xref: /openjdk9/jdk/src/java.base/share/classes/java/math/BigInteger.java

/*
 * Copyright (c) 1996, 2016, 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.
 */

/*
 * Portions Copyright (c) 1995  Colin Plumb.  All rights reserved.
 */

package java.math;

import java.io.IOException;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.io.ObjectStreamField;
import java.util.Arrays;
import java.util.Objects;
import java.util.Random;
import java.util.concurrent.ThreadLocalRandom;

import jdk.internal.math.DoubleConsts;
import jdk.internal.math.FloatConsts;
import jdk.internal.HotSpotIntrinsicCandidate;

/**
 * Immutable arbitrary-precision integers.  All operations behave as if
 * BigIntegers were represented in two's-complement notation (like Java's
 * primitive integer types).  BigInteger provides analogues to all of Java's
 * primitive integer operators, and all relevant methods from java.lang.Math.
 * Additionally, BigInteger provides operations for modular arithmetic, GCD
 * calculation, primality testing, prime generation, bit manipulation,
 * and a few other miscellaneous operations.
 *
 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
 * For example, division by zero throws an {@code ArithmeticException}, and
 * division of a negative by a positive yields a negative (or zero) remainder.
 * All of the details in the Spec concerning overflow are ignored, as
 * BigIntegers are made as large as necessary to accommodate the results of an
 * operation.
 *
 * <p>Semantics of shift operations extend those of Java's shift operators
 * to allow for negative shift distances.  A right-shift with a negative
 * shift distance results in a left shift, and vice-versa.  The unsigned
 * right shift operator ({@code >>>}) is omitted, as this operation makes
 * little sense in combination with the "infinite word size" abstraction
 * provided by this class.
 *
 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
 * bitwise integer operators.  The binary operators ({@code and},
 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
 * of the two operands prior to performing the operation.
 *
 * <p>Comparison operations perform signed integer comparisons, analogous to
 * those performed by Java's relational and equality operators.
 *
 * <p>Modular arithmetic operations are provided to compute residues, perform
 * exponentiation, and compute multiplicative inverses.  These methods always
 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
 * inclusive.
 *
 * <p>Bit operations operate on a single bit of the two's-complement
 * representation of their operand.  If necessary, the operand is sign-
 * extended so that it contains the designated bit.  None of the single-bit
 * operations can produce a BigInteger with a different sign from the
 * BigInteger being operated on, as they affect only a single bit, and the
 * "infinite word size" abstraction provided by this class ensures that there
 * are infinitely many "virtual sign bits" preceding each BigInteger.
 *
 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
 * descriptions of BigInteger methods.  The pseudo-code expression
 * {@code (i + j)} is shorthand for "a BigInteger whose value is
 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
 * The pseudo-code expression {@code (i == j)} is shorthand for
 * "{@code true} if and only if the BigInteger {@code i} represents the same
 * value as the BigInteger {@code j}."  Other pseudo-code expressions are
 * interpreted similarly.
 *
 * <p>All methods and constructors in this class throw
 * {@code NullPointerException} when passed
 * a null object reference for any input parameter.
 *
 * BigInteger must support values in the range
 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive)
 * and may support values outside of that range.
 *
 * The range of probable prime values is limited and may be less than
 * the full supported positive range of {@code BigInteger}.
 * The range must be at least 1 to 2<sup>500000000</sup>.
 *
 * @implNote
 * BigInteger constructors and operations throw {@code ArithmeticException} when
 * the result is out of the supported range of
 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive).
 *
 * @see     BigDecimal
 * @jls     4.2.2 Integer Operations
 * @author  Josh Bloch
 * @author  Michael McCloskey
 * @author  Alan Eliasen
 * @author  Timothy Buktu
 * @since 1.1
 */

public class BigInteger extends Number implements Comparable<BigInteger> {
    /**
     * The signum of this BigInteger: -1 for negative, 0 for zero, or
     * 1 for positive.  Note that the BigInteger zero <em>must</em> have
     * a signum of 0.  This is necessary to ensures that there is exactly one
     * representation for each BigInteger value.
     */
    final int signum;

    /**
     * The magnitude of this BigInteger, in <i>big-endian</i> order: the
     * zeroth element of this array is the most-significant int of the
     * magnitude.  The magnitude must be "minimal" in that the most-significant
     * int ({@code mag[0]}) must be non-zero.  This is necessary to
     * ensure that there is exactly one representation for each BigInteger
     * value.  Note that this implies that the BigInteger zero has a
     * zero-length mag array.
     */
    final int[] mag;

    // The following fields are stable variables. A stable variable's value
    // changes at most once from the default zero value to a non-zero stable
    // value. A stable value is calculated lazily on demand.

    /**
     * One plus the bitCount of this BigInteger. This is a stable variable.
     *
     * @see #bitCount
     */
    private int bitCountPlusOne;

    /**
     * One plus the bitLength of this BigInteger. This is a stable variable.
     * (either value is acceptable).
     *
     * @see #bitLength()
     */
    private int bitLengthPlusOne;

    /**
     * Two plus the lowest set bit of this BigInteger. This is a stable variable.
     *
     * @see #getLowestSetBit
     */
    private int lowestSetBitPlusTwo;

    /**
     * Two plus the index of the lowest-order int in the magnitude of this
     * BigInteger that contains a nonzero int. This is a stable variable. The
     * least significant int has int-number 0, the next int in order of
     * increasing significance has int-number 1, and so forth.
     *
     * <p>Note: never used for a BigInteger with a magnitude of zero.
     *
     * @see #firstNonzeroIntNum()
     */
    private int firstNonzeroIntNumPlusTwo;

    /**
     * This mask is used to obtain the value of an int as if it were unsigned.
     */
    static final long LONG_MASK = 0xffffffffL;

    /**
     * This constant limits {@code mag.length} of BigIntegers to the supported
     * range.
     */
    private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)

    /**
     * Bit lengths larger than this constant can cause overflow in searchLen
     * calculation and in BitSieve.singleSearch method.
     */
    private static final  int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;

    /**
     * The threshold value for using Karatsuba multiplication.  If the number
     * of ints in both mag arrays are greater than this number, then
     * Karatsuba multiplication will be used.   This value is found
     * experimentally to work well.
     */
    private static final int KARATSUBA_THRESHOLD = 80;

    /**
     * The threshold value for using 3-way Toom-Cook multiplication.
     * If the number of ints in each mag array is greater than the
     * Karatsuba threshold, and the number of ints in at least one of
     * the mag arrays is greater than this threshold, then Toom-Cook
     * multiplication will be used.
     */
    private static final int TOOM_COOK_THRESHOLD = 240;

    /**
     * The threshold value for using Karatsuba squaring.  If the number
     * of ints in the number are larger than this value,
     * Karatsuba squaring will be used.   This value is found
     * experimentally to work well.
     */
    private static final int KARATSUBA_SQUARE_THRESHOLD = 128;

    /**
     * The threshold value for using Toom-Cook squaring.  If the number
     * of ints in the number are larger than this value,
     * Toom-Cook squaring will be used.   This value is found
     * experimentally to work well.
     */
    private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;

    /**
     * The threshold value for using Burnikel-Ziegler division.  If the number
     * of ints in the divisor are larger than this value, Burnikel-Ziegler
     * division may be used.  This value is found experimentally to work well.
     */
    static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;

    /**
     * The offset value for using Burnikel-Ziegler division.  If the number
     * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
     * number of ints in the dividend is greater than the number of ints in the
     * divisor plus this value, Burnikel-Ziegler division will be used.  This
     * value is found experimentally to work well.
     */
    static final int BURNIKEL_ZIEGLER_OFFSET = 40;

    /**
     * The threshold value for using Schoenhage recursive base conversion. If
     * the number of ints in the number are larger than this value,
     * the Schoenhage algorithm will be used.  In practice, it appears that the
     * Schoenhage routine is faster for any threshold down to 2, and is
     * relatively flat for thresholds between 2-25, so this choice may be
     * varied within this range for very small effect.
     */
    private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;

    /**
     * The threshold value for using squaring code to perform multiplication
     * of a {@code BigInteger} instance by itself.  If the number of ints in
     * the number are larger than this value, {@code multiply(this)} will
     * return {@code square()}.
     */
    private static final int MULTIPLY_SQUARE_THRESHOLD = 20;

    /**
     * The threshold for using an intrinsic version of
     * implMontgomeryXXX to perform Montgomery multiplication.  If the
     * number of ints in the number is more than this value we do not
     * use the intrinsic.
     */
    private static final int MONTGOMERY_INTRINSIC_THRESHOLD = 512;


    // Constructors

    /**
     * Translates a byte sub-array containing the two's-complement binary
     * representation of a BigInteger into a BigInteger.  The sub-array is
     * specified via an offset into the array and a length.  The sub-array is
     * assumed to be in <i>big-endian</i> byte-order: the most significant
     * byte is the element at index {@code off}.  The {@code val} array is
     * assumed to be unchanged for the duration of the constructor call.
     *
     * An {@code IndexOutOfBoundsException} is thrown if the length of the array
     * {@code val} is non-zero and either {@code off} is negative, {@code len}
     * is negative, or {@code off+len} is greater than the length of
     * {@code val}.
     *
     * @param  val byte array containing a sub-array which is the big-endian
     *         two's-complement binary representation of a BigInteger.
     * @param  off the start offset of the binary representation.
     * @param  len the number of bytes to use.
     * @throws NumberFormatException {@code val} is zero bytes long.
     * @throws IndexOutOfBoundsException if the provided array offset and
     *         length would cause an index into the byte array to be
     *         negative or greater than or equal to the array length.
     * @since 9
     */
    public BigInteger(byte[] val, int off, int len) {
        if (val.length == 0) {
            throw new NumberFormatException("Zero length BigInteger");
        } else if ((off < 0) || (off >= val.length) || (len < 0) ||
                   (len > val.length - off)) { // 0 <= off < val.length
            throw new IndexOutOfBoundsException();
        }

        if (val[off] < 0) {
            mag = makePositive(val, off, len);
            signum = -1;
        } else {
            mag = stripLeadingZeroBytes(val, off, len);
            signum = (mag.length == 0 ? 0 : 1);
        }
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * Translates a byte array containing the two's-complement binary
     * representation of a BigInteger into a BigInteger.  The input array is
     * assumed to be in <i>big-endian</i> byte-order: the most significant
     * byte is in the zeroth element.  The {@code val} array is assumed to be
     * unchanged for the duration of the constructor call.
     *
     * @param  val big-endian two's-complement binary representation of a
     *         BigInteger.
     * @throws NumberFormatException {@code val} is zero bytes long.
     */
    public BigInteger(byte[] val) {
        this(val, 0, val.length);
    }

    /**
     * This private constructor translates an int array containing the
     * two's-complement binary representation of a BigInteger into a
     * BigInteger. The input array is assumed to be in <i>big-endian</i>
     * int-order: the most significant int is in the zeroth element.  The
     * {@code val} array is assumed to be unchanged for the duration of
     * the constructor call.
     */
    private BigInteger(int[] val) {
        if (val.length == 0)
            throw new NumberFormatException("Zero length BigInteger");

        if (val[0] < 0) {
            mag = makePositive(val);
            signum = -1;
        } else {
            mag = trustedStripLeadingZeroInts(val);
            signum = (mag.length == 0 ? 0 : 1);
        }
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * Translates the sign-magnitude representation of a BigInteger into a
     * BigInteger.  The sign is represented as an integer signum value: -1 for
     * negative, 0 for zero, or 1 for positive.  The magnitude is a sub-array of
     * a byte array in <i>big-endian</i> byte-order: the most significant byte
     * is the element at index {@code off}.  A zero value of the length
     * {@code len} is permissible, and will result in a BigInteger value of 0,
     * whether signum is -1, 0 or 1.  The {@code magnitude} array is assumed to
     * be unchanged for the duration of the constructor call.
     *
     * An {@code IndexOutOfBoundsException} is thrown if the length of the array
     * {@code magnitude} is non-zero and either {@code off} is negative,
     * {@code len} is negative, or {@code off+len} is greater than the length of
     * {@code magnitude}.
     *
     * @param  signum signum of the number (-1 for negative, 0 for zero, 1
     *         for positive).
     * @param  magnitude big-endian binary representation of the magnitude of
     *         the number.
     * @param  off the start offset of the binary representation.
     * @param  len the number of bytes to use.
     * @throws NumberFormatException {@code signum} is not one of the three
     *         legal values (-1, 0, and 1), or {@code signum} is 0 and
     *         {@code magnitude} contains one or more non-zero bytes.
     * @throws IndexOutOfBoundsException if the provided array offset and
     *         length would cause an index into the byte array to be
     *         negative or greater than or equal to the array length.
     * @since 9
     */
    public BigInteger(int signum, byte[] magnitude, int off, int len) {
        if (signum < -1 || signum > 1) {
            throw(new NumberFormatException("Invalid signum value"));
        } else if ((off < 0) || (len < 0) ||
            (len > 0 &&
                ((off >= magnitude.length) ||
                 (len > magnitude.length - off)))) { // 0 <= off < magnitude.length
            throw new IndexOutOfBoundsException();
        }

