/src/libressl/crypto/ec/ecp_smpl.c

Source (jump to first uncovered line)
/* $OpenBSD: ecp_smpl.c,v 1.34 2022/01/20 11:02:44 inoguchi Exp $ */
/* Includes code written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
 * for the OpenSSL project.
 * Includes code written by Bodo Moeller for the OpenSSL project.
*/
/* ====================================================================
 * Copyright (c) 1998-2002 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 this software developed by SUN MICROSYSTEMS, INC.,
 * and contributed to the OpenSSL project.
 */

#include <openssl/err.h>

#include "bn_lcl.h"
#include "ec_lcl.h"

const EC_METHOD *
EC_GFp_simple_method(void)
{
  static const EC_METHOD ret = {
    .flags = EC_FLAGS_DEFAULT_OCT,
    .field_type = NID_X9_62_prime_field,
    .group_init = ec_GFp_simple_group_init,
    .group_finish = ec_GFp_simple_group_finish,
    .group_clear_finish = ec_GFp_simple_group_clear_finish,
    .group_copy = ec_GFp_simple_group_copy,
    .group_set_curve = ec_GFp_simple_group_set_curve,
    .group_get_curve = ec_GFp_simple_group_get_curve,
    .group_get_degree = ec_GFp_simple_group_get_degree,
    .group_order_bits = ec_group_simple_order_bits,
    .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 =
        ec_GFp_simple_set_Jprojective_coordinates,
    .point_get_Jprojective_coordinates =
        ec_GFp_simple_get_Jprojective_coordinates,
    .point_set_affine_coordinates =
        ec_GFp_simple_point_set_affine_coordinates,
    .point_get_affine_coordinates =
        ec_GFp_simple_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_generator_ct = ec_GFp_simple_mul_generator_ct,
    .mul_single_ct = ec_GFp_simple_mul_single_ct,
    .mul_double_nonct = ec_GFp_simple_mul_double_nonct,
    .field_mul = ec_GFp_simple_field_mul,
    .field_sqr = ec_GFp_simple_field_sqr,
    .blind_coordinates = ec_GFp_simple_blind_coordinates,
  };

  return &ret;
}


/* 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 ec_GFp_simple_points_make_affine() and
 * ec_GFp_simple_point_get_affine_coordinates() 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);
  BN_init(&group->a);
  BN_init(&group->b);
  group->a_is_minus3 = 0;
  return 1;
}


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


void
ec_GFp_simple_group_clear_finish(EC_GROUP * group)
{
  BN_clear_free(&group->field);
  BN_clear_free(&group->a);
  BN_clear_free(&group->b);
}


int
ec_GFp_simple_group_copy(EC_GROUP * dest, const EC_GROUP * src)
{
  if (!BN_copy(&dest->field, &src->field))
    return 0;
  if (!BN_copy(&dest->a, &src->a))
    return 0;
  if (!BN_copy(&dest->b, &src->b))
    return 0;

  dest->a_is_minus3 = src->a_is_minus3;

  return 1;
}


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;
  BIGNUM *tmp_a;

  /* p must be a prime > 3 */
  if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
    ECerror(EC_R_INVALID_FIELD);
    return 0;
  }
  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((tmp_a = BN_CTX_get(ctx)) == NULL)
    goto err;

  /* group->field */
  if (!BN_copy(&group->field, p))
    goto err;
  BN_set_negative(&group->field, 0);

  /* group->a */
  if (!BN_nnmod(tmp_a, a, p, ctx))
    goto err;
  if (group->meth->field_encode) {
    if (!group->meth->field_encode(group, &group->a, tmp_a, ctx))
      goto err;
  } else if (!BN_copy(&group->a, tmp_a))
    goto err;

  /* group->b */
  if (!BN_nnmod(&group->b, b, p, ctx))
    goto err;
  if (group->meth->field_encode)
    if (!group->meth->field_encode(group, &group->b, &group->b, ctx))
      goto err;

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

  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, BN_CTX * ctx)
{
  int ret = 0;
  BN_CTX *new_ctx = NULL;

  if (p != NULL) {
    if (!BN_copy(p, &group->field))
      return 0;
  }
  if (a != NULL || b != NULL) {
    if (group->meth->field_decode) {
      if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
          return 0;
      }
      if (a != NULL) {
        if (!group->meth->field_decode(group, a, &group->a, ctx))
          goto err;
      }
      if (b != NULL) {
        if (!group->meth->field_decode(group, b, &group->b, ctx))
          goto err;
      }
    } else {
      if (a != NULL) {
        if (!BN_copy(a, &group->a))
          goto err;
      }
      if (b != NULL) {
        if (!BN_copy(b, &group->b))
          goto err;
      }
    }
  }
  ret = 1;

 err:
  BN_CTX_free(new_ctx);
  return ret;
}


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


int
ec_GFp_simple_group_check_discriminant(const EC_GROUP * group, BN_CTX * ctx)
{
  int ret = 0;
  BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
  const BIGNUM *p = &group->field;
  BN_CTX *new_ctx = NULL;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL) {
      ECerror(ERR_R_MALLOC_FAILURE);
      goto err;
    }
  }
  BN_CTX_start(ctx);
  if ((a = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((b = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((tmp_1 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((tmp_2 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((order = BN_CTX_get(ctx)) == NULL)
    goto err;

  if (group->meth->field_decode) {
    if (!group->meth->field_decode(group, a, &group->a, ctx))
      goto err;
    if (!group->meth->field_decode(group, b, &group->b, ctx))
      goto err;
  } else {
    if (!BN_copy(a, &group->a))
      goto err;
    if (!BN_copy(b, &group->b))
      goto err;
  }

