text: /* mpihelp-div.c  -  MPI helper functions
   *	Copyright (C) 1994, 1996 Free Software Foundation, Inc.
   *	Copyright (C) 1998, 1999 Free Software Foundation, Inc.
   *
   * This file is part of GnuPG.
   *
   * GnuPG is free software; you can redistribute it and/or modify
   * it under the terms of the GNU General Public License as published by
   * the Free Software Foundation; either version 2 of the License, or
   * (at your option) any later version.
   *
   * GnuPG 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 General Public License for more details.
   *
   * You should have received a copy of the GNU General Public License
   * along with this program; if not, write to the Free Software
   * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
   *
   * Note: This code is heavily based on the GNU MP Library.
   *	 Actually it's the same code with only minor changes in the
   *	 way the data is stored; this is to support the abstraction
   *	 of an optional secure memory allocation which may be used
   *	 to avoid revealing of sensitive data due to paging etc.
   *	 The GNU MP Library itself is published under the LGPL;
   *	 however I decided to publish this code under the plain GPL.
   */
  
  #include "mpi-internal.h"
  #include "longlong.h"
  
  #ifndef UMUL_TIME
  #define UMUL_TIME 1
  #endif
  #ifndef UDIV_TIME
  #define UDIV_TIME UMUL_TIME
  #endif
  
  /* FIXME: We should be using invert_limb (or invert_normalized_limb)
   * here (not udiv_qrnnd).
   */
  
  mpi_limb_t
  mpihelp_mod_1(mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
  	      mpi_limb_t divisor_limb)
  {
  	mpi_size_t i;
  	mpi_limb_t n1, n0, r;
  	int dummy;
  
  	/* Botch: Should this be handled at all?  Rely on callers?  */
  	if (!dividend_size)
  		return 0;
  
  	/* If multiplication is much faster than division, and the
  	 * dividend is large, pre-invert the divisor, and use
  	 * only multiplications in the inner loop.
  	 *
  	 * This test should be read:
  	 *   Does it ever help to use udiv_qrnnd_preinv?
  	 *     && Does what we save compensate for the inversion overhead?
  	 */
  	if (UDIV_TIME > (2 * UMUL_TIME + 6)
  	    && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
  		int normalization_steps;
  
  		count_leading_zeros(normalization_steps, divisor_limb);
  		if (normalization_steps) {
  			mpi_limb_t divisor_limb_inverted;
  
  			divisor_limb <<= normalization_steps;
  
  			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
  			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
  			 * most significant bit (with weight 2**N) implicit.
  			 *
  			 * Special case for DIVISOR_LIMB == 100...000.
  			 */
  			if (!(divisor_limb << 1))
  				divisor_limb_inverted = ~(mpi_limb_t) 0;
  			else
  				udiv_qrnnd(divisor_limb_inverted, dummy,
  					   -divisor_limb, 0, divisor_limb);
  
  			n1 = dividend_ptr[dividend_size - 1];
  			r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);
  
  			/* Possible optimization:
  			 * if (r == 0
  			 * && divisor_limb > ((n1 << normalization_steps)
  			 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
  			 * ...one division less...
  			 */
  			for (i = dividend_size - 2; i >= 0; i--) {
  				n0 = dividend_ptr[i];
  				UDIV_QRNND_PREINV(dummy, r, r,
  						  ((n1 << normalization_steps)
  						   | (n0 >>
  						      (BITS_PER_MPI_LIMB -
  						       normalization_steps))),
  						  divisor_limb,
  						  divisor_limb_inverted);
  				n1 = n0;
  			}
  			UDIV_QRNND_PREINV(dummy, r, r,
  					  n1 << normalization_steps,
  					  divisor_limb, divisor_limb_inverted);
  			return r >> normalization_steps;
  		} else {
  			mpi_limb_t divisor_limb_inverted;
  
  			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
  			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
  			 * most significant bit (with weight 2**N) implicit.
  			 *
  			 * Special case for DIVISOR_LIMB == 100...000.
  			 */
  			if (!(divisor_limb << 1))
  				divisor_limb_inverted = ~(mpi_limb_t) 0;
  			else
  				udiv_qrnnd(divisor_limb_inverted, dummy,
  					   -divisor_limb, 0, divisor_limb);
  
  			i = dividend_size - 1;
  			r = dividend_ptr[i];
  
  			if (r >= divisor_limb)
  				r = 0;
  			else
  				i--;
  
  			for (; i >= 0; i--) {
  				n0 = dividend_ptr[i];
  				UDIV_QRNND_PREINV(dummy, r, r,
  						  n0, divisor_limb,
  						  divisor_limb_inverted);
  			}
  			return r;
  		}
  	} else {
  		if (UDIV_NEEDS_NORMALIZATION) {
  			int normalization_steps;
  
  			count_leading_zeros(normalization_steps, divisor_limb);
  			if (normalization_steps) {
  				divisor_limb <<= normalization_steps;
  
  				n1 = dividend_ptr[dividend_size - 1];
  				r = n1 >> (BITS_PER_MPI_LIMB -
  					   normalization_steps);
  
