Fast estimation of number representation lengths.

[u/mdw/catacomb] / mpx.h

/* -*-c-*-
 *
 * $Id: mpx.h,v 1.14 2002/10/09 00:36:03 mdw Exp $
 *
 * Low level multiprecision arithmetic
 *
 * (c) 1999 Straylight/Edgeware
 */

/*----- Licensing notice --------------------------------------------------* 
 *
 * This file is part of Catacomb.
 *
 * Catacomb is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Library General Public License as
 * published by the Free Software Foundation; either version 2 of the
 * License, or (at your option) any later version.
 * 
 * Catacomb 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 Library General Public License for more details.
 * 
 * You should have received a copy of the GNU Library General Public
 * License along with Catacomb; if not, write to the Free
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA 02111-1307, USA.
 */

/*----- Revision history --------------------------------------------------* 
 *
 * $Log: mpx.h,v $
 * Revision 1.14  2002/10/09 00:36:03  mdw
 * Fix bounds on workspace for Karatsuba operations.
 *
 * Revision 1.13  2002/10/06 22:52:50  mdw
 * Pile of changes for supporting two's complement properly.
 *
 * Revision 1.12  2001/04/03 19:36:05  mdw
 * Add some simple bitwise operations so that Perl can use them.
 *
 * Revision 1.11  2000/10/08 15:48:35  mdw
 * Rename Karatsuba constants now that we have @gfx_kmul@ too.
 *
 * Revision 1.10  2000/10/08 12:06:12  mdw
 * Provide @mpx_ueq@ for rapidly testing equality of two integers.
 *
 * Revision 1.9  1999/12/22 15:49:07  mdw
 * New function for division by a small integer.
 *
 * Revision 1.8  1999/12/11 10:57:43  mdw
 * Karatsuba squaring algorithm.
 *
 * Revision 1.7  1999/12/11 01:51:28  mdw
 * Change Karatsuba parameters slightly.
 *
 * Revision 1.6  1999/12/10 23:23:51  mdw
 * Karatsuba-Ofman multiplication algorithm.
 *
 * Revision 1.5  1999/11/20 22:23:27  mdw
 * Add function versions of some low-level macros with wider use.
 *
 * Revision 1.4  1999/11/17 18:04:43  mdw
 * Add two's complement support.  Fix a bug in MPX_UMLAN.
 *
 * Revision 1.3  1999/11/13 01:51:29  mdw
 * Minor interface changes.  Should be stable now.
 *
 * Revision 1.2  1999/11/11 17:47:55  mdw
 * Minor changes for different `mptypes.h' format.
 *
 * Revision 1.1  1999/09/03 08:41:12  mdw
 * Initial import.
 *
 */

#ifndef CATACOMB_MPX_H
#define CATACOMB_MPX_H

#ifdef __cplusplus
  extern "C" {
#endif

/*----- The idea ----------------------------------------------------------*
 *
 * This file provides functions and macros which work on vectors of words as
 * unsigned multiprecision integers.  The interface works in terms of base
 * and limit pointers (i.e., a pointer to the start of a vector, and a
 * pointer just past its end) rather than base pointer and length, because
 * that requires more arithmetic and state to work on.
 *
 * The interfaces are slightly bizarre in other ways.  Try to use the
 * higher-level functions where you can: they're rather better designed to
 * actually be friendly and useful.
 */

/*----- Header files ------------------------------------------------------*/

#include <string.h>

#ifndef CATACOMB_MPW_H
#  include "mpw.h"
#endif

/*----- General manipulation ----------------------------------------------*/

/* --- @MPX_SHRINK@ --- *
 *
 * Arguments:	@const mpw *v@ = pointer to vector of words
 *		@const mpw *vl@ = (updated) current limit of vector
 *
 * Use:		Shrinks down the limit of a multiprecision integer vector.
 */

#define MPX_SHRINK(v, vl) do {						\
  const mpw *_vv = (v), *_vvl = (vl);					\
  while (_vvl > _vv && !_vvl[-1])					\
    _vvl--;								\
  (vl) = (mpw *)_vvl;							\
} while (0)

/* --- @MPX_BITS@ --- *
 *
 * Arguments:	@unsigned long b@ = result variable
 *		@const mpw *v@ = pointer to array of words
 *		@const mpw *vl@ = limit of vector (from @MPX_SHRINK@)
 *
 * Use:		Calculates the number of bits in a multiprecision value.
 */

