/*

  * Copyright (c) 1999, 2007, Oracle and/or its affiliates. All rights reserved.

  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.

  *

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

  * under the terms of the GNU General Public License version 2 only, as

  * published by the Free Software Foundation.  Oracle designates this

  * particular file as subject to the "Classpath" exception as provided

  * by Oracle in the LICENSE file that accompanied this code.

  *

  * This code 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

  * version 2 for more details (a copy is included in the LICENSE file that

  * accompanied this code).

  *

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

  * 2 along with this work; if not, write to the Free Software Foundation,

  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.

  *

  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA

  * or visit www.oracle.com if you need additional information or have any

  * questions.

  */

 package java.math;

 /**

  * A class used to represent multiprecision integers that makes efficient

  * use of allocated space by allowing a number to occupy only part of

  * an array so that the arrays do not have to be reallocated as often.

  * When performing an operation with many iterations the array used to

  * hold a number is only reallocated when necessary and does not have to

  * be the same size as the number it represents. A mutable number allows

  * calculations to occur on the same number without having to create

  * a new number for every step of the calculation as occurs with

  * BigIntegers.

  *

  * @see     BigInteger

  * @author  Michael McCloskey

  * @since   1.3

  */

 import java.util.Arrays;

 import static java.math.BigInteger.LONG_MASK;

 import static java.math.BigDecimal.INFLATED;

 class MutableBigInteger {

     /**

      * Holds the magnitude of this MutableBigInteger in big endian order.

      * The magnitude may start at an offset into the value array, and it may

      * end before the length of the value array.

      */

     int[] value;

     /**

      * The number of ints of the value array that are currently used

      * to hold the magnitude of this MutableBigInteger. The magnitude starts

      * at an offset and offset + intLen may be less than value.length.

      */

     int intLen;

     /**

      * The offset into the value array where the magnitude of this

      * MutableBigInteger begins.

      */

     int offset = 0;

     // Constants

     /**

      * MutableBigInteger with one element value array with the value 1. Used by

      * BigDecimal divideAndRound to increment the quotient. Use this constant

      * only when the method is not going to modify this object.

      */

     static final MutableBigInteger ONE = new MutableBigInteger(1);

     // Constructors

     /**

      * The default constructor. An empty MutableBigInteger is created with

      * a one word capacity.

      */

     MutableBigInteger() {

         value = new int[1];

         intLen = 0;

     }

     /**

      * Construct a new MutableBigInteger with a magnitude specified by

      * the int val.

      */

     MutableBigInteger(int val) {

         value = new int[1];

         intLen = 1;

         value[0] = val;

     }

     /**

      * Construct a new MutableBigInteger with the specified value array

      * up to the length of the array supplied.

      */

     MutableBigInteger(int[] val) {

         value = val;

         intLen = val.length;

     }

     /**

      * Construct a new MutableBigInteger with a magnitude equal to the

      * specified BigInteger.

      */

     MutableBigInteger(BigInteger b) {

         intLen = b.mag.length;

         value = Arrays.copyOf(b.mag, intLen);

     }

     /**

      * Construct a new MutableBigInteger with a magnitude equal to the

      * specified MutableBigInteger.

      */

     MutableBigInteger(MutableBigInteger val) {

         intLen = val.intLen;

         value = Arrays.copyOfRange(val.value, val.offset, val.offset + intLen);

     }

     /**

      * Internal helper method to return the magnitude array. The caller is not

      * supposed to modify the returned array.

      */

     private int[] getMagnitudeArray() {

         if (offset > 0 || value.length != intLen)

             return Arrays.copyOfRange(value, offset, offset + intLen);

         return value;

     }

     /**

      * Convert this MutableBigInteger to a long value. The caller has to make

      * sure this MutableBigInteger can be fit into long.

      */

     private long toLong() {

         assert (intLen <= 2) : "this MutableBigInteger exceeds the range of long";

         if (intLen == 0)

             return 0;

         long d = value[offset] & LONG_MASK;

         return (intLen == 2) ? d << 32 | (value[offset + 1] & LONG_MASK) : d;

     }

     /**

      * Convert this MutableBigInteger to a BigInteger object.

      */

     BigInteger toBigInteger(int sign) {

         if (intLen == 0 || sign == 0)

             return BigInteger.ZERO;

         return new BigInteger(getMagnitudeArray(), sign);

     }

     /**

      * Convert this MutableBigInteger to BigDecimal object with the specified sign

      * and scale.

      */

     BigDecimal toBigDecimal(int sign, int scale) {

         if (intLen == 0 || sign == 0)

             return BigDecimal.valueOf(0, scale);

         int[] mag = getMagnitudeArray();

         int len = mag.length;

         int d = mag[0];

         // If this MutableBigInteger can't be fit into long, we need to

         // make a BigInteger object for the resultant BigDecimal object.

         if (len > 2 || (d < 0 && len == 2))

             return new BigDecimal(new BigInteger(mag, sign), INFLATED, scale, 0);

         long v = (len == 2) ?

             ((mag[1] & LONG_MASK) | (d & LONG_MASK) << 32) :

             d & LONG_MASK;

         return new BigDecimal(null, sign == -1 ? -v : v, scale, 0);

     }

     /**

      * Clear out a MutableBigInteger for reuse.

      */

     void clear() {

         offset = intLen = 0;

         for (int index=0, n=value.length; index < n; index++)

             value[index] = 0;

     }

     /**

      * Set a MutableBigInteger to zero, removing its offset.