        // stripLeadingZeroBytes() returns a zero length array if len == 0
        this.mag = stripLeadingZeroBytes(magnitude, off, len);

        if (this.mag.length == 0) {
            this.signum = 0;
        } else {
            if (signum == 0)
                throw(new NumberFormatException("signum-magnitude mismatch"));
            this.signum = signum;
        }
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * Translates the sign-magnitude representation of a BigInteger into a
     * BigInteger.  The sign is represented as an integer signum value: -1 for
     * negative, 0 for zero, or 1 for positive.  The magnitude is a byte array
     * in <i>big-endian</i> byte-order: the most significant byte is the
     * zeroth element.  A zero-length magnitude array is permissible, and will
     * result in a BigInteger value of 0, whether signum is -1, 0 or 1.  The
     * {@code magnitude} array is assumed to be unchanged for the duration of
     * the constructor call.
     *
     * @param  signum signum of the number (-1 for negative, 0 for zero, 1
     *         for positive).
     * @param  magnitude big-endian binary representation of the magnitude of
     *         the number.
     * @throws NumberFormatException {@code signum} is not one of the three
     *         legal values (-1, 0, and 1), or {@code signum} is 0 and
     *         {@code magnitude} contains one or more non-zero bytes.
     */
    public BigInteger(int signum, byte[] magnitude) {
         this(signum, magnitude, 0, magnitude.length);
    }

    /**
     * A constructor for internal use that translates the sign-magnitude
     * representation of a BigInteger into a BigInteger. It checks the
     * arguments and copies the magnitude so this constructor would be
     * safe for external use.  The {@code magnitude} array is assumed to be
     * unchanged for the duration of the constructor call.
     */
    private BigInteger(int signum, int[] magnitude) {
        this.mag = stripLeadingZeroInts(magnitude);

        if (signum < -1 || signum > 1)
            throw(new NumberFormatException("Invalid signum value"));

        if (this.mag.length == 0) {
            this.signum = 0;
        } else {
            if (signum == 0)
                throw(new NumberFormatException("signum-magnitude mismatch"));
            this.signum = signum;
        }
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * Translates the String representation of a BigInteger in the
     * specified radix into a BigInteger.  The String representation
     * consists of an optional minus or plus sign followed by a
     * sequence of one or more digits in the specified radix.  The
     * character-to-digit mapping is provided by {@code
     * Character.digit}.  The String may not contain any extraneous
     * characters (whitespace, for example).
     *
     * @param val String representation of BigInteger.
     * @param radix radix to be used in interpreting {@code val}.
     * @throws NumberFormatException {@code val} is not a valid representation
     *         of a BigInteger in the specified radix, or {@code radix} is
     *         outside the range from {@link Character#MIN_RADIX} to
     *         {@link Character#MAX_RADIX}, inclusive.
     * @see    Character#digit
     */
    public BigInteger(String val, int radix) {
        int cursor = 0, numDigits;
        final int len = val.length();

        if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
            throw new NumberFormatException("Radix out of range");
        if (len == 0)
            throw new NumberFormatException("Zero length BigInteger");

        // Check for at most one leading sign
        int sign = 1;
        int index1 = val.lastIndexOf('-');
        int index2 = val.lastIndexOf('+');
        if (index1 >= 0) {
            if (index1 != 0 || index2 >= 0) {
                throw new NumberFormatException("Illegal embedded sign character");
            }
            sign = -1;
            cursor = 1;
        } else if (index2 >= 0) {
            if (index2 != 0) {
                throw new NumberFormatException("Illegal embedded sign character");
            }
            cursor = 1;
        }
        if (cursor == len)
            throw new NumberFormatException("Zero length BigInteger");

        // Skip leading zeros and compute number of digits in magnitude
        while (cursor < len &&
               Character.digit(val.charAt(cursor), radix) == 0) {
            cursor++;
        }

        if (cursor == len) {
            signum = 0;
            mag = ZERO.mag;
            return;
        }

        numDigits = len - cursor;
        signum = sign;

        // Pre-allocate array of expected size. May be too large but can
        // never be too small. Typically exact.
        long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
        if (numBits + 31 >= (1L << 32)) {
            reportOverflow();
        }
        int numWords = (int) (numBits + 31) >>> 5;
        int[] magnitude = new int[numWords];

        // Process first (potentially short) digit group
        int firstGroupLen = numDigits % digitsPerInt[radix];
        if (firstGroupLen == 0)
            firstGroupLen = digitsPerInt[radix];
        String group = val.substring(cursor, cursor += firstGroupLen);
        magnitude[numWords - 1] = Integer.parseInt(group, radix);
        if (magnitude[numWords - 1] < 0)
            throw new NumberFormatException("Illegal digit");

        // Process remaining digit groups
        int superRadix = intRadix[radix];
        int groupVal = 0;
        while (cursor < len) {
            group = val.substring(cursor, cursor += digitsPerInt[radix]);
            groupVal = Integer.parseInt(group, radix);
            if (groupVal < 0)
                throw new NumberFormatException("Illegal digit");
            destructiveMulAdd(magnitude, superRadix, groupVal);
        }
        // Required for cases where the array was overallocated.
        mag = trustedStripLeadingZeroInts(magnitude);
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /*
     * Constructs a new BigInteger using a char array with radix=10.
     * Sign is precalculated outside and not allowed in the val. The {@code val}
     * array is assumed to be unchanged for the duration of the constructor
     * call.
     */
    BigInteger(char[] val, int sign, int len) {
        int cursor = 0, numDigits;

        // Skip leading zeros and compute number of digits in magnitude
        while (cursor < len && Character.digit(val[cursor], 10) == 0) {
            cursor++;
        }
        if (cursor == len) {
            signum = 0;
            mag = ZERO.mag;
            return;
        }

        numDigits = len - cursor;
        signum = sign;
        // Pre-allocate array of expected size
        int numWords;
        if (len < 10) {
            numWords = 1;
        } else {
            long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
            if (numBits + 31 >= (1L << 32)) {
                reportOverflow();
            }
            numWords = (int) (numBits + 31) >>> 5;
        }
        int[] magnitude = new int[numWords];

        // Process first (potentially short) digit group
        int firstGroupLen = numDigits % digitsPerInt[10];
        if (firstGroupLen == 0)
            firstGroupLen = digitsPerInt[10];
        magnitude[numWords - 1] = parseInt(val, cursor,  cursor += firstGroupLen);

        // Process remaining digit groups
        while (cursor < len) {
            int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
            destructiveMulAdd(magnitude, intRadix[10], groupVal);
        }
        mag = trustedStripLeadingZeroInts(magnitude);
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    // Create an integer with the digits between the two indexes
    // Assumes start < end. The result may be negative, but it
    // is to be treated as an unsigned value.
    private int parseInt(char[] source, int start, int end) {
        int result = Character.digit(source[start++], 10);
        if (result == -1)
            throw new NumberFormatException(new String(source));

        for (int index = start; index < end; index++) {
            int nextVal = Character.digit(source[index], 10);
            if (nextVal == -1)
                throw new NumberFormatException(new String(source));
            result = 10*result + nextVal;
        }

        return result;
    }

    // bitsPerDigit in the given radix times 1024
    // Rounded up to avoid underallocation.
    private static long bitsPerDigit[] = { 0, 0,
        1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
        3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
        4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
                                           5253, 5295};

    // Multiply x array times word y in place, and add word z
    private static void destructiveMulAdd(int[] x, int y, int z) {
        // Perform the multiplication word by word
        long ylong = y & LONG_MASK;
        long zlong = z & LONG_MASK;
        int len = x.length;

        long product = 0;
        long carry = 0;
        for (int i = len-1; i >= 0; i--) {
            product = ylong * (x[i] & LONG_MASK) + carry;
            x[i] = (int)product;
            carry = product >>> 32;
        }

        // Perform the addition
        long sum = (x[len-1] & LONG_MASK) + zlong;
        x[len-1] = (int)sum;
        carry = sum >>> 32;
        for (int i = len-2; i >= 0; i--) {
            sum = (x[i] & LONG_MASK) + carry;
            x[i] = (int)sum;
            carry = sum >>> 32;
        }
    }

    /**
     * Translates the decimal String representation of a BigInteger into a
     * BigInteger.  The String representation consists of an optional minus
     * sign followed by a sequence of one or more decimal digits.  The
     * character-to-digit mapping is provided by {@code Character.digit}.
     * The String may not contain any extraneous characters (whitespace, for
     * example).
     *
     * @param val decimal String representation of BigInteger.
     * @throws NumberFormatException {@code val} is not a valid representation
     *         of a BigInteger.
     * @see    Character#digit
     */
    public BigInteger(String val) {
        this(val, 10);
    }

    /**
     * Constructs a randomly generated BigInteger, uniformly distributed over
     * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
     * The uniformity of the distribution assumes that a fair source of random
     * bits is provided in {@code rnd}.  Note that this constructor always
     * constructs a non-negative BigInteger.
     *
     * @param  numBits maximum bitLength of the new BigInteger.
     * @param  rnd source of randomness to be used in computing the new
     *         BigInteger.
     * @throws IllegalArgumentException {@code numBits} is negative.
     * @see #bitLength()
     */
    public BigInteger(int numBits, Random rnd) {
        this(1, randomBits(numBits, rnd));
    }

    private static byte[] randomBits(int numBits, Random rnd) {
        if (numBits < 0)
            throw new IllegalArgumentException("numBits must be non-negative");
        int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
        byte[] randomBits = new byte[numBytes];

        // Generate random bytes and mask out any excess bits
        if (numBytes > 0) {
            rnd.nextBytes(randomBits);
            int excessBits = 8*numBytes - numBits;
            randomBits[0] &= (1 << (8-excessBits)) - 1;
        }
        return randomBits;
    }

    /**
     * Constructs a randomly generated positive BigInteger that is probably
     * prime, with the specified bitLength.
     *
     * @apiNote It is recommended that the {@link #probablePrime probablePrime}
     * method be used in preference to this constructor unless there
     * is a compelling need to specify a certainty.
     *
     * @param  bitLength bitLength of the returned BigInteger.
     * @param  certainty a measure of the uncertainty that the caller is
     *         willing to tolerate.  The probability that the new BigInteger
     *         represents a prime number will exceed
     *         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
     *         this constructor is proportional to the value of this parameter.
     * @param  rnd source of random bits used to select candidates to be
     *         tested for primality.
     * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
     * @see    #bitLength()
     */
    public BigInteger(int bitLength, int certainty, Random rnd) {
        BigInteger prime;

        if (bitLength < 2)
            throw new ArithmeticException("bitLength < 2");
        prime = (bitLength < SMALL_PRIME_THRESHOLD
                                ? smallPrime(bitLength, certainty, rnd)
                                : largePrime(bitLength, certainty, rnd));
        signum = 1;
        mag = prime.mag;
    }

    // Minimum size in bits that the requested prime number has
    // before we use the large prime number generating algorithms.
    // The cutoff of 95 was chosen empirically for best performance.
    private static final int SMALL_PRIME_THRESHOLD = 95;

    // Certainty required to meet the spec of probablePrime
    private static final int DEFAULT_PRIME_CERTAINTY = 100;

    /**
     * Returns a positive BigInteger that is probably prime, with the
     * specified bitLength. The probability that a BigInteger returned
     * by this method is composite does not exceed 2<sup>-100</sup>.
     *
     * @param  bitLength bitLength of the returned BigInteger.
     * @param  rnd source of random bits used to select candidates to be
     *         tested for primality.
     * @return a BigInteger of {@code bitLength} bits that is probably prime
     * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
     * @see    #bitLength()
     * @since 1.4
     */
    public static BigInteger probablePrime(int bitLength, Random rnd) {
        if (bitLength < 2)
            throw new ArithmeticException("bitLength < 2");

        return (bitLength < SMALL_PRIME_THRESHOLD ?
                smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
                largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
    }

    /**
     * Find a random number of the specified bitLength that is probably prime.
     * This method is used for smaller primes, its performance degrades on
     * larger bitlengths.
     *
     * This method assumes bitLength > 1.
     */
    private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
        int magLen = (bitLength + 31) >>> 5;
        int temp[] = new int[magLen];
        int highBit = 1 << ((bitLength+31) & 0x1f);  // High bit of high int
        int highMask = (highBit << 1) - 1;  // Bits to keep in high int

        while (true) {
            // Construct a candidate
            for (int i=0; i < magLen; i++)
                temp[i] = rnd.nextInt();
            temp[0] = (temp[0] & highMask) | highBit;  // Ensure exact length
            if (bitLength > 2)
                temp[magLen-1] |= 1;  // Make odd if bitlen > 2

            BigInteger p = new BigInteger(temp, 1);

            // Do cheap "pre-test" if applicable
            if (bitLength > 6) {
                long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
                if ((r%3==0)  || (r%5==0)  || (r%7==0)  || (r%11==0) ||
                    (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
                    (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
                    continue; // Candidate is composite; try another
            }

            // All candidates of bitLength 2 and 3 are prime by this point
            if (bitLength < 4)
                return p;

            // Do expensive test if we survive pre-test (or it's inapplicable)
            if (p.primeToCertainty(certainty, rnd))
                return p;
        }
    }

    private static final BigInteger SMALL_PRIME_PRODUCT
                       = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);

    /**
     * Find a random number of the specified bitLength that is probably prime.
     * This method is more appropriate for larger bitlengths since it uses
     * a sieve to eliminate most composites before using a more expensive
     * test.
     */
    private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
        BigInteger p;
        p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
        p.mag[p.mag.length-1] &= 0xfffffffe;