  				/* Possible optimization:
  				 * if (r == 0
  				 * && divisor_limb > ((n1 << normalization_steps)
  				 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
  				 * ...one division less...
  				 */
  				for (i = dividend_size - 2; i >= 0; i--) {
  					n0 = dividend_ptr[i];
  					udiv_qrnnd(dummy, r, r,
  						   ((n1 << normalization_steps)
  						    | (n0 >>
  						       (BITS_PER_MPI_LIMB -
  							normalization_steps))),
  						   divisor_limb);
  					n1 = n0;
  				}
  				udiv_qrnnd(dummy, r, r,
  					   n1 << normalization_steps,
  					   divisor_limb);
  				return r >> normalization_steps;
  			}
  		}
  		/* No normalization needed, either because udiv_qrnnd doesn't require
  		 * it, or because DIVISOR_LIMB is already normalized.  */
  		i = dividend_size - 1;
  		r = dividend_ptr[i];
  
  		if (r >= divisor_limb)
  			r = 0;
  		else
  			i--;
  
  		for (; i >= 0; i--) {
  			n0 = dividend_ptr[i];
  			udiv_qrnnd(dummy, r, r, n0, divisor_limb);
  		}
  		return r;
  	}
  }
  
  /* Divide num (NP/NSIZE) by den (DP/DSIZE) and write
   * the NSIZE-DSIZE least significant quotient limbs at QP
   * and the DSIZE long remainder at NP.	If QEXTRA_LIMBS is
   * non-zero, generate that many fraction bits and append them after the
   * other quotient limbs.
   * Return the most significant limb of the quotient, this is always 0 or 1.
   *
   * Preconditions:
   * 0. NSIZE >= DSIZE.
   * 1. The most significant bit of the divisor must be set.
   * 2. QP must either not overlap with the input operands at all, or
   *    QP + DSIZE >= NP must hold true.	(This means that it's
   *    possible to put the quotient in the high part of NUM, right after the
   *    remainder in NUM.
   * 3. NSIZE >= DSIZE, even if QEXTRA_LIMBS is non-zero.
   */
  
  mpi_limb_t
  mpihelp_divrem(mpi_ptr_t qp, mpi_size_t qextra_limbs,
  	       mpi_ptr_t np, mpi_size_t nsize, mpi_ptr_t dp, mpi_size_t dsize)
  {
  	mpi_limb_t most_significant_q_limb = 0;
  
  	switch (dsize) {
  	case 0:
  		/* We are asked to divide by zero, so go ahead and do it!  (To make
  		   the compiler not remove this statement, return the value.)  */
  		/*
  		 * existing clients of this function have been modified
  		 * not to call it with dsize == 0, so this should not happen
  		 */
  		return 1 / dsize;
  
  	case 1:
  		{
  			mpi_size_t i;
  			mpi_limb_t n1;
  			mpi_limb_t d;
  
  			d = dp[0];
  			n1 = np[nsize - 1];
  
  			if (n1 >= d) {
  				n1 -= d;
  				most_significant_q_limb = 1;
  			}
  
  			qp += qextra_limbs;
  			for (i = nsize - 2; i >= 0; i--)
  				udiv_qrnnd(qp[i], n1, n1, np[i], d);
  			qp -= qextra_limbs;
  
  			for (i = qextra_limbs - 1; i >= 0; i--)
  				udiv_qrnnd(qp[i], n1, n1, 0, d);
  
  			np[0] = n1;
  		}
  		break;
  
  	case 2:
  		{
  			mpi_size_t i;
  			mpi_limb_t n1, n0, n2;
  			mpi_limb_t d1, d0;
  
  			np += nsize - 2;
  			d1 = dp[1];
  			d0 = dp[0];
  			n1 = np[1];
  			n0 = np[0];
  
  			if (n1 >= d1 && (n1 > d1 || n0 >= d0)) {
  				sub_ddmmss(n1, n0, n1, n0, d1, d0);
  				most_significant_q_limb = 1;
  			}
  
  			for (i = qextra_limbs + nsize - 2 - 1; i >= 0; i--) {
  				mpi_limb_t q;
  				mpi_limb_t r;
  