      */

     void reset() {

         offset = intLen = 0;

     }

     /**

      * Compare the magnitude of two MutableBigIntegers. Returns -1, 0 or 1

      * as this MutableBigInteger is numerically less than, equal to, or

      * greater than <tt>b</tt>.

      */

     final int compare(MutableBigInteger b) {

         int blen = b.intLen;

         if (intLen < blen)

             return -1;

         if (intLen > blen)

            return 1;

         // Add Integer.MIN_VALUE to make the comparison act as unsigned integer

         // comparison.

         int[] bval = b.value;

         for (int i = offset, j = b.offset; i < intLen + offset; i++, j++) {

             int b1 = value[i] + 0x80000000;

             int b2 = bval[j]  + 0x80000000;

             if (b1 < b2)

                 return -1;

             if (b1 > b2)

                 return 1;

         }

         return 0;

     }

     /**

      * Compare this against half of a MutableBigInteger object (Needed for

      * remainder tests).

      * Assumes no leading unnecessary zeros, which holds for results

      * from divide().

      */

     final int compareHalf(MutableBigInteger b) {

         int blen = b.intLen;

         int len = intLen;

         if (len <= 0)

             return blen <=0 ? 0 : -1;

         if (len > blen)

             return 1;

         if (len < blen - 1)

             return -1;

         int[] bval = b.value;

         int bstart = 0;

         int carry = 0;

         // Only 2 cases left:len == blen or len == blen - 1

         if (len != blen) { // len == blen - 1

             if (bval[bstart] == 1) {

                 ++bstart;

                 carry = 0x80000000;

             } else

                 return -1;

         }

         // compare values with right-shifted values of b,

         // carrying shifted-out bits across words

         int[] val = value;

         for (int i = offset, j = bstart; i < len + offset;) {

             int bv = bval[j++];

             long hb = ((bv >>> 1) + carry) & LONG_MASK;

             long v = val[i++] & LONG_MASK;

             if (v != hb)

                 return v < hb ? -1 : 1;

             carry = (bv & 1) << 31; // carray will be either 0x80000000 or 0

         }

         return carry == 0? 0 : -1;

     }

     /**

      * Return the index of the lowest set bit in this MutableBigInteger. If the

      * magnitude of this MutableBigInteger is zero, -1 is returned.

      */

     private final int getLowestSetBit() {

         if (intLen == 0)

             return -1;

         int j, b;

         for (j=intLen-1; (j>0) && (value[j+offset]==0); j--)

             ;

         b = value[j+offset];

         if (b==0)

             return -1;

         return ((intLen-1-j)<<5) + Integer.numberOfTrailingZeros(b);

     }

     /**

      * Return the int in use in this MutableBigInteger at the specified

      * index. This method is not used because it is not inlined on all

      * platforms.

      */

     private final int getInt(int index) {

         return value[offset+index];

     }

     /**

      * Return a long which is equal to the unsigned value of the int in

      * use in this MutableBigInteger at the specified index. This method is

      * not used because it is not inlined on all platforms.

      */

     private final long getLong(int index) {

         return value[offset+index] & LONG_MASK;

     }

     /**

      * Ensure that the MutableBigInteger is in normal form, specifically

      * making sure that there are no leading zeros, and that if the

      * magnitude is zero, then intLen is zero.

      */

     final void normalize() {

         if (intLen == 0) {

             offset = 0;

             return;

         }

         int index = offset;

         if (value[index] != 0)

             return;

         int indexBound = index+intLen;

         do {

             index++;

         } while(index < indexBound && value[index]==0);

         int numZeros = index - offset;

         intLen -= numZeros;

         offset = (intLen==0 ?  0 : offset+numZeros);

     }

     /**

      * If this MutableBigInteger cannot hold len words, increase the size

      * of the value array to len words.

      */

     private final void ensureCapacity(int len) {

         if (value.length < len) {

             value = new int[len];

             offset = 0;

             intLen = len;

         }

     }

     /**

      * Convert this MutableBigInteger into an int array with no leading

      * zeros, of a length that is equal to this MutableBigInteger's intLen.

      */

     int[] toIntArray() {

         int[] result = new int[intLen];

         for(int i=0; i<intLen; i++)

             result[i] = value[offset+i];

         return result;

     }

     /**

      * Sets the int at index+offset in this MutableBigInteger to val.

      * This does not get inlined on all platforms so it is not used

      * as often as originally intended.

      */

     void setInt(int index, int val) {

         value[offset + index] = val;

     }

     /**

      * Sets this MutableBigInteger's value array to the specified array.

      * The intLen is set to the specified length.

      */

     void setValue(int[] val, int length) {

         value = val;

         intLen = length;

         offset = 0;

     }

     /**

      * Sets this MutableBigInteger's value array to a copy of the specified

      * array. The intLen is set to the length of the new array.

      */

     void copyValue(MutableBigInteger src) {

         int len = src.intLen;

         if (value.length < len)

             value = new int[len];

         System.arraycopy(src.value, src.offset, value, 0, len);

         intLen = len;

         offset = 0;

     }

     /**

      * Sets this MutableBigInteger's value array to a copy of the specified

      * array. The intLen is set to the length of the specified array.

      */

     void copyValue(int[] val) {

         int len = val.length;

         if (value.length < len)

             value = new int[len];

         System.arraycopy(val, 0, value, 0, len);

         intLen = len;

         offset = 0;

     }

     /**

      * Returns true iff this MutableBigInteger has a value of one.