  /*
   * check the discriminant: y^2 = x^3 + a*x + b is an elliptic curve
   * <=> 4*a^3 + 27*b^2 != 0 (mod p) 0 =< a, b < p
   */
  if (BN_is_zero(a)) {
    if (BN_is_zero(b))
      goto err;
  } else if (!BN_is_zero(b)) {
    if (!BN_mod_sqr(tmp_1, a, p, ctx))
      goto err;
    if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
      goto err;
    if (!BN_lshift(tmp_1, tmp_2, 2))
      goto err;
    /* tmp_1 = 4*a^3 */

    if (!BN_mod_sqr(tmp_2, b, p, ctx))
      goto err;
    if (!BN_mul_word(tmp_2, 27))
      goto err;
    /* tmp_2 = 27*b^2 */

    if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
      goto err;
    if (BN_is_zero(a))
      goto err;
  }
  ret = 1;

 err:
  if (ctx != NULL)
    BN_CTX_end(ctx);
  BN_CTX_free(new_ctx);
  return ret;
}


int
ec_GFp_simple_point_init(EC_POINT * point)
{
  BN_init(&point->X);
  BN_init(&point->Y);
  BN_init(&point->Z);
  point->Z_is_one = 0;

  return 1;
}


void
ec_GFp_simple_point_finish(EC_POINT * point)
{
  BN_free(&point->X);
  BN_free(&point->Y);
  BN_free(&point->Z);
}


void
ec_GFp_simple_point_clear_finish(EC_POINT * point)
{
  BN_clear_free(&point->X);
  BN_clear_free(&point->Y);
  BN_clear_free(&point->Z);
  point->Z_is_one = 0;
}


int
ec_GFp_simple_point_copy(EC_POINT * dest, const EC_POINT * src)
{
  if (!BN_copy(&dest->X, &src->X))
    return 0;
  if (!BN_copy(&dest->Y, &src->Y))
    return 0;
  if (!BN_copy(&dest->Z, &src->Z))
    return 0;
  dest->Z_is_one = src->Z_is_one;

  return 1;
}


int
ec_GFp_simple_point_set_to_infinity(const EC_GROUP * group, EC_POINT * point)
{
  point->Z_is_one = 0;
  BN_zero(&point->Z);
  return 1;
}


int
ec_GFp_simple_set_Jprojective_coordinates(const EC_GROUP *group,
    EC_POINT *point, const BIGNUM *x, const BIGNUM *y, const BIGNUM *z,
    BN_CTX *ctx)
{
  BN_CTX *new_ctx = NULL;
  int ret = 0;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  if (x != NULL) {
    if (!BN_nnmod(&point->X, x, &group->field, ctx))
      goto err;
    if (group->meth->field_encode) {
      if (!group->meth->field_encode(group, &point->X, &point->X, ctx))
        goto err;
    }
  }
  if (y != NULL) {
    if (!BN_nnmod(&point->Y, y, &group->field, ctx))
      goto err;
    if (group->meth->field_encode) {
      if (!group->meth->field_encode(group, &point->Y, &point->Y, ctx))
        goto err;
    }
  }
  if (z != NULL) {
    int Z_is_one;

    if (!BN_nnmod(&point->Z, z, &group->field, ctx))
      goto err;
    Z_is_one = BN_is_one(&point->Z);
    if (group->meth->field_encode) {
      if (Z_is_one && (group->meth->field_set_to_one != 0)) {
        if (!group->meth->field_set_to_one(group, &point->Z, ctx))
          goto err;
      } else {
        if (!group->meth->field_encode(group, &point->Z, &point->Z, ctx))
          goto err;
      }
    }
    point->Z_is_one = Z_is_one;
  }
  ret = 1;

 err:
  BN_CTX_free(new_ctx);
  return ret;
}

int
ec_GFp_simple_get_Jprojective_coordinates(const EC_GROUP *group,
    const EC_POINT *point, BIGNUM *x, BIGNUM *y, BIGNUM *z, BN_CTX *ctx)
{
  BN_CTX *new_ctx = NULL;
  int ret = 0;

  if (group->meth->field_decode != 0) {
    if (ctx == NULL) {
      ctx = new_ctx = BN_CTX_new();
      if (ctx == NULL)
        return 0;
    }
    if (x != NULL) {
      if (!group->meth->field_decode(group, x, &point->X, ctx))
        goto err;
    }
    if (y != NULL) {
      if (!group->meth->field_decode(group, y, &point->Y, ctx))
        goto err;
    }
    if (z != NULL) {
      if (!group->meth->field_decode(group, z, &point->Z, ctx))
        goto err;
    }
  } else {
    if (x != NULL) {
      if (!BN_copy(x, &point->X))
        goto err;
    }
    if (y != NULL) {
      if (!BN_copy(y, &point->Y))
        goto err;
    }
    if (z != NULL) {
      if (!BN_copy(z, &point->Z))
        goto err;
    }
  }

  ret = 1;

 err:
  BN_CTX_free(new_ctx);
  return ret;
}

int
ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP * group, EC_POINT * point,
    const BIGNUM * x, const BIGNUM * y, BN_CTX * ctx)
{
  if (x == NULL || y == NULL) {
    /* unlike for projective coordinates, we do not tolerate this */
    ECerror(ERR_R_PASSED_NULL_PARAMETER);
    return 0;
  }
  return EC_POINT_set_Jprojective_coordinates(group, point, x, y,
      BN_value_one(), ctx);
}

int
ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP * group, const EC_POINT * point,
    BIGNUM * x, BIGNUM * y, BN_CTX * ctx)
{
  BN_CTX *new_ctx = NULL;
  BIGNUM *Z, *Z_1, *Z_2, *Z_3;
  const BIGNUM *Z_;
  int ret = 0;

  if (EC_POINT_is_at_infinity(group, point) > 0) {
    ECerror(EC_R_POINT_AT_INFINITY);
    return 0;
  }
  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((Z = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((Z_1 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((Z_2 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((Z_3 = BN_CTX_get(ctx)) == NULL)
    goto err;

