boringssl/crypto/bn/generic.c

/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young (eay@cryptsoft.com).
 * The implementation was written so as to conform with Netscapes SSL.
 *
 * This library is free for commercial and non-commercial use as long as
 * the following conditions are aheared to.  The following conditions
 * apply to all code found in this distribution, be it the RC4, RSA,
 * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
 * included with this distribution is covered by the same copyright terms
 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
 *
 * Copyright remains Eric Young's, and as such any Copyright notices in
 * the code are not to be removed.
 * If this package is used in a product, Eric Young should be given attribution
 * as the author of the parts of the library used.
 * This can be in the form of a textual message at program startup or
 * in documentation (online or textual) provided with the package.
 *
 * 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 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 acknowledgement:
 *    "This product includes cryptographic software written by
 *     Eric Young (eay@cryptsoft.com)"
 *    The word 'cryptographic' can be left out if the rouines from the library
 *    being used are not cryptographic related :-).
 * 4. If you include any Windows specific code (or a derivative thereof) from
 *    the apps directory (application code) you must include an acknowledgement:
 *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
 *
 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
 * ANY EXPRESS 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 AUTHOR OR 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.
 *
 * The licence and distribution terms for any publically available version or
 * derivative of this code cannot be changed.  i.e. this code cannot simply be
 * copied and put under another distribution licence
 * [including the GNU Public Licence.] */

#include <openssl/bn.h>

#include <assert.h>

#include "internal.h"


/* Generic implementations of most operations are needed for:
 * - Configurations without inline assembly.
 * - Architectures other than x86 or x86_64.
 * - Windows x84_64; x86_64-gcc.c does not build on MSVC. */
#if defined(OPENSSL_NO_ASM) || \
    (!defined(OPENSSL_X86_64) && !defined(OPENSSL_X86)) || \
    (defined(OPENSSL_X86_64) && defined(OPENSSL_WINDOWS))

#if defined(OPENSSL_WINDOWS)
#define alloca _alloca
#else
#include <alloca.h>
#endif

#ifdef BN_LLONG
#define mul_add(r, a, w, c)             \
  {                                     \
    BN_ULLONG t;                        \
    t = (BN_ULLONG)w * (a) + (r) + (c); \
    (r) = Lw(t);                        \
    (c) = Hw(t);                        \
  }

#define mul(r, a, w, c)           \
  {                               \
    BN_ULLONG t;                  \
    t = (BN_ULLONG)w * (a) + (c); \
    (r) = Lw(t);                  \
    (c) = Hw(t);                  \
  }

#define sqr(r0, r1, a)        \
  {                           \
    BN_ULLONG t;              \
    t = (BN_ULLONG)(a) * (a); \
    (r0) = Lw(t);             \
    (r1) = Hw(t);             \
  }

#elif defined(BN_UMULT_LOHI)
#define mul_add(r, a, w, c)             \
  {                                     \
    BN_ULONG high, low, ret, tmp = (a); \
    ret = (r);                          \
    BN_UMULT_LOHI(low, high, w, tmp);   \
    ret += (c);                         \
    (c) = (ret < (c)) ? 1 : 0;          \
    (c) += high;                        \
    ret += low;                         \
    (c) += (ret < low) ? 1 : 0;         \
    (r) = ret;                          \
  }

#define mul(r, a, w, c)                \
  {                                    \
    BN_ULONG high, low, ret, ta = (a); \
    BN_UMULT_LOHI(low, high, w, ta);   \
    ret = low + (c);                   \
    (c) = high;                        \
    (c) += (ret < low) ? 1 : 0;        \
    (r) = ret;                         \
  }

#define sqr(r0, r1, a)               \
  {                                  \
    BN_ULONG tmp = (a);              \
    BN_UMULT_LOHI(r0, r1, tmp, tmp); \
  }

#else

/*************************************************************
 * No long long type
 */

#define LBITS(a) ((a) & BN_MASK2l)
#define HBITS(a) (((a) >> BN_BITS4) & BN_MASK2l)
#define L2HBITS(a) (((a) << BN_BITS4) & BN_MASK2)

#define LLBITS(a) ((a) & BN_MASKl)
#define LHBITS(a) (((a) >> BN_BITS2) & BN_MASKl)
#define LL2HBITS(a) ((BN_ULLONG)((a) & BN_MASKl) << BN_BITS2)

#define mul64(l, h, bl, bh)       \
  {                               \
    BN_ULONG m, m1, lt, ht;       \
                                  \
    lt = l;                       \
    ht = h;                       \
    m = (bh) * (lt);              \
    lt = (bl) * (lt);             \
    m1 = (bl) * (ht);             \
    ht = (bh) * (ht);             \
    m = (m + m1) & BN_MASK2;      \
    if (m < m1)                   \
      ht += L2HBITS((BN_ULONG)1); \
    ht += HBITS(m);               \
    m1 = L2HBITS(m);              \
    lt = (lt + m1) & BN_MASK2;    \
    if (lt < m1)                  \
      ht++;                       \
    (l) = lt;                     \
    (h) = ht;                     \
  }

