boringssl/crypto/fipsmodule/ec/simple.c

/* Originally written by Bodo Moeller for the OpenSSL project.
 * ====================================================================
 * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com).
 *
 */
/* ====================================================================
 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
 *
 * Portions of the attached software ("Contribution") are developed by
 * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
 *
 * The Contribution is licensed pursuant to the OpenSSL open source
 * license provided above.
 *
 * The elliptic curve binary polynomial software is originally written by
 * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
 * Laboratories. */

#include <openssl/ec.h>

#include <string.h>

#include <openssl/bn.h>
#include <openssl/err.h>
#include <openssl/mem.h>

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


// Most method functions in this file are designed to work with non-trivial
// representations of field elements if necessary (see ecp_mont.c): while
// standard modular addition and subtraction are used, the field_mul and
// field_sqr methods will be used for multiplication, and field_encode and
// field_decode (if defined) will be used for converting between
// representations.
//
// Functions here specifically assume that if a non-trivial representation is
// used, it is a Montgomery representation (i.e. 'encoding' means multiplying
// by some factor R).

int ec_GFp_simple_group_init(EC_GROUP *group) {
  BN_init(&group->field);
  group->a_is_minus3 = 0;
  return 1;
}

void ec_GFp_simple_group_finish(EC_GROUP *group) {
  BN_free(&group->field);
}

int ec_GFp_simple_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;

  // p must be a prime > 3
  if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
    OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD);
    return 0;
  }

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL) {
      return 0;
    }
  }

  BN_CTX_start(ctx);
  BIGNUM *tmp = BN_CTX_get(ctx);
  if (tmp == NULL) {
    goto err;
  }

  // group->field
  if (!BN_copy(&group->field, p)) {
    goto err;
  }
  BN_set_negative(&group->field, 0);
  // Store the field in minimal form, so it can be used with |BN_ULONG| arrays.
  bn_set_minimal_width(&group->field);

  // group->a
  if (!BN_nnmod(tmp, a, &group->field, ctx) ||
      !ec_bignum_to_felem(group, &group->a, tmp)) {
    goto err;
  }

  // group->a_is_minus3
  if (!BN_add_word(tmp, 3)) {
    goto err;
  }
  group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field));

  // group->b
  if (!BN_nnmod(tmp, b, &group->field, ctx) ||
      !ec_bignum_to_felem(group, &group->b, tmp)) {
    goto err;
  }

  if (!ec_bignum_to_felem(group, &group->one, BN_value_one())) {
    goto err;
  }

  ret = 1;

err:
  BN_CTX_end(ctx);
  BN_CTX_free(new_ctx);
  return ret;
}

int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
                                  BIGNUM *b) {
  if ((p != NULL && !BN_copy(p, &group->field)) ||
      (a != NULL && !ec_felem_to_bignum(group, a, &group->a)) ||
      (b != NULL && !ec_felem_to_bignum(group, b, &group->b))) {
    return 0;
  }
  return 1;
}

unsigned ec_GFp_simple_group_get_degree(const EC_GROUP *group) {
  return BN_num_bits(&group->field);
}

void ec_GFp_simple_point_init(EC_RAW_POINT *point) {
  OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM));
  OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM));
  OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM));
}

void ec_GFp_simple_point_copy(EC_RAW_POINT *dest, const EC_RAW_POINT *src) {
  OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM));
  OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM));
  OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM));
}

void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
                                         EC_RAW_POINT *point) {
  // Although it is strictly only necessary to zero Z, we zero the entire point
  // in case |point| was stack-allocated and yet to be initialized.
  ec_GFp_simple_point_init(point);
}

int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
                                               EC_RAW_POINT *point,
                                               const BIGNUM *x,
                                               const BIGNUM *y) {
  if (x == NULL || y == NULL) {
    OPENSSL_PUT_ERROR(EC, ERR_R_PASSED_NULL_PARAMETER);
    return 0;
  }

  if (!ec_bignum_to_felem(group, &point->X, x) ||
      !ec_bignum_to_felem(group, &point->Y, y)) {
    return 0;
  }
  OPENSSL_memcpy(&point->Z, &group->one, sizeof(EC_FELEM));

  return 1;
}

void ec_GFp_simple_add(const EC_GROUP *group, EC_RAW_POINT *out,
                       const EC_RAW_POINT *a, const EC_RAW_POINT *b) {
  if (a == b) {
    ec_GFp_simple_dbl(group, out, a);
    return;
  }


  // The method is taken from:
  //   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl
  //
  // Coq transcription and correctness proof:
  // <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L467>
  // <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L544>
  void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                          const EC_FELEM *b) = group->meth->felem_mul;
  void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
      group->meth->felem_sqr;