#define MPX_BITS(b, v, vl) do {						\
  const mpw *_v = (v), *_vl = (vl);					\
  MPX_SHRINK(_v, _vl);							\
  if (_v == _vl)							\
    (b) = 0;								\
  else {								\
    unsigned long _b = MPW_BITS * (_vl - _v - 1) + 1;	       		\
    mpw _w = _vl[-1];							\
    unsigned _k = MPW_BITS / 2;						\
    while (_k) {							\
      if (_w >> _k) {							\
	_w >>= _k;							\
	_b += _k;							\
      }									\
      _k >>= 1;								\
    }									\
    (b) = _b;								\
  }									\
} while (0)

/* --- @MPX_OCTETS@ --- *
 *
 * Arguments:	@size_t o@ = result variable
 *		@const mpw *v, *vl@ = pointer to array of words
 *
 * Use:		Calculates the number of octets in a multiprecision value.
 */

#define MPX_OCTETS(o, v, vl) do {					\
  unsigned long _bb;							\
  MPX_BITS(_bb, (v), (vl));						\
  (o) = (_bb + 7) >> 3;							\
} while (0)

/* --- @MPX_OCTETS2C@ --- *
 *
 * Arguments:	@size_t o@ = result variable
 *		@const mpw *v, *vl@ = pointer to array of words
 *
 * Use:		Calculates the number of octets in a multiprecision value, if
 *		you represent it as two's complement.
 */

#define MPX_OCTETS2C(o, v, vl) do {					\
  unsigned long _bb;							\
  MPX_BITS(_bb, (v), (vl));						\
  (o) = (_bb >> 3) + 1;							\
} while (0)

/* --- @MPX_COPY@ --- *
 *
 * Arguments:	@dv, dvl@ = destination vector base and limit
 *		@av, avl@ = source vector base and limit
 *
 * Use:		Copies a multiprecision integer.
 */

#define MPX_COPY(dv, dvl, av, avl) do {					\
  mpw *_dv = (dv), *_dvl = (dvl);					\
  size_t _dn = _dvl - _dv;						\
  const mpw *_av = (av), *_avl = (avl);					\
  size_t _an = _avl - _av;						\
  if (_av == _dv) {							\
    if (_dvl > _avl)							\
      memset(_dv, 0, MPWS(_dn - _an));					\
  } else if (_an >= _dn)						\
    memmove(_dv, _av, MPWS(_dn));					\
  else {								\
    memmove(_dv, _av, MPWS(_an));					\
    memset(_dv + _an, 0, MPWS(_dn - _an));				\
  }									\
} while (0)

/* --- @MPX_ZERO@ --- *
 *
 * Arguments:	@v, vl@ = base and limit of vector to clear
 *
 * Use:		Zeroes the area between the two vector pointers.
 */

#define MPX_ZERO(v, vl) do {						\
  mpw *_v = (v), *_vl = (vl);						\
  if (_v < _vl)								\
    memset(_v, 0, MPWS(_vl - _v));					\
} while (0)

/*----- Loading and storing -----------------------------------------------*/

/* --- @mpx_storel@ --- *
 *
 * Arguments:	@const mpw *v, *vl@ = base and limit of source vector
 *		@void *p@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Stores an MP in an octet array, least significant octet
 *		first.  High-end octets are silently discarded if there
 *		isn't enough space for them.
 */

extern void mpx_storel(const mpw */*v*/, const mpw */*vl*/,
		       void */*p*/, size_t /*sz*/);

/* --- @mpx_loadl@ --- *
 *
 * Arguments:	@mpw *v, *vl@ = base and limit of destination vector
 *		@const void *p@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Loads an MP in an octet array, least significant octet
 *		first.  High-end octets are ignored if there isn't enough
 *		space for them.
 */

extern void mpx_loadl(mpw */*v*/, mpw */*vl*/,
		      const void */*p*/, size_t /*sz*/);

/* --- @mpx_storeb@ --- *
 *
 * Arguments:	@const mpw *v, *vl@ = base and limit of source vector
 *		@void *p@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Stores an MP in an octet array, most significant octet
 *		first.  High-end octets are silently discarded if there
 *		isn't enough space for them.
 */

extern void mpx_storeb(const mpw */*v*/, const mpw */*vl*/,
		       void */*p*/, size_t /*sz*/);