#define sqr64(lo, ho, in)                    \
  {                                          \
    BN_ULONG l, h, m;                        \
                                             \
    h = (in);                                \
    l = LBITS(h);                            \
    h = HBITS(h);                            \
    m = (l) * (h);                           \
    l *= l;                                  \
    h *= h;                                  \
    h += (m & BN_MASK2h1) >> (BN_BITS4 - 1); \
    m = (m & BN_MASK2l) << (BN_BITS4 + 1);   \
    l = (l + m) & BN_MASK2;                  \
    if (l < m)                               \
      h++;                                   \
    (lo) = l;                                \
    (ho) = h;                                \
  }

#define mul_add(r, a, bl, bh, c) \
  {                              \
    BN_ULONG l, h;               \
                                 \
    h = (a);                     \
    l = LBITS(h);                \
    h = HBITS(h);                \
    mul64(l, h, (bl), (bh));     \
                                 \
    /* non-multiply part */      \
    l = (l + (c)) & BN_MASK2;    \
    if (l < (c))                 \
      h++;                       \
    (c) = (r);                   \
    l = (l + (c)) & BN_MASK2;    \
    if (l < (c))                 \
      h++;                       \
    (c) = h & BN_MASK2;          \
    (r) = l;                     \
  }

#define mul(r, a, bl, bh, c)  \
  {                           \
    BN_ULONG l, h;            \
                              \
    h = (a);                  \
    l = LBITS(h);             \
    h = HBITS(h);             \
    mul64(l, h, (bl), (bh));  \
                              \
    /* non-multiply part */   \
    l += (c);                 \
    if ((l & BN_MASK2) < (c)) \
      h++;                    \
    (c) = h & BN_MASK2;       \
    (r) = l & BN_MASK2;       \
  }
#endif /* !BN_LLONG */

#if defined(BN_LLONG) || defined(BN_UMULT_HIGH)

BN_ULONG bn_mul_add_words(BN_ULONG *rp, const BN_ULONG *ap, int num,
                          BN_ULONG w) {
  BN_ULONG c1 = 0;

  assert(num >= 0);
  if (num <= 0) {
    return c1;
  }

  while (num & ~3) {
    mul_add(rp[0], ap[0], w, c1);
    mul_add(rp[1], ap[1], w, c1);
    mul_add(rp[2], ap[2], w, c1);
    mul_add(rp[3], ap[3], w, c1);
    ap += 4;
    rp += 4;
    num -= 4;
  }

  while (num) {
    mul_add(rp[0], ap[0], w, c1);
    ap++;
    rp++;
    num--;
  }

  return c1;
}

BN_ULONG bn_mul_words(BN_ULONG *rp, const BN_ULONG *ap, int num, BN_ULONG w) {
  BN_ULONG c1 = 0;

  assert(num >= 0);
  if (num <= 0) {
    return c1;
  }

  while (num & ~3) {
    mul(rp[0], ap[0], w, c1);
    mul(rp[1], ap[1], w, c1);
    mul(rp[2], ap[2], w, c1);
    mul(rp[3], ap[3], w, c1);
    ap += 4;
    rp += 4;
    num -= 4;
  }
  while (num) {
    mul(rp[0], ap[0], w, c1);
    ap++;
    rp++;
    num--;
  }
  return c1;
}

void bn_sqr_words(BN_ULONG *r, const BN_ULONG *a, int n) {
  assert(n >= 0);
  if (n <= 0) {
    return;
  }

  while (n & ~3) {
    sqr(r[0], r[1], a[0]);
    sqr(r[2], r[3], a[1]);
    sqr(r[4], r[5], a[2]);
    sqr(r[6], r[7], a[3]);
    a += 4;
    r += 8;
    n -= 4;
  }
  while (n) {
    sqr(r[0], r[1], a[0]);
    a++;
    r += 2;
    n--;
  }
}

#else /* !(defined(BN_LLONG) || defined(BN_UMULT_HIGH)) */

BN_ULONG bn_mul_add_words(BN_ULONG *rp, const BN_ULONG *ap, int num,
                          BN_ULONG w) {
  BN_ULONG c = 0;
  BN_ULONG bl, bh;

  assert(num >= 0);
  if (num <= 0) {
    return (BN_ULONG)0;
  }

  bl = LBITS(w);
  bh = HBITS(w);

  while (num & ~3) {
    mul_add(rp[0], ap[0], bl, bh, c);
    mul_add(rp[1], ap[1], bl, bh, c);
    mul_add(rp[2], ap[2], bl, bh, c);
    mul_add(rp[3], ap[3], bl, bh, c);
    ap += 4;
    rp += 4;
    num -= 4;
  }
  while (num) {
    mul_add(rp[0], ap[0], bl, bh, c);
    ap++;
    rp++;
    num--;
  }
  return c;
}