        // Use a sieve length likely to contain the next prime number
        int searchLen = getPrimeSearchLen(bitLength);
        BitSieve searchSieve = new BitSieve(p, searchLen);
        BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);

        while ((candidate == null) || (candidate.bitLength() != bitLength)) {
            p = p.add(BigInteger.valueOf(2*searchLen));
            if (p.bitLength() != bitLength)
                p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
            p.mag[p.mag.length-1] &= 0xfffffffe;
            searchSieve = new BitSieve(p, searchLen);
            candidate = searchSieve.retrieve(p, certainty, rnd);
        }
        return candidate;
    }

   /**
    * Returns the first integer greater than this {@code BigInteger} that
    * is probably prime.  The probability that the number returned by this
    * method is composite does not exceed 2<sup>-100</sup>. This method will
    * never skip over a prime when searching: if it returns {@code p}, there
    * is no prime {@code q} such that {@code this < q < p}.
    *
    * @return the first integer greater than this {@code BigInteger} that
    *         is probably prime.
    * @throws ArithmeticException {@code this < 0} or {@code this} is too large.
    * @since 1.5
    */
    public BigInteger nextProbablePrime() {
        if (this.signum < 0)
            throw new ArithmeticException("start < 0: " + this);

        // Handle trivial cases
        if ((this.signum == 0) || this.equals(ONE))
            return TWO;

        BigInteger result = this.add(ONE);

        // Fastpath for small numbers
        if (result.bitLength() < SMALL_PRIME_THRESHOLD) {

            // Ensure an odd number
            if (!result.testBit(0))
                result = result.add(ONE);

            while (true) {
                // Do cheap "pre-test" if applicable
                if (result.bitLength() > 6) {
                    long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
                    if ((r%3==0)  || (r%5==0)  || (r%7==0)  || (r%11==0) ||
                        (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
                        (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
                        result = result.add(TWO);
                        continue; // Candidate is composite; try another
                    }
                }

                // All candidates of bitLength 2 and 3 are prime by this point
                if (result.bitLength() < 4)
                    return result;

                // The expensive test
                if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
                    return result;

                result = result.add(TWO);
            }
        }

        // Start at previous even number
        if (result.testBit(0))
            result = result.subtract(ONE);

        // Looking for the next large prime
        int searchLen = getPrimeSearchLen(result.bitLength());

        while (true) {
           BitSieve searchSieve = new BitSieve(result, searchLen);
           BigInteger candidate = searchSieve.retrieve(result,
                                                 DEFAULT_PRIME_CERTAINTY, null);
           if (candidate != null)
               return candidate;
           result = result.add(BigInteger.valueOf(2 * searchLen));
        }
    }

    private static int getPrimeSearchLen(int bitLength) {
        if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
            throw new ArithmeticException("Prime search implementation restriction on bitLength");
        }
        return bitLength / 20 * 64;
    }

    /**
     * Returns {@code true} if this BigInteger is probably prime,
     * {@code false} if it's definitely composite.
     *
     * This method assumes bitLength > 2.
     *
     * @param  certainty a measure of the uncertainty that the caller is
     *         willing to tolerate: if the call returns {@code true}
     *         the probability that this BigInteger is prime exceeds
     *         {@code (1 - 1/2<sup>certainty</sup>)}.  The execution time of
     *         this method is proportional to the value of this parameter.
     * @return {@code true} if this BigInteger is probably prime,
     *         {@code false} if it's definitely composite.
     */
    boolean primeToCertainty(int certainty, Random random) {
        int rounds = 0;
        int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;

        // The relationship between the certainty and the number of rounds
        // we perform is given in the draft standard ANSI X9.80, "PRIME
        // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
        int sizeInBits = this.bitLength();
        if (sizeInBits < 100) {
            rounds = 50;
            rounds = n < rounds ? n : rounds;
            return passesMillerRabin(rounds, random);
        }

        if (sizeInBits < 256) {
            rounds = 27;
        } else if (sizeInBits < 512) {
            rounds = 15;
        } else if (sizeInBits < 768) {
            rounds = 8;
        } else if (sizeInBits < 1024) {
            rounds = 4;
        } else {
            rounds = 2;
        }
        rounds = n < rounds ? n : rounds;

        return passesMillerRabin(rounds, random) && passesLucasLehmer();
    }

    /**
     * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
     *
     * The following assumptions are made:
     * This BigInteger is a positive, odd number.
     */
    private boolean passesLucasLehmer() {
        BigInteger thisPlusOne = this.add(ONE);

        // Step 1
        int d = 5;
        while (jacobiSymbol(d, this) != -1) {
            // 5, -7, 9, -11, ...
            d = (d < 0) ? Math.abs(d)+2 : -(d+2);
        }

        // Step 2
        BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);

        // Step 3
        return u.mod(this).equals(ZERO);
    }

    /**
     * Computes Jacobi(p,n).
     * Assumes n positive, odd, n>=3.
     */
    private static int jacobiSymbol(int p, BigInteger n) {
        if (p == 0)
            return 0;

        // Algorithm and comments adapted from Colin Plumb's C library.
        int j = 1;
        int u = n.mag[n.mag.length-1];

        // Make p positive
        if (p < 0) {
            p = -p;
            int n8 = u & 7;
            if ((n8 == 3) || (n8 == 7))
                j = -j; // 3 (011) or 7 (111) mod 8
        }

        // Get rid of factors of 2 in p
        while ((p & 3) == 0)
            p >>= 2;
        if ((p & 1) == 0) {
            p >>= 1;
            if (((u ^ (u>>1)) & 2) != 0)
                j = -j; // 3 (011) or 5 (101) mod 8
        }
        if (p == 1)
            return j;
        // Then, apply quadratic reciprocity
        if ((p & u & 2) != 0)   // p = u = 3 (mod 4)?
            j = -j;
        // And reduce u mod p
        u = n.mod(BigInteger.valueOf(p)).intValue();

        // Now compute Jacobi(u,p), u < p
        while (u != 0) {
            while ((u & 3) == 0)
                u >>= 2;
            if ((u & 1) == 0) {
                u >>= 1;
                if (((p ^ (p>>1)) & 2) != 0)
                    j = -j;     // 3 (011) or 5 (101) mod 8
            }
            if (u == 1)
                return j;
            // Now both u and p are odd, so use quadratic reciprocity
            assert (u < p);
            int t = u; u = p; p = t;
            if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
                j = -j;
            // Now u >= p, so it can be reduced
            u %= p;
        }
        return 0;
    }

    private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
        BigInteger d = BigInteger.valueOf(z);
        BigInteger u = ONE; BigInteger u2;
        BigInteger v = ONE; BigInteger v2;

        for (int i=k.bitLength()-2; i >= 0; i--) {
            u2 = u.multiply(v).mod(n);

            v2 = v.square().add(d.multiply(u.square())).mod(n);
            if (v2.testBit(0))
                v2 = v2.subtract(n);

            v2 = v2.shiftRight(1);

            u = u2; v = v2;
            if (k.testBit(i)) {
                u2 = u.add(v).mod(n);
                if (u2.testBit(0))
                    u2 = u2.subtract(n);

                u2 = u2.shiftRight(1);
                v2 = v.add(d.multiply(u)).mod(n);
                if (v2.testBit(0))
                    v2 = v2.subtract(n);
                v2 = v2.shiftRight(1);

                u = u2; v = v2;
            }
        }
        return u;
    }

    /**
     * Returns true iff this BigInteger passes the specified number of
     * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
     * 186-2).
     *
     * The following assumptions are made:
     * This BigInteger is a positive, odd number greater than 2.
     * iterations<=50.
     */
    private boolean passesMillerRabin(int iterations, Random rnd) {
        // Find a and m such that m is odd and this == 1 + 2**a * m
        BigInteger thisMinusOne = this.subtract(ONE);
        BigInteger m = thisMinusOne;
        int a = m.getLowestSetBit();
        m = m.shiftRight(a);

        // Do the tests
        if (rnd == null) {
            rnd = ThreadLocalRandom.current();
        }
        for (int i=0; i < iterations; i++) {
            // Generate a uniform random on (1, this)
            BigInteger b;
            do {
                b = new BigInteger(this.bitLength(), rnd);
            } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);

            int j = 0;
            BigInteger z = b.modPow(m, this);
            while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
                if (j > 0 && z.equals(ONE) || ++j == a)
                    return false;
                z = z.modPow(TWO, this);
            }
        }
        return true;
    }

    /**
     * This internal constructor differs from its public cousin
     * with the arguments reversed in two ways: it assumes that its
     * arguments are correct, and it doesn't copy the magnitude array.
     */
    BigInteger(int[] magnitude, int signum) {
        this.signum = (magnitude.length == 0 ? 0 : signum);
        this.mag = magnitude;
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * This private constructor is for internal use and assumes that its
     * arguments are correct.  The {@code magnitude} array is assumed to be
     * unchanged for the duration of the constructor call.
     */
    private BigInteger(byte[] magnitude, int signum) {
        this.signum = (magnitude.length == 0 ? 0 : signum);
        this.mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
        if (mag.length >= MAX_MAG_LENGTH) {
            checkRange();
        }
    }

    /**
     * Throws an {@code ArithmeticException} if the {@code BigInteger} would be
     * out of the supported range.
     *
     * @throws ArithmeticException if {@code this} exceeds the supported range.
     */
    private void checkRange() {
        if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
            reportOverflow();
        }
    }

    private static void reportOverflow() {
        throw new ArithmeticException("BigInteger would overflow supported range");
    }

    //Static Factory Methods

    /**
     * Returns a BigInteger whose value is equal to that of the
     * specified {@code long}.
     *
     * @apiNote This static factory method is provided in preference
     * to a ({@code long}) constructor because it allows for reuse of
     * frequently used BigIntegers.
     *
     * @param  val value of the BigInteger to return.
     * @return a BigInteger with the specified value.
     */
    public static BigInteger valueOf(long val) {
        // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
        if (val == 0)
            return ZERO;
        if (val > 0 && val <= MAX_CONSTANT)
            return posConst[(int) val];
        else if (val < 0 && val >= -MAX_CONSTANT)
            return negConst[(int) -val];

        return new BigInteger(val);
    }

    /**
     * Constructs a BigInteger with the specified value, which may not be zero.
     */
    private BigInteger(long val) {
        if (val < 0) {
            val = -val;
            signum = -1;
        } else {
            signum = 1;
        }

        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            mag = new int[1];
            mag[0] = (int)val;
        } else {
            mag = new int[2];
            mag[0] = highWord;
            mag[1] = (int)val;
        }
    }

    /**
     * Returns a BigInteger with the given two's complement representation.
     * Assumes that the input array will not be modified (the returned
     * BigInteger will reference the input array if feasible).
     */
    private static BigInteger valueOf(int val[]) {
        return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
    }

    // Constants

    /**
     * Initialize static constant array when class is loaded.
     */
    private static final int MAX_CONSTANT = 16;
    private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
    private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];

    /**
     * The cache of powers of each radix.  This allows us to not have to
     * recalculate powers of radix^(2^n) more than once.  This speeds
     * Schoenhage recursive base conversion significantly.
     */
    private static volatile BigInteger[][] powerCache;

    /** The cache of logarithms of radices for base conversion. */
    private static final double[] logCache;

    /** The natural log of 2.  This is used in computing cache indices. */
    private static final double LOG_TWO = Math.log(2.0);

    static {
        for (int i = 1; i <= MAX_CONSTANT; i++) {
            int[] magnitude = new int[1];
            magnitude[0] = i;
            posConst[i] = new BigInteger(magnitude,  1);
            negConst[i] = new BigInteger(magnitude, -1);
        }

        /*
         * Initialize the cache of radix^(2^x) values used for base conversion
         * with just the very first value.  Additional values will be created
         * on demand.
         */
        powerCache = new BigInteger[Character.MAX_RADIX+1][];
        logCache = new double[Character.MAX_RADIX+1];

        for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
            powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
            logCache[i] = Math.log(i);
        }
    }

    /**
     * The BigInteger constant zero.
     *
     * @since   1.2
     */
    public static final BigInteger ZERO = new BigInteger(new int[0], 0);

    /**
     * The BigInteger constant one.
     *
     * @since   1.2
     */
    public static final BigInteger ONE = valueOf(1);

    /**
     * The BigInteger constant two.
     *
     * @since   9
     */
    public static final BigInteger TWO = valueOf(2);

    /**
     * The BigInteger constant -1.  (Not exported.)
     */
    private static final BigInteger NEGATIVE_ONE = valueOf(-1);

    /**
     * The BigInteger constant ten.
     *
     * @since   1.5
     */
    public static final BigInteger TEN = valueOf(10);

    // Arithmetic Operations

    /**
     * Returns a BigInteger whose value is {@code (this + val)}.
     *
     * @param  val value to be added to this BigInteger.
     * @return {@code this + val}
     */
    public BigInteger add(BigInteger val) {
        if (val.signum == 0)
            return this;
        if (signum == 0)
            return val;
        if (val.signum == signum)
            return new BigInteger(add(mag, val.mag), signum);

        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);

        return new BigInteger(resultMag, cmp == signum ? 1 : -1);
    }

    /**
     * Package private methods used by BigDecimal code to add a BigInteger
     * with a long. Assumes val is not equal to INFLATED.
     */
    BigInteger add(long val) {
        if (val == 0)
            return this;
        if (signum == 0)
            return valueOf(val);
        if (Long.signum(val) == signum)
            return new BigInteger(add(mag, Math.abs(val)), signum);
        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);
        return new BigInteger(resultMag, cmp == signum ? 1 : -1);
    }