/* --- @mpx_loadb@ --- *
 *
 * Arguments:	@mpw *v, *vl@ = base and limit of destination vector
 *		@const void *p@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Loads an MP in an octet array, most significant octet
 *		first.  High-end octets are ignored if there isn't enough
 *		space for them.
 */

extern void mpx_loadb(mpw */*v*/, mpw */*vl*/,
		      const void */*p*/, size_t /*sz*/);

/* --- @mpx_storel2cn@ --- *
 *
 * Arguments:	@const mpw *v, *vl@ = base and limit of source vector
 *		@void *pp@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Stores a negative MP in an octet array, least significant
 *		octet first, as two's complement.  High-end octets are
 *		silently discarded if there isn't enough space for them.
 *		This obviously makes the output bad.
 */

extern void mpx_storel2cn(const mpw */*v*/, const mpw */*vl*/,
			  void */*p*/, size_t /*sz*/);

/* --- @mpx_loadl2cn@ --- *
 *
 * Arguments:	@mpw *v, *vl@ = base and limit of destination vector
 *		@const void *pp@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Loads a negative MP in an octet array, least significant
 *		octet first, as two's complement.  High-end octets are
 *		ignored if there isn't enough space for them.  This probably
 *		means you made the wrong choice coming here.
 */

extern void mpx_loadl2cn(mpw */*v*/, mpw */*vl*/,
			 const void */*p*/, size_t /*sz*/);

/* --- @mpx_storeb2cn@ --- *
 *
 * Arguments:	@const mpw *v, *vl@ = base and limit of source vector
 *		@void *pp@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Stores a negative MP in an octet array, most significant
 *		octet first, as two's complement.  High-end octets are
 *		silently discarded if there isn't enough space for them,
 *		which probably isn't what you meant.
 */

extern void mpx_storeb2cn(const mpw */*v*/, const mpw */*vl*/,
			  void */*p*/, size_t /*sz*/);

/* --- @mpx_loadb2cn@ --- *
 *
 * Arguments:	@mpw *v, *vl@ = base and limit of destination vector
 *		@const void *pp@ = pointer to octet array
 *		@size_t sz@ = size of octet array
 *
 * Returns:	---
 *
 * Use:		Loads a negative MP in an octet array, most significant octet
 *		first as two's complement.  High-end octets are ignored if
 *		there isn't enough space for them.  This probably means you
 *		chose this function wrongly.
 */

extern void mpx_loadb2cn(mpw */*v*/, mpw */*vl*/,
			 const void */*p*/, size_t /*sz*/);


/*----- Logical shifting --------------------------------------------------*/

/* --- @mpx_lsl@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = source vector base and limit
 *		@size_t n@ = number of bit positions to shift by
 *
 * Returns:	---
 *
 * Use:		Performs a logical shift left operation on an integer.
 */

extern void mpx_lsl(mpw */*dv*/, mpw */*dvl*/,
		    const mpw */*av*/, const mpw */*avl*/,
		    size_t /*n*/);

/* --- @mpx_lsr@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = source vector base and limit
 *		@size_t n@ = number of bit positions to shift by
 *
 * Returns:	---
 *
 * Use:		Performs a logical shift right operation on an integer.
 */

extern void mpx_lsr(mpw */*dv*/, mpw */*dvl*/,
		    const mpw */*av*/, const mpw */*avl*/,
		    size_t /*n*/);

/*----- Bitwise operations ------------------------------------------------*/

/* --- How to implement them --- *
 *
 *   x:  0011
 *   y:  0101
 */

#define MPX_B0000(x, y) (0u)
#define MPX_B0001(x, y) ((x) & (y))
#define MPX_B0010(x, y) ((x) & ~(y))
#define MPX_B0011(x, y) (x)
#define MPX_B0100(x, y) (~(x) & ~(y))
#define MPX_B0101(x, y) (y)
#define MPX_B0110(x, y) ((x) ^ (y))
#define MPX_B0111(x, y) ((x) | (y))
#define MPX_B1000(x, y) (~((x) | (y)))
#define MPX_B1001(x, y) (~((x) ^ (y)))
#define MPX_B1010(x, y) (~(y))
#define MPX_B1011(x, y) ((x) | ~(y))
#define MPX_B1100(x, y) (~(x))
#define MPX_B1101(x, y) (~(x) | (y))
#define MPX_B1110(x, y) (~((x) & (y)))
#define MPX_B1111(x, y) (~0u)