BN_ULONG bn_mul_words(BN_ULONG *rp, const BN_ULONG *ap, int num, BN_ULONG w) {
  BN_ULONG carry = 0;
  BN_ULONG bl, bh;

  assert(num >= 0);
  if (num <= 0) {
    return (BN_ULONG)0;
  }

  bl = LBITS(w);
  bh = HBITS(w);

  while (num & ~3) {
    mul(rp[0], ap[0], bl, bh, carry);
    mul(rp[1], ap[1], bl, bh, carry);
    mul(rp[2], ap[2], bl, bh, carry);
    mul(rp[3], ap[3], bl, bh, carry);
    ap += 4;
    rp += 4;
    num -= 4;
  }
  while (num) {
    mul(rp[0], ap[0], bl, bh, carry);
    ap++;
    rp++;
    num--;
  }
  return carry;
}

void bn_sqr_words(BN_ULONG *r, const BN_ULONG *a, int n) {
  assert(n >= 0);
  if (n <= 0) {
    return;
  }

  while (n & ~3) {
    sqr64(r[0], r[1], a[0]);
    sqr64(r[2], r[3], a[1]);
    sqr64(r[4], r[5], a[2]);
    sqr64(r[6], r[7], a[3]);
    a += 4;
    r += 8;
    n -= 4;
  }
  while (n) {
    sqr64(r[0], r[1], a[0]);
    a++;
    r += 2;
    n--;
  }
}

#endif /* !(defined(BN_LLONG) || defined(BN_UMULT_HIGH)) */

#if defined(BN_LLONG)

BN_ULONG bn_div_words(BN_ULONG h, BN_ULONG l, BN_ULONG d) {
  return (BN_ULONG)(((((BN_ULLONG)h) << BN_BITS2) | l) / (BN_ULLONG)d);
}

#else

/* Divide h,l by d and return the result. */
BN_ULONG bn_div_words(BN_ULONG h, BN_ULONG l, BN_ULONG d) {
  BN_ULONG dh, dl, q, ret = 0, th, tl, t;
  int i, count = 2;

  if (d == 0) {
    return BN_MASK2;
  }

  i = BN_num_bits_word(d);
  assert((i == BN_BITS2) || (h <= (BN_ULONG)1 << i));

  i = BN_BITS2 - i;
  if (h >= d) {
    h -= d;
  }

  if (i) {
    d <<= i;
    h = (h << i) | (l >> (BN_BITS2 - i));
    l <<= i;
  }
  dh = (d & BN_MASK2h) >> BN_BITS4;
  dl = (d & BN_MASK2l);
  for (;;) {
    if ((h >> BN_BITS4) == dh) {
      q = BN_MASK2l;
    } else {
      q = h / dh;
    }

    th = q * dh;
    tl = dl * q;
    for (;;) {
      t = h - th;
      if ((t & BN_MASK2h) ||
          ((tl) <= ((t << BN_BITS4) | ((l & BN_MASK2h) >> BN_BITS4)))) {
        break;
      }
      q--;
      th -= dh;
      tl -= dl;
    }
    t = (tl >> BN_BITS4);
    tl = (tl << BN_BITS4) & BN_MASK2h;
    th += t;

    if (l < tl) {
      th++;
    }
    l -= tl;
    if (h < th) {
      h += d;
      q--;
    }
    h -= th;

    if (--count == 0) {
      break;
    }

    ret = q << BN_BITS4;
    h = ((h << BN_BITS4) | (l >> BN_BITS4)) & BN_MASK2;
    l = (l & BN_MASK2l) << BN_BITS4;
  }

  ret |= q;
  return ret;
}

#endif /* !defined(BN_LLONG) */

#ifdef BN_LLONG
BN_ULONG bn_add_words(BN_ULONG *r, const BN_ULONG *a, const BN_ULONG *b,
                      int n) {
  BN_ULLONG ll = 0;

  assert(n >= 0);
  if (n <= 0) {
    return (BN_ULONG)0;
  }

  while (n & ~3) {
    ll += (BN_ULLONG)a[0] + b[0];
    r[0] = (BN_ULONG)ll & BN_MASK2;
    ll >>= BN_BITS2;
    ll += (BN_ULLONG)a[1] + b[1];
    r[1] = (BN_ULONG)ll & BN_MASK2;
    ll >>= BN_BITS2;
    ll += (BN_ULLONG)a[2] + b[2];
    r[2] = (BN_ULONG)ll & BN_MASK2;
    ll >>= BN_BITS2;
    ll += (BN_ULLONG)a[3] + b[3];
    r[3] = (BN_ULONG)ll & BN_MASK2;
    ll >>= BN_BITS2;
    a += 4;
    b += 4;
    r += 4;
    n -= 4;
  }
  while (n) {
    ll += (BN_ULLONG)a[0] + b[0];
    r[0] = (BN_ULONG)ll & BN_MASK2;
    ll >>= BN_BITS2;
    a++;
    b++;
    r++;
    n--;
  }
  return (BN_ULONG)ll;
}

