/* mpi-inv.c  -  MPI functions

 *  Copyright (C) 1998, 2001, 2002, 2003 Free Software Foundation, Inc.

 *

 * This file is part of Libgcrypt.

 *

 * Libgcrypt is free software; you can redistribute it and/or modify

 * it under the terms of the GNU Lesser General Public License as

 * published by the Free Software Foundation; either version 2.1 of

 * the License, or (at your option) any later version.

 *

 * Libgcrypt is distributed in the hope that it will be useful,

 * but WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the

 * GNU Lesser General Public License for more details.

 *

 * You should have received a copy of the GNU Lesser General Public

 * License along with this program; if not, see <http://www.gnu.org/licenses/>.

 */

#include <config.h>

#include <stdio.h>

#include <stdlib.h>

#include "mpi-internal.h"

#include "g10lib.h"

/*

 * This uses a modular inversion algorithm designed by Niels Möller

 * which was implemented in Nettle.  The same algorithm was later also

 * adapted to GMP in mpn_sec_invert.

 *

 * For the description of the algorithm, see Algorithm 5 in Appendix A

 * of "Fast Software Polynomial Multiplication on ARM Processors using

 * the NEON Engine" by Danilo Câmara, Conrado P. L. Gouvêa, Julio

 * López, and Ricardo Dahab:

 *   https://hal.inria.fr/hal-01506572/document

 *

 * Note that in the reference above, at the line 2 of Algorithm 5,

 * initial value of V was described as V:=1 wrongly.  It must be V:=0.

 */

static mpi_ptr_t

mpih_invm_odd (mpi_ptr_t ap, mpi_ptr_t np, mpi_size_t nsize)

{

  int secure;

  unsigned int iterations;

  mpi_ptr_t n1hp;

  mpi_ptr_t bp;

  mpi_ptr_t up, vp;

  secure = _gcry_is_secure (ap);

  up = mpi_alloc_limb_space (nsize, secure);

  MPN_ZERO (up, nsize);

  up[0] = 1;

  vp = mpi_alloc_limb_space (nsize, secure);

  MPN_ZERO (vp, nsize);

  secure = _gcry_is_secure (np);

  bp = mpi_alloc_limb_space (nsize, secure);

  MPN_COPY (bp, np, nsize);

  n1hp = mpi_alloc_limb_space (nsize, secure);

  MPN_COPY (n1hp, np, nsize);

  _gcry_mpih_rshift (n1hp, n1hp, nsize, 1);

  _gcry_mpih_add_1 (n1hp, n1hp, nsize, 1);

  iterations = 2 * nsize * BITS_PER_MPI_LIMB;

  while (iterations-- > 0)

    {

      mpi_limb_t odd_a, odd_u, underflow, borrow;

      odd_a = ap[0] & 1;

      underflow = mpih_sub_n_cond (ap, ap, bp, nsize, odd_a);

      mpih_add_n_cond (bp, bp, ap, nsize, underflow);

      mpih_abs_cond (ap, ap, nsize, underflow);

      mpih_swap_cond (up, vp, nsize, underflow);

      _gcry_mpih_rshift (ap, ap, nsize, 1);

      borrow = mpih_sub_n_cond (up, up, vp, nsize, odd_a);

      mpih_add_n_cond (up, up, np, nsize, borrow);

      odd_u = _gcry_mpih_rshift (up, up, nsize, 1) != 0;

      mpih_add_n_cond (up, up, n1hp, nsize, odd_u);

    }

  _gcry_mpi_free_limb_space (n1hp, nsize);

  _gcry_mpi_free_limb_space (up, nsize);

  if (_gcry_mpih_cmp_ui (bp, nsize, 1) == 0)

    {

      /* Inverse exists.  */

      _gcry_mpi_free_limb_space (bp, nsize);

      return vp;

    }

  else

    {

      _gcry_mpi_free_limb_space (bp, nsize);

      _gcry_mpi_free_limb_space (vp, nsize);

      return NULL;

    }

}

/*

 * Calculate the multiplicative inverse X of A mod 2^K

 * A must be positive.