  /* transform  (X, Y, Z)  into  (x, y) := (X/Z^2, Y/Z^3) */

  if (group->meth->field_decode) {
    if (!group->meth->field_decode(group, Z, &point->Z, ctx))
      goto err;
    Z_ = Z;
  } else {
    Z_ = &point->Z;
  }

  if (BN_is_one(Z_)) {
    if (group->meth->field_decode) {
      if (x != NULL) {
        if (!group->meth->field_decode(group, x, &point->X, ctx))
          goto err;
      }
      if (y != NULL) {
        if (!group->meth->field_decode(group, y, &point->Y, ctx))
          goto err;
      }
    } else {
      if (x != NULL) {
        if (!BN_copy(x, &point->X))
          goto err;
      }
      if (y != NULL) {
        if (!BN_copy(y, &point->Y))
          goto err;
      }
    }
  } else {
    if (BN_mod_inverse_ct(Z_1, Z_, &group->field, ctx) == NULL) {
      ECerror(ERR_R_BN_LIB);
      goto err;
    }
    if (group->meth->field_encode == 0) {
      /* field_sqr works on standard representation */
      if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
        goto err;
    } else {
      if (!BN_mod_sqr(Z_2, Z_1, &group->field, ctx))
        goto err;
    }

    if (x != NULL) {
      /*
       * in the Montgomery case, field_mul will cancel out
       * Montgomery factor in X:
       */
      if (!group->meth->field_mul(group, x, &point->X, Z_2, ctx))
        goto err;
    }
    if (y != NULL) {
      if (group->meth->field_encode == 0) {
        /* field_mul works on standard representation */
        if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
          goto err;
      } else {
        if (!BN_mod_mul(Z_3, Z_2, Z_1, &group->field, ctx))
          goto err;
      }

      /*
       * in the Montgomery case, field_mul will cancel out
       * Montgomery factor in Y:
       */
      if (!group->meth->field_mul(group, y, &point->Y, Z_3, ctx))
        goto err;
    }
  }

  ret = 1;

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

int
ec_GFp_simple_add(const EC_GROUP * group, EC_POINT * r, const EC_POINT * a, const EC_POINT * b, BN_CTX * ctx)
{
  int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
  int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
  const BIGNUM *p;
  BN_CTX *new_ctx = NULL;
  BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
  int ret = 0;

  if (a == b)
    return EC_POINT_dbl(group, r, a, ctx);
  if (EC_POINT_is_at_infinity(group, a) > 0)
    return EC_POINT_copy(r, b);
  if (EC_POINT_is_at_infinity(group, b) > 0)
    return EC_POINT_copy(r, a);

  field_mul = group->meth->field_mul;
  field_sqr = group->meth->field_sqr;
  p = &group->field;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((n0 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n1 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n2 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n3 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n4 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n5 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((n6 = BN_CTX_get(ctx)) == NULL)
    goto end;

  /*
   * Note that in this function we must not read components of 'a' or
   * 'b' once we have written the corresponding components of 'r'. ('r'
   * might be one of 'a' or 'b'.)
   */

  /* n1, n2 */
  if (b->Z_is_one) {
    if (!BN_copy(n1, &a->X))
      goto end;
    if (!BN_copy(n2, &a->Y))
      goto end;
    /* n1 = X_a */
    /* n2 = Y_a */
  } else {
    if (!field_sqr(group, n0, &b->Z, ctx))
      goto end;
    if (!field_mul(group, n1, &a->X, n0, ctx))
      goto end;
    /* n1 = X_a * Z_b^2 */

    if (!field_mul(group, n0, n0, &b->Z, ctx))
      goto end;
    if (!field_mul(group, n2, &a->Y, n0, ctx))
      goto end;
    /* n2 = Y_a * Z_b^3 */
  }

  /* n3, n4 */
  if (a->Z_is_one) {
    if (!BN_copy(n3, &b->X))
      goto end;
    if (!BN_copy(n4, &b->Y))
      goto end;
    /* n3 = X_b */
    /* n4 = Y_b */
  } else {
    if (!field_sqr(group, n0, &a->Z, ctx))
      goto end;
    if (!field_mul(group, n3, &b->X, n0, ctx))
      goto end;
    /* n3 = X_b * Z_a^2 */

    if (!field_mul(group, n0, n0, &a->Z, ctx))
      goto end;
    if (!field_mul(group, n4, &b->Y, n0, ctx))
      goto end;
    /* n4 = Y_b * Z_a^3 */
  }

  /* n5, n6 */
  if (!BN_mod_sub_quick(n5, n1, n3, p))
    goto end;
  if (!BN_mod_sub_quick(n6, n2, n4, p))
    goto end;
  /* n5 = n1 - n3 */
  /* n6 = n2 - n4 */

  if (BN_is_zero(n5)) {
    if (BN_is_zero(n6)) {
      /* a is the same point as b */
      BN_CTX_end(ctx);
      ret = EC_POINT_dbl(group, r, a, ctx);
      ctx = NULL;
      goto end;
    } else {
      /* a is the inverse of b */
      BN_zero(&r->Z);
      r->Z_is_one = 0;
      ret = 1;
      goto end;
    }
  }
  /* 'n7', 'n8' */
  if (!BN_mod_add_quick(n1, n1, n3, p))
    goto end;
  if (!BN_mod_add_quick(n2, n2, n4, p))
    goto end;
  /* 'n7' = n1 + n3 */
  /* 'n8' = n2 + n4 */