/* --- @mpx_bitop@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector
 *		@const mpw *av, *avl@ = first source vector
 *		@const mpw *bv, *bvl@ = second source vector
 *
 * Returns:	---
 *
 * Use:		Provide the dyadic boolean functions.  The functions are
 *		named after the truth table they generate:
 *
 *			a:	0011
 *			b:	0101
 *			@mpx_bitXXXX@
 */

#define MPX_DOBIN(what)							\
  what(0000) what(0001) what(0010) what(0011)				\
  what(0100) what(0101) what(0110) what(0111)				\
  what(1000) what(1001) what(1010) what(1011)				\
  what(1100) what(1101) what(1110) what(1111)

#define MPX_BITDECL(string)						\
  extern void mpx_bit##string(mpw */*dv*/, mpw */*dvl*/,		\
			      const mpw */*av*/, const mpw */*avl*/,	\
			      const mpw */*bv*/, const mpw */*bvl*/);
MPX_DOBIN(MPX_BITDECL)

/* --- @mpx_[n]and@, @mpx_[n]or@, @mpx_xor@ --- *
 *
 * Synonyms for the commonly-used functions above.
 */

#define mpx_and  mpx_bit0001
#define mpx_or   mpx_bit0111
#define mpx_nand mpx_bit1110
#define mpx_nor  mpx_bit1000
#define mpx_xor  mpx_bit0110

/* --- @mpx_not@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector
 *		@const mpw *av, *avl@ = first source vector
 *
 * Returns:	---
 *
 * Use;		Bitwise NOT.
 */

extern void mpx_not(mpw */*dv*/, mpw */*dvl*/,
		    const mpw */*av*/, const mpw */*avl*/);

/*----- Unsigned arithmetic -----------------------------------------------*/

/* --- @mpx_2c@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector
 *		@const mpw *v, *vl@ = source vector
 *
 * Returns:	---
 *
 * Use:		Calculates the two's complement of @v@.
 */

extern void mpx_2c(mpw */*dv*/, mpw */*dvl*/,
		   const mpw */*v*/, const mpw */*vl*/);

/* --- @mpx_ueq@ --- *
 *
 * Arguments:	@const mpw *av, *avl@ = first argument vector base and limit
 *		@const mpw *bv, *bvl@ = second argument vector base and limit
 *
 * Returns:	Nonzero if the two vectors are equal.
 *
 * Use:		Performs an unsigned integer test for equality.
 */

extern int mpx_ueq(const mpw */*av*/, const mpw */*avl*/,
		   const mpw */*bv*/, const mpw */*bvl*/);

/* --- @mpx_ucmp@ --- *
 *
 * Arguments:	@const mpw *av, *avl@ = first argument vector base and limit
 *		@const mpw *bv, *bvl@ = second argument vector base and limit
 *
 * Returns:	Less than, equal to, or greater than zero depending on
 *		whether @a@ is less than, equal to or greater than @b@,
 *		respectively.
 *
 * Use:		Performs an unsigned integer comparison.
 */

#define MPX_UCMP(av, avl, op, dv, dvl)					\
  (mpx_ucmp((av), (avl), (dv), (dvl)) op 0)

extern int mpx_ucmp(const mpw */*av*/, const mpw */*avl*/,
		    const mpw */*bv*/, const mpw */*bvl*/);

/* --- @mpx_uadd@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = first addend vector base and limit
 *		@const mpw *bv, *bvl@ = second addend vector base and limit
 *
 * Returns:	---
 *
 * Use:		Performs unsigned integer addition.  If the result overflows
 *		the destination vector, high-order bits are discarded.  This
 *		means that two's complement addition happens more or less for
 *		free, although that's more a side-effect than anything else.
 *		The result vector may be equal to either or both source
 *		vectors, but may not otherwise overlap them.
 */

extern void mpx_uadd(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/,
		     const mpw */*bv*/, const mpw */*bvl*/);

/* --- @mpx_uaddn@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = source and destination base and limit
 *		@mpw n@ = other addend
 *
 * Returns:	---
 *
 * Use:		Adds a small integer to a multiprecision number.
 */