 *

 * See section 7 in "A New Algorithm for Inversion mod p^k" by Çetin

 * Kaya Koç: https://eprint.iacr.org/2017/411.pdf

 */

static mpi_ptr_t

mpih_invm_pow2 (mpi_ptr_t ap, mpi_size_t asize, unsigned int k)

{

  int secure = _gcry_is_secure (ap);

  mpi_size_t i;

  unsigned int iterations;

  mpi_ptr_t xp, wp, up, vp;

  mpi_size_t usize;

  if (!(ap[0] & 1))

    return NULL;

  iterations = ((k + BITS_PER_MPI_LIMB - 1) / BITS_PER_MPI_LIMB)

    * BITS_PER_MPI_LIMB;

  usize = iterations / BITS_PER_MPI_LIMB;

  up = mpi_alloc_limb_space (usize, secure);

  MPN_ZERO (up, usize);

  up[0] = 1;

  vp = mpi_alloc_limb_space (usize, secure);

  for (i = 0; i < (usize < asize ? usize : asize); i++)

    vp[i] = ap[i];

  for (; i < usize; i++)

    vp[i] = 0;

  if ((k % BITS_PER_MPI_LIMB))

    for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)

      vp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);

  wp = mpi_alloc_limb_space (usize, secure);

  MPN_COPY (wp, up, usize);

  xp = mpi_alloc_limb_space (usize, secure);

  MPN_ZERO (xp, usize);

  /*

   * It can be considered that overflow at _gcry_mpih_sub_n results

   * adding 2^(USIZE*BITS_PER_MPI_LIMB), which is no problem in modulo

   * 2^K computation.

   */

  for (i = 0; i < iterations; i++)

    {

      int b0 = (up[0] & 1);

      xp[i/BITS_PER_MPI_LIMB] |= ((mpi_limb_t)b0<<(i%BITS_PER_MPI_LIMB));

      _gcry_mpih_sub_n (wp, up, vp, usize);

      mpih_set_cond (up, wp, usize, b0);

      _gcry_mpih_rshift (up, up, usize, 1);

    }

  if ((k % BITS_PER_MPI_LIMB))

    for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)

      xp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);

  _gcry_mpi_free_limb_space (up, usize);

  _gcry_mpi_free_limb_space (vp, usize);

  _gcry_mpi_free_limb_space (wp, usize);

  return xp;

}

/****************

 * Calculate the multiplicative inverse X of A mod N

 * That is: Find the solution x for

 *    1 = (a*x) mod n

 */

static int

mpi_invm_generic (gcry_mpi_t x, gcry_mpi_t a, gcry_mpi_t n)

{

    int is_gcd_one;

#if 0

    /* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X) */

    gcry_mpi_t u, v, u1, u2, u3, v1, v2, v3, q, t1, t2, t3;

    u = mpi_copy(a);

    v = mpi_copy(n);

    u1 = mpi_alloc_set_ui(1);

    u2 = mpi_alloc_set_ui(0);

    u3 = mpi_copy(u);

    v1 = mpi_alloc_set_ui(0);

    v2 = mpi_alloc_set_ui(1);

    v3 = mpi_copy(v);

    q  = mpi_alloc( mpi_get_nlimbs(u)+1 );

    t1 = mpi_alloc( mpi_get_nlimbs(u)+1 );

    t2 = mpi_alloc( mpi_get_nlimbs(u)+1 );

    t3 = mpi_alloc( mpi_get_nlimbs(u)+1 );

    while( mpi_cmp_ui( v3, 0 ) ) {

  mpi_fdiv_q( q, u3, v3 );

  mpi_mul(t1, v1, q); mpi_mul(t2, v2, q); mpi_mul(t3, v3, q);

  mpi_sub(t1, u1, t1); mpi_sub(t2, u2, t2); mpi_sub(t3, u3, t3);

  mpi_set(u1, v1); mpi_set(u2, v2); mpi_set(u3, v3);

  mpi_set(v1, t1); mpi_set(v2, t2); mpi_set(v3, t3);

    }

    /*  log_debug("result:\n");

  log_mpidump("q =", q );

  log_mpidump("u1=", u1);

  log_mpidump("u2=", u2);

  log_mpidump("u3=", u3);

  log_mpidump("v1=", v1);

  log_mpidump("v2=", v2); */

    mpi_set(x, u1);

    is_gcd_one = (mpi_cmp_ui (u3, 1) == 0);

    mpi_free(u1);

    mpi_free(u2);

    mpi_free(u3);

    mpi_free(v1);

    mpi_free(v2);

    mpi_free(v3);

    mpi_free(q);

    mpi_free(t1);

    mpi_free(t2);

    mpi_free(t3);

    mpi_free(u);

    mpi_free(v);

#elif 0

    /* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X)

     * modified according to Michael Penk's solution for Exercise 35

     * (in the first edition)

     * In the third edition, it's Exercise 39, and it is described in

     * page 646 of ANSWERS TO EXERCISES chapter.