    /**
     * Adds the contents of the int array x and long value val. This
     * method allocates a new int array to hold the answer and returns
     * a reference to that array.  Assumes x.length &gt; 0 and val is
     * non-negative
     */
    private static int[] add(int[] x, long val) {
        int[] y;
        long sum = 0;
        int xIndex = x.length;
        int[] result;
        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            result = new int[xIndex];
            sum = (x[--xIndex] & LONG_MASK) + val;
            result[xIndex] = (int)sum;
        } else {
            if (xIndex == 1) {
                result = new int[2];
                sum = val  + (x[0] & LONG_MASK);
                result[1] = (int)sum;
                result[0] = (int)(sum >>> 32);
                return result;
            } else {
                result = new int[xIndex];
                sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
                result[xIndex] = (int)sum;
                sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
                result[xIndex] = (int)sum;
            }
        }
        // Copy remainder of longer number while carry propagation is required
        boolean carry = (sum >>> 32 != 0);
        while (xIndex > 0 && carry)
            carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
        // Copy remainder of longer number
        while (xIndex > 0)
            result[--xIndex] = x[xIndex];
        // Grow result if necessary
        if (carry) {
            int bigger[] = new int[result.length + 1];
            System.arraycopy(result, 0, bigger, 1, result.length);
            bigger[0] = 0x01;
            return bigger;
        }
        return result;
    }

    /**
     * Adds the contents of the int arrays x and y. This method allocates
     * a new int array to hold the answer and returns a reference to that
     * array.
     */
    private static int[] add(int[] x, int[] y) {
        // If x is shorter, swap the two arrays
        if (x.length < y.length) {
            int[] tmp = x;
            x = y;
            y = tmp;
        }

        int xIndex = x.length;
        int yIndex = y.length;
        int result[] = new int[xIndex];
        long sum = 0;
        if (yIndex == 1) {
            sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
            result[xIndex] = (int)sum;
        } else {
            // Add common parts of both numbers
            while (yIndex > 0) {
                sum = (x[--xIndex] & LONG_MASK) +
                      (y[--yIndex] & LONG_MASK) + (sum >>> 32);
                result[xIndex] = (int)sum;
            }
        }
        // Copy remainder of longer number while carry propagation is required
        boolean carry = (sum >>> 32 != 0);
        while (xIndex > 0 && carry)
            carry = ((result[--xIndex] = x[xIndex] + 1) == 0);

        // Copy remainder of longer number
        while (xIndex > 0)
            result[--xIndex] = x[xIndex];

        // Grow result if necessary
        if (carry) {
            int bigger[] = new int[result.length + 1];
            System.arraycopy(result, 0, bigger, 1, result.length);
            bigger[0] = 0x01;
            return bigger;
        }
        return result;
    }

    private static int[] subtract(long val, int[] little) {
        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            int result[] = new int[1];
            result[0] = (int)(val - (little[0] & LONG_MASK));
            return result;
        } else {
            int result[] = new int[2];
            if (little.length == 1) {
                long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
                result[1] = (int)difference;
                // Subtract remainder of longer number while borrow propagates
                boolean borrow = (difference >> 32 != 0);
                if (borrow) {
                    result[0] = highWord - 1;
                } else {        // Copy remainder of longer number
                    result[0] = highWord;
                }
                return result;
            } else { // little.length == 2
                long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
                result[1] = (int)difference;
                difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
                result[0] = (int)difference;
                return result;
            }
        }
    }

    /**
     * Subtracts the contents of the second argument (val) from the
     * first (big).  The first int array (big) must represent a larger number
     * than the second.  This method allocates the space necessary to hold the
     * answer.
     * assumes val &gt;= 0
     */
    private static int[] subtract(int[] big, long val) {
        int highWord = (int)(val >>> 32);
        int bigIndex = big.length;
        int result[] = new int[bigIndex];
        long difference = 0;

        if (highWord == 0) {
            difference = (big[--bigIndex] & LONG_MASK) - val;
            result[bigIndex] = (int)difference;
        } else {
            difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
            result[bigIndex] = (int)difference;
            difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
            result[bigIndex] = (int)difference;
        }

        // Subtract remainder of longer number while borrow propagates
        boolean borrow = (difference >> 32 != 0);
        while (bigIndex > 0 && borrow)
            borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);

        // Copy remainder of longer number
        while (bigIndex > 0)
            result[--bigIndex] = big[bigIndex];

        return result;
    }

    /**
     * Returns a BigInteger whose value is {@code (this - val)}.
     *
     * @param  val value to be subtracted from this BigInteger.
     * @return {@code this - val}
     */
    public BigInteger subtract(BigInteger val) {
        if (val.signum == 0)
            return this;
        if (signum == 0)
            return val.negate();
        if (val.signum != signum)
            return new BigInteger(add(mag, val.mag), signum);

        int cmp = compareMagnitude(val);
        if (cmp == 0)
            return ZERO;
        int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);
        return new BigInteger(resultMag, cmp == signum ? 1 : -1);
    }

    /**
     * Subtracts the contents of the second int arrays (little) from the
     * first (big).  The first int array (big) must represent a larger number
     * than the second.  This method allocates the space necessary to hold the
     * answer.
     */
    private static int[] subtract(int[] big, int[] little) {
        int bigIndex = big.length;
        int result[] = new int[bigIndex];
        int littleIndex = little.length;
        long difference = 0;

        // Subtract common parts of both numbers
        while (littleIndex > 0) {
            difference = (big[--bigIndex] & LONG_MASK) -
                         (little[--littleIndex] & LONG_MASK) +
                         (difference >> 32);
            result[bigIndex] = (int)difference;
        }

        // Subtract remainder of longer number while borrow propagates
        boolean borrow = (difference >> 32 != 0);
        while (bigIndex > 0 && borrow)
            borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);

        // Copy remainder of longer number
        while (bigIndex > 0)
            result[--bigIndex] = big[bigIndex];

        return result;
    }

    /**
     * Returns a BigInteger whose value is {@code (this * val)}.
     *
     * @implNote An implementation may offer better algorithmic
     * performance when {@code val == this}.
     *
     * @param  val value to be multiplied by this BigInteger.
     * @return {@code this * val}
     */
    public BigInteger multiply(BigInteger val) {
        if (val.signum == 0 || signum == 0)
            return ZERO;

        int xlen = mag.length;

        if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
            return square();
        }

        int ylen = val.mag.length;

        if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
            int resultSign = signum == val.signum ? 1 : -1;
            if (val.mag.length == 1) {
                return multiplyByInt(mag,val.mag[0], resultSign);
            }
            if (mag.length == 1) {
                return multiplyByInt(val.mag,mag[0], resultSign);
            }
            int[] result = multiplyToLen(mag, xlen,
                                         val.mag, ylen, null);
            result = trustedStripLeadingZeroInts(result);
            return new BigInteger(result, resultSign);
        } else {
            if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
                return multiplyKaratsuba(this, val);
            } else {
                return multiplyToomCook3(this, val);
            }
        }
    }

    private static BigInteger multiplyByInt(int[] x, int y, int sign) {
        if (Integer.bitCount(y) == 1) {
            return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
        }
        int xlen = x.length;
        int[] rmag =  new int[xlen + 1];
        long carry = 0;
        long yl = y & LONG_MASK;
        int rstart = rmag.length - 1;
        for (int i = xlen - 1; i >= 0; i--) {
            long product = (x[i] & LONG_MASK) * yl + carry;
            rmag[rstart--] = (int)product;
            carry = product >>> 32;
        }
        if (carry == 0L) {
            rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
        } else {
            rmag[rstart] = (int)carry;
        }
        return new BigInteger(rmag, sign);
    }

    /**
     * Package private methods used by BigDecimal code to multiply a BigInteger
     * with a long. Assumes v is not equal to INFLATED.
     */
    BigInteger multiply(long v) {
        if (v == 0 || signum == 0)
          return ZERO;
        if (v == BigDecimal.INFLATED)
            return multiply(BigInteger.valueOf(v));
        int rsign = (v > 0 ? signum : -signum);
        if (v < 0)
            v = -v;
        long dh = v >>> 32;      // higher order bits
        long dl = v & LONG_MASK; // lower order bits

        int xlen = mag.length;
        int[] value = mag;
        int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
        long carry = 0;
        int rstart = rmag.length - 1;
        for (int i = xlen - 1; i >= 0; i--) {
            long product = (value[i] & LONG_MASK) * dl + carry;
            rmag[rstart--] = (int)product;
            carry = product >>> 32;
        }
        rmag[rstart] = (int)carry;
        if (dh != 0L) {
            carry = 0;
            rstart = rmag.length - 2;
            for (int i = xlen - 1; i >= 0; i--) {
                long product = (value[i] & LONG_MASK) * dh +
                    (rmag[rstart] & LONG_MASK) + carry;
                rmag[rstart--] = (int)product;
                carry = product >>> 32;
            }
            rmag[0] = (int)carry;
        }
        if (carry == 0L)
            rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
        return new BigInteger(rmag, rsign);
    }

    /**
     * Multiplies int arrays x and y to the specified lengths and places
     * the result into z. There will be no leading zeros in the resultant array.
     */
    private static int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
        multiplyToLenCheck(x, xlen);
        multiplyToLenCheck(y, ylen);
        return implMultiplyToLen(x, xlen, y, ylen, z);
    }

    @HotSpotIntrinsicCandidate
    private static int[] implMultiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
        int xstart = xlen - 1;
        int ystart = ylen - 1;

        if (z == null || z.length < (xlen+ ylen))
            z = new int[xlen+ylen];

        long carry = 0;
        for (int j=ystart, k=ystart+1+xstart; j >= 0; j--, k--) {
            long product = (y[j] & LONG_MASK) *
                           (x[xstart] & LONG_MASK) + carry;
            z[k] = (int)product;
            carry = product >>> 32;
        }
        z[xstart] = (int)carry;

        for (int i = xstart-1; i >= 0; i--) {
            carry = 0;
            for (int j=ystart, k=ystart+1+i; j >= 0; j--, k--) {
                long product = (y[j] & LONG_MASK) *
                               (x[i] & LONG_MASK) +
                               (z[k] & LONG_MASK) + carry;
                z[k] = (int)product;
                carry = product >>> 32;
            }
            z[i] = (int)carry;
        }
        return z;
    }

    private static void multiplyToLenCheck(int[] array, int length) {
        if (length <= 0) {
            return;  // not an error because multiplyToLen won't execute if len <= 0
        }

        Objects.requireNonNull(array);

        if (length > array.length) {
            throw new ArrayIndexOutOfBoundsException(length - 1);
        }
    }

    /**
     * Multiplies two BigIntegers using the Karatsuba multiplication
     * algorithm.  This is a recursive divide-and-conquer algorithm which is
     * more efficient for large numbers than what is commonly called the
     * "grade-school" algorithm used in multiplyToLen.  If the numbers to be
     * multiplied have length n, the "grade-school" algorithm has an
     * asymptotic complexity of O(n^2).  In contrast, the Karatsuba algorithm
     * has complexity of O(n^(log2(3))), or O(n^1.585).  It achieves this
     * increased performance by doing 3 multiplies instead of 4 when
     * evaluating the product.  As it has some overhead, should be used when
     * both numbers are larger than a certain threshold (found
     * experimentally).
     *
     * See:  http://en.wikipedia.org/wiki/Karatsuba_algorithm
     */
    private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
        int xlen = x.mag.length;
        int ylen = y.mag.length;

        // The number of ints in each half of the number.
        int half = (Math.max(xlen, ylen)+1) / 2;

        // xl and yl are the lower halves of x and y respectively,
        // xh and yh are the upper halves.
        BigInteger xl = x.getLower(half);
        BigInteger xh = x.getUpper(half);
        BigInteger yl = y.getLower(half);
        BigInteger yh = y.getUpper(half);

        BigInteger p1 = xh.multiply(yh);  // p1 = xh*yh
        BigInteger p2 = xl.multiply(yl);  // p2 = xl*yl

        // p3=(xh+xl)*(yh+yl)
        BigInteger p3 = xh.add(xl).multiply(yh.add(yl));

        // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
        BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);

        if (x.signum != y.signum) {
            return result.negate();
        } else {
            return result;
        }
    }

    /**
     * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
     * algorithm.  This is a recursive divide-and-conquer algorithm which is
     * more efficient for large numbers than what is commonly called the
     * "grade-school" algorithm used in multiplyToLen.  If the numbers to be
     * multiplied have length n, the "grade-school" algorithm has an
     * asymptotic complexity of O(n^2).  In contrast, 3-way Toom-Cook has a
     * complexity of about O(n^1.465).  It achieves this increased asymptotic
     * performance by breaking each number into three parts and by doing 5
     * multiplies instead of 9 when evaluating the product.  Due to overhead
     * (additions, shifts, and one division) in the Toom-Cook algorithm, it
     * should only be used when both numbers are larger than a certain
     * threshold (found experimentally).  This threshold is generally larger
     * than that for Karatsuba multiplication, so this algorithm is generally
     * only used when numbers become significantly larger.
     *
     * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
     * by Marco Bodrato.
     *
     *  See: http://bodrato.it/toom-cook/
     *       http://bodrato.it/papers/#WAIFI2007
     *
     * "Towards Optimal Toom-Cook Multiplication for Univariate and
     * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
     * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
     * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
     *
     */
    private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
        int alen = a.mag.length;
        int blen = b.mag.length;

        int largest = Math.max(alen, blen);

        // k is the size (in ints) of the lower-order slices.
        int k = (largest+2)/3;   // Equal to ceil(largest/3)

        // r is the size (in ints) of the highest-order slice.
        int r = largest - 2*k;