#else /* !BN_LLONG */

BN_ULONG bn_add_words(BN_ULONG *r, const BN_ULONG *a, const BN_ULONG *b,
                      int n) {
  BN_ULONG c, l, t;

  assert(n >= 0);
  if (n <= 0) {
    return (BN_ULONG)0;
  }

  c = 0;
  while (n & ~3) {
    t = a[0];
    t = (t + c) & BN_MASK2;
    c = (t < c);
    l = (t + b[0]) & BN_MASK2;
    c += (l < t);
    r[0] = l;
    t = a[1];
    t = (t + c) & BN_MASK2;
    c = (t < c);
    l = (t + b[1]) & BN_MASK2;
    c += (l < t);
    r[1] = l;
    t = a[2];
    t = (t + c) & BN_MASK2;
    c = (t < c);
    l = (t + b[2]) & BN_MASK2;
    c += (l < t);
    r[2] = l;
    t = a[3];
    t = (t + c) & BN_MASK2;
    c = (t < c);
    l = (t + b[3]) & BN_MASK2;
    c += (l < t);
    r[3] = l;
    a += 4;
    b += 4;
    r += 4;
    n -= 4;
  }
  while (n) {
    t = a[0];
    t = (t + c) & BN_MASK2;
    c = (t < c);
    l = (t + b[0]) & BN_MASK2;
    c += (l < t);
    r[0] = l;
    a++;
    b++;
    r++;
    n--;
  }
  return (BN_ULONG)c;
}

#endif /* !BN_LLONG */

BN_ULONG bn_sub_words(BN_ULONG *r, const BN_ULONG *a, const BN_ULONG *b,
                      int n) {
  BN_ULONG t1, t2;
  int c = 0;

  assert(n >= 0);
  if (n <= 0) {
    return (BN_ULONG)0;
  }

  while (n & ~3) {
    t1 = a[0];
    t2 = b[0];
    r[0] = (t1 - t2 - c) & BN_MASK2;
    if (t1 != t2)
      c = (t1 < t2);
    t1 = a[1];
    t2 = b[1];
    r[1] = (t1 - t2 - c) & BN_MASK2;
    if (t1 != t2)
      c = (t1 < t2);
    t1 = a[2];
    t2 = b[2];
    r[2] = (t1 - t2 - c) & BN_MASK2;
    if (t1 != t2)
      c = (t1 < t2);
    t1 = a[3];
    t2 = b[3];
    r[3] = (t1 - t2 - c) & BN_MASK2;
    if (t1 != t2)
      c = (t1 < t2);
    a += 4;
    b += 4;
    r += 4;
    n -= 4;
  }
  while (n) {
    t1 = a[0];
    t2 = b[0];
    r[0] = (t1 - t2 - c) & BN_MASK2;
    if (t1 != t2)
      c = (t1 < t2);
    a++;
    b++;
    r++;
    n--;
  }
  return c;
}

/* mul_add_c(a,b,c0,c1,c2)  -- c+=a*b for three word number c=(c2,c1,c0) */
/* mul_add_c2(a,b,c0,c1,c2) -- c+=2*a*b for three word number c=(c2,c1,c0) */
/* sqr_add_c(a,i,c0,c1,c2)  -- c+=a[i]^2 for three word number c=(c2,c1,c0) */
/* sqr_add_c2(a,i,c0,c1,c2) -- c+=2*a[i]*a[j] for three word number c=(c2,c1,c0) */

#ifdef BN_LLONG

/* Keep in mind that additions to multiplication result can not overflow,
 * because its high half cannot be all-ones. */
#define mul_add_c(a, b, c0, c1, c2)     \
  do {                                  \
    BN_ULONG hi;                        \
    BN_ULLONG t = (BN_ULLONG)(a) * (b); \
    t += c0; /* no carry */             \
    c0 = (BN_ULONG)Lw(t);               \
    hi = (BN_ULONG)Hw(t);               \
    c1 = (c1 + hi) & BN_MASK2;          \
    if (c1 < hi)                        \
      c2++;                             \
  } while (0)

#define mul_add_c2(a, b, c0, c1, c2)      \
  do {                                    \
    BN_ULONG hi;                          \
    BN_ULLONG t = (BN_ULLONG)(a) * (b);   \
    BN_ULLONG tt = t + c0; /* no carry */ \
    c0 = (BN_ULONG)Lw(tt);                \
    hi = (BN_ULONG)Hw(tt);                \
    c1 = (c1 + hi) & BN_MASK2;            \
    if (c1 < hi)                          \
      c2++;                               \
    t += c0; /* no carry */               \
    c0 = (BN_ULONG)Lw(t);                 \
    hi = (BN_ULONG)Hw(t);                 \
    c1 = (c1 + hi) & BN_MASK2;            \
    if (c1 < hi)                          \
      c2++;                               \
  } while (0)