      */

     boolean isOne() {

         return (intLen == 1) && (value[offset] == 1);

     }

     /**

      * Returns true iff this MutableBigInteger has a value of zero.

      */

     boolean isZero() {

         return (intLen == 0);

     }

     /**

      * Returns true iff this MutableBigInteger is even.

      */

     boolean isEven() {

         return (intLen == 0) || ((value[offset + intLen - 1] & 1) == 0);

     }

     /**

      * Returns true iff this MutableBigInteger is odd.

      */

     boolean isOdd() {

         return isZero() ? false : ((value[offset + intLen - 1] & 1) == 1);

     }

     /**

      * Returns true iff this MutableBigInteger is in normal form. A

      * MutableBigInteger is in normal form if it has no leading zeros

      * after the offset, and intLen + offset <= value.length.

      */

     boolean isNormal() {

         if (intLen + offset > value.length)

             return false;

         if (intLen ==0)

             return true;

         return (value[offset] != 0);

     }

     /**

      * Returns a String representation of this MutableBigInteger in radix 10.

      */

     public String toString() {

         BigInteger b = toBigInteger(1);

         return b.toString();

     }

     /**

      * Right shift this MutableBigInteger n bits. The MutableBigInteger is left

      * in normal form.

      */

     void rightShift(int n) {

         if (intLen == 0)

             return;

         int nInts = n >>> 5;

         int nBits = n & 0x1F;

         this.intLen -= nInts;

         if (nBits == 0)

             return;

         int bitsInHighWord = BigInteger.bitLengthForInt(value[offset]);

         if (nBits >= bitsInHighWord) {

             this.primitiveLeftShift(32 - nBits);

             this.intLen--;

         } else {

             primitiveRightShift(nBits);

         }

     }

     /**

      * Left shift this MutableBigInteger n bits.

      */

     void leftShift(int n) {

         /*

          * If there is enough storage space in this MutableBigInteger already

          * the available space will be used. Space to the right of the used

          * ints in the value array is faster to utilize, so the extra space

          * will be taken from the right if possible.

          */

         if (intLen == 0)

            return;

         int nInts = n >>> 5;

         int nBits = n&0x1F;

         int bitsInHighWord = BigInteger.bitLengthForInt(value[offset]);

         // If shift can be done without moving words, do so

         if (n <= (32-bitsInHighWord)) {

             primitiveLeftShift(nBits);

             return;

         }

         int newLen = intLen + nInts +1;

         if (nBits <= (32-bitsInHighWord))

             newLen--;

         if (value.length < newLen) {

             // The array must grow

             int[] result = new int[newLen];

             for (int i=0; i<intLen; i++)

                 result[i] = value[offset+i];

             setValue(result, newLen);

         } else if (value.length - offset >= newLen) {

             // Use space on right

             for(int i=0; i<newLen - intLen; i++)

                 value[offset+intLen+i] = 0;

         } else {

             // Must use space on left

             for (int i=0; i<intLen; i++)

                 value[i] = value[offset+i];

             for (int i=intLen; i<newLen; i++)

                 value[i] = 0;

             offset = 0;

         }

         intLen = newLen;

         if (nBits == 0)

             return;

         if (nBits <= (32-bitsInHighWord))

             primitiveLeftShift(nBits);

         else

             primitiveRightShift(32 -nBits);

     }

     /**

      * A primitive used for division. This method adds in one multiple of the

      * divisor a back to the dividend result at a specified offset. It is used

      * when qhat was estimated too large, and must be adjusted.

      */

     private int divadd(int[] a, int[] result, int offset) {

         long carry = 0;

         for (int j=a.length-1; j >= 0; j--) {

             long sum = (a[j] & LONG_MASK) +

                        (result[j+offset] & LONG_MASK) + carry;

             result[j+offset] = (int)sum;

             carry = sum >>> 32;

         }

         return (int)carry;

     }

     /**

      * This method is used for division. It multiplies an n word input a by one

      * word input x, and subtracts the n word product from q. This is needed

      * when subtracting qhat*divisor from dividend.

      */

     private int mulsub(int[] q, int[] a, int x, int len, int offset) {

         long xLong = x & LONG_MASK;

         long carry = 0;

         offset += len;

         for (int j=len-1; j >= 0; j--) {

             long product = (a[j] & LONG_MASK) * xLong + carry;

             long difference = q[offset] - product;

             q[offset--] = (int)difference;

             carry = (product >>> 32)

                      + (((difference & LONG_MASK) >

                          (((~(int)product) & LONG_MASK))) ? 1:0);

         }

         return (int)carry;

     }

     /**

      * Right shift this MutableBigInteger n bits, where n is

      * less than 32.

      * Assumes that intLen > 0, n > 0 for speed

      */

     private final void primitiveRightShift(int n) {

         int[] val = value;

         int n2 = 32 - n;

         for (int i=offset+intLen-1, c=val[i]; i>offset; i--) {

             int b = c;

             c = val[i-1];

             val[i] = (c << n2) | (b >>> n);

         }

         val[offset] >>>= n;

     }

     /**

      * Left shift this MutableBigInteger n bits, where n is

      * less than 32.

      * Assumes that intLen > 0, n > 0 for speed

      */

     private final void primitiveLeftShift(int n) {

         int[] val = value;

         int n2 = 32 - n;

         for (int i=offset, c=val[i], m=i+intLen-1; i<m; i++) {

             int b = c;

             c = val[i+1];

             val[i] = (b << n) | (c >>> n2);

         }

         val[offset+intLen-1] <<= n;

     }

     /**

      * Adds the contents of two MutableBigInteger objects.The result

      * is placed within this MutableBigInteger.