#define MPX_UADDN(dv, dvl, n) do {					\
  mpw *_ddv = (dv), *_ddvl = (dvl);					\
  mpw _c = (n);								\
									\
  while (_c && _ddv < _ddvl) {						\
    mpd _x = (mpd)*_ddv + (mpd)_c;					\
    *_ddv++ = MPW(_x);							\
    _c = _x >> MPW_BITS;						\
  }									\
} while (0)

extern void mpx_uaddn(mpw */*dv*/, mpw */*dvl*/, mpw /*n*/);

/* --- @mpx_usub@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = first argument vector base and limit
 *		@const mpw *bv, *bvl@ = second argument vector base and limit
 *
 * Returns:	---
 *
 * Use:		Performs unsigned integer subtraction.  If the result
 *		overflows the destination vector, high-order bits are
 *		discarded.  This means that two's complement subtraction
 *		happens more or less for free, although that's more a side-
 *		effect than anything else.  The result vector may be equal to
 *		either or both source vectors, but may not otherwise overlap
 *		them.
 */

extern void mpx_usub(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/,
		     const mpw */*bv*/, const mpw */*bvl*/);

/* --- @mpx_usubn@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = source and destination base and limit
 *		@n@ = subtrahend
 *
 * Returns:	---
 *
 * Use:		Subtracts a small integer from a multiprecision number.
 */

#define MPX_USUBN(dv, dvl, n) do {					\
  mpw *_ddv = (dv), *_ddvl = (dvl);					\
  mpw _c = (n);								\
									\
  while (_ddv < _ddvl) {						\
    mpd _x = (mpd)*_ddv - (mpd)_c;					\
    *_ddv++ = MPW(_x);							\
    if (_x >> MPW_BITS)							\
      _c = 1;								\
    else								\
      break;								\
  }									\
} while (0)

extern void mpx_usubn(mpw */*dv*/, mpw */*dvl*/, mpw /*n*/);

/* --- @mpx_umul@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = multiplicand vector base and limit
 *		@const mpw *bv, *bvl@ = multiplier vector base and limit
 *
 * Returns:	---
 *
 * Use:		Performs unsigned integer multiplication.  If the result
 *		overflows the desination vector, high-order bits are
 *		discarded.  The result vector may not overlap the argument
 *		vectors in any way.
 */

extern void mpx_umul(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/,
		     const mpw */*bv*/, const mpw */*bvl*/);

/* --- @mpx_umuln@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *avl@ = multiplicand vector base and limit
 *		@mpw m@ = multiplier
 *
 * Returns:	---
 *
 * Use:		Multiplies a multiprecision integer by a single-word value.
 *		The destination and source may be equal.  The destination
 *		is completely cleared after use.
 */

#define MPX_UMULN(dv, dvl, av, avl, m) do {				\
  mpw *_dv = (dv), *_dvl = (dvl);					\
  const mpw *_av = (av), *_avl = (avl);					\
  mpw _c = 0;								\
  mpd _m = (m);								\
									\
  while (_av < _avl) {							\
    mpd _x;								\
    if (_dv >= _dvl)							\
      break;								\
    _x = (mpd)_m * (mpd)*_av++ + _c;					\
    *_dv++ = MPW(_x);							\
    _c = _x >> MPW_BITS;						\
  }									\
  if (_dv < _dvl) {							\
    *_dv++ = MPW(_c);							\
    MPX_ZERO(_dv, _dvl);						\
  }									\
} while (0)

extern void mpx_umuln(mpw */*dv*/, mpw */*dvl*/,
		      const mpw */*av*/, const mpw */*avl*/, mpw m);

/* --- @mpx_umlan@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination/accumulator base and limit
 *		@const mpw *av, *avl@ = multiplicand vector base and limit
 *		@mpw m@ = multiplier
 *
 * Returns:	---
 *
 * Use:		Multiplies a multiprecision integer by a single-word value
 *		and adds the result to an accumulator.
 */

#define MPX_UMLAN(dv, dvl, av, avl, m) do {				\
  mpw *_dv = (dv), *_dvl = (dvl);					\
  const mpw *_av = (av), *_avl = (avl);					\
  mpw _cc = 0;								\
  mpd _m = (m);								\
									\
  while (_dv < _dvl && _av < _avl) {					\
    mpd _x;								\
    _x = (mpd)*_dv + (mpd)_m * (mpd)*_av++ + _cc;			\
    *_dv++ = MPW(_x);							\
    _cc = _x >> MPW_BITS;						\
  }									\
  MPX_UADDN(_dv, _dvl, _cc);						\
} while (0)

extern void mpx_umlan(mpw */*dv*/, mpw */*dvl*/,
		      const mpw */*av*/, const mpw */*avl*/, mpw m);