  /* Z_r */
  if (a->Z_is_one && b->Z_is_one) {
    if (!BN_copy(&r->Z, n5))
      goto end;
  } else {
    if (a->Z_is_one) {
      if (!BN_copy(n0, &b->Z))
        goto end;
    } else if (b->Z_is_one) {
      if (!BN_copy(n0, &a->Z))
        goto end;
    } else {
      if (!field_mul(group, n0, &a->Z, &b->Z, ctx))
        goto end;
    }
    if (!field_mul(group, &r->Z, n0, n5, ctx))
      goto end;
  }
  r->Z_is_one = 0;
  /* Z_r = Z_a * Z_b * n5 */

  /* X_r */
  if (!field_sqr(group, n0, n6, ctx))
    goto end;
  if (!field_sqr(group, n4, n5, ctx))
    goto end;
  if (!field_mul(group, n3, n1, n4, ctx))
    goto end;
  if (!BN_mod_sub_quick(&r->X, n0, n3, p))
    goto end;
  /* X_r = n6^2 - n5^2 * 'n7' */

  /* 'n9' */
  if (!BN_mod_lshift1_quick(n0, &r->X, p))
    goto end;
  if (!BN_mod_sub_quick(n0, n3, n0, p))
    goto end;
  /* n9 = n5^2 * 'n7' - 2 * X_r */

  /* Y_r */
  if (!field_mul(group, n0, n0, n6, ctx))
    goto end;
  if (!field_mul(group, n5, n4, n5, ctx))
    goto end; /* now n5 is n5^3 */
  if (!field_mul(group, n1, n2, n5, ctx))
    goto end;
  if (!BN_mod_sub_quick(n0, n0, n1, p))
    goto end;
  if (BN_is_odd(n0))
    if (!BN_add(n0, n0, p))
      goto end;
  /* now  0 <= n0 < 2*p,  and n0 is even */
  if (!BN_rshift1(&r->Y, n0))
    goto end;
  /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */

  ret = 1;

 end:
  if (ctx)    /* otherwise we already called BN_CTX_end */
    BN_CTX_end(ctx);
  BN_CTX_free(new_ctx);
  return ret;
}


int
ec_GFp_simple_dbl(const EC_GROUP * group, EC_POINT * r, const EC_POINT * a, BN_CTX * ctx)
{
  int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
  int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
  const BIGNUM *p;
  BN_CTX *new_ctx = NULL;
  BIGNUM *n0, *n1, *n2, *n3;
  int ret = 0;

  if (EC_POINT_is_at_infinity(group, a) > 0) {
    BN_zero(&r->Z);
    r->Z_is_one = 0;
    return 1;
  }
  field_mul = group->meth->field_mul;
  field_sqr = group->meth->field_sqr;
  p = &group->field;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((n0 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((n1 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((n2 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((n3 = BN_CTX_get(ctx)) == NULL)
    goto err;

  /*
   * Note that in this function we must not read components of 'a' once
   * we have written the corresponding components of 'r'. ('r' might
   * the same as 'a'.)
   */

  /* n1 */
  if (a->Z_is_one) {
    if (!field_sqr(group, n0, &a->X, ctx))
      goto err;
    if (!BN_mod_lshift1_quick(n1, n0, p))
      goto err;
    if (!BN_mod_add_quick(n0, n0, n1, p))
      goto err;
    if (!BN_mod_add_quick(n1, n0, &group->a, p))
      goto err;
    /* n1 = 3 * X_a^2 + a_curve */
  } else if (group->a_is_minus3) {
    if (!field_sqr(group, n1, &a->Z, ctx))
      goto err;
    if (!BN_mod_add_quick(n0, &a->X, n1, p))
      goto err;
    if (!BN_mod_sub_quick(n2, &a->X, n1, p))
      goto err;
    if (!field_mul(group, n1, n0, n2, ctx))
      goto err;
    if (!BN_mod_lshift1_quick(n0, n1, p))
      goto err;
    if (!BN_mod_add_quick(n1, n0, n1, p))
      goto err;
    /*
     * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2) = 3 * X_a^2 - 3 *
     * Z_a^4
     */
  } else {
    if (!field_sqr(group, n0, &a->X, ctx))
      goto err;
    if (!BN_mod_lshift1_quick(n1, n0, p))
      goto err;
    if (!BN_mod_add_quick(n0, n0, n1, p))
      goto err;
    if (!field_sqr(group, n1, &a->Z, ctx))
      goto err;
    if (!field_sqr(group, n1, n1, ctx))
      goto err;
    if (!field_mul(group, n1, n1, &group->a, ctx))
      goto err;
    if (!BN_mod_add_quick(n1, n1, n0, p))
      goto err;
    /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
  }

  /* Z_r */
  if (a->Z_is_one) {
    if (!BN_copy(n0, &a->Y))
      goto err;
  } else {
    if (!field_mul(group, n0, &a->Y, &a->Z, ctx))
      goto err;
  }
  if (!BN_mod_lshift1_quick(&r->Z, n0, p))
    goto err;
  r->Z_is_one = 0;
  /* Z_r = 2 * Y_a * Z_a */

  /* n2 */
  if (!field_sqr(group, n3, &a->Y, ctx))
    goto err;
  if (!field_mul(group, n2, &a->X, n3, ctx))
    goto err;
  if (!BN_mod_lshift_quick(n2, n2, 2, p))
    goto err;
  /* n2 = 4 * X_a * Y_a^2 */

  /* X_r */
  if (!BN_mod_lshift1_quick(n0, n2, p))
    goto err;
  if (!field_sqr(group, &r->X, n1, ctx))
    goto err;
  if (!BN_mod_sub_quick(&r->X, &r->X, n0, p))
    goto err;
  /* X_r = n1^2 - 2 * n2 */

  /* n3 */
  if (!field_sqr(group, n0, n3, ctx))
    goto err;
  if (!BN_mod_lshift_quick(n3, n0, 3, p))
    goto err;
  /* n3 = 8 * Y_a^4 */