     */

    /* FIXME: we can simplify this in most cases (see Knuth) */

    gcry_mpi_t u, v, u1, u2, u3, v1, v2, v3, t1, t2, t3;

    unsigned k;

    int sign;

    u = mpi_copy(a);

    v = mpi_copy(n);

    for(k=0; !mpi_test_bit(u,0) && !mpi_test_bit(v,0); k++ ) {

  mpi_rshift(u, u, 1);

  mpi_rshift(v, v, 1);

    }

    u1 = mpi_alloc_set_ui(1);

    u2 = mpi_alloc_set_ui(0);

    u3 = mpi_copy(u);

    v1 = mpi_copy(v);          /* !-- used as const 1 */

    v2 = mpi_alloc( mpi_get_nlimbs(u) ); mpi_sub( v2, u1, u );

    v3 = mpi_copy(v);

    if( mpi_test_bit(u, 0) ) { /* u is odd */

  t1 = mpi_alloc_set_ui(0);

  t2 = mpi_alloc_set_ui(1); t2->sign = 1;

  t3 = mpi_copy(v); t3->sign = !t3->sign;

  goto Y4;

    }

    else {

  t1 = mpi_alloc_set_ui(1);

  t2 = mpi_alloc_set_ui(0);

  t3 = mpi_copy(u);

    }

    do {

  do {

      if( mpi_test_bit(t1, 0) || mpi_test_bit(t2, 0) ) { /* one is odd */

    mpi_add(t1, t1, v);

    mpi_sub(t2, t2, u);

      }

      mpi_rshift(t1, t1, 1);

      mpi_rshift(t2, t2, 1);

      mpi_rshift(t3, t3, 1);

    Y4:

      ;

  } while( !mpi_test_bit( t3, 0 ) ); /* while t3 is even */

  if( !t3->sign ) {

      mpi_set(u1, t1);

      mpi_set(u2, t2);

      mpi_set(u3, t3);

  }

  else {

      mpi_sub(v1, v, t1);

      sign = u->sign; u->sign = !u->sign;

      mpi_sub(v2, u, t2);

      u->sign = sign;

      sign = t3->sign; t3->sign = !t3->sign;

      mpi_set(v3, t3);

      t3->sign = sign;

  }

  mpi_sub(t1, u1, v1);

  mpi_sub(t2, u2, v2);

  mpi_sub(t3, u3, v3);

  if( t1->sign ) {

      mpi_add(t1, t1, v);

      mpi_sub(t2, t2, u);

  }

    } while( mpi_cmp_ui( t3, 0 ) ); /* while t3 != 0 */

    /* mpi_lshift( u3, u3, k ); */

    is_gcd_one = (k == 0 && mpi_cmp_ui (u3, 1) == 0);

    mpi_set(x, u1);

    mpi_free(u1);

    mpi_free(u2);

    mpi_free(u3);

    mpi_free(v1);

    mpi_free(v2);

    mpi_free(v3);

    mpi_free(t1);

    mpi_free(t2);

    mpi_free(t3);

#else

    /* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X)

     * modified according to Michael Penk's solution for Exercise 35

     * with further enhancement */

    /* The reference in the comment above is for the first edition.

     * In the third edition, it's Exercise 39, and it is described in

     * page 646 of ANSWERS TO EXERCISES chapter.