        // Obtain slices of the numbers. a2 and b2 are the most significant
        // bits of the numbers a and b, and a0 and b0 the least significant.
        BigInteger a0, a1, a2, b0, b1, b2;
        a2 = a.getToomSlice(k, r, 0, largest);
        a1 = a.getToomSlice(k, r, 1, largest);
        a0 = a.getToomSlice(k, r, 2, largest);
        b2 = b.getToomSlice(k, r, 0, largest);
        b1 = b.getToomSlice(k, r, 1, largest);
        b0 = b.getToomSlice(k, r, 2, largest);

        BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;

        v0 = a0.multiply(b0);
        da1 = a2.add(a0);
        db1 = b2.add(b0);
        vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
        da1 = da1.add(a1);
        db1 = db1.add(b1);
        v1 = da1.multiply(db1);
        v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
             db1.add(b2).shiftLeft(1).subtract(b0));
        vinf = a2.multiply(b2);

        // The algorithm requires two divisions by 2 and one by 3.
        // All divisions are known to be exact, that is, they do not produce
        // remainders, and all results are positive.  The divisions by 2 are
        // implemented as right shifts which are relatively efficient, leaving
        // only an exact division by 3, which is done by a specialized
        // linear-time algorithm.
        t2 = v2.subtract(vm1).exactDivideBy3();
        tm1 = v1.subtract(vm1).shiftRight(1);
        t1 = v1.subtract(v0);
        t2 = t2.subtract(t1).shiftRight(1);
        t1 = t1.subtract(tm1).subtract(vinf);
        t2 = t2.subtract(vinf.shiftLeft(1));
        tm1 = tm1.subtract(t2);

        // Number of bits to shift left.
        int ss = k*32;

        BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);

        if (a.signum != b.signum) {
            return result.negate();
        } else {
            return result;
        }
    }


    /**
     * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
     *
     * @param lowerSize The size of the lower-order bit slices.
     * @param upperSize The size of the higher-order bit slices.
     * @param slice The index of which slice is requested, which must be a
     * number from 0 to size-1. Slice 0 is the highest-order bits, and slice
     * size-1 are the lowest-order bits. Slice 0 may be of different size than
     * the other slices.
     * @param fullsize The size of the larger integer array, used to align
     * slices to the appropriate position when multiplying different-sized
     * numbers.
     */
    private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
                                    int fullsize) {
        int start, end, sliceSize, len, offset;

        len = mag.length;
        offset = fullsize - len;

        if (slice == 0) {
            start = 0 - offset;
            end = upperSize - 1 - offset;
        } else {
            start = upperSize + (slice-1)*lowerSize - offset;
            end = start + lowerSize - 1;
        }

        if (start < 0) {
            start = 0;
        }
        if (end < 0) {
           return ZERO;
        }

        sliceSize = (end-start) + 1;

        if (sliceSize <= 0) {
            return ZERO;
        }

        // While performing Toom-Cook, all slices are positive and
        // the sign is adjusted when the final number is composed.
        if (start == 0 && sliceSize >= len) {
            return this.abs();
        }

        int intSlice[] = new int[sliceSize];
        System.arraycopy(mag, start, intSlice, 0, sliceSize);

        return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
    }

    /**
     * Does an exact division (that is, the remainder is known to be zero)
     * of the specified number by 3.  This is used in Toom-Cook
     * multiplication.  This is an efficient algorithm that runs in linear
     * time.  If the argument is not exactly divisible by 3, results are
     * undefined.  Note that this is expected to be called with positive
     * arguments only.
     */
    private BigInteger exactDivideBy3() {
        int len = mag.length;
        int[] result = new int[len];
        long x, w, q, borrow;
        borrow = 0L;
        for (int i=len-1; i >= 0; i--) {
            x = (mag[i] & LONG_MASK);
            w = x - borrow;
            if (borrow > x) {      // Did we make the number go negative?
                borrow = 1L;
            } else {
                borrow = 0L;
            }

            // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32).  Thus,
            // the effect of this is to divide by 3 (mod 2^32).
            // This is much faster than division on most architectures.
            q = (w * 0xAAAAAAABL) & LONG_MASK;
            result[i] = (int) q;

            // Now check the borrow. The second check can of course be
            // eliminated if the first fails.
            if (q >= 0x55555556L) {
                borrow++;
                if (q >= 0xAAAAAAABL)
                    borrow++;
            }
        }
        result = trustedStripLeadingZeroInts(result);
        return new BigInteger(result, signum);
    }

    /**
     * Returns a new BigInteger representing n lower ints of the number.
     * This is used by Karatsuba multiplication and Karatsuba squaring.
     */
    private BigInteger getLower(int n) {
        int len = mag.length;

        if (len <= n) {
            return abs();
        }

        int lowerInts[] = new int[n];
        System.arraycopy(mag, len-n, lowerInts, 0, n);

        return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
    }

    /**
     * Returns a new BigInteger representing mag.length-n upper
     * ints of the number.  This is used by Karatsuba multiplication and
     * Karatsuba squaring.
     */
    private BigInteger getUpper(int n) {
        int len = mag.length;

        if (len <= n) {
            return ZERO;
        }

        int upperLen = len - n;
        int upperInts[] = new int[upperLen];
        System.arraycopy(mag, 0, upperInts, 0, upperLen);

        return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
    }

    // Squaring

    /**
     * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
     *
     * @return {@code this<sup>2</sup>}
     */
    private BigInteger square() {
        if (signum == 0) {
            return ZERO;
        }
        int len = mag.length;

        if (len < KARATSUBA_SQUARE_THRESHOLD) {
            int[] z = squareToLen(mag, len, null);
            return new BigInteger(trustedStripLeadingZeroInts(z), 1);
        } else {
            if (len < TOOM_COOK_SQUARE_THRESHOLD) {
                return squareKaratsuba();
            } else {
                return squareToomCook3();
            }
        }
    }

    /**
     * Squares the contents of the int array x. The result is placed into the
     * int array z.  The contents of x are not changed.
     */
    private static final int[] squareToLen(int[] x, int len, int[] z) {
         int zlen = len << 1;
         if (z == null || z.length < zlen)
             z = new int[zlen];

         // Execute checks before calling intrinsified method.
         implSquareToLenChecks(x, len, z, zlen);
         return implSquareToLen(x, len, z, zlen);
     }

     /**
      * Parameters validation.
      */
     private static void implSquareToLenChecks(int[] x, int len, int[] z, int zlen) throws RuntimeException {
         if (len < 1) {
             throw new IllegalArgumentException("invalid input length: " + len);
         }
         if (len > x.length) {
             throw new IllegalArgumentException("input length out of bound: " +
                                        len + " > " + x.length);
         }
         if (len * 2 > z.length) {
             throw new IllegalArgumentException("input length out of bound: " +
                                        (len * 2) + " > " + z.length);
         }
         if (zlen < 1) {
             throw new IllegalArgumentException("invalid input length: " + zlen);
         }
         if (zlen > z.length) {
             throw new IllegalArgumentException("input length out of bound: " +
                                        len + " > " + z.length);
         }
     }

     /**
      * Java Runtime may use intrinsic for this method.
      */
     @HotSpotIntrinsicCandidate
     private static final int[] implSquareToLen(int[] x, int len, int[] z, int zlen) {
        /*
         * The algorithm used here is adapted from Colin Plumb's C library.
         * Technique: Consider the partial products in the multiplication
         * of "abcde" by itself:
         *
         *               a  b  c  d  e
         *            *  a  b  c  d  e
         *          ==================
         *              ae be ce de ee
         *           ad bd cd dd de
         *        ac bc cc cd ce
         *     ab bb bc bd be
         *  aa ab ac ad ae
         *
         * Note that everything above the main diagonal:
         *              ae be ce de = (abcd) * e
         *           ad bd cd       = (abc) * d
         *        ac bc             = (ab) * c
         *     ab                   = (a) * b
         *
         * is a copy of everything below the main diagonal:
         *                       de
         *                 cd ce
         *           bc bd be
         *     ab ac ad ae
         *
         * Thus, the sum is 2 * (off the diagonal) + diagonal.
         *
         * This is accumulated beginning with the diagonal (which
         * consist of the squares of the digits of the input), which is then
         * divided by two, the off-diagonal added, and multiplied by two
         * again.  The low bit is simply a copy of the low bit of the
         * input, so it doesn't need special care.
         */

        // Store the squares, right shifted one bit (i.e., divided by 2)
        int lastProductLowWord = 0;
        for (int j=0, i=0; j < len; j++) {
            long piece = (x[j] & LONG_MASK);
            long product = piece * piece;
            z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
            z[i++] = (int)(product >>> 1);
            lastProductLowWord = (int)product;
        }

        // Add in off-diagonal sums
        for (int i=len, offset=1; i > 0; i--, offset+=2) {
            int t = x[i-1];
            t = mulAdd(z, x, offset, i-1, t);
            addOne(z, offset-1, i, t);
        }

        // Shift back up and set low bit
        primitiveLeftShift(z, zlen, 1);
        z[zlen-1] |= x[len-1] & 1;

        return z;
    }

    /**
     * Squares a BigInteger using the Karatsuba squaring algorithm.  It should
     * be used when both numbers are larger than a certain threshold (found
     * experimentally).  It is a recursive divide-and-conquer algorithm that
     * has better asymptotic performance than the algorithm used in
     * squareToLen.
     */
    private BigInteger squareKaratsuba() {
        int half = (mag.length+1) / 2;

        BigInteger xl = getLower(half);
        BigInteger xh = getUpper(half);

        BigInteger xhs = xh.square();  // xhs = xh^2
        BigInteger xls = xl.square();  // xls = xl^2

        // xh^2 << 64  +  (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
        return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
    }

    /**
     * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm.  It
     * should be used when both numbers are larger than a certain threshold
     * (found experimentally).  It is a recursive divide-and-conquer algorithm
     * that has better asymptotic performance than the algorithm used in
     * squareToLen or squareKaratsuba.
     */
    private BigInteger squareToomCook3() {
        int len = mag.length;

        // k is the size (in ints) of the lower-order slices.
        int k = (len+2)/3;   // Equal to ceil(largest/3)

        // r is the size (in ints) of the highest-order slice.
        int r = len - 2*k;

        // Obtain slices of the numbers. a2 is the most significant
        // bits of the number, and a0 the least significant.
        BigInteger a0, a1, a2;
        a2 = getToomSlice(k, r, 0, len);
        a1 = getToomSlice(k, r, 1, len);
        a0 = getToomSlice(k, r, 2, len);
        BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;

        v0 = a0.square();
        da1 = a2.add(a0);
        vm1 = da1.subtract(a1).square();
        da1 = da1.add(a1);
        v1 = da1.square();
        vinf = a2.square();
        v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();

        // The algorithm requires two divisions by 2 and one by 3.
        // All divisions are known to be exact, that is, they do not produce
        // remainders, and all results are positive.  The divisions by 2 are
        // implemented as right shifts which are relatively efficient, leaving
        // only a division by 3.
        // The division by 3 is done by an optimized algorithm for this case.
        t2 = v2.subtract(vm1).exactDivideBy3();
        tm1 = v1.subtract(vm1).shiftRight(1);
        t1 = v1.subtract(v0);
        t2 = t2.subtract(t1).shiftRight(1);
        t1 = t1.subtract(tm1).subtract(vinf);
        t2 = t2.subtract(vinf.shiftLeft(1));
        tm1 = tm1.subtract(t2);

        // Number of bits to shift left.
        int ss = k*32;

        return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
    }

    // Division

    /**
     * Returns a BigInteger whose value is {@code (this / val)}.
     *
     * @param  val value by which this BigInteger is to be divided.
     * @return {@code this / val}
     * @throws ArithmeticException if {@code val} is zero.
     */
    public BigInteger divide(BigInteger val) {
        if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return divideKnuth(val);
        } else {
            return divideBurnikelZiegler(val);
        }
    }

    /**
     * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
     *
     * @param  val value by which this BigInteger is to be divided.
     * @return {@code this / val}
     * @throws ArithmeticException if {@code val} is zero.
     * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
     */
    private BigInteger divideKnuth(BigInteger val) {
        MutableBigInteger q = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);

        a.divideKnuth(b, q, false);
        return q.toBigInteger(this.signum * val.signum);
    }

    /**
     * Returns an array of two BigIntegers containing {@code (this / val)}
     * followed by {@code (this % val)}.
     *
     * @param  val value by which this BigInteger is to be divided, and the
     *         remainder computed.
     * @return an array of two BigIntegers: the quotient {@code (this / val)}
     *         is the initial element, and the remainder {@code (this % val)}
     *         is the final element.
     * @throws ArithmeticException if {@code val} is zero.
     */
    public BigInteger[] divideAndRemainder(BigInteger val) {
        if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return divideAndRemainderKnuth(val);
        } else {
            return divideAndRemainderBurnikelZiegler(val);
        }
    }

    /** Long division */
    private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
        BigInteger[] result = new BigInteger[2];
        MutableBigInteger q = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);
        MutableBigInteger r = a.divideKnuth(b, q);
        result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
        result[1] = r.toBigInteger(this.signum);
        return result;
    }

    /**
     * Returns a BigInteger whose value is {@code (this % val)}.
     *
     * @param  val value by which this BigInteger is to be divided, and the
     *         remainder computed.
     * @return {@code this % val}
     * @throws ArithmeticException if {@code val} is zero.
     */
    public BigInteger remainder(BigInteger val) {
        if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
                mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
            return remainderKnuth(val);
        } else {
            return remainderBurnikelZiegler(val);
        }
    }