  EC_FELEM x_out, y_out, z_out;
  BN_ULONG z1nz = ec_felem_non_zero_mask(group, &a->Z);
  BN_ULONG z2nz = ec_felem_non_zero_mask(group, &b->Z);

  // z1z1 = z1z1 = z1**2
  EC_FELEM z1z1;
  felem_sqr(group, &z1z1, &a->Z);

  // z2z2 = z2**2
  EC_FELEM z2z2;
  felem_sqr(group, &z2z2, &b->Z);

  // u1 = x1*z2z2
  EC_FELEM u1;
  felem_mul(group, &u1, &a->X, &z2z2);

  // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2
  EC_FELEM two_z1z2;
  ec_felem_add(group, &two_z1z2, &a->Z, &b->Z);
  felem_sqr(group, &two_z1z2, &two_z1z2);
  ec_felem_sub(group, &two_z1z2, &two_z1z2, &z1z1);
  ec_felem_sub(group, &two_z1z2, &two_z1z2, &z2z2);

  // s1 = y1 * z2**3
  EC_FELEM s1;
  felem_mul(group, &s1, &b->Z, &z2z2);
  felem_mul(group, &s1, &s1, &a->Y);

  // u2 = x2*z1z1
  EC_FELEM u2;
  felem_mul(group, &u2, &b->X, &z1z1);

  // h = u2 - u1
  EC_FELEM h;
  ec_felem_sub(group, &h, &u2, &u1);

  BN_ULONG xneq = ec_felem_non_zero_mask(group, &h);

  // z_out = two_z1z2 * h
  felem_mul(group, &z_out, &h, &two_z1z2);

  // z1z1z1 = z1 * z1z1
  EC_FELEM z1z1z1;
  felem_mul(group, &z1z1z1, &a->Z, &z1z1);

  // s2 = y2 * z1**3
  EC_FELEM s2;
  felem_mul(group, &s2, &b->Y, &z1z1z1);

  // r = (s2 - s1)*2
  EC_FELEM r;
  ec_felem_sub(group, &r, &s2, &s1);
  ec_felem_add(group, &r, &r, &r);

  BN_ULONG yneq = ec_felem_non_zero_mask(group, &r);

  // TODO(davidben): Analyze how case relates to timing considerations for the
  // supported curves which hit it (P-224, P-384, and P-521) and the
  // to-be-written constant-time generic multiplication implementation.
  if (!xneq && !yneq && z1nz && z2nz) {
    ec_GFp_simple_dbl(group, out, a);
    return;
  }

  // I = (2h)**2
  EC_FELEM i;
  ec_felem_add(group, &i, &h, &h);
  felem_sqr(group, &i, &i);

  // J = h * I
  EC_FELEM j;
  felem_mul(group, &j, &h, &i);

  // V = U1 * I
  EC_FELEM v;
  felem_mul(group, &v, &u1, &i);

  // x_out = r**2 - J - 2V
  felem_sqr(group, &x_out, &r);
  ec_felem_sub(group, &x_out, &x_out, &j);
  ec_felem_sub(group, &x_out, &x_out, &v);
  ec_felem_sub(group, &x_out, &x_out, &v);

  // y_out = r(V-x_out) - 2 * s1 * J
  ec_felem_sub(group, &y_out, &v, &x_out);
  felem_mul(group, &y_out, &y_out, &r);
  EC_FELEM s1j;
  felem_mul(group, &s1j, &s1, &j);
  ec_felem_sub(group, &y_out, &y_out, &s1j);
  ec_felem_sub(group, &y_out, &y_out, &s1j);

  ec_felem_select(group, &x_out, z1nz, &x_out, &b->X);
  ec_felem_select(group, &out->X, z2nz, &x_out, &a->X);
  ec_felem_select(group, &y_out, z1nz, &y_out, &b->Y);
  ec_felem_select(group, &out->Y, z2nz, &y_out, &a->Y);
  ec_felem_select(group, &z_out, z1nz, &z_out, &b->Z);
  ec_felem_select(group, &out->Z, z2nz, &z_out, &a->Z);
}

void ec_GFp_simple_dbl(const EC_GROUP *group, EC_RAW_POINT *r,
                       const EC_RAW_POINT *a) {
  void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                          const EC_FELEM *b) = group->meth->felem_mul;
  void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
      group->meth->felem_sqr;

  if (group->a_is_minus3) {
    // The method is taken from:
    //   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
    //
    // Coq transcription and correctness proof:
    // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93>
    // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201>
    EC_FELEM delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta;
    // delta = z^2
    felem_sqr(group, &delta, &a->Z);
    // gamma = y^2
    felem_sqr(group, &gamma, &a->Y);
    // beta = x*gamma
    felem_mul(group, &beta, &a->X, &gamma);