#define sqr_add_c(a, i, c0, c1, c2)       \
  do {                                    \
    BN_ULONG hi;                          \
    BN_ULLONG t = (BN_ULLONG)a[i] * a[i]; \
    t += c0; /* no carry */               \
    c0 = (BN_ULONG)Lw(t);                 \
    hi = (BN_ULONG)Hw(t);                 \
    c1 = (c1 + hi) & BN_MASK2;            \
    if (c1 < hi)                          \
      c2++;                               \
  } while (0)

#define sqr_add_c2(a, i, j, c0, c1, c2) mul_add_c2((a)[i], (a)[j], c0, c1, c2)

#elif defined(BN_UMULT_LOHI)

/* Keep in mind that additions to hi can not overflow, because the high word of
 * a multiplication result cannot be all-ones. */
#define mul_add_c(a, b, c0, c1, c2) \
  do {                              \
    BN_ULONG ta = (a), tb = (b);    \
    BN_ULONG lo, hi;                \
    BN_UMULT_LOHI(lo, hi, ta, tb);  \
    c0 += lo;                       \
    hi += (c0 < lo) ? 1 : 0;        \
    c1 += hi;                       \
    c2 += (c1 < hi) ? 1 : 0;        \
  } while (0)

#define mul_add_c2(a, b, c0, c1, c2) \
  do {                               \
    BN_ULONG ta = (a), tb = (b);     \
    BN_ULONG lo, hi, tt;             \
    BN_UMULT_LOHI(lo, hi, ta, tb);   \
    c0 += lo;                        \
    tt = hi + ((c0 < lo) ? 1 : 0);   \
    c1 += tt;                        \
    c2 += (c1 < tt) ? 1 : 0;         \
    c0 += lo;                        \
    hi += (c0 < lo) ? 1 : 0;         \
    c1 += hi;                        \
    c2 += (c1 < hi) ? 1 : 0;         \
  } while (0)

#define sqr_add_c(a, i, c0, c1, c2) \
  do {                              \
    BN_ULONG ta = (a)[i];           \
    BN_ULONG lo, hi;                \
    BN_UMULT_LOHI(lo, hi, ta, ta);  \
    c0 += lo;                       \
    hi += (c0 < lo) ? 1 : 0;        \
    c1 += hi;                       \
    c2 += (c1 < hi) ? 1 : 0;        \
  } while (0)

#define sqr_add_c2(a, i, j, c0, c1, c2) mul_add_c2((a)[i], (a)[j], c0, c1, c2)

#else /* !BN_LLONG */

/* Keep in mind that additions to hi can not overflow, because
 * the high word of a multiplication result cannot be all-ones. */

#define mul_add_c(a, b, c0, c1, c2)        \
  do {                                     \
    BN_ULONG lo = LBITS(a), hi = HBITS(a); \
    BN_ULONG bl = LBITS(b), bh = HBITS(b); \
    mul64(lo, hi, bl, bh);                 \
    c0 = (c0 + lo) & BN_MASK2;             \
    if (c0 < lo)                           \
      hi++;                                \
    c1 = (c1 + hi) & BN_MASK2;             \
    if (c1 < hi)                           \
      c2++;                                \
  } while (0)

#define mul_add_c2(a, b, c0, c1, c2)       \
  do {                                     \
    BN_ULONG tt;                           \
    BN_ULONG lo = LBITS(a), hi = HBITS(a); \
    BN_ULONG bl = LBITS(b), bh = HBITS(b); \
    mul64(lo, hi, bl, bh);                 \
    tt = hi;                               \
    c0 = (c0 + lo) & BN_MASK2;             \
    if (c0 < lo)                           \
      tt++;                                \
    c1 = (c1 + tt) & BN_MASK2;             \
    if (c1 < tt)                           \
      c2++;                                \
    c0 = (c0 + lo) & BN_MASK2;             \
    if (c0 < lo)                           \
      hi++;                                \
    c1 = (c1 + hi) & BN_MASK2;             \
    if (c1 < hi)                           \
      c2++;                                \
  } while (0)

#define sqr_add_c(a, i, c0, c1, c2) \
  do {                              \
    BN_ULONG lo, hi;                \
    sqr64(lo, hi, (a)[i]);          \
    c0 = (c0 + lo) & BN_MASK2;      \
    if (c0 < lo)                    \
      hi++;                         \
    c1 = (c1 + hi) & BN_MASK2;      \
    if (c1 < hi)                    \
      c2++;                         \
  } while (0)