  /* Y_r */
  if (!BN_mod_sub_quick(n0, n2, &r->X, p))
    goto err;
  if (!field_mul(group, n0, n1, n0, ctx))
    goto err;
  if (!BN_mod_sub_quick(&r->Y, n0, n3, p))
    goto err;
  /* Y_r = n1 * (n2 - X_r) - n3 */

  ret = 1;

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


int
ec_GFp_simple_invert(const EC_GROUP * group, EC_POINT * point, BN_CTX * ctx)
{
  if (EC_POINT_is_at_infinity(group, point) > 0 || BN_is_zero(&point->Y))
    /* point is its own inverse */
    return 1;

  return BN_usub(&point->Y, &group->field, &point->Y);
}


int
ec_GFp_simple_is_at_infinity(const EC_GROUP * group, const EC_POINT * point)
{
  return BN_is_zero(&point->Z);
}


int
ec_GFp_simple_is_on_curve(const EC_GROUP * group, const EC_POINT * point, BN_CTX * ctx)
{
  int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
  int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
  const BIGNUM *p;
  BN_CTX *new_ctx = NULL;
  BIGNUM *rh, *tmp, *Z4, *Z6;
  int ret = -1;

  if (EC_POINT_is_at_infinity(group, point) > 0)
    return 1;

  field_mul = group->meth->field_mul;
  field_sqr = group->meth->field_sqr;
  p = &group->field;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return -1;
  }
  BN_CTX_start(ctx);
  if ((rh = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((tmp = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((Z4 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((Z6 = BN_CTX_get(ctx)) == NULL)
    goto err;

  /*
   * 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'.
   */

  /* rh := X^2 */
  if (!field_sqr(group, rh, &point->X, ctx))
    goto err;

  if (!point->Z_is_one) {
    if (!field_sqr(group, tmp, &point->Z, ctx))
      goto err;
    if (!field_sqr(group, Z4, tmp, ctx))
      goto err;
    if (!field_mul(group, Z6, Z4, tmp, ctx))
      goto err;

    /* rh := (rh + a*Z^4)*X */
    if (group->a_is_minus3) {
      if (!BN_mod_lshift1_quick(tmp, Z4, p))
        goto err;
      if (!BN_mod_add_quick(tmp, tmp, Z4, p))
        goto err;
      if (!BN_mod_sub_quick(rh, rh, tmp, p))
        goto err;
      if (!field_mul(group, rh, rh, &point->X, ctx))
        goto err;
    } else {
      if (!field_mul(group, tmp, Z4, &group->a, ctx))
        goto err;
      if (!BN_mod_add_quick(rh, rh, tmp, p))
        goto err;
      if (!field_mul(group, rh, rh, &point->X, ctx))
        goto err;
    }

    /* rh := rh + b*Z^6 */
    if (!field_mul(group, tmp, &group->b, Z6, ctx))
      goto err;
    if (!BN_mod_add_quick(rh, rh, tmp, p))
      goto err;
  } else {
    /* point->Z_is_one */

    /* rh := (rh + a)*X */
    if (!BN_mod_add_quick(rh, rh, &group->a, p))
      goto err;
    if (!field_mul(group, rh, rh, &point->X, ctx))
      goto err;
    /* rh := rh + b */
    if (!BN_mod_add_quick(rh, rh, &group->b, p))
      goto err;
  }

  /* 'lh' := Y^2 */
  if (!field_sqr(group, tmp, &point->Y, ctx))
    goto err;

  ret = (0 == BN_ucmp(tmp, rh));

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


int
ec_GFp_simple_cmp(const EC_GROUP * group, const EC_POINT * a, const EC_POINT * b, BN_CTX * ctx)
{
  /*
   * return values: -1   error 0   equal (in affine coordinates) 1
   * not equal
   */

  int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
  int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
  BN_CTX *new_ctx = NULL;
  BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
  const BIGNUM *tmp1_, *tmp2_;
  int ret = -1;

  if (EC_POINT_is_at_infinity(group, a) > 0) {
    return EC_POINT_is_at_infinity(group, b) > 0 ? 0 : 1;
  }
  if (EC_POINT_is_at_infinity(group, b) > 0)
    return 1;

  if (a->Z_is_one && b->Z_is_one) {
    return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
  }
  field_mul = group->meth->field_mul;
  field_sqr = group->meth->field_sqr;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return -1;
  }
  BN_CTX_start(ctx);
  if ((tmp1 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((tmp2 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((Za23 = BN_CTX_get(ctx)) == NULL)
    goto end;
  if ((Zb23 = BN_CTX_get(ctx)) == NULL)
    goto end;

  /*
   * 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).
   */

  if (!b->Z_is_one) {
    if (!field_sqr(group, Zb23, &b->Z, ctx))
      goto end;
    if (!field_mul(group, tmp1, &a->X, Zb23, ctx))
      goto end;
    tmp1_ = tmp1;
  } else
    tmp1_ = &a->X;
  if (!a->Z_is_one) {
    if (!field_sqr(group, Za23, &a->Z, ctx))
      goto end;
    if (!field_mul(group, tmp2, &b->X, Za23, ctx))
      goto end;
    tmp2_ = tmp2;
  } else
    tmp2_ = &b->X;

  /* compare  X_a*Z_b^2  with  X_b*Z_a^2 */
  if (BN_cmp(tmp1_, tmp2_) != 0) {
    ret = 1;  /* points differ */
    goto end;
  }
  if (!b->Z_is_one) {
    if (!field_mul(group, Zb23, Zb23, &b->Z, ctx))
      goto end;
    if (!field_mul(group, tmp1, &a->Y, Zb23, ctx))
      goto end;
    /* tmp1_ = tmp1 */
  } else
    tmp1_ = &a->Y;
  if (!a->Z_is_one) {
    if (!field_mul(group, Za23, Za23, &a->Z, ctx))
      goto end;
    if (!field_mul(group, tmp2, &b->Y, Za23, ctx))
      goto end;
    /* tmp2_ = tmp2 */
  } else
    tmp2_ = &b->Y;