/* --- @mpx_usqr@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = destination vector base and limit
 *		@const mpw *av, *av@ = source vector base and limit
 *
 * Returns:	---
 *
 * Use:		Performs unsigned integer squaring.  The result vector must
 *		not overlap the source vector in any way.
 */

extern void mpx_usqr(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/);

/* --- @mpx_udiv@ --- *
 *
 * Arguments:	@mpw *qv, *qvl@ = quotient vector base and limit
 *		@mpw *rv, *rvl@ = dividend/remainder vector base and limit
 *		@const mpw *dv, *dvl@ = divisor vector base and limit
 *		@mpw *sv, *svl@ = scratch workspace 
 *
 * Returns:	---
 *
 * Use:		Performs unsigned integer division.  If the result overflows
 *		the quotient vector, high-order bits are discarded.  (Clearly
 *		the remainder vector can't overflow.)  The various vectors
 *		may not overlap in any way.  Yes, I know it's a bit odd
 *		requiring the dividend to be in the result position but it
 *		does make some sense really.  The remainder must have
 *		headroom for at least two extra words.  The scratch space
 *		must be at least one word larger than the divisor.
 */

extern void mpx_udiv(mpw */*qv*/, mpw */*qvl*/, mpw */*rv*/, mpw */*rvl*/,
		     const mpw */*dv*/, const mpw */*dvl*/,
		     mpw */*sv*/, mpw */*svl*/);

/* --- @mpx_udivn@ --- *
 *
 * Arguments:	@mpw *qv, *qvl@ = storage for the quotient (may overlap
 *			dividend)
 *		@const mpw *rv, *rvl@ = dividend
 *		@mpw d@ = single-precision divisor
 *
 * Returns:	Remainder after divison.
 *
 * Use:		Performs a single-precision division operation.
 */

extern mpw mpx_udivn(mpw */*qv*/, mpw */*qvl*/,
		     const mpw */*rv*/, const mpw */*rvl*/, mpw /*d*/);

/*----- Karatsuba multiplication algorithms -------------------------------*/

/* --- @MPK_THRESH@ --- *
 *
 * This is the limiting length for using Karatsuba algorithms.  It's best to
 * use the simpler classical multiplication method on numbers smaller than
 * this.  It is unsafe to make this constant less than four (i.e., the
 * algorithms will fail).
 */

#define MPK_THRESH 16

/* --- @mpx_kmul@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = pointer to destination buffer
 *		@const mpw *av, *avl@ = pointer to first argument
 *		@const mpw *bv, *bvl@ = pointer to second argument
 *		@mpw *sv, *svl@ = pointer to scratch workspace
 *
 * Returns:	---
 *
 * Use:		Multiplies two multiprecision integers using Karatsuba's
 *		algorithm.  This is rather faster than traditional long
 *		multiplication (e.g., @mpx_umul@) on large numbers, although
 *		more expensive on small ones.
 *
 *		The destination must be three times as large as the larger
 *		argument.  The scratch space must be five times as large as
 *		the larger argument.
 */

extern void mpx_kmul(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/,
		     const mpw */*bv*/, const mpw */*bvl*/,
		     mpw */*sv*/, mpw */*svl*/);

/* --- @mpx_ksqr@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = pointer to destination buffer
 *		@const mpw *av, *avl@ = pointer to first argument
 *		@mpw *sv, *svl@ = pointer to scratch workspace
 *
 * Returns:	---
 *
 * Use:		Squares a multiprecision integers using something similar to
 *		Karatsuba's multiplication algorithm.  This is rather faster
 *		than traditional long multiplication (e.g., @mpx_umul@) on
 *		large numbers, although more expensive on small ones, and
 *		rather simpler than full-blown Karatsuba multiplication.
 *
 *		The destination must be three times as large as the larger
 *		argument.  The scratch space must be five times as large as
 *		the larger argument.
 */

extern void mpx_ksqr(mpw */*dv*/, mpw */*dvl*/,
		     const mpw */*av*/, const mpw */*avl*/,
		     mpw */*sv*/, mpw */*svl*/);

/*----- That's all, folks -------------------------------------------------*/

#ifdef __cplusplus
  }
#endif

#endif