#define sqr_add_c2(a, i, j, c0, c1, c2) mul_add_c2((a)[i], (a)[j], c0, c1, c2)
#endif /* !BN_LLONG */

void bn_mul_comba8(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b) {
  BN_ULONG c1, c2, c3;

  c1 = 0;
  c2 = 0;
  c3 = 0;
  mul_add_c(a[0], b[0], c1, c2, c3);
  r[0] = c1;
  c1 = 0;
  mul_add_c(a[0], b[1], c2, c3, c1);
  mul_add_c(a[1], b[0], c2, c3, c1);
  r[1] = c2;
  c2 = 0;
  mul_add_c(a[2], b[0], c3, c1, c2);
  mul_add_c(a[1], b[1], c3, c1, c2);
  mul_add_c(a[0], b[2], c3, c1, c2);
  r[2] = c3;
  c3 = 0;
  mul_add_c(a[0], b[3], c1, c2, c3);
  mul_add_c(a[1], b[2], c1, c2, c3);
  mul_add_c(a[2], b[1], c1, c2, c3);
  mul_add_c(a[3], b[0], c1, c2, c3);
  r[3] = c1;
  c1 = 0;
  mul_add_c(a[4], b[0], c2, c3, c1);
  mul_add_c(a[3], b[1], c2, c3, c1);
  mul_add_c(a[2], b[2], c2, c3, c1);
  mul_add_c(a[1], b[3], c2, c3, c1);
  mul_add_c(a[0], b[4], c2, c3, c1);
  r[4] = c2;
  c2 = 0;
  mul_add_c(a[0], b[5], c3, c1, c2);
  mul_add_c(a[1], b[4], c3, c1, c2);
  mul_add_c(a[2], b[3], c3, c1, c2);
  mul_add_c(a[3], b[2], c3, c1, c2);
  mul_add_c(a[4], b[1], c3, c1, c2);
  mul_add_c(a[5], b[0], c3, c1, c2);
  r[5] = c3;
  c3 = 0;
  mul_add_c(a[6], b[0], c1, c2, c3);
  mul_add_c(a[5], b[1], c1, c2, c3);
  mul_add_c(a[4], b[2], c1, c2, c3);
  mul_add_c(a[3], b[3], c1, c2, c3);
  mul_add_c(a[2], b[4], c1, c2, c3);
  mul_add_c(a[1], b[5], c1, c2, c3);
  mul_add_c(a[0], b[6], c1, c2, c3);
  r[6] = c1;
  c1 = 0;
  mul_add_c(a[0], b[7], c2, c3, c1);
  mul_add_c(a[1], b[6], c2, c3, c1);
  mul_add_c(a[2], b[5], c2, c3, c1);
  mul_add_c(a[3], b[4], c2, c3, c1);
  mul_add_c(a[4], b[3], c2, c3, c1);
  mul_add_c(a[5], b[2], c2, c3, c1);
  mul_add_c(a[6], b[1], c2, c3, c1);
  mul_add_c(a[7], b[0], c2, c3, c1);
  r[7] = c2;
  c2 = 0;
  mul_add_c(a[7], b[1], c3, c1, c2);
  mul_add_c(a[6], b[2], c3, c1, c2);
  mul_add_c(a[5], b[3], c3, c1, c2);
  mul_add_c(a[4], b[4], c3, c1, c2);
  mul_add_c(a[3], b[5], c3, c1, c2);
  mul_add_c(a[2], b[6], c3, c1, c2);
  mul_add_c(a[1], b[7], c3, c1, c2);
  r[8] = c3;
  c3 = 0;
  mul_add_c(a[2], b[7], c1, c2, c3);
  mul_add_c(a[3], b[6], c1, c2, c3);
  mul_add_c(a[4], b[5], c1, c2, c3);
  mul_add_c(a[5], b[4], c1, c2, c3);
  mul_add_c(a[6], b[3], c1, c2, c3);
  mul_add_c(a[7], b[2], c1, c2, c3);
  r[9] = c1;
  c1 = 0;
  mul_add_c(a[7], b[3], c2, c3, c1);
  mul_add_c(a[6], b[4], c2, c3, c1);
  mul_add_c(a[5], b[5], c2, c3, c1);
  mul_add_c(a[4], b[6], c2, c3, c1);
  mul_add_c(a[3], b[7], c2, c3, c1);
  r[10] = c2;
  c2 = 0;
  mul_add_c(a[4], b[7], c3, c1, c2);
  mul_add_c(a[5], b[6], c3, c1, c2);
  mul_add_c(a[6], b[5], c3, c1, c2);
  mul_add_c(a[7], b[4], c3, c1, c2);
  r[11] = c3;
  c3 = 0;
  mul_add_c(a[7], b[5], c1, c2, c3);
  mul_add_c(a[6], b[6], c1, c2, c3);
  mul_add_c(a[5], b[7], c1, c2, c3);
  r[12] = c1;
  c1 = 0;
  mul_add_c(a[6], b[7], c2, c3, c1);
  mul_add_c(a[7], b[6], c2, c3, c1);
  r[13] = c2;
  c2 = 0;
  mul_add_c(a[7], b[7], c3, c1, c2);
  r[14] = c3;
  r[15] = c1;
}