    // alpha = 3*(x-delta)*(x+delta)
    ec_felem_sub(group, &ftmp, &a->X, &delta);
    ec_felem_add(group, &ftmp2, &a->X, &delta);

    ec_felem_add(group, &tmptmp, &ftmp2, &ftmp2);
    ec_felem_add(group, &ftmp2, &ftmp2, &tmptmp);
    felem_mul(group, &alpha, &ftmp, &ftmp2);

    // x' = alpha^2 - 8*beta
    felem_sqr(group, &r->X, &alpha);
    ec_felem_add(group, &fourbeta, &beta, &beta);
    ec_felem_add(group, &fourbeta, &fourbeta, &fourbeta);
    ec_felem_add(group, &tmptmp, &fourbeta, &fourbeta);
    ec_felem_sub(group, &r->X, &r->X, &tmptmp);

    // z' = (y + z)^2 - gamma - delta
    ec_felem_add(group, &delta, &gamma, &delta);
    ec_felem_add(group, &ftmp, &a->Y, &a->Z);
    felem_sqr(group, &r->Z, &ftmp);
    ec_felem_sub(group, &r->Z, &r->Z, &delta);

    // y' = alpha*(4*beta - x') - 8*gamma^2
    ec_felem_sub(group, &r->Y, &fourbeta, &r->X);
    ec_felem_add(group, &gamma, &gamma, &gamma);
    felem_sqr(group, &gamma, &gamma);
    felem_mul(group, &r->Y, &alpha, &r->Y);
    ec_felem_add(group, &gamma, &gamma, &gamma);
    ec_felem_sub(group, &r->Y, &r->Y, &gamma);
  } else {
    // The method is taken from:
    //   http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl
    //
    // Coq transcription and correctness proof:
    // <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L102>
    // <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L534>
    EC_FELEM xx, yy, yyyy, zz;
    felem_sqr(group, &xx, &a->X);
    felem_sqr(group, &yy, &a->Y);
    felem_sqr(group, &yyyy, &yy);
    felem_sqr(group, &zz, &a->Z);

    // s = 2*((x_in + yy)^2 - xx - yyyy)
    EC_FELEM s;
    ec_felem_add(group, &s, &a->X, &yy);
    felem_sqr(group, &s, &s);
    ec_felem_sub(group, &s, &s, &xx);
    ec_felem_sub(group, &s, &s, &yyyy);
    ec_felem_add(group, &s, &s, &s);

    // m = 3*xx + a*zz^2
    EC_FELEM m;
    felem_sqr(group, &m, &zz);
    felem_mul(group, &m, &group->a, &m);
    ec_felem_add(group, &m, &m, &xx);
    ec_felem_add(group, &m, &m, &xx);
    ec_felem_add(group, &m, &m, &xx);

    // x_out = m^2 - 2*s
    felem_sqr(group, &r->X, &m);
    ec_felem_sub(group, &r->X, &r->X, &s);
    ec_felem_sub(group, &r->X, &r->X, &s);

    // z_out = (y_in + z_in)^2 - yy - zz
    ec_felem_add(group, &r->Z, &a->Y, &a->Z);
    felem_sqr(group, &r->Z, &r->Z);
    ec_felem_sub(group, &r->Z, &r->Z, &yy);
    ec_felem_sub(group, &r->Z, &r->Z, &zz);

    // y_out = m*(s-x_out) - 8*yyyy
    ec_felem_add(group, &yyyy, &yyyy, &yyyy);
    ec_felem_add(group, &yyyy, &yyyy, &yyyy);
    ec_felem_add(group, &yyyy, &yyyy, &yyyy);
    ec_felem_sub(group, &r->Y, &s, &r->X);
    felem_mul(group, &r->Y, &r->Y, &m);
    ec_felem_sub(group, &r->Y, &r->Y, &yyyy);
  }
}

void ec_GFp_simple_invert(const EC_GROUP *group, EC_RAW_POINT *point) {
  ec_felem_neg(group, &point->Y, &point->Y);
}

int ec_GFp_simple_is_at_infinity(const EC_GROUP *group,
                                 const EC_RAW_POINT *point) {
  return ec_felem_non_zero_mask(group, &point->Z) == 0;
}

int ec_GFp_simple_is_on_curve(const EC_GROUP *group,
                              const EC_RAW_POINT *point) {
  if (ec_GFp_simple_is_at_infinity(group, point)) {
    return 1;
  }