  /* compare  Y_a*Z_b^3  with  Y_b*Z_a^3 */
  if (BN_cmp(tmp1_, tmp2_) != 0) {
    ret = 1;  /* points differ */
    goto end;
  }
  /* points are equal */
  ret = 0;

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


int
ec_GFp_simple_make_affine(const EC_GROUP * group, EC_POINT * point, BN_CTX * ctx)
{
  BN_CTX *new_ctx = NULL;
  BIGNUM *x, *y;
  int ret = 0;

  if (point->Z_is_one || EC_POINT_is_at_infinity(group, point) > 0)
    return 1;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((x = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((y = BN_CTX_get(ctx)) == NULL)
    goto err;

  if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx))
    goto err;
  if (!EC_POINT_set_affine_coordinates(group, point, x, y, ctx))
    goto err;
  if (!point->Z_is_one) {
    ECerror(ERR_R_INTERNAL_ERROR);
    goto err;
  }
  ret = 1;

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


int
ec_GFp_simple_points_make_affine(const EC_GROUP * group, size_t num, EC_POINT * points[], BN_CTX * ctx)
{
  BN_CTX *new_ctx = NULL;
  BIGNUM *tmp0, *tmp1;
  size_t pow2 = 0;
  BIGNUM **heap = NULL;
  size_t i;
  int ret = 0;

  if (num == 0)
    return 1;

  if (ctx == NULL) {
    ctx = new_ctx = BN_CTX_new();
    if (ctx == NULL)
      return 0;
  }
  BN_CTX_start(ctx);
  if ((tmp0 = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((tmp1 = BN_CTX_get(ctx)) == NULL)
    goto err;

  /*
   * Before converting the individual points, compute inverses of all Z
   * values. Modular inversion is rather slow, but luckily we can do
   * with a single explicit inversion, plus about 3 multiplications per
   * input value.
   */

  pow2 = 1;
  while (num > pow2)
    pow2 <<= 1;
  /*
   * Now pow2 is the smallest power of 2 satifsying pow2 >= num. We
   * need twice that.
   */
  pow2 <<= 1;

  heap = reallocarray(NULL, pow2, sizeof heap[0]);
  if (heap == NULL)
    goto err;

  /*
   * The array is used as a binary tree, exactly as in heapsort:
   *
   * heap[1] heap[2]                     heap[3] heap[4]       heap[5]
   * heap[6]       heap[7] heap[8]heap[9] heap[10]heap[11]
   * heap[12]heap[13] heap[14] heap[15]
   *
   * We put the Z's in the last line; then we set each other node to the
   * product of its two child-nodes (where empty or 0 entries are
   * treated as ones); then we invert heap[1]; then we invert each
   * other node by replacing it by the product of its parent (after
   * inversion) and its sibling (before inversion).
   */
  heap[0] = NULL;
  for (i = pow2 / 2 - 1; i > 0; i--)
    heap[i] = NULL;
  for (i = 0; i < num; i++)
    heap[pow2 / 2 + i] = &points[i]->Z;
  for (i = pow2 / 2 + num; i < pow2; i++)
    heap[i] = NULL;

  /* set each node to the product of its children */
  for (i = pow2 / 2 - 1; i > 0; i--) {
    heap[i] = BN_new();
    if (heap[i] == NULL)
      goto err;

    if (heap[2 * i] != NULL) {
      if ((heap[2 * i + 1] == NULL) || BN_is_zero(heap[2 * i + 1])) {
        if (!BN_copy(heap[i], heap[2 * i]))
          goto err;
      } else {
        if (BN_is_zero(heap[2 * i])) {
          if (!BN_copy(heap[i], heap[2 * i + 1]))
            goto err;
        } else {
          if (!group->meth->field_mul(group, heap[i],
            heap[2 * i], heap[2 * i + 1], ctx))
            goto err;
        }
      }
    }
  }

  /* invert heap[1] */
  if (!BN_is_zero(heap[1])) {
    if (BN_mod_inverse_ct(heap[1], heap[1], &group->field, ctx) == NULL) {
      ECerror(ERR_R_BN_LIB);
      goto err;
    }
  }
  if (group->meth->field_encode != 0) {
    /*
     * in the Montgomery case, we just turned  R*H  (representing
     * H) into  1/(R*H),  but we need  R*(1/H)  (representing
     * 1/H); i.e. we have need to multiply by the Montgomery
     * factor twice
     */
    if (!group->meth->field_encode(group, heap[1], heap[1], ctx))
      goto err;
    if (!group->meth->field_encode(group, heap[1], heap[1], ctx))
      goto err;
  }
  /* set other heap[i]'s to their inverses */
  for (i = 2; i < pow2 / 2 + num; i += 2) {
    /* i is even */
    if ((heap[i + 1] != NULL) && !BN_is_zero(heap[i + 1])) {
      if (!group->meth->field_mul(group, tmp0, heap[i / 2], heap[i + 1], ctx))
        goto err;
      if (!group->meth->field_mul(group, tmp1, heap[i / 2], heap[i], ctx))
        goto err;
      if (!BN_copy(heap[i], tmp0))
        goto err;
      if (!BN_copy(heap[i + 1], tmp1))
        goto err;
    } else {
      if (!BN_copy(heap[i], heap[i / 2]))
        goto err;
    }
  }