  				if (i >= qextra_limbs)
  					np--;
  				else
  					np[0] = 0;
  
  				if (n1 == d1) {
  					/* Q should be either 111..111 or 111..110.  Need special
  					 * treatment of this rare case as normal division would
  					 * give overflow.  */
  					q = ~(mpi_limb_t) 0;
  
  					r = n0 + d1;
  					if (r < d1) {	/* Carry in the addition? */
  						add_ssaaaa(n1, n0, r - d0,
  							   np[0], 0, d0);
  						qp[i] = q;
  						continue;
  					}
  					n1 = d0 - (d0 != 0 ? 1 : 0);
  					n0 = -d0;
  				} else {
  					udiv_qrnnd(q, r, n1, n0, d1);
  					umul_ppmm(n1, n0, d0, q);
  				}
  
  				n2 = np[0];
  q_test:
  				if (n1 > r || (n1 == r && n0 > n2)) {
  					/* The estimated Q was too large.  */
  					q--;
  					sub_ddmmss(n1, n0, n1, n0, 0, d0);
  					r += d1;
  					if (r >= d1)	/* If not carry, test Q again.  */
  						goto q_test;
  				}
  
  				qp[i] = q;
  				sub_ddmmss(n1, n0, r, n2, n1, n0);
  			}
  			np[1] = n1;
  			np[0] = n0;
  		}
  		break;
  
  	default:
  		{
  			mpi_size_t i;
  			mpi_limb_t dX, d1, n0;
  
  			np += nsize - dsize;
  			dX = dp[dsize - 1];
  			d1 = dp[dsize - 2];
  			n0 = np[dsize - 1];
  
  			if (n0 >= dX) {
  				if (n0 > dX
  				    || mpihelp_cmp(np, dp, dsize - 1) >= 0) {
  					mpihelp_sub_n(np, np, dp, dsize);
  					n0 = np[dsize - 1];
  					most_significant_q_limb = 1;
  				}
  			}
  
  			for (i = qextra_limbs + nsize - dsize - 1; i >= 0; i--) {
  				mpi_limb_t q;
  				mpi_limb_t n1, n2;
  				mpi_limb_t cy_limb;
  
  				if (i >= qextra_limbs) {
  					np--;
  					n2 = np[dsize];
  				} else {
  					n2 = np[dsize - 1];
  					MPN_COPY_DECR(np + 1, np, dsize - 1);
  					np[0] = 0;
  				}
  
  				if (n0 == dX) {
  					/* This might over-estimate q, but it's probably not worth
  					 * the extra code here to find out.  */
  					q = ~(mpi_limb_t) 0;
  				} else {
  					mpi_limb_t r;
  
  					udiv_qrnnd(q, r, n0, np[dsize - 1], dX);
  					umul_ppmm(n1, n0, d1, q);
  
  					while (n1 > r
  					       || (n1 == r
  						   && n0 > np[dsize - 2])) {
  						q--;
  						r += dX;
  						if (r < dX)	/* I.e. "carry in previous addition?" */
  							break;
  						n1 -= n0 < d1;
  						n0 -= d1;
  					}
  				}
  
  				/* Possible optimization: We already have (q * n0) and (1 * n1)
  				 * after the calculation of q.  Taking advantage of that, we
  				 * could make this loop make two iterations less.  */
  				cy_limb = mpihelp_submul_1(np, dp, dsize, q);
  
  				if (n2 != cy_limb) {
  					mpihelp_add_n(np, np, dp, dsize);
  					q--;
  				}
  
  				qp[i] = q;
  				n0 = np[dsize - 1];
  			}
  		}
  	}
  
  	return most_significant_q_limb;
  }
  
  /****************
   * Divide (DIVIDEND_PTR,,DIVIDEND_SIZE) by DIVISOR_LIMB.
   * Write DIVIDEND_SIZE limbs of quotient at QUOT_PTR.
   * Return the single-limb remainder.
   * There are no constraints on the value of the divisor.
   *
   * QUOT_PTR and DIVIDEND_PTR might point to the same limb.
   */
  
  mpi_limb_t
  mpihelp_divmod_1(mpi_ptr_t quot_ptr,
  		 mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
  		 mpi_limb_t divisor_limb)
  {
  	mpi_size_t i;
  	mpi_limb_t n1, n0, r;
  	int dummy;
  
  	if (!dividend_size)
  		return 0;
  