  // We have a curve defined by a Weierstrass equation
  //      y^2 = x^3 + a*x + b.
  // The point to consider is given in Jacobian projective coordinates
  // where  (X, Y, Z)  represents  (x, y) = (X/Z^2, Y/Z^3).
  // Substituting this and multiplying by  Z^6  transforms the above equation
  // into
  //      Y^2 = X^3 + a*X*Z^4 + b*Z^6.
  // To test this, we add up the right-hand side in 'rh'.

  void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                          const EC_FELEM *b) = group->meth->felem_mul;
  void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
      group->meth->felem_sqr;

  // rh := X^2
  EC_FELEM rh;
  felem_sqr(group, &rh, &point->X);

  EC_FELEM tmp, Z4, Z6;
  if (!ec_felem_equal(group, &point->Z, &group->one)) {
    felem_sqr(group, &tmp, &point->Z);
    felem_sqr(group, &Z4, &tmp);
    felem_mul(group, &Z6, &Z4, &tmp);

    // rh := (rh + a*Z^4)*X
    if (group->a_is_minus3) {
      ec_felem_add(group, &tmp, &Z4, &Z4);
      ec_felem_add(group, &tmp, &tmp, &Z4);
      ec_felem_sub(group, &rh, &rh, &tmp);
      felem_mul(group, &rh, &rh, &point->X);
    } else {
      felem_mul(group, &tmp, &Z4, &group->a);
      ec_felem_add(group, &rh, &rh, &tmp);
      felem_mul(group, &rh, &rh, &point->X);
    }

    // rh := rh + b*Z^6
    felem_mul(group, &tmp, &group->b, &Z6);
    ec_felem_add(group, &rh, &rh, &tmp);
  } else {
    // rh := (rh + a)*X
    ec_felem_add(group, &rh, &rh, &group->a);
    felem_mul(group, &rh, &rh, &point->X);
    // rh := rh + b
    ec_felem_add(group, &rh, &rh, &group->b);
  }

  // 'lh' := Y^2
  felem_sqr(group, &tmp, &point->Y);
  return ec_felem_equal(group, &tmp, &rh);
}

int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_RAW_POINT *a,
                      const EC_RAW_POINT *b) {
  // Note this function returns zero if |a| and |b| are equal and 1 if they are
  // not equal.
  if (ec_GFp_simple_is_at_infinity(group, a)) {
    return ec_GFp_simple_is_at_infinity(group, b) ? 0 : 1;
  }

  if (ec_GFp_simple_is_at_infinity(group, b)) {
    return 1;
  }

  int a_Z_is_one = ec_felem_equal(group, &a->Z, &group->one);
  int b_Z_is_one = ec_felem_equal(group, &b->Z, &group->one);

  if (a_Z_is_one && b_Z_is_one) {
    return !ec_felem_equal(group, &a->X, &b->X) ||
           !ec_felem_equal(group, &a->Y, &b->Y);
  }

  void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                          const EC_FELEM *b) = group->meth->felem_mul;
  void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
      group->meth->felem_sqr;

  // We have to decide whether
  //     (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
  // or equivalently, whether
  //     (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).

  EC_FELEM tmp1, tmp2, Za23, Zb23;
  const EC_FELEM *tmp1_, *tmp2_;
  if (!b_Z_is_one) {
    felem_sqr(group, &Zb23, &b->Z);
    felem_mul(group, &tmp1, &a->X, &Zb23);
    tmp1_ = &tmp1;
  } else {
    tmp1_ = &a->X;
  }
  if (!a_Z_is_one) {
    felem_sqr(group, &Za23, &a->Z);
    felem_mul(group, &tmp2, &b->X, &Za23);
    tmp2_ = &tmp2;
  } else {
    tmp2_ = &b->X;
  }

  // Compare  X_a*Z_b^2  with  X_b*Z_a^2.
  if (!ec_felem_equal(group, tmp1_, tmp2_)) {
    return 1;  // The points differ.
  }

  if (!b_Z_is_one) {
    felem_mul(group, &Zb23, &Zb23, &b->Z);
    felem_mul(group, &tmp1, &a->Y, &Zb23);
    // tmp1_ = &tmp1
  } else {
    tmp1_ = &a->Y;
  }
  if (!a_Z_is_one) {
    felem_mul(group, &Za23, &Za23, &a->Z);
    felem_mul(group, &tmp2, &b->Y, &Za23);
    // tmp2_ = &tmp2
  } else {
    tmp2_ = &b->Y;
  }

  // Compare  Y_a*Z_b^3  with  Y_b*Z_a^3.
  if (!ec_felem_equal(group, tmp1_, tmp2_)) {
    return 1;  // The points differ.
  }

  // The points are equal.
  return 0;
}