      * The contents of the addend are not changed.

      */

     void add(MutableBigInteger addend) {

         int x = intLen;

         int y = addend.intLen;

         int resultLen = (intLen > addend.intLen ? intLen : addend.intLen);

         int[] result = (value.length < resultLen ? new int[resultLen] : value);

         int rstart = result.length-1;

         long sum;

         long carry = 0;

         // Add common parts of both numbers

         while(x>0 && y>0) {

             x--; y--;

             sum = (value[x+offset] & LONG_MASK) +

                 (addend.value[y+addend.offset] & LONG_MASK) + carry;

             result[rstart--] = (int)sum;

             carry = sum >>> 32;

         }

         // Add remainder of the longer number

         while(x>0) {

             x--;

             if (carry == 0 && result == value && rstart == (x + offset))

                 return;

             sum = (value[x+offset] & LONG_MASK) + carry;

             result[rstart--] = (int)sum;

             carry = sum >>> 32;

         }

         while(y>0) {

             y--;

             sum = (addend.value[y+addend.offset] & LONG_MASK) + carry;

             result[rstart--] = (int)sum;

             carry = sum >>> 32;

         }

         if (carry > 0) { // Result must grow in length

             resultLen++;

             if (result.length < resultLen) {

                 int temp[] = new int[resultLen];

                 // Result one word longer from carry-out; copy low-order

                 // bits into new result.

                 System.arraycopy(result, 0, temp, 1, result.length);

                 temp[0] = 1;

                 result = temp;

             } else {

                 result[rstart--] = 1;

             }

         }

         value = result;

         intLen = resultLen;

         offset = result.length - resultLen;

     }

     /**

      * Subtracts the smaller of this and b from the larger and places the

      * result into this MutableBigInteger.

      */

     int subtract(MutableBigInteger b) {

         MutableBigInteger a = this;

         int[] result = value;

         int sign = a.compare(b);

         if (sign == 0) {

             reset();

             return 0;

         }

         if (sign < 0) {

             MutableBigInteger tmp = a;

             a = b;

             b = tmp;

         }

         int resultLen = a.intLen;

         if (result.length < resultLen)

             result = new int[resultLen];

         long diff = 0;

         int x = a.intLen;

         int y = b.intLen;

         int rstart = result.length - 1;

         // Subtract common parts of both numbers

         while (y>0) {

             x--; y--;

             diff = (a.value[x+a.offset] & LONG_MASK) -

                    (b.value[y+b.offset] & LONG_MASK) - ((int)-(diff>>32));

             result[rstart--] = (int)diff;

         }

         // Subtract remainder of longer number

         while (x>0) {

             x--;

             diff = (a.value[x+a.offset] & LONG_MASK) - ((int)-(diff>>32));

             result[rstart--] = (int)diff;

         }

         value = result;

         intLen = resultLen;

         offset = value.length - resultLen;

         normalize();

         return sign;

     }

     /**

      * Subtracts the smaller of a and b from the larger and places the result

      * into the larger. Returns 1 if the answer is in a, -1 if in b, 0 if no

      * operation was performed.

      */

     private int difference(MutableBigInteger b) {

         MutableBigInteger a = this;

         int sign = a.compare(b);

         if (sign ==0)

             return 0;

         if (sign < 0) {

             MutableBigInteger tmp = a;

             a = b;

             b = tmp;

         }

         long diff = 0;

         int x = a.intLen;

         int y = b.intLen;

         // Subtract common parts of both numbers

         while (y>0) {

             x--; y--;

             diff = (a.value[a.offset+ x] & LONG_MASK) -

                 (b.value[b.offset+ y] & LONG_MASK) - ((int)-(diff>>32));

             a.value[a.offset+x] = (int)diff;

         }

         // Subtract remainder of longer number

         while (x>0) {

             x--;

             diff = (a.value[a.offset+ x] & LONG_MASK) - ((int)-(diff>>32));

             a.value[a.offset+x] = (int)diff;

         }

         a.normalize();

         return sign;

     }

     /**

      * Multiply the contents of two MutableBigInteger objects. The result is

      * placed into MutableBigInteger z. The contents of y are not changed.

      */

     void multiply(MutableBigInteger y, MutableBigInteger z) {

         int xLen = intLen;

         int yLen = y.intLen;

         int newLen = xLen + yLen;

         // Put z into an appropriate state to receive product

         if (z.value.length < newLen)

             z.value = new int[newLen];

         z.offset = 0;

         z.intLen = newLen;

         // The first iteration is hoisted out of the loop to avoid extra add

         long carry = 0;

         for (int j=yLen-1, k=yLen+xLen-1; j >= 0; j--, k--) {

                 long product = (y.value[j+y.offset] & LONG_MASK) *

                                (value[xLen-1+offset] & LONG_MASK) + carry;

                 z.value[k] = (int)product;

                 carry = product >>> 32;

         }

         z.value[xLen-1] = (int)carry;

         // Perform the multiplication word by word

         for (int i = xLen-2; i >= 0; i--) {

             carry = 0;

             for (int j=yLen-1, k=yLen+i; j >= 0; j--, k--) {

                 long product = (y.value[j+y.offset] & LONG_MASK) *

                                (value[i+offset] & LONG_MASK) +

                                (z.value[k] & LONG_MASK) + carry;

                 z.value[k] = (int)product;

                 carry = product >>> 32;

             }

             z.value[i] = (int)carry;

         }

         // Remove leading zeros from product

         z.normalize();

     }

     /**

      * Multiply the contents of this MutableBigInteger by the word y. The

      * result is placed into z.