  /*
   * we have replaced all non-zero Z's by their inverses, now fix up
   * all the points
   */
  for (i = 0; i < num; i++) {
    EC_POINT *p = points[i];

    if (!BN_is_zero(&p->Z)) {
      /* turn  (X, Y, 1/Z)  into  (X/Z^2, Y/Z^3, 1) */

      if (!group->meth->field_sqr(group, tmp1, &p->Z, ctx))
        goto err;
      if (!group->meth->field_mul(group, &p->X, &p->X, tmp1, ctx))
        goto err;

      if (!group->meth->field_mul(group, tmp1, tmp1, &p->Z, ctx))
        goto err;
      if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp1, ctx))
        goto err;

      if (group->meth->field_set_to_one != 0) {
        if (!group->meth->field_set_to_one(group, &p->Z, ctx))
          goto err;
      } else {
        if (!BN_one(&p->Z))
          goto err;
      }
      p->Z_is_one = 1;
    }
  }

  ret = 1;

 err:
  BN_CTX_end(ctx);
  BN_CTX_free(new_ctx);
  if (heap != NULL) {
    /*
     * heap[pow2/2] .. heap[pow2-1] have not been allocated
     * locally!
     */
    for (i = pow2 / 2 - 1; i > 0; i--) {
      BN_clear_free(heap[i]);
    }
    free(heap);
  }
  return ret;
}


int
ec_GFp_simple_field_mul(const EC_GROUP * group, BIGNUM * r, const BIGNUM * a, const BIGNUM * b, BN_CTX * ctx)
{
  return BN_mod_mul(r, a, b, &group->field, ctx);
}

int
ec_GFp_simple_field_sqr(const EC_GROUP * group, BIGNUM * r, const BIGNUM * a, BN_CTX * ctx)
{
  return BN_mod_sqr(r, a, &group->field, ctx);
}

/*
 * Apply randomization of EC point projective coordinates:
 *
 *  (X, Y, Z) = (lambda^2 * X, lambda^3 * Y, lambda * Z)
 *
 * where lambda is in the interval [1, group->field).
 */
int
ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p, BN_CTX *ctx)
{
  BIGNUM *lambda = NULL;
  BIGNUM *tmp = NULL;
  int ret = 0;

  BN_CTX_start(ctx);
  if ((lambda = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((tmp = BN_CTX_get(ctx)) == NULL)
    goto err;

  /* Generate lambda in [1, group->field - 1] */
  if (!bn_rand_interval(lambda, BN_value_one(), &group->field))
    goto err;

  if (group->meth->field_encode != NULL &&
      !group->meth->field_encode(group, lambda, lambda, ctx))
    goto err;

  /* Z = lambda * Z */
  if (!group->meth->field_mul(group, &p->Z, lambda, &p->Z, ctx))
    goto err;

  /* tmp = lambda^2 */
  if (!group->meth->field_sqr(group, tmp, lambda, ctx))
    goto err;

  /* X = lambda^2 * X */
  if (!group->meth->field_mul(group, &p->X, tmp, &p->X, ctx))
    goto err;

  /* tmp = lambda^3 */
  if (!group->meth->field_mul(group, tmp, tmp, lambda, ctx))
    goto err;

  /* Y = lambda^3 * Y */
  if (!group->meth->field_mul(group, &p->Y, tmp, &p->Y, ctx))
    goto err;

  /* Disable optimized arithmetics after replacing Z by lambda * Z. */
  p->Z_is_one = 0;

  ret = 1;

 err:
  BN_CTX_end(ctx);
  return ret;
}


#define EC_POINT_BN_set_flags(P, flags) do {       \
  BN_set_flags(&(P)->X, (flags));               \
  BN_set_flags(&(P)->Y, (flags));               \
  BN_set_flags(&(P)->Z, (flags));               \
} while(0)

#define EC_POINT_CSWAP(c, a, b, w, t) do {           \
  if (!BN_swap_ct(c, &(a)->X, &(b)->X, w)  ||      \
      !BN_swap_ct(c, &(a)->Y, &(b)->Y, w)  ||      \
      !BN_swap_ct(c, &(a)->Z, &(b)->Z, w))     \
    goto err;           \
  t = ((a)->Z_is_one ^ (b)->Z_is_one) & (c);      \
  (a)->Z_is_one ^= (t);           \
  (b)->Z_is_one ^= (t);           \
} while(0)

/*
 * This function computes (in constant time) a point multiplication over the
 * EC group.
 *
 * At a high level, it is Montgomery ladder with conditional swaps.
 *
 * It performs either a fixed point multiplication
 *          (scalar * generator)
 * when point is NULL, or a variable point multiplication
 *          (scalar * point)
 * when point is not NULL.
 *
 * scalar should be in the range [0,n) otherwise all constant time bets are off.
 *
 * NB: This says nothing about EC_POINT_add and EC_POINT_dbl,
 * which of course are not constant time themselves.
 *
 * The product is stored in r.
 *
 * Returns 1 on success, 0 otherwise.
 */
static int
ec_GFp_simple_mul_ct(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
    const EC_POINT *point, BN_CTX *ctx)
{
  int i, cardinality_bits, group_top, kbit, pbit, Z_is_one;
  EC_POINT *s = NULL;
  BIGNUM *k = NULL;
  BIGNUM *lambda = NULL;
  BIGNUM *cardinality = NULL;
  BN_CTX *new_ctx = NULL;
  int ret = 0;

  if (ctx == NULL && (ctx = new_ctx = BN_CTX_new()) == NULL)
    return 0;

  BN_CTX_start(ctx);

  if ((s = EC_POINT_new(group)) == NULL)
    goto err;

  if (point == NULL) {
    if (!EC_POINT_copy(s, group->generator))
      goto err;
  } else {
    if (!EC_POINT_copy(s, point))
      goto err;
  }

  EC_POINT_BN_set_flags(s, BN_FLG_CONSTTIME);

  if ((cardinality = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((lambda = BN_CTX_get(ctx)) == NULL)
    goto err;
  if ((k = BN_CTX_get(ctx)) == NULL)
    goto err;
  if (!BN_mul(cardinality, &group->order, &group->cofactor, ctx))
    goto err;