     */

    gcry_mpi_t u, v, u1, u2=NULL, u3, v1, v2=NULL, v3, t1, t2=NULL, t3;

    unsigned k;

    int sign;

    int odd ;

    u = mpi_copy(a);

    v = mpi_copy(n);

    for(k=0; !mpi_test_bit(u,0) && !mpi_test_bit(v,0); k++ ) {

  mpi_rshift(u, u, 1);

  mpi_rshift(v, v, 1);

    }

    odd = mpi_test_bit(v,0);

    u1 = mpi_alloc_set_ui(1);

    if( !odd )

  u2 = mpi_alloc_set_ui(0);

    u3 = mpi_copy(u);

    v1 = mpi_copy(v);

    if( !odd ) {

  v2 = mpi_alloc( mpi_get_nlimbs(u) );

  mpi_sub( v2, u1, u ); /* U is used as const 1 */

    }

    v3 = mpi_copy(v);

    if( mpi_test_bit(u, 0) ) { /* u is odd */

  t1 = mpi_alloc_set_ui(0);

  if( !odd ) {

      t2 = mpi_alloc_set_ui(1); t2->sign = 1;

  }

  t3 = mpi_copy(v); t3->sign = !t3->sign;

  goto Y4;

    }

    else {

  t1 = mpi_alloc_set_ui(1);

  if( !odd )

      t2 = mpi_alloc_set_ui(0);

  t3 = mpi_copy(u);

    }

    do {

  do {

      if( !odd ) {

    if( mpi_test_bit(t1, 0) || mpi_test_bit(t2, 0) ) { /* one is odd */

        mpi_add(t1, t1, v);

        mpi_sub(t2, t2, u);

    }

    mpi_rshift(t1, t1, 1);

    mpi_rshift(t2, t2, 1);

    mpi_rshift(t3, t3, 1);

      }

      else {

    if( mpi_test_bit(t1, 0) )

        mpi_add(t1, t1, v);

    mpi_rshift(t1, t1, 1);

    mpi_rshift(t3, t3, 1);

      }

    Y4:

      ;

  } while( !mpi_test_bit( t3, 0 ) ); /* while t3 is even */

  if( !t3->sign ) {

      mpi_set(u1, t1);

      if( !odd )

    mpi_set(u2, t2);

      mpi_set(u3, t3);

  }

  else {

      mpi_sub(v1, v, t1);

      sign = u->sign; u->sign = !u->sign;

      if( !odd )

    mpi_sub(v2, u, t2);

      u->sign = sign;

      sign = t3->sign; t3->sign = !t3->sign;

      mpi_set(v3, t3);

      t3->sign = sign;

  }

  mpi_sub(t1, u1, v1);

  if( !odd )

      mpi_sub(t2, u2, v2);

  mpi_sub(t3, u3, v3);

  if( t1->sign ) {

      mpi_add(t1, t1, v);

      if( !odd )

    mpi_sub(t2, t2, u);

  }

    } while( mpi_cmp_ui( t3, 0 ) ); /* while t3 != 0 */

    /* mpi_lshift( u3, u3, k ); */

    is_gcd_one = (k == 0 && mpi_cmp_ui (u3, 1) == 0);

    mpi_set(x, u1);

    mpi_free(u1);

    mpi_free(v1);

    mpi_free(t1);

    if( !odd ) {

  mpi_free(u2);

  mpi_free(v2);

  mpi_free(t2);

    }

    mpi_free(u3);

    mpi_free(v3);

    mpi_free(t3);

    mpi_free(u);

    mpi_free(v);

#endif

    return is_gcd_one;

}

/*

 * Set X to the multiplicative inverse of A mod M.  Return true if the

 * inverse exists.

 */

int

_gcry_mpi_invm (gcry_mpi_t x, gcry_mpi_t a, gcry_mpi_t n)

{

  mpi_ptr_t ap, xp;

  if (!mpi_cmp_ui (a, 0))

    return 0; /* Inverse does not exists.  */

  if (!mpi_cmp_ui (n, 1))

    return 0; /* Inverse does not exists.  */

  if (mpi_test_bit (n, 0))

    {

      if (a->nlimbs <= n->nlimbs)

        {

          ap = mpi_alloc_limb_space (n->nlimbs, _gcry_is_secure (a->d));

          MPN_ZERO (ap, n->nlimbs);

          MPN_COPY (ap, a->d, a->nlimbs);

        }

      else

        ap = _gcry_mpih_mod (a->d, a->nlimbs, n->d, n->nlimbs);

      xp = mpih_invm_odd (ap, n->d, n->nlimbs);