      */

     void mul(int y, MutableBigInteger z) {

         if (y == 1) {

             z.copyValue(this);

             return;

         }

         if (y == 0) {

             z.clear();

             return;

         }

         // Perform the multiplication word by word

         long ylong = y & LONG_MASK;

         int[] zval = (z.value.length<intLen+1 ? new int[intLen + 1]

                                               : z.value);

         long carry = 0;

         for (int i = intLen-1; i >= 0; i--) {

             long product = ylong * (value[i+offset] & LONG_MASK) + carry;

             zval[i+1] = (int)product;

             carry = product >>> 32;

         }

         if (carry == 0) {

             z.offset = 1;

             z.intLen = intLen;

         } else {

             z.offset = 0;

             z.intLen = intLen + 1;

             zval[0] = (int)carry;

         }

         z.value = zval;

     }

      /**

      * This method is used for division of an n word dividend by a one word

      * divisor. The quotient is placed into quotient. The one word divisor is

      * specified by divisor.

      *

      * @return the remainder of the division is returned.

      *

      */

     int divideOneWord(int divisor, MutableBigInteger quotient) {

         long divisorLong = divisor & LONG_MASK;

         // Special case of one word dividend

         if (intLen == 1) {

             long dividendValue = value[offset] & LONG_MASK;

             int q = (int) (dividendValue / divisorLong);

             int r = (int) (dividendValue - q * divisorLong);

             quotient.value[0] = q;

             quotient.intLen = (q == 0) ? 0 : 1;

             quotient.offset = 0;

             return r;

         }

         if (quotient.value.length < intLen)

             quotient.value = new int[intLen];

         quotient.offset = 0;

         quotient.intLen = intLen;

         // Normalize the divisor

         int shift = Integer.numberOfLeadingZeros(divisor);

         int rem = value[offset];

         long remLong = rem & LONG_MASK;

         if (remLong < divisorLong) {

             quotient.value[0] = 0;

         } else {

             quotient.value[0] = (int)(remLong / divisorLong);

             rem = (int) (remLong - (quotient.value[0] * divisorLong));

             remLong = rem & LONG_MASK;

         }

         int xlen = intLen;

         int[] qWord = new int[2];

         while (--xlen > 0) {

             long dividendEstimate = (remLong<<32) |

                 (value[offset + intLen - xlen] & LONG_MASK);

             if (dividendEstimate >= 0) {

                 qWord[0] = (int) (dividendEstimate / divisorLong);

                 qWord[1] = (int) (dividendEstimate - qWord[0] * divisorLong);

             } else {

                 divWord(qWord, dividendEstimate, divisor);

             }

             quotient.value[intLen - xlen] = qWord[0];

             rem = qWord[1];

             remLong = rem & LONG_MASK;

         }

         quotient.normalize();

         // Unnormalize

         if (shift > 0)

             return rem % divisor;

         else

             return rem;

     }

     /**

      * Calculates the quotient of this div b and places the quotient in the

      * provided MutableBigInteger objects and the remainder object is returned.

      *

      * Uses Algorithm D in Knuth section 4.3.1.

      * Many optimizations to that algorithm have been adapted from the Colin

      * Plumb C library.

      * It special cases one word divisors for speed. The content of b is not

      * changed.

      *

      */

     MutableBigInteger divide(MutableBigInteger b, MutableBigInteger quotient) {

         if (b.intLen == 0)

             throw new ArithmeticException("BigInteger divide by zero");

         // Dividend is zero

         if (intLen == 0) {

             quotient.intLen = quotient.offset;

             return new MutableBigInteger();

         }

         int cmp = compare(b);

         // Dividend less than divisor

         if (cmp < 0) {

             quotient.intLen = quotient.offset = 0;

             return new MutableBigInteger(this);

         }

         // Dividend equal to divisor

         if (cmp == 0) {

             quotient.value[0] = quotient.intLen = 1;

             quotient.offset = 0;

             return new MutableBigInteger();

         }

         quotient.clear();

         // Special case one word divisor

         if (b.intLen == 1) {

             int r = divideOneWord(b.value[b.offset], quotient);

             if (r == 0)

                 return new MutableBigInteger();

             return new MutableBigInteger(r);

         }

         // Copy divisor value to protect divisor

         int[] div = Arrays.copyOfRange(b.value, b.offset, b.offset + b.intLen);

         return divideMagnitude(div, quotient);

     }

     /**

      * Internally used  to calculate the quotient of this div v and places the

      * quotient in the provided MutableBigInteger object and the remainder is

      * returned.

      *

      * @return the remainder of the division will be returned.

      */

     long divide(long v, MutableBigInteger quotient) {

         if (v == 0)

             throw new ArithmeticException("BigInteger divide by zero");

         // Dividend is zero

         if (intLen == 0) {

             quotient.intLen = quotient.offset = 0;

             return 0;

         }

         if (v < 0)

             v = -v;

         int d = (int)(v >>> 32);

         quotient.clear();

         // Special case on word divisor

         if (d == 0)

             return divideOneWord((int)v, quotient) & LONG_MASK;

         else {

             int[] div = new int[]{ d, (int)(v & LONG_MASK) };

             return divideMagnitude(div, quotient).toLong();

         }

     }

     /**

      * Divide this MutableBigInteger by the divisor represented by its magnitude

      * array. The quotient will be placed into the provided quotient object &

      * the remainder object is returned.