  /*
   * Group cardinalities are often on a word boundary.
   * So when we pad the scalar, some timing diff might
   * pop if it needs to be expanded due to carries.
   * So expand ahead of time.
   */
  cardinality_bits = BN_num_bits(cardinality);
  group_top = cardinality->top;
  if ((bn_wexpand(k, group_top + 2) == NULL) ||
      (bn_wexpand(lambda, group_top + 2) == NULL))
    goto err;

  if (!BN_copy(k, scalar))
    goto err;

  BN_set_flags(k, BN_FLG_CONSTTIME);

  if (BN_num_bits(k) > cardinality_bits || BN_is_negative(k)) {
    /*
     * This is an unusual input, and we don't guarantee
     * constant-timeness
     */
    if (!BN_nnmod(k, k, cardinality, ctx))
      goto err;
  }

  if (!BN_add(lambda, k, cardinality))
    goto err;
  BN_set_flags(lambda, BN_FLG_CONSTTIME);
  if (!BN_add(k, lambda, cardinality))
    goto err;
  /*
   * lambda := scalar + cardinality
   * k := scalar + 2*cardinality
   */
  kbit = BN_is_bit_set(lambda, cardinality_bits);
  if (!BN_swap_ct(kbit, k, lambda, group_top + 2))
    goto err;

  group_top = group->field.top;
  if ((bn_wexpand(&s->X, group_top) == NULL) ||
      (bn_wexpand(&s->Y, group_top) == NULL) ||
      (bn_wexpand(&s->Z, group_top) == NULL) ||
      (bn_wexpand(&r->X, group_top) == NULL) ||
      (bn_wexpand(&r->Y, group_top) == NULL) ||
      (bn_wexpand(&r->Z, group_top) == NULL))
    goto err;

  /*
   * Apply coordinate blinding for EC_POINT if the underlying EC_METHOD
   * implements it.
   */
  if (!ec_point_blind_coordinates(group, s, ctx))
    goto err;

  /* top bit is a 1, in a fixed pos */
  if (!EC_POINT_copy(r, s))
    goto err;

  EC_POINT_BN_set_flags(r, BN_FLG_CONSTTIME);

  if (!EC_POINT_dbl(group, s, s, ctx))
    goto err;

  pbit = 0;

  /*
   * The ladder step, with branches, is
   *
   * k[i] == 0: S = add(R, S), R = dbl(R)
   * k[i] == 1: R = add(S, R), S = dbl(S)
   *
   * Swapping R, S conditionally on k[i] leaves you with state
   *
   * k[i] == 0: T, U = R, S
   * k[i] == 1: T, U = S, R
   *
   * Then perform the ECC ops.
   *
   * U = add(T, U)
   * T = dbl(T)
   *
   * Which leaves you with state
   *
   * k[i] == 0: U = add(R, S), T = dbl(R)
   * k[i] == 1: U = add(S, R), T = dbl(S)
   *
   * Swapping T, U conditionally on k[i] leaves you with state
   *
   * k[i] == 0: R, S = T, U
   * k[i] == 1: R, S = U, T
   *
   * Which leaves you with state
   *
   * k[i] == 0: S = add(R, S), R = dbl(R)
   * k[i] == 1: R = add(S, R), S = dbl(S)
   *
   * So we get the same logic, but instead of a branch it's a
   * conditional swap, followed by ECC ops, then another conditional swap.
   *
   * Optimization: The end of iteration i and start of i-1 looks like
   *
   * ...
   * CSWAP(k[i], R, S)
   * ECC
   * CSWAP(k[i], R, S)
   * (next iteration)
   * CSWAP(k[i-1], R, S)
   * ECC
   * CSWAP(k[i-1], R, S)
   * ...
   *
   * So instead of two contiguous swaps, you can merge the condition
   * bits and do a single swap.
   *
   * k[i]   k[i-1]    Outcome
   * 0      0         No Swap
   * 0      1         Swap
   * 1      0         Swap
   * 1      1         No Swap
   *
   * This is XOR. pbit tracks the previous bit of k.
   */

  for (i = cardinality_bits - 1; i >= 0; i--) {
    kbit = BN_is_bit_set(k, i) ^ pbit;
    EC_POINT_CSWAP(kbit, r, s, group_top, Z_is_one);
    if (!EC_POINT_add(group, s, r, s, ctx))
      goto err;
    if (!EC_POINT_dbl(group, r, r, ctx))
      goto err;
    /*
     * pbit logic merges this cswap with that of the
     * next iteration
     */
    pbit ^= kbit;
  }
  /* one final cswap to move the right value into r */
  EC_POINT_CSWAP(pbit, r, s, group_top, Z_is_one);

  ret = 1;

 err:
  EC_POINT_free(s);
  if (ctx != NULL)
    BN_CTX_end(ctx);
  BN_CTX_free(new_ctx);

  return ret;
}

#undef EC_POINT_BN_set_flags
#undef EC_POINT_CSWAP

int
ec_GFp_simple_mul_generator_ct(const EC_GROUP *group, EC_POINT *r,
    const BIGNUM *scalar, BN_CTX *ctx)
{
  return ec_GFp_simple_mul_ct(group, r, scalar, NULL, ctx);
}

int
ec_GFp_simple_mul_single_ct(const EC_GROUP *group, EC_POINT *r,
    const BIGNUM *scalar, const EC_POINT *point, BN_CTX *ctx)
{
  return ec_GFp_simple_mul_ct(group, r, scalar, point, ctx);
}

int
ec_GFp_simple_mul_double_nonct(const EC_GROUP *group, EC_POINT *r,
    const BIGNUM *g_scalar, const BIGNUM *p_scalar, const EC_POINT *point,
    BN_CTX *ctx)
{
  return ec_wNAF_mul(group, r, g_scalar, 1, &point, &p_scalar, ctx);
}

Coverage Report

Created: 2022-08-24 06:30