    /** Long division */
    private BigInteger remainderKnuth(BigInteger val) {
        MutableBigInteger q = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);

        return a.divideKnuth(b, q).toBigInteger(this.signum);
    }

    /**
     * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
     * @param  val the divisor
     * @return {@code this / val}
     */
    private BigInteger divideBurnikelZiegler(BigInteger val) {
        return divideAndRemainderBurnikelZiegler(val)[0];
    }

    /**
     * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
     * @param val the divisor
     * @return {@code this % val}
     */
    private BigInteger remainderBurnikelZiegler(BigInteger val) {
        return divideAndRemainderBurnikelZiegler(val)[1];
    }

    /**
     * Computes {@code this / val} and {@code this % val} using the
     * Burnikel-Ziegler algorithm.
     * @param val the divisor
     * @return an array containing the quotient and remainder
     */
    private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
        MutableBigInteger q = new MutableBigInteger();
        MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
        BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
        BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
        return new BigInteger[] {qBigInt, rBigInt};
    }

    /**
     * Returns a BigInteger whose value is <code>(this<sup>exponent</sup>)</code>.
     * Note that {@code exponent} is an integer rather than a BigInteger.
     *
     * @param  exponent exponent to which this BigInteger is to be raised.
     * @return <code>this<sup>exponent</sup></code>
     * @throws ArithmeticException {@code exponent} is negative.  (This would
     *         cause the operation to yield a non-integer value.)
     */
    public BigInteger pow(int exponent) {
        if (exponent < 0) {
            throw new ArithmeticException("Negative exponent");
        }
        if (signum == 0) {
            return (exponent == 0 ? ONE : this);
        }

        BigInteger partToSquare = this.abs();

        // Factor out powers of two from the base, as the exponentiation of
        // these can be done by left shifts only.
        // The remaining part can then be exponentiated faster.  The
        // powers of two will be multiplied back at the end.
        int powersOfTwo = partToSquare.getLowestSetBit();
        long bitsToShift = (long)powersOfTwo * exponent;
        if (bitsToShift > Integer.MAX_VALUE) {
            reportOverflow();
        }

        int remainingBits;

        // Factor the powers of two out quickly by shifting right, if needed.
        if (powersOfTwo > 0) {
            partToSquare = partToSquare.shiftRight(powersOfTwo);
            remainingBits = partToSquare.bitLength();
            if (remainingBits == 1) {  // Nothing left but +/- 1?
                if (signum < 0 && (exponent&1) == 1) {
                    return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
                } else {
                    return ONE.shiftLeft(powersOfTwo*exponent);
                }
            }
        } else {
            remainingBits = partToSquare.bitLength();
            if (remainingBits == 1) { // Nothing left but +/- 1?
                if (signum < 0  && (exponent&1) == 1) {
                    return NEGATIVE_ONE;
                } else {
                    return ONE;
                }
            }
        }

        // This is a quick way to approximate the size of the result,
        // similar to doing log2[n] * exponent.  This will give an upper bound
        // of how big the result can be, and which algorithm to use.
        long scaleFactor = (long)remainingBits * exponent;

        // Use slightly different algorithms for small and large operands.
        // See if the result will safely fit into a long. (Largest 2^63-1)
        if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
            // Small number algorithm.  Everything fits into a long.
            int newSign = (signum <0  && (exponent&1) == 1 ? -1 : 1);
            long result = 1;
            long baseToPow2 = partToSquare.mag[0] & LONG_MASK;

            int workingExponent = exponent;

            // Perform exponentiation using repeated squaring trick
            while (workingExponent != 0) {
                if ((workingExponent & 1) == 1) {
                    result = result * baseToPow2;
                }

                if ((workingExponent >>>= 1) != 0) {
                    baseToPow2 = baseToPow2 * baseToPow2;
                }
            }

            // Multiply back the powers of two (quickly, by shifting left)
            if (powersOfTwo > 0) {
                if (bitsToShift + scaleFactor <= 62) { // Fits in long?
                    return valueOf((result << bitsToShift) * newSign);
                } else {
                    return valueOf(result*newSign).shiftLeft((int) bitsToShift);
                }
            }
            else {
                return valueOf(result*newSign);
            }
        } else {
            // Large number algorithm.  This is basically identical to
            // the algorithm above, but calls multiply() and square()
            // which may use more efficient algorithms for large numbers.
            BigInteger answer = ONE;

            int workingExponent = exponent;
            // Perform exponentiation using repeated squaring trick
            while (workingExponent != 0) {
                if ((workingExponent & 1) == 1) {
                    answer = answer.multiply(partToSquare);
                }

                if ((workingExponent >>>= 1) != 0) {
                    partToSquare = partToSquare.square();
                }
            }
            // Multiply back the (exponentiated) powers of two (quickly,
            // by shifting left)
            if (powersOfTwo > 0) {
                answer = answer.shiftLeft(powersOfTwo*exponent);
            }

            if (signum < 0 && (exponent&1) == 1) {
                return answer.negate();
            } else {
                return answer;
            }
        }
    }

    /**
     * Returns the integer square root of this BigInteger.  The integer square
     * root of the corresponding mathematical integer {@code n} is the largest
     * mathematical integer {@code s} such that {@code s*s <= n}.  It is equal
     * to the value of {@code floor(sqrt(n))}, where {@code sqrt(n)} denotes the
     * real square root of {@code n} treated as a real.  Note that the integer
     * square root will be less than the real square root if the latter is not
     * representable as an integral value.
     *
     * @return the integer square root of {@code this}
     * @throws ArithmeticException if {@code this} is negative.  (The square
     *         root of a negative integer {@code val} is
     *         {@code (i * sqrt(-val))} where <i>i</i> is the
     *         <i>imaginary unit</i> and is equal to
     *         {@code sqrt(-1)}.)
     * @since  9
     */
    public BigInteger sqrt() {
        if (this.signum < 0) {
            throw new ArithmeticException("Negative BigInteger");
        }

        return new MutableBigInteger(this.mag).sqrt().toBigInteger();
    }

    /**
     * Returns an array of two BigIntegers containing the integer square root
     * {@code s} of {@code this} and its remainder {@code this - s*s},
     * respectively.
     *
     * @return an array of two BigIntegers with the integer square root at
     *         offset 0 and the remainder at offset 1
     * @throws ArithmeticException if {@code this} is negative.  (The square
     *         root of a negative integer {@code val} is
     *         {@code (i * sqrt(-val))} where <i>i</i> is the
     *         <i>imaginary unit</i> and is equal to
     *         {@code sqrt(-1)}.)
     * @see #sqrt()
     * @since  9
     */
    public BigInteger[] sqrtAndRemainder() {
        BigInteger s = sqrt();
        BigInteger r = this.subtract(s.square());
        assert r.compareTo(BigInteger.ZERO) >= 0;
        return new BigInteger[] {s, r};
    }

    /**
     * Returns a BigInteger whose value is the greatest common divisor of
     * {@code abs(this)} and {@code abs(val)}.  Returns 0 if
     * {@code this == 0 && val == 0}.
     *
     * @param  val value with which the GCD is to be computed.
     * @return {@code GCD(abs(this), abs(val))}
     */
    public BigInteger gcd(BigInteger val) {
        if (val.signum == 0)
            return this.abs();
        else if (this.signum == 0)
            return val.abs();

        MutableBigInteger a = new MutableBigInteger(this);
        MutableBigInteger b = new MutableBigInteger(val);

        MutableBigInteger result = a.hybridGCD(b);

        return result.toBigInteger(1);
    }

    /**
     * Package private method to return bit length for an integer.
     */
    static int bitLengthForInt(int n) {
        return 32 - Integer.numberOfLeadingZeros(n);
    }

    /**
     * Left shift int array a up to len by n bits. Returns the array that
     * results from the shift since space may have to be reallocated.
     */
    private static int[] leftShift(int[] a, int len, int n) {
        int nInts = n >>> 5;
        int nBits = n&0x1F;
        int bitsInHighWord = bitLengthForInt(a[0]);

        // If shift can be done without recopy, do so
        if (n <= (32-bitsInHighWord)) {
            primitiveLeftShift(a, len, nBits);
            return a;
        } else { // Array must be resized
            if (nBits <= (32-bitsInHighWord)) {
                int result[] = new int[nInts+len];
                System.arraycopy(a, 0, result, 0, len);
                primitiveLeftShift(result, result.length, nBits);
                return result;
            } else {
                int result[] = new int[nInts+len+1];
                System.arraycopy(a, 0, result, 0, len);
                primitiveRightShift(result, result.length, 32 - nBits);
                return result;
            }
        }
    }

    // shifts a up to len right n bits assumes no leading zeros, 0<n<32
    static void primitiveRightShift(int[] a, int len, int n) {
        int n2 = 32 - n;
        for (int i=len-1, c=a[i]; i > 0; i--) {
            int b = c;
            c = a[i-1];
            a[i] = (c << n2) | (b >>> n);
        }
        a[0] >>>= n;
    }

    // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
    static void primitiveLeftShift(int[] a, int len, int n) {
        if (len == 0 || n == 0)
            return;

        int n2 = 32 - n;
        for (int i=0, c=a[i], m=i+len-1; i < m; i++) {
            int b = c;
            c = a[i+1];
            a[i] = (b << n) | (c >>> n2);
        }
        a[len-1] <<= n;
    }

    /**
     * Calculate bitlength of contents of the first len elements an int array,
     * assuming there are no leading zero ints.
     */
    private static int bitLength(int[] val, int len) {
        if (len == 0)
            return 0;
        return ((len - 1) << 5) + bitLengthForInt(val[0]);
    }

    /**
     * Returns a BigInteger whose value is the absolute value of this
     * BigInteger.
     *
     * @return {@code abs(this)}
     */
    public BigInteger abs() {
        return (signum >= 0 ? this : this.negate());
    }

    /**
     * Returns a BigInteger whose value is {@code (-this)}.
     *
     * @return {@code -this}
     */
    public BigInteger negate() {
        return new BigInteger(this.mag, -this.signum);
    }

    /**
     * Returns the signum function of this BigInteger.
     *
     * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
     *         positive.
     */
    public int signum() {
        return this.signum;
    }

    // Modular Arithmetic Operations

    /**
     * Returns a BigInteger whose value is {@code (this mod m}).  This method
     * differs from {@code remainder} in that it always returns a
     * <i>non-negative</i> BigInteger.
     *
     * @param  m the modulus.
     * @return {@code this mod m}
     * @throws ArithmeticException {@code m} &le; 0
     * @see    #remainder
     */
    public BigInteger mod(BigInteger m) {
        if (m.signum <= 0)
            throw new ArithmeticException("BigInteger: modulus not positive");

        BigInteger result = this.remainder(m);
        return (result.signum >= 0 ? result : result.add(m));
    }

    /**
     * Returns a BigInteger whose value is
     * <code>(this<sup>exponent</sup> mod m)</code>.  (Unlike {@code pow}, this
     * method permits negative exponents.)
     *
     * @param  exponent the exponent.
     * @param  m the modulus.
     * @return <code>this<sup>exponent</sup> mod m</code>
     * @throws ArithmeticException {@code m} &le; 0 or the exponent is
     *         negative and this BigInteger is not <i>relatively
     *         prime</i> to {@code m}.
     * @see    #modInverse
     */
    public BigInteger modPow(BigInteger exponent, BigInteger m) {
        if (m.signum <= 0)
            throw new ArithmeticException("BigInteger: modulus not positive");

        // Trivial cases
        if (exponent.signum == 0)
            return (m.equals(ONE) ? ZERO : ONE);

        if (this.equals(ONE))
            return (m.equals(ONE) ? ZERO : ONE);

        if (this.equals(ZERO) && exponent.signum >= 0)
            return ZERO;

        if (this.equals(negConst[1]) && (!exponent.testBit(0)))
            return (m.equals(ONE) ? ZERO : ONE);

        boolean invertResult;
        if ((invertResult = (exponent.signum < 0)))
            exponent = exponent.negate();

        BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
                           ? this.mod(m) : this);
        BigInteger result;
        if (m.testBit(0)) { // odd modulus
            result = base.oddModPow(exponent, m);
        } else {
            /*
             * Even modulus.  Tear it into an "odd part" (m1) and power of two
             * (m2), exponentiate mod m1, manually exponentiate mod m2, and
             * use Chinese Remainder Theorem to combine results.
             */

            // Tear m apart into odd part (m1) and power of 2 (m2)
            int p = m.getLowestSetBit();   // Max pow of 2 that divides m

            BigInteger m1 = m.shiftRight(p);  // m/2**p
            BigInteger m2 = ONE.shiftLeft(p); // 2**p

            // Calculate new base from m1
            BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
                                ? this.mod(m1) : this);

            // Caculate (base ** exponent) mod m1.
            BigInteger a1 = (m1.equals(ONE) ? ZERO :
                             base2.oddModPow(exponent, m1));

            // Calculate (this ** exponent) mod m2
            BigInteger a2 = base.modPow2(exponent, p);