      */

     private MutableBigInteger divideMagnitude(int[] divisor,

                                               MutableBigInteger quotient) {

         // Remainder starts as dividend with space for a leading zero

         MutableBigInteger rem = new MutableBigInteger(new int[intLen + 1]);

         System.arraycopy(value, offset, rem.value, 1, intLen);

         rem.intLen = intLen;

         rem.offset = 1;

         int nlen = rem.intLen;

         // Set the quotient size

         int dlen = divisor.length;

         int limit = nlen - dlen + 1;

         if (quotient.value.length < limit) {

             quotient.value = new int[limit];

             quotient.offset = 0;

         }

         quotient.intLen = limit;

         int[] q = quotient.value;

         // D1 normalize the divisor

         int shift = Integer.numberOfLeadingZeros(divisor[0]);

         if (shift > 0) {

             // First shift will not grow array

             BigInteger.primitiveLeftShift(divisor, dlen, shift);

             // But this one might

             rem.leftShift(shift);

         }

         // Must insert leading 0 in rem if its length did not change

         if (rem.intLen == nlen) {

             rem.offset = 0;

             rem.value[0] = 0;

             rem.intLen++;

         }

         int dh = divisor[0];

         long dhLong = dh & LONG_MASK;

         int dl = divisor[1];

         int[] qWord = new int[2];

         // D2 Initialize j

         for(int j=0; j<limit; j++) {

             // D3 Calculate qhat

             // estimate qhat

             int qhat = 0;

             int qrem = 0;

             boolean skipCorrection = false;

             int nh = rem.value[j+rem.offset];

             int nh2 = nh + 0x80000000;

             int nm = rem.value[j+1+rem.offset];

             if (nh == dh) {

                 qhat = ~0;

                 qrem = nh + nm;

                 skipCorrection = qrem + 0x80000000 < nh2;

             } else {

                 long nChunk = (((long)nh) << 32) | (nm & LONG_MASK);

                 if (nChunk >= 0) {

                     qhat = (int) (nChunk / dhLong);

                     qrem = (int) (nChunk - (qhat * dhLong));

                 } else {

                     divWord(qWord, nChunk, dh);

                     qhat = qWord[0];

                     qrem = qWord[1];

                 }

             }

             if (qhat == 0)

                 continue;

             if (!skipCorrection) { // Correct qhat

                 long nl = rem.value[j+2+rem.offset] & LONG_MASK;

                 long rs = ((qrem & LONG_MASK) << 32) | nl;

                 long estProduct = (dl & LONG_MASK) * (qhat & LONG_MASK);

                 if (unsignedLongCompare(estProduct, rs)) {

                     qhat--;

                     qrem = (int)((qrem & LONG_MASK) + dhLong);

                     if ((qrem & LONG_MASK) >=  dhLong) {

                         estProduct -= (dl & LONG_MASK);

                         rs = ((qrem & LONG_MASK) << 32) | nl;

                         if (unsignedLongCompare(estProduct, rs))

                             qhat--;

                     }

                 }

             }

             // D4 Multiply and subtract

             rem.value[j+rem.offset] = 0;

             int borrow = mulsub(rem.value, divisor, qhat, dlen, j+rem.offset);

             // D5 Test remainder

             if (borrow + 0x80000000 > nh2) {

                 // D6 Add back

                 divadd(divisor, rem.value, j+1+rem.offset);

                 qhat--;

             }

             // Store the quotient digit

             q[j] = qhat;

         } // D7 loop on j

         // D8 Unnormalize

         if (shift > 0)

             rem.rightShift(shift);

         quotient.normalize();

         rem.normalize();

         return rem;

     }

     /**

      * Compare two longs as if they were unsigned.

      * Returns true iff one is bigger than two.

      */

     private boolean unsignedLongCompare(long one, long two) {

         return (one+Long.MIN_VALUE) > (two+Long.MIN_VALUE);

     }

     /**

      * This method divides a long quantity by an int to estimate

      * qhat for two multi precision numbers. It is used when

      * the signed value of n is less than zero.

      */

     private void divWord(int[] result, long n, int d) {

         long dLong = d & LONG_MASK;

         if (dLong == 1) {

             result[0] = (int)n;

             result[1] = 0;

             return;

         }

         // Approximate the quotient and remainder

         long q = (n >>> 1) / (dLong >>> 1);

         long r = n - q*dLong;

         // Correct the approximation

         while (r < 0) {

             r += dLong;

             q--;

         }

         while (r >= dLong) {

             r -= dLong;

             q++;

         }

         // n - q*dlong == r && 0 <= r <dLong, hence we're done.

         result[0] = (int)q;

         result[1] = (int)r;

     }

     /**

      * Calculate GCD of this and b. This and b are changed by the computation.

      */

     MutableBigInteger hybridGCD(MutableBigInteger b) {

         // Use Euclid's algorithm until the numbers are approximately the

         // same length, then use the binary GCD algorithm to find the GCD.

         MutableBigInteger a = this;

         MutableBigInteger q = new MutableBigInteger();

         while (b.intLen != 0) {

             if (Math.abs(a.intLen - b.intLen) < 2)

                 return a.binaryGCD(b);

             MutableBigInteger r = a.divide(b, q);

             a = b;

             b = r;

         }

         return a;

     }

     /**

      * Calculate GCD of this and v.