void bn_mul_comba4(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b) {
  BN_ULONG c1, c2, c3;

  c1 = 0;
  c2 = 0;
  c3 = 0;
  mul_add_c(a[0], b[0], c1, c2, c3);
  r[0] = c1;
  c1 = 0;
  mul_add_c(a[0], b[1], c2, c3, c1);
  mul_add_c(a[1], b[0], c2, c3, c1);
  r[1] = c2;
  c2 = 0;
  mul_add_c(a[2], b[0], c3, c1, c2);
  mul_add_c(a[1], b[1], c3, c1, c2);
  mul_add_c(a[0], b[2], c3, c1, c2);
  r[2] = c3;
  c3 = 0;
  mul_add_c(a[0], b[3], c1, c2, c3);
  mul_add_c(a[1], b[2], c1, c2, c3);
  mul_add_c(a[2], b[1], c1, c2, c3);
  mul_add_c(a[3], b[0], c1, c2, c3);
  r[3] = c1;
  c1 = 0;
  mul_add_c(a[3], b[1], c2, c3, c1);
  mul_add_c(a[2], b[2], c2, c3, c1);
  mul_add_c(a[1], b[3], c2, c3, c1);
  r[4] = c2;
  c2 = 0;
  mul_add_c(a[2], b[3], c3, c1, c2);
  mul_add_c(a[3], b[2], c3, c1, c2);
  r[5] = c3;
  c3 = 0;
  mul_add_c(a[3], b[3], c1, c2, c3);
  r[6] = c1;
  r[7] = c2;
}

void bn_sqr_comba8(BN_ULONG *r, const BN_ULONG *a) {
  BN_ULONG c1, c2, c3;

  c1 = 0;
  c2 = 0;
  c3 = 0;
  sqr_add_c(a, 0, c1, c2, c3);
  r[0] = c1;
  c1 = 0;
  sqr_add_c2(a, 1, 0, c2, c3, c1);
  r[1] = c2;
  c2 = 0;
  sqr_add_c(a, 1, c3, c1, c2);
  sqr_add_c2(a, 2, 0, c3, c1, c2);
  r[2] = c3;
  c3 = 0;
  sqr_add_c2(a, 3, 0, c1, c2, c3);
  sqr_add_c2(a, 2, 1, c1, c2, c3);
  r[3] = c1;
  c1 = 0;
  sqr_add_c(a, 2, c2, c3, c1);
  sqr_add_c2(a, 3, 1, c2, c3, c1);
  sqr_add_c2(a, 4, 0, c2, c3, c1);
  r[4] = c2;
  c2 = 0;
  sqr_add_c2(a, 5, 0, c3, c1, c2);
  sqr_add_c2(a, 4, 1, c3, c1, c2);
  sqr_add_c2(a, 3, 2, c3, c1, c2);
  r[5] = c3;
  c3 = 0;
  sqr_add_c(a, 3, c1, c2, c3);
  sqr_add_c2(a, 4, 2, c1, c2, c3);
  sqr_add_c2(a, 5, 1, c1, c2, c3);
  sqr_add_c2(a, 6, 0, c1, c2, c3);
  r[6] = c1;
  c1 = 0;
  sqr_add_c2(a, 7, 0, c2, c3, c1);
  sqr_add_c2(a, 6, 1, c2, c3, c1);
  sqr_add_c2(a, 5, 2, c2, c3, c1);
  sqr_add_c2(a, 4, 3, c2, c3, c1);
  r[7] = c2;
  c2 = 0;
  sqr_add_c(a, 4, c3, c1, c2);
  sqr_add_c2(a, 5, 3, c3, c1, c2);
  sqr_add_c2(a, 6, 2, c3, c1, c2);
  sqr_add_c2(a, 7, 1, c3, c1, c2);
  r[8] = c3;
  c3 = 0;
  sqr_add_c2(a, 7, 2, c1, c2, c3);
  sqr_add_c2(a, 6, 3, c1, c2, c3);
  sqr_add_c2(a, 5, 4, c1, c2, c3);
  r[9] = c1;
  c1 = 0;
  sqr_add_c(a, 5, c2, c3, c1);
  sqr_add_c2(a, 6, 4, c2, c3, c1);
  sqr_add_c2(a, 7, 3, c2, c3, c1);
  r[10] = c2;
  c2 = 0;
  sqr_add_c2(a, 7, 4, c3, c1, c2);
  sqr_add_c2(a, 6, 5, c3, c1, c2);
  r[11] = c3;
  c3 = 0;
  sqr_add_c(a, 6, c1, c2, c3);
  sqr_add_c2(a, 7, 5, c1, c2, c3);
  r[12] = c1;
  c1 = 0;
  sqr_add_c2(a, 7, 6, c2, c3, c1);
  r[13] = c2;
  c2 = 0;
  sqr_add_c(a, 7, c3, c1, c2);
  r[14] = c3;
  r[15] = c1;
}