  	/* If multiplication is much faster than division, and the
  	 * dividend is large, pre-invert the divisor, and use
  	 * only multiplications in the inner loop.
  	 *
  	 * This test should be read:
  	 * Does it ever help to use udiv_qrnnd_preinv?
  	 * && Does what we save compensate for the inversion overhead?
  	 */
  	if (UDIV_TIME > (2 * UMUL_TIME + 6)
  	    && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
  		int normalization_steps;
  
  		count_leading_zeros(normalization_steps, divisor_limb);
  		if (normalization_steps) {
  			mpi_limb_t divisor_limb_inverted;
  
  			divisor_limb <<= normalization_steps;
  
  			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
  			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
  			 * most significant bit (with weight 2**N) implicit.
  			 */
  			/* Special case for DIVISOR_LIMB == 100...000.  */
  			if (!(divisor_limb << 1))
  				divisor_limb_inverted = ~(mpi_limb_t) 0;
  			else
  				udiv_qrnnd(divisor_limb_inverted, dummy,
  					   -divisor_limb, 0, divisor_limb);
  
  			n1 = dividend_ptr[dividend_size - 1];
  			r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);
  
  			/* Possible optimization:
  			 * if (r == 0
  			 * && divisor_limb > ((n1 << normalization_steps)
  			 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
  			 * ...one division less...
  			 */
  			for (i = dividend_size - 2; i >= 0; i--) {
  				n0 = dividend_ptr[i];
  				UDIV_QRNND_PREINV(quot_ptr[i + 1], r, r,
  						  ((n1 << normalization_steps)
  						   | (n0 >>
  						      (BITS_PER_MPI_LIMB -
  						       normalization_steps))),
  						  divisor_limb,
  						  divisor_limb_inverted);
  				n1 = n0;
  			}
  			UDIV_QRNND_PREINV(quot_ptr[0], r, r,
  					  n1 << normalization_steps,
  					  divisor_limb, divisor_limb_inverted);
  			return r >> normalization_steps;
  		} else {
  			mpi_limb_t divisor_limb_inverted;
  
  			/* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB.  The
  			 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
  			 * most significant bit (with weight 2**N) implicit.
  			 */
  			/* Special case for DIVISOR_LIMB == 100...000.  */
  			if (!(divisor_limb << 1))
  				divisor_limb_inverted = ~(mpi_limb_t) 0;
  			else
  				udiv_qrnnd(divisor_limb_inverted, dummy,
  					   -divisor_limb, 0, divisor_limb);
  
  			i = dividend_size - 1;
  			r = dividend_ptr[i];
  
  			if (r >= divisor_limb)
  				r = 0;
  			else
  				quot_ptr[i--] = 0;
  
  			for (; i >= 0; i--) {
  				n0 = dividend_ptr[i];
  				UDIV_QRNND_PREINV(quot_ptr[i], r, r,
  						  n0, divisor_limb,
  						  divisor_limb_inverted);
  			}
  			return r;
  		}
  	} else {
  		if (UDIV_NEEDS_NORMALIZATION) {
  			int normalization_steps;
  
  			count_leading_zeros(normalization_steps, divisor_limb);
  			if (normalization_steps) {
  				divisor_limb <<= normalization_steps;
  
  				n1 = dividend_ptr[dividend_size - 1];
  				r = n1 >> (BITS_PER_MPI_LIMB -
  					   normalization_steps);
  
  				/* Possible optimization:
  				 * if (r == 0
  				 * && divisor_limb > ((n1 << normalization_steps)
  				 *                 | (dividend_ptr[dividend_size - 2] >> ...)))
  				 * ...one division less...
  				 */
  				for (i = dividend_size - 2; i >= 0; i--) {
  					n0 = dividend_ptr[i];
  					udiv_qrnnd(quot_ptr[i + 1], r, r,
  						   ((n1 << normalization_steps)
  						    | (n0 >>
  						       (BITS_PER_MPI_LIMB -
  							normalization_steps))),
  						   divisor_limb);
  					n1 = n0;
  				}
  				udiv_qrnnd(quot_ptr[0], r, r,
  					   n1 << normalization_steps,
  					   divisor_limb);
  				return r >> normalization_steps;
  			}
  		}
  		/* No normalization needed, either because udiv_qrnnd doesn't require
  		 * it, or because DIVISOR_LIMB is already normalized.  */
  		i = dividend_size - 1;
  		r = dividend_ptr[i];
  
  		if (r >= divisor_limb)
  			r = 0;
  		else
  			quot_ptr[i--] = 0;
  
  		for (; i >= 0; i--) {
  			n0 = dividend_ptr[i];
  			udiv_qrnnd(quot_ptr[i], r, r, n0, divisor_limb);
  		}
  		return r;
  	}
  }
  