      * Assumes that this and v are not zero.

      */

     private MutableBigInteger binaryGCD(MutableBigInteger v) {

         // Algorithm B from Knuth section 4.5.2

         MutableBigInteger u = this;

         MutableBigInteger r = new MutableBigInteger();

         // step B1

         int s1 = u.getLowestSetBit();

         int s2 = v.getLowestSetBit();

         int k = (s1 < s2) ? s1 : s2;

         if (k != 0) {

             u.rightShift(k);

             v.rightShift(k);

         }

         // step B2

         boolean uOdd = (k==s1);

         MutableBigInteger t = uOdd ? v: u;

         int tsign = uOdd ? -1 : 1;

         int lb;

         while ((lb = t.getLowestSetBit()) >= 0) {

             // steps B3 and B4

             t.rightShift(lb);

             // step B5

             if (tsign > 0)

                 u = t;

             else

                 v = t;

             // Special case one word numbers

             if (u.intLen < 2 && v.intLen < 2) {

                 int x = u.value[u.offset];

                 int y = v.value[v.offset];

                 x  = binaryGcd(x, y);

                 r.value[0] = x;

                 r.intLen = 1;

                 r.offset = 0;

                 if (k > 0)

                     r.leftShift(k);

                 return r;

             }

             // step B6

             if ((tsign = u.difference(v)) == 0)

                 break;

             t = (tsign >= 0) ? u : v;

         }

         if (k > 0)

             u.leftShift(k);

         return u;

     }

     /**

      * Calculate GCD of a and b interpreted as unsigned integers.

      */

     static int binaryGcd(int a, int b) {

         if (b==0)

             return a;

         if (a==0)

             return b;

         // Right shift a & b till their last bits equal to 1.

         int aZeros = Integer.numberOfTrailingZeros(a);

         int bZeros = Integer.numberOfTrailingZeros(b);

         a >>>= aZeros;

         b >>>= bZeros;

         int t = (aZeros < bZeros ? aZeros : bZeros);

         while (a != b) {

             if ((a+0x80000000) > (b+0x80000000)) {  // a > b as unsigned

                 a -= b;

                 a >>>= Integer.numberOfTrailingZeros(a);

             } else {

                 b -= a;

                 b >>>= Integer.numberOfTrailingZeros(b);

             }

         }

         return a<<t;

     }

     /**

      * Returns the modInverse of this mod p. This and p are not affected by

      * the operation.

      */

     MutableBigInteger mutableModInverse(MutableBigInteger p) {

         // Modulus is odd, use Schroeppel's algorithm

         if (p.isOdd())

             return modInverse(p);

         // Base and modulus are even, throw exception

         if (isEven())

             throw new ArithmeticException("BigInteger not invertible.");

         // Get even part of modulus expressed as a power of 2

         int powersOf2 = p.getLowestSetBit();

         // Construct odd part of modulus

         MutableBigInteger oddMod = new MutableBigInteger(p);

         oddMod.rightShift(powersOf2);

         if (oddMod.isOne())

             return modInverseMP2(powersOf2);

         // Calculate 1/a mod oddMod

         MutableBigInteger oddPart = modInverse(oddMod);

         // Calculate 1/a mod evenMod

         MutableBigInteger evenPart = modInverseMP2(powersOf2);

         // Combine the results using Chinese Remainder Theorem

         MutableBigInteger y1 = modInverseBP2(oddMod, powersOf2);

         MutableBigInteger y2 = oddMod.modInverseMP2(powersOf2);

         MutableBigInteger temp1 = new MutableBigInteger();

         MutableBigInteger temp2 = new MutableBigInteger();

         MutableBigInteger result = new MutableBigInteger();

         oddPart.leftShift(powersOf2);

         oddPart.multiply(y1, result);

         evenPart.multiply(oddMod, temp1);

         temp1.multiply(y2, temp2);

         result.add(temp2);

         return result.divide(p, temp1);

     }

     /*

      * Calculate the multiplicative inverse of this mod 2^k.

      */

     MutableBigInteger modInverseMP2(int k) {

         if (isEven())

             throw new ArithmeticException("Non-invertible. (GCD != 1)");

         if (k > 64)

             return euclidModInverse(k);

         int t = inverseMod32(value[offset+intLen-1]);

         if (k < 33) {

             t = (k == 32 ? t : t & ((1 << k) - 1));

             return new MutableBigInteger(t);

         }

         long pLong = (value[offset+intLen-1] & LONG_MASK);

         if (intLen > 1)

             pLong |=  ((long)value[offset+intLen-2] << 32);

         long tLong = t & LONG_MASK;

         tLong = tLong * (2 - pLong * tLong);  // 1 more Newton iter step

         tLong = (k == 64 ? tLong : tLong & ((1L << k) - 1));

         MutableBigInteger result = new MutableBigInteger(new int[2]);

         result.value[0] = (int)(tLong >>> 32);

         result.value[1] = (int)tLong;

         result.intLen = 2;

         result.normalize();

         return result;

     }

     /*

      * Returns the multiplicative inverse of val mod 2^32.  Assumes val is odd.

      */

     static int inverseMod32(int val) {

         // Newton's iteration!

         int t = val;

         t *= 2 - val*t;

         t *= 2 - val*t;

         t *= 2 - val*t;

         t *= 2 - val*t;

         return t;

     }

     /*

      * Calculate the multiplicative inverse of 2^k mod mod, where mod is odd.