            // Combine results using Chinese Remainder Theorem
            BigInteger y1 = m2.modInverse(m1);
            BigInteger y2 = m1.modInverse(m2);

            if (m.mag.length < MAX_MAG_LENGTH / 2) {
                result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
            } else {
                MutableBigInteger t1 = new MutableBigInteger();
                new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
                MutableBigInteger t2 = new MutableBigInteger();
                new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
                t1.add(t2);
                MutableBigInteger q = new MutableBigInteger();
                result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
            }
        }

        return (invertResult ? result.modInverse(m) : result);
    }

    // Montgomery multiplication.  These are wrappers for
    // implMontgomeryXX routines which are expected to be replaced by
    // virtual machine intrinsics.  We don't use the intrinsics for
    // very large operands: MONTGOMERY_INTRINSIC_THRESHOLD should be
    // larger than any reasonable crypto key.
    private static int[] montgomeryMultiply(int[] a, int[] b, int[] n, int len, long inv,
                                            int[] product) {
        implMontgomeryMultiplyChecks(a, b, n, len, product);
        if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
            // Very long argument: do not use an intrinsic
            product = multiplyToLen(a, len, b, len, product);
            return montReduce(product, n, len, (int)inv);
        } else {
            return implMontgomeryMultiply(a, b, n, len, inv, materialize(product, len));
        }
    }
    private static int[] montgomerySquare(int[] a, int[] n, int len, long inv,
                                          int[] product) {
        implMontgomeryMultiplyChecks(a, a, n, len, product);
        if (len > MONTGOMERY_INTRINSIC_THRESHOLD) {
            // Very long argument: do not use an intrinsic
            product = squareToLen(a, len, product);
            return montReduce(product, n, len, (int)inv);
        } else {
            return implMontgomerySquare(a, n, len, inv, materialize(product, len));
        }
    }

    // Range-check everything.
    private static void implMontgomeryMultiplyChecks
        (int[] a, int[] b, int[] n, int len, int[] product) throws RuntimeException {
        if (len % 2 != 0) {
            throw new IllegalArgumentException("input array length must be even: " + len);
        }

        if (len < 1) {
            throw new IllegalArgumentException("invalid input length: " + len);
        }

        if (len > a.length ||
            len > b.length ||
            len > n.length ||
            (product != null && len > product.length)) {
            throw new IllegalArgumentException("input array length out of bound: " + len);
        }
    }

    // Make sure that the int array z (which is expected to contain
    // the result of a Montgomery multiplication) is present and
    // sufficiently large.
    private static int[] materialize(int[] z, int len) {
         if (z == null || z.length < len)
             z = new int[len];
         return z;
    }

    // These methods are intended to be be replaced by virtual machine
    // intrinsics.
    @HotSpotIntrinsicCandidate
    private static int[] implMontgomeryMultiply(int[] a, int[] b, int[] n, int len,
                                         long inv, int[] product) {
        product = multiplyToLen(a, len, b, len, product);
        return montReduce(product, n, len, (int)inv);
    }
    @HotSpotIntrinsicCandidate
    private static int[] implMontgomerySquare(int[] a, int[] n, int len,
                                       long inv, int[] product) {
        product = squareToLen(a, len, product);
        return montReduce(product, n, len, (int)inv);
    }

    static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
                                                Integer.MAX_VALUE}; // Sentinel

    /**
     * Returns a BigInteger whose value is x to the power of y mod z.
     * Assumes: z is odd && x < z.
     */
    private BigInteger oddModPow(BigInteger y, BigInteger z) {
    /*
     * The algorithm is adapted from Colin Plumb's C library.
     *
     * The window algorithm:
     * The idea is to keep a running product of b1 = n^(high-order bits of exp)
     * and then keep appending exponent bits to it.  The following patterns
     * apply to a 3-bit window (k = 3):
     * To append   0: square
     * To append   1: square, multiply by n^1
     * To append  10: square, multiply by n^1, square
     * To append  11: square, square, multiply by n^3
     * To append 100: square, multiply by n^1, square, square
     * To append 101: square, square, square, multiply by n^5
     * To append 110: square, square, multiply by n^3, square
     * To append 111: square, square, square, multiply by n^7
     *
     * Since each pattern involves only one multiply, the longer the pattern
     * the better, except that a 0 (no multiplies) can be appended directly.
     * We precompute a table of odd powers of n, up to 2^k, and can then
     * multiply k bits of exponent at a time.  Actually, assuming random
     * exponents, there is on average one zero bit between needs to
     * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
     * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
     * you have to do one multiply per k+1 bits of exponent.
     *
     * The loop walks down the exponent, squaring the result buffer as
     * it goes.  There is a wbits+1 bit lookahead buffer, buf, that is
     * filled with the upcoming exponent bits.  (What is read after the
     * end of the exponent is unimportant, but it is filled with zero here.)
     * When the most-significant bit of this buffer becomes set, i.e.
     * (buf & tblmask) != 0, we have to decide what pattern to multiply
     * by, and when to do it.  We decide, remember to do it in future
     * after a suitable number of squarings have passed (e.g. a pattern
     * of "100" in the buffer requires that we multiply by n^1 immediately;
     * a pattern of "110" calls for multiplying by n^3 after one more
     * squaring), clear the buffer, and continue.
     *
     * When we start, there is one more optimization: the result buffer
     * is implcitly one, so squaring it or multiplying by it can be
     * optimized away.  Further, if we start with a pattern like "100"
     * in the lookahead window, rather than placing n into the buffer
     * and then starting to square it, we have already computed n^2
     * to compute the odd-powers table, so we can place that into
     * the buffer and save a squaring.
     *
     * This means that if you have a k-bit window, to compute n^z,
     * where z is the high k bits of the exponent, 1/2 of the time
     * it requires no squarings.  1/4 of the time, it requires 1
     * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
     * And the remaining 1/2^(k-1) of the time, the top k bits are a
     * 1 followed by k-1 0 bits, so it again only requires k-2
     * squarings, not k-1.  The average of these is 1.  Add that
     * to the one squaring we have to do to compute the table,
     * and you'll see that a k-bit window saves k-2 squarings
     * as well as reducing the multiplies.  (It actually doesn't
     * hurt in the case k = 1, either.)
     */
        // Special case for exponent of one
        if (y.equals(ONE))
            return this;

        // Special case for base of zero
        if (signum == 0)
            return ZERO;

        int[] base = mag.clone();
        int[] exp = y.mag;
        int[] mod = z.mag;
        int modLen = mod.length;

        // Make modLen even. It is conventional to use a cryptographic
        // modulus that is 512, 768, 1024, or 2048 bits, so this code
        // will not normally be executed. However, it is necessary for
        // the correct functioning of the HotSpot intrinsics.
        if ((modLen & 1) != 0) {
            int[] x = new int[modLen + 1];
            System.arraycopy(mod, 0, x, 1, modLen);
            mod = x;
            modLen++;
        }

        // Select an appropriate window size
        int wbits = 0;
        int ebits = bitLength(exp, exp.length);
        // if exponent is 65537 (0x10001), use minimum window size
        if ((ebits != 17) || (exp[0] != 65537)) {
            while (ebits > bnExpModThreshTable[wbits]) {
                wbits++;
            }
        }

        // Calculate appropriate table size
        int tblmask = 1 << wbits;

        // Allocate table for precomputed odd powers of base in Montgomery form
        int[][] table = new int[tblmask][];
        for (int i=0; i < tblmask; i++)
            table[i] = new int[modLen];

        // Compute the modular inverse of the least significant 64-bit
        // digit of the modulus
        long n0 = (mod[modLen-1] & LONG_MASK) + ((mod[modLen-2] & LONG_MASK) << 32);
        long inv = -MutableBigInteger.inverseMod64(n0);

        // Convert base to Montgomery form
        int[] a = leftShift(base, base.length, modLen << 5);

        MutableBigInteger q = new MutableBigInteger(),
                          a2 = new MutableBigInteger(a),
                          b2 = new MutableBigInteger(mod);
        b2.normalize(); // MutableBigInteger.divide() assumes that its
                        // divisor is in normal form.

        MutableBigInteger r= a2.divide(b2, q);
        table[0] = r.toIntArray();

        // Pad table[0] with leading zeros so its length is at least modLen
        if (table[0].length < modLen) {
           int offset = modLen - table[0].length;
           int[] t2 = new int[modLen];
           System.arraycopy(table[0], 0, t2, offset, table[0].length);
           table[0] = t2;
        }

        // Set b to the square of the base
        int[] b = montgomerySquare(table[0], mod, modLen, inv, null);

        // Set t to high half of b
        int[] t = Arrays.copyOf(b, modLen);

        // Fill in the table with odd powers of the base
        for (int i=1; i < tblmask; i++) {
            table[i] = montgomeryMultiply(t, table[i-1], mod, modLen, inv, null);
        }

        // Pre load the window that slides over the exponent
        int bitpos = 1 << ((ebits-1) & (32-1));

        int buf = 0;
        int elen = exp.length;
        int eIndex = 0;
        for (int i = 0; i <= wbits; i++) {
            buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0);
            bitpos >>>= 1;
            if (bitpos == 0) {
                eIndex++;
                bitpos = 1 << (32-1);
                elen--;
            }
        }

        int multpos = ebits;

        // The first iteration, which is hoisted out of the main loop
        ebits--;
        boolean isone = true;

        multpos = ebits - wbits;
        while ((buf & 1) == 0) {
            buf >>>= 1;
            multpos++;
        }

        int[] mult = table[buf >>> 1];

        buf = 0;
        if (multpos == ebits)
            isone = false;

        // The main loop
        while (true) {
            ebits--;
            // Advance the window
            buf <<= 1;

            if (elen != 0) {
                buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
                bitpos >>>= 1;
                if (bitpos == 0) {
                    eIndex++;
                    bitpos = 1 << (32-1);
                    elen--;
                }
            }

            // Examine the window for pending multiplies
            if ((buf & tblmask) != 0) {
                multpos = ebits - wbits;
                while ((buf & 1) == 0) {
                    buf >>>= 1;
                    multpos++;
                }
                mult = table[buf >>> 1];
                buf = 0;
            }

            // Perform multiply
            if (ebits == multpos) {
                if (isone) {
                    b = mult.clone();
                    isone = false;
                } else {
                    t = b;
                    a = montgomeryMultiply(t, mult, mod, modLen, inv, a);
                    t = a; a = b; b = t;
                }
            }

            // Check if done
            if (ebits == 0)
                break;

            // Square the input
            if (!isone) {
                t = b;
                a = montgomerySquare(t, mod, modLen, inv, a);
                t = a; a = b; b = t;
            }
        }

        // Convert result out of Montgomery form and return
        int[] t2 = new int[2*modLen];
        System.arraycopy(b, 0, t2, modLen, modLen);

        b = montReduce(t2, mod, modLen, (int)inv);

        t2 = Arrays.copyOf(b, modLen);

        return new BigInteger(1, t2);
    }

    /**
     * Montgomery reduce n, modulo mod.  This reduces modulo mod and divides
     * by 2^(32*mlen). Adapted from Colin Plumb's C library.
     */
    private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
        int c=0;
        int len = mlen;
        int offset=0;

        do {
            int nEnd = n[n.length-1-offset];
            int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
            c += addOne(n, offset, mlen, carry);
            offset++;
        } while (--len > 0);

        while (c > 0)
            c += subN(n, mod, mlen);

        while (intArrayCmpToLen(n, mod, mlen) >= 0)
            subN(n, mod, mlen);

        return n;
    }


    /*
     * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
     * equal to, or greater than arg2 up to length len.
     */
    private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
        for (int i=0; i < len; i++) {
            long b1 = arg1[i] & LONG_MASK;
            long b2 = arg2[i] & LONG_MASK;
            if (b1 < b2)
                return -1;
            if (b1 > b2)
                return 1;
        }
        return 0;
    }

    /**
     * Subtracts two numbers of same length, returning borrow.
     */
    private static int subN(int[] a, int[] b, int len) {
        long sum = 0;

        while (--len >= 0) {
            sum = (a[len] & LONG_MASK) -
                 (b[len] & LONG_MASK) + (sum >> 32);
            a[len] = (int)sum;
        }

        return (int)(sum >> 32);
    }

    /**
     * Multiply an array by one word k and add to result, return the carry
     */
    static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
        implMulAddCheck(out, in, offset, len, k);
        return implMulAdd(out, in, offset, len, k);
    }

    /**
     * Parameters validation.
     */
    private static void implMulAddCheck(int[] out, int[] in, int offset, int len, int k) {
        if (len > in.length) {
            throw new IllegalArgumentException("input length is out of bound: " + len + " > " + in.length);
        }
        if (offset < 0) {
            throw new IllegalArgumentException("input offset is invalid: " + offset);
        }
        if (offset > (out.length - 1)) {
            throw new IllegalArgumentException("input offset is out of bound: " + offset + " > " + (out.length - 1));
        }
        if (len > (out.length - offset)) {
            throw new IllegalArgumentException("input len is out of bound: " + len + " > " + (out.length - offset));
        }
    }

    /**
     * Java Runtime may use intrinsic for this method.
     */
    @HotSpotIntrinsicCandidate
    private static int implMulAdd(int[] out, int[] in, int offset, int len, int k) {
        long kLong = k & LONG_MASK;
        long carry = 0;

        offset = out.length-offset - 1;
        for (int j=len-1; j >= 0; j--) {
            long product = (in[j] & LONG_MASK) * kLong +
                           (out[offset] & LONG_MASK) + carry;
            out[offset--] = (int)product;
            carry = product >>> 32;
        }
        return (int)carry;
    }

    /**
     * Add one word to the number a mlen words into a. Return the resulting
     * carry.
     */
    static int addOne(int[] a, int offset, int mlen, int carry) {
        offset = a.length-1-mlen-offset;
        long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);

        a[offset] = (int)t;
        if ((t >>> 32) == 0)
            return 0;
        while (--mlen >= 0) {
            if (--offset < 0) { // Carry out of number
                return 1;
            } else {
                a[offset]++;
                if (a[offset] != 0)
                    return 0;
            }
        }
        return 1;
    }

