p->Y);

  p224_generic_to_felem(out[1][2], &p->Z);

  for (size_t j = 2; j <= 16; ++j) {

    if (j & 1) {

      p224_point_add(out[j][0], out[j][1], out[j][2], out[1][0], out[1][1],

                     out[1][2], 0, out[j - 1][0], out[j - 1][1], out[j - 1][2]);

    } else {

      p224_point_double(out[j][0], out[j][1], out[j][2], out[j / 2][0],

                        out[j / 2][1], out[j / 2][2]);

    }

  }

}

static void ec_GFp_nistp224_point_mul(const EC_GROUP *group, EC_JACOBIAN *r,

                                      const EC_JACOBIAN *p,

                                      const EC_SCALAR *scalar) {

  p224_felem p_pre_comp[17][3];

  ec_GFp_nistp224_make_precomp(p_pre_comp, p);

  // Set nq to the point at infinity.

  p224_felem nq[3], tmp[4];

  OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));

  int skip = 1;  // Save two point operations in the first round.

  for (size_t i = 220; i < 221; i--) {

    if (!skip) {

      p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);

    }

    // Add every 5 doublings.

    if (i % 5 == 0) {

      crypto_word_t bits = p224_get_bit(scalar, i + 4) << 5;

      bits |= p224_get_bit(scalar, i + 3) << 4;

      bits |= p224_get_bit(scalar, i + 2) << 3;

      bits |= p224_get_bit(scalar, i + 1) << 2;

      bits |= p224_get_bit(scalar, i) << 1;

      bits |= p224_get_bit(scalar, i - 1);

      crypto_word_t sign, digit;

      ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);

      // Select the point to add or subtract.

      p224_select_point(digit, 17, (const p224_felem(*)[3])p_pre_comp, tmp);

      p224_felem_neg(tmp[3], tmp[1]);  // (X, -Y, Z) is the negative point

      p224_copy_conditional(tmp[1], tmp[3], sign);

      if (!skip) {

        p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */,

                       tmp[0], tmp[1], tmp[2]);

      } else {

        OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));

        skip = 0;

      }

    }

  }

  // Reduce the output to its unique minimal representation.

  p224_felem_to_generic(&r->X, nq[0]);

  p224_felem_to_generic(&r->Y, nq[1]);

  p224_felem_to_generic(&r->Z, nq[2]);

}

static void ec_GFp_nistp224_point_mul_base(const EC_GROUP *group,

                                           EC_JACOBIAN *r,

                                           const EC_SCALAR *scalar) {

  // Set nq to the point at infinity.

  p224_felem nq[3], tmp[3];

  OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));

  int skip = 1;  // Save two point operations in the first round.

  for (size_t i = 27; i < 28; i--) {

    // double

    if (!skip) {

      p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);

    }

    // First, look 28 bits upwards.

    crypto_word_t bits = p224_get_bit(scalar, i + 196) << 3;

    bits |= p224_get_bit(scalar, i + 140) << 2;

    bits |= p224_get_bit(scalar, i + 84) << 1;

    bits |= p224_get_bit(scalar, i + 28);

    // Select the point to add, in constant time.

    p224_select_point(bits, 16, g_p224_pre_comp[1], tmp);

    if (!skip) {

      p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,

                     tmp[0], tmp[1], tmp[2]);

    } else {

      OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem));

      skip = 0;

    }

    // Second, look at the current position/

    bits = p224_get_bit(scalar, i + 168) << 3;

    bits |= p224_get_bit(scalar, i + 112) << 2;

    bits |= p224_get_bit(scalar, i + 56) << 1;

    bits |= p224_get_bit(scalar, i);

    // Select the point to add, in constant time.

    p224_select_point(bits, 16, g_p224_pre_comp[0], tmp);

    p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,

                   tmp[0], tmp[1], tmp[2]);

  }

  // Reduce the output to its unique minimal representation.

  p224_felem_to_generic(&r->X, nq[0]);

  p224_felem_to_generic(&r->Y, nq[1]);

  p224_felem_to_generic(&r->Z, nq[2]);

}

static void ec_GFp_nistp224_point_mul_public(const EC_GROUP *group,

                                             EC_JACOBIAN *r,

                                             const EC_SCALAR *g_scalar,

                                             const EC_JACOBIAN *p,

                                             const EC_SCALAR *p_scalar) {

  // TODO(davidben): If P-224 ECDSA verify performance ever matters, using

  // |ec_compute_wNAF| for |p_scalar| would likely be an easy improvement.

  p224_felem p_pre_comp[17][3];

  ec_GFp_nistp224_make_precomp(p_pre_comp, p);

  // Set nq to the point at infinity.

  p224_felem nq[3], tmp[3];

  OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem));

  // Loop over both scalars msb-to-lsb, interleaving additions of multiples of