      _gcry_mpi_free_limb_space (ap, n->nlimbs);

      if (xp)

        {

          _gcry_mpi_assign_limb_space (x, xp, n->nlimbs);

          x->nlimbs = n->nlimbs;

          return 1;

        }

      else

        return 0; /* Inverse does not exists.  */

    }

  else if (!a->sign && !n->sign)

    {

      unsigned int k = mpi_trailing_zeros (n);

      mpi_size_t x1size = ((k + BITS_PER_MPI_LIMB - 1) / BITS_PER_MPI_LIMB);

      mpi_size_t hsize;

      gcry_mpi_t q;

      mpi_ptr_t x1p, x2p, q_invp, hp, diffp;

      mpi_size_t i;

      if (k == _gcry_mpi_get_nbits (n) - 1)

        {

          x1p = mpih_invm_pow2 (a->d, a->nlimbs, k);

          if (x1p)

            {

              _gcry_mpi_assign_limb_space (x, x1p, x1size);

              x->nlimbs = x1size;

              return 1;

            }

          else

            return 0; /* Inverse does not exists.  */

        }

      /* N can be expressed as P * Q, where P = 2^K.  P and Q are coprime.  */

      /*

       * Compute X1 = invm (A, P) and X2 = invm (A, Q), and combine

       * them by Garner's formula, to get X = invm (A, P*Q).

       * A special case of Chinese Remainder Theorem.

       */

      /* X1 = invm (A, P) */

      x1p = mpih_invm_pow2 (a->d, a->nlimbs, k);

      if (!x1p)

        return 0;               /* Inverse does not exists.  */

      /* Q = N / P          */

      q = mpi_new (0);

      mpi_rshift (q, n, k);

      /* X2 = invm (A%Q, Q) */

      ap = _gcry_mpih_mod (a->d, a->nlimbs, q->d, q->nlimbs);

      x2p = mpih_invm_odd (ap, q->d, q->nlimbs);

      _gcry_mpi_free_limb_space (ap, q->nlimbs);

      if (!x2p)

        {

          _gcry_mpi_free_limb_space (x1p, x1size);

          mpi_free (q);

          return 0;             /* Inverse does not exists.  */

        }

      /* Q_inv = Q^(-1) = invm (Q, P) */

      q_invp = mpih_invm_pow2 (q->d, q->nlimbs, k);

      /* H = (X1 - X2) * Q_inv % P */

      diffp = mpi_alloc_limb_space (x1size, _gcry_is_secure (a->d));

      if (x1size >= q->nlimbs)

        _gcry_mpih_sub (diffp, x1p, x1size, x2p, q->nlimbs);

      else

  _gcry_mpih_sub_n (diffp, x1p, x2p, x1size);

      _gcry_mpi_free_limb_space (x1p, x1size);

      if ((k % BITS_PER_MPI_LIMB))

        for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)

          diffp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);

      hsize = x1size * 2;

      hp = mpi_alloc_limb_space (hsize, _gcry_is_secure (a->d));

      _gcry_mpih_mul_n (hp, diffp, q_invp, x1size);

      _gcry_mpi_free_limb_space (diffp, x1size);

      _gcry_mpi_free_limb_space (q_invp, x1size);

      for (i = x1size; i < hsize; i++)

        hp[i] = 0;

      if ((k % BITS_PER_MPI_LIMB))

        for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)

          hp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);

      xp = mpi_alloc_limb_space (x1size + q->nlimbs, _gcry_is_secure (a->d));

      if (x1size >= q->nlimbs)

        _gcry_mpih_mul (xp, hp, x1size, q->d, q->nlimbs);

      else

        _gcry_mpih_mul (xp, q->d, q->nlimbs, hp, x1size);

      _gcry_mpi_free_limb_space (hp, hsize);

      _gcry_mpih_add (xp, xp, x1size + q->nlimbs, x2p, q->nlimbs);

      _gcry_mpi_free_limb_space (x2p, q->nlimbs);

      _gcry_mpi_assign_limb_space (x, xp, x1size + q->nlimbs);

      x->nlimbs = x1size + q->nlimbs;

      mpi_free (q);

      return 1;

    }

  else

    return mpi_invm_generic (x, a, n);

}