      */

     static MutableBigInteger modInverseBP2(MutableBigInteger mod, int k) {

         // Copy the mod to protect original

         return fixup(new MutableBigInteger(1), new MutableBigInteger(mod), k);

     }

     /**

      * Calculate the multiplicative inverse of this mod mod, where mod is odd.

      * This and mod are not changed by the calculation.

      *

      * This method implements an algorithm due to Richard Schroeppel, that uses

      * the same intermediate representation as Montgomery Reduction

      * ("Montgomery Form").  The algorithm is described in an unpublished

      * manuscript entitled "Fast Modular Reciprocals."

      */

     private MutableBigInteger modInverse(MutableBigInteger mod) {

         MutableBigInteger p = new MutableBigInteger(mod);

         MutableBigInteger f = new MutableBigInteger(this);

         MutableBigInteger g = new MutableBigInteger(p);

         SignedMutableBigInteger c = new SignedMutableBigInteger(1);

         SignedMutableBigInteger d = new SignedMutableBigInteger();

         MutableBigInteger temp = null;

         SignedMutableBigInteger sTemp = null;

         int k = 0;

         // Right shift f k times until odd, left shift d k times

         if (f.isEven()) {

             int trailingZeros = f.getLowestSetBit();

             f.rightShift(trailingZeros);

             d.leftShift(trailingZeros);

             k = trailingZeros;

         }

         // The Almost Inverse Algorithm

         while(!f.isOne()) {

             // If gcd(f, g) != 1, number is not invertible modulo mod

             if (f.isZero())

                 throw new ArithmeticException("BigInteger not invertible.");

             // If f < g exchange f, g and c, d

             if (f.compare(g) < 0) {

                 temp = f; f = g; g = temp;

                 sTemp = d; d = c; c = sTemp;

             }

             // If f == g (mod 4)

             if (((f.value[f.offset + f.intLen - 1] ^

                  g.value[g.offset + g.intLen - 1]) & 3) == 0) {

                 f.subtract(g);

                 c.signedSubtract(d);

             } else { // If f != g (mod 4)

                 f.add(g);

                 c.signedAdd(d);

             }

             // Right shift f k times until odd, left shift d k times

             int trailingZeros = f.getLowestSetBit();

             f.rightShift(trailingZeros);

             d.leftShift(trailingZeros);

             k += trailingZeros;

         }

         while (c.sign < 0)

            c.signedAdd(p);

         return fixup(c, p, k);

     }

     /*

      * The Fixup Algorithm

      * Calculates X such that X = C * 2^(-k) (mod P)

      * Assumes C<P and P is odd.

      */

     static MutableBigInteger fixup(MutableBigInteger c, MutableBigInteger p,

                                                                       int k) {

         MutableBigInteger temp = new MutableBigInteger();

         // Set r to the multiplicative inverse of p mod 2^32

         int r = -inverseMod32(p.value[p.offset+p.intLen-1]);

         for(int i=0, numWords = k >> 5; i<numWords; i++) {

             // V = R * c (mod 2^j)

             int  v = r * c.value[c.offset + c.intLen-1];

             // c = c + (v * p)

             p.mul(v, temp);

             c.add(temp);

             // c = c / 2^j

             c.intLen--;

         }

         int numBits = k & 0x1f;

         if (numBits != 0) {

             // V = R * c (mod 2^j)

             int v = r * c.value[c.offset + c.intLen-1];

             v &= ((1<<numBits) - 1);

             // c = c + (v * p)

             p.mul(v, temp);

             c.add(temp);

             // c = c / 2^j

             c.rightShift(numBits);

         }

         // In theory, c may be greater than p at this point (Very rare!)

         while (c.compare(p) >= 0)

             c.subtract(p);

         return c;

     }

     /**

      * Uses the extended Euclidean algorithm to compute the modInverse of base

      * mod a modulus that is a power of 2. The modulus is 2^k.

      */

     MutableBigInteger euclidModInverse(int k) {

         MutableBigInteger b = new MutableBigInteger(1);

         b.leftShift(k);

         MutableBigInteger mod = new MutableBigInteger(b);

         MutableBigInteger a = new MutableBigInteger(this);

         MutableBigInteger q = new MutableBigInteger();

         MutableBigInteger r = b.divide(a, q);

         MutableBigInteger swapper = b;

         // swap b & r

         b = r;

         r = swapper;

         MutableBigInteger t1 = new MutableBigInteger(q);

         MutableBigInteger t0 = new MutableBigInteger(1);

         MutableBigInteger temp = new MutableBigInteger();

         while (!b.isOne()) {

             r = a.divide(b, q);

             if (r.intLen == 0)

                 throw new ArithmeticException("BigInteger not invertible.");

             swapper = r;

             a = swapper;

             if (q.intLen == 1)

                 t1.mul(q.value[q.offset], temp);

             else

                 q.multiply(t1, temp);

             swapper = q;

             q = temp;

             temp = swapper;

             t0.add(q);

             if (a.isOne())

                 return t0;

             r = b.divide(a, q);

             if (r.intLen == 0)

                 throw new ArithmeticException("BigInteger not invertible.");

             swapper = b;

             b =  r;

             if (q.intLen == 1)

                 t0.mul(q.value[q.offset], temp);

             else

                 q.multiply(t0, temp);

             swapper = q; q = temp; temp = swapper;

             t1.add(q);

         }

         mod.subtract(t1);

         return mod;

     }

 }