void bn_sqr_comba4(BN_ULONG *r, const BN_ULONG *a) {
  BN_ULONG c1, c2, c3;

  c1 = 0;
  c2 = 0;
  c3 = 0;
  sqr_add_c(a, 0, c1, c2, c3);
  r[0] = c1;
  c1 = 0;
  sqr_add_c2(a, 1, 0, c2, c3, c1);
  r[1] = c2;
  c2 = 0;
  sqr_add_c(a, 1, c3, c1, c2);
  sqr_add_c2(a, 2, 0, c3, c1, c2);
  r[2] = c3;
  c3 = 0;
  sqr_add_c2(a, 3, 0, c1, c2, c3);
  sqr_add_c2(a, 2, 1, c1, c2, c3);
  r[3] = c1;
  c1 = 0;
  sqr_add_c(a, 2, c2, c3, c1);
  sqr_add_c2(a, 3, 1, c2, c3, c1);
  r[4] = c2;
  c2 = 0;
  sqr_add_c2(a, 3, 2, c3, c1, c2);
  r[5] = c3;
  c3 = 0;
  sqr_add_c(a, 3, c1, c2, c3);
  r[6] = c1;
  r[7] = c2;
}

#if defined(OPENSSL_NO_ASM) || (!defined(OPENSSL_ARM) && !defined(OPENSSL_X86_64))
/* This is essentially reference implementation, which may or may not
 * result in performance improvement. E.g. on IA-32 this routine was
 * observed to give 40% faster rsa1024 private key operations and 10%
 * faster rsa4096 ones, while on AMD64 it improves rsa1024 sign only
 * by 10% and *worsens* rsa4096 sign by 15%. Once again, it's a
 * reference implementation, one to be used as starting point for
 * platform-specific assembler. Mentioned numbers apply to compiler
 * generated code compiled with and without -DOPENSSL_BN_ASM_MONT and
 * can vary not only from platform to platform, but even for compiler
 * versions. Assembler vs. assembler improvement coefficients can
 * [and are known to] differ and are to be documented elsewhere. */
int bn_mul_mont(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
                const BN_ULONG *np, const BN_ULONG *n0p, int num) {
  BN_ULONG c0, c1, ml, *tp, n0;
#ifdef mul64
  BN_ULONG mh;
#endif
  volatile BN_ULONG *vp;
  int i = 0, j;

#if 0 /* template for platform-specific implementation */
	if (ap==bp)	return bn_sqr_mont(rp,ap,np,n0p,num);
#endif
  vp = tp = alloca((num + 2) * sizeof(BN_ULONG));

  n0 = *n0p;

  c0 = 0;
  ml = bp[0];
#ifdef mul64
  mh = HBITS(ml);
  ml = LBITS(ml);
  for (j = 0; j < num; ++j)
    mul(tp[j], ap[j], ml, mh, c0);
#else
  for (j = 0; j < num; ++j)
    mul(tp[j], ap[j], ml, c0);
#endif

  tp[num] = c0;
  tp[num + 1] = 0;
  goto enter;

  for (i = 0; i < num; i++) {
    c0 = 0;
    ml = bp[i];
#ifdef mul64
    mh = HBITS(ml);
    ml = LBITS(ml);
    for (j = 0; j < num; ++j)
      mul_add(tp[j], ap[j], ml, mh, c0);
#else
    for (j = 0; j < num; ++j)
      mul_add(tp[j], ap[j], ml, c0);
#endif
    c1 = (tp[num] + c0) & BN_MASK2;
    tp[num] = c1;
    tp[num + 1] = (c1 < c0 ? 1 : 0);
  enter:
    c1 = tp[0];
    ml = (c1 * n0) & BN_MASK2;
    c0 = 0;
#ifdef mul64
    mh = HBITS(ml);
    ml = LBITS(ml);
    mul_add(c1, np[0], ml, mh, c0);
#else
    mul_add(c1, ml, np[0], c0);
#endif
    for (j = 1; j < num; j++) {
      c1 = tp[j];
#ifdef mul64
      mul_add(c1, np[j], ml, mh, c0);
#else
      mul_add(c1, ml, np[j], c0);
#endif
      tp[j - 1] = c1 & BN_MASK2;
    }
    c1 = (tp[num] + c0) & BN_MASK2;
    tp[num - 1] = c1;
    tp[num] = tp[num + 1] + (c1 < c0 ? 1 : 0);
  }

  if (tp[num] != 0 || tp[num - 1] >= np[num - 1]) {
    c0 = bn_sub_words(rp, tp, np, num);
    if (tp[num] != 0 || c0 == 0) {
      for (i = 0; i < num + 2; i++)
        vp[i] = 0;
      return 1;
    }
  }
  for (i = 0; i < num; i++)
    rp[i] = tp[i], vp[i] = 0;
  vp[num] = 0;
  vp[num + 1] = 0;
  return 1;
}
#endif

#endif