    /**
     * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
     */
    private BigInteger modPow2(BigInteger exponent, int p) {
        /*
         * Perform exponentiation using repeated squaring trick, chopping off
         * high order bits as indicated by modulus.
         */
        BigInteger result = ONE;
        BigInteger baseToPow2 = this.mod2(p);
        int expOffset = 0;

        int limit = exponent.bitLength();

        if (this.testBit(0))
           limit = (p-1) < limit ? (p-1) : limit;

        while (expOffset < limit) {
            if (exponent.testBit(expOffset))
                result = result.multiply(baseToPow2).mod2(p);
            expOffset++;
            if (expOffset < limit)
                baseToPow2 = baseToPow2.square().mod2(p);
        }

        return result;
    }

    /**
     * Returns a BigInteger whose value is this mod(2**p).
     * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
     */
    private BigInteger mod2(int p) {
        if (bitLength() <= p)
            return this;

        // Copy remaining ints of mag
        int numInts = (p + 31) >>> 5;
        int[] mag = new int[numInts];
        System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);

        // Mask out any excess bits
        int excessBits = (numInts << 5) - p;
        mag[0] &= (1L << (32-excessBits)) - 1;

        return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
    }

    /**
     * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
     *
     * @param  m the modulus.
     * @return {@code this}<sup>-1</sup> {@code mod m}.
     * @throws ArithmeticException {@code  m} &le; 0, or this BigInteger
     *         has no multiplicative inverse mod m (that is, this BigInteger
     *         is not <i>relatively prime</i> to m).
     */
    public BigInteger modInverse(BigInteger m) {
        if (m.signum != 1)
            throw new ArithmeticException("BigInteger: modulus not positive");

        if (m.equals(ONE))
            return ZERO;

        // Calculate (this mod m)
        BigInteger modVal = this;
        if (signum < 0 || (this.compareMagnitude(m) >= 0))
            modVal = this.mod(m);

        if (modVal.equals(ONE))
            return ONE;

        MutableBigInteger a = new MutableBigInteger(modVal);
        MutableBigInteger b = new MutableBigInteger(m);

        MutableBigInteger result = a.mutableModInverse(b);
        return result.toBigInteger(1);
    }

    // Shift Operations

    /**
     * Returns a BigInteger whose value is {@code (this << n)}.
     * The shift distance, {@code n}, may be negative, in which case
     * this method performs a right shift.
     * (Computes <code>floor(this * 2<sup>n</sup>)</code>.)
     *
     * @param  n shift distance, in bits.
     * @return {@code this << n}
     * @see #shiftRight
     */
    public BigInteger shiftLeft(int n) {
        if (signum == 0)
            return ZERO;
        if (n > 0) {
            return new BigInteger(shiftLeft(mag, n), signum);
        } else if (n == 0) {
            return this;
        } else {
            // Possible int overflow in (-n) is not a trouble,
            // because shiftRightImpl considers its argument unsigned
            return shiftRightImpl(-n);
        }
    }

    /**
     * Returns a magnitude array whose value is {@code (mag << n)}.
     * The shift distance, {@code n}, is considered unnsigned.
     * (Computes <code>this * 2<sup>n</sup></code>.)
     *
     * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
     * @param  n unsigned shift distance, in bits.
     * @return {@code mag << n}
     */
    private static int[] shiftLeft(int[] mag, int n) {
        int nInts = n >>> 5;
        int nBits = n & 0x1f;
        int magLen = mag.length;
        int newMag[] = null;

        if (nBits == 0) {
            newMag = new int[magLen + nInts];
            System.arraycopy(mag, 0, newMag, 0, magLen);
        } else {
            int i = 0;
            int nBits2 = 32 - nBits;
            int highBits = mag[0] >>> nBits2;
            if (highBits != 0) {
                newMag = new int[magLen + nInts + 1];
                newMag[i++] = highBits;
            } else {
                newMag = new int[magLen + nInts];
            }
            int j=0;
            while (j < magLen-1)
                newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
            newMag[i] = mag[j] << nBits;
        }
        return newMag;
    }

    /**
     * Returns a BigInteger whose value is {@code (this >> n)}.  Sign
     * extension is performed.  The shift distance, {@code n}, may be
     * negative, in which case this method performs a left shift.
     * (Computes <code>floor(this / 2<sup>n</sup>)</code>.)
     *
     * @param  n shift distance, in bits.
     * @return {@code this >> n}
     * @see #shiftLeft
     */
    public BigInteger shiftRight(int n) {
        if (signum == 0)
            return ZERO;
        if (n > 0) {
            return shiftRightImpl(n);
        } else if (n == 0) {
            return this;
        } else {
            // Possible int overflow in {@code -n} is not a trouble,
            // because shiftLeft considers its argument unsigned
            return new BigInteger(shiftLeft(mag, -n), signum);
        }
    }

    /**
     * Returns a BigInteger whose value is {@code (this >> n)}. The shift
     * distance, {@code n}, is considered unsigned.
     * (Computes <code>floor(this * 2<sup>-n</sup>)</code>.)
     *
     * @param  n unsigned shift distance, in bits.
     * @return {@code this >> n}
     */
    private BigInteger shiftRightImpl(int n) {
        int nInts = n >>> 5;
        int nBits = n & 0x1f;
        int magLen = mag.length;
        int newMag[] = null;

        // Special case: entire contents shifted off the end
        if (nInts >= magLen)
            return (signum >= 0 ? ZERO : negConst[1]);

        if (nBits == 0) {
            int newMagLen = magLen - nInts;
            newMag = Arrays.copyOf(mag, newMagLen);
        } else {
            int i = 0;
            int highBits = mag[0] >>> nBits;
            if (highBits != 0) {
                newMag = new int[magLen - nInts];
                newMag[i++] = highBits;
            } else {
                newMag = new int[magLen - nInts -1];
            }

            int nBits2 = 32 - nBits;
            int j=0;
            while (j < magLen - nInts - 1)
                newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
        }

        if (signum < 0) {
            // Find out whether any one-bits were shifted off the end.
            boolean onesLost = false;
            for (int i=magLen-1, j=magLen-nInts; i >= j && !onesLost; i--)
                onesLost = (mag[i] != 0);
            if (!onesLost && nBits != 0)
                onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);

            if (onesLost)
                newMag = javaIncrement(newMag);
        }

        return new BigInteger(newMag, signum);
    }

    int[] javaIncrement(int[] val) {
        int lastSum = 0;
        for (int i=val.length-1;  i >= 0 && lastSum == 0; i--)
            lastSum = (val[i] += 1);
        if (lastSum == 0) {
            val = new int[val.length+1];
            val[0] = 1;
        }
        return val;
    }

    // Bitwise Operations

    /**
     * Returns a BigInteger whose value is {@code (this & val)}.  (This
     * method returns a negative BigInteger if and only if this and val are
     * both negative.)
     *
     * @param val value to be AND'ed with this BigInteger.
     * @return {@code this & val}
     */
    public BigInteger and(BigInteger val) {
        int[] result = new int[Math.max(intLength(), val.intLength())];
        for (int i=0; i < result.length; i++)
            result[i] = (getInt(result.length-i-1)
                         & val.getInt(result.length-i-1));

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is {@code (this | val)}.  (This method
     * returns a negative BigInteger if and only if either this or val is
     * negative.)
     *
     * @param val value to be OR'ed with this BigInteger.
     * @return {@code this | val}
     */
    public BigInteger or(BigInteger val) {
        int[] result = new int[Math.max(intLength(), val.intLength())];
        for (int i=0; i < result.length; i++)
            result[i] = (getInt(result.length-i-1)
                         | val.getInt(result.length-i-1));

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is {@code (this ^ val)}.  (This method
     * returns a negative BigInteger if and only if exactly one of this and
     * val are negative.)
     *
     * @param val value to be XOR'ed with this BigInteger.
     * @return {@code this ^ val}
     */
    public BigInteger xor(BigInteger val) {
        int[] result = new int[Math.max(intLength(), val.intLength())];
        for (int i=0; i < result.length; i++)
            result[i] = (getInt(result.length-i-1)
                         ^ val.getInt(result.length-i-1));

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is {@code (~this)}.  (This method
     * returns a negative value if and only if this BigInteger is
     * non-negative.)
     *
     * @return {@code ~this}
     */
    public BigInteger not() {
        int[] result = new int[intLength()];
        for (int i=0; i < result.length; i++)
            result[i] = ~getInt(result.length-i-1);

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is {@code (this & ~val)}.  This
     * method, which is equivalent to {@code and(val.not())}, is provided as
     * a convenience for masking operations.  (This method returns a negative
     * BigInteger if and only if {@code this} is negative and {@code val} is
     * positive.)
     *
     * @param val value to be complemented and AND'ed with this BigInteger.
     * @return {@code this & ~val}
     */
    public BigInteger andNot(BigInteger val) {
        int[] result = new int[Math.max(intLength(), val.intLength())];
        for (int i=0; i < result.length; i++)
            result[i] = (getInt(result.length-i-1)
                         & ~val.getInt(result.length-i-1));

        return valueOf(result);
    }


    // Single Bit Operations

    /**
     * Returns {@code true} if and only if the designated bit is set.
     * (Computes {@code ((this & (1<<n)) != 0)}.)
     *
     * @param  n index of bit to test.
     * @return {@code true} if and only if the designated bit is set.
     * @throws ArithmeticException {@code n} is negative.
     */
    public boolean testBit(int n) {
        if (n < 0)
            throw new ArithmeticException("Negative bit address");

        return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
    }

    /**
     * Returns a BigInteger whose value is equivalent to this BigInteger
     * with the designated bit set.  (Computes {@code (this | (1<<n))}.)
     *
     * @param  n index of bit to set.
     * @return {@code this | (1<<n)}
     * @throws ArithmeticException {@code n} is negative.
     */
    public BigInteger setBit(int n) {
        if (n < 0)
            throw new ArithmeticException("Negative bit address");

        int intNum = n >>> 5;
        int[] result = new int[Math.max(intLength(), intNum+2)];

        for (int i=0; i < result.length; i++)
            result[result.length-i-1] = getInt(i);

        result[result.length-intNum-1] |= (1 << (n & 31));

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is equivalent to this BigInteger
     * with the designated bit cleared.
     * (Computes {@code (this & ~(1<<n))}.)
     *
     * @param  n index of bit to clear.
     * @return {@code this & ~(1<<n)}
     * @throws ArithmeticException {@code n} is negative.
     */
    public BigInteger clearBit(int n) {
        if (n < 0)
            throw new ArithmeticException("Negative bit address");

        int intNum = n >>> 5;
        int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];

        for (int i=0; i < result.length; i++)
            result[result.length-i-1] = getInt(i);

        result[result.length-intNum-1] &= ~(1 << (n & 31));

        return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is equivalent to this BigInteger
     * with the designated bit flipped.
     * (Computes {@code (this ^ (1<<n))}.)
     *
     * @param  n index of bit to flip.
     * @return {@code this ^ (1<<n)}
     * @throws ArithmeticException {@code n} is negative.
     */
    public BigInteger flipBit(int n) {
        if (n < 0)
            throw new ArithmeticException("Negative bit address");

        int intNum = n >>> 5;
        int[] result = new int[Math.max(intLength(), intNum+2)];

        for (int i=0; i < result.length; i++)
            result[result.length-i-1] = getInt(i);

        result[result.length-intNum-1] ^= (1 << (n & 31));

        return valueOf(result);
    }

    /**
     * Returns the index of the rightmost (lowest-order) one bit in this
     * BigInteger (the number of zero bits to the right of the rightmost
     * one bit).  Returns -1 if this BigInteger contains no one bits.
     * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
     *
     * @return index of the rightmost one bit in this BigInteger.
     */
    public int getLowestSetBit() {
        int lsb = lowestSetBitPlusTwo - 2;
        if (lsb == -2) {  // lowestSetBit not initialized yet
            lsb = 0;
            if (signum == 0) {
                lsb -= 1;
            } else {
                // Search for lowest order nonzero int
                int i,b;
                for (i=0; (b = getInt(i)) == 0; i++)
                    ;
                lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
            }
            lowestSetBitPlusTwo = lsb + 2;
        }
        return lsb;
    }


    // Miscellaneous Bit Operations

    /**
     * Returns the number of bits in the minimal two's-complement
     * representation of this BigInteger, <em>excluding</em> a sign bit.
     * For positive BigIntegers, this is equivalent to the number of bits in
     * the ordinary binary representation.  (Computes
     * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
     *
     * @return number of bits in the minimal two's-complement
     *         representation of this BigInteger, <em>excluding</em> a sign bit.
     */
    public int bitLength() {
        int n = bitLengthPlusOne - 1;
        if (n == -1) { // bitLength not initialized yet
            int[] m = mag;
            int len = m.length;
            if (len == 0) {
                n = 0; // offset by one to initialize
            }  else {
                // Calculate the bit length of the magnitude
                int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
                 if (signum < 0) {
                     // Check if magnitude is a power of two
                     boolean pow2 = (Integer.bitCount(mag[0]) == 1);
                     for (int i=1; i< len && pow2; i++)
                         pow2 = (mag[i] == 0);

                     n = (pow2 ? magBitLength -1 : magBitLength);
                 } else {
                     n = magBitLength;
                 }
            }
            bitLengthPlusOne = n + 1;
        }
        return n;
    }

    /**
     * Returns the number of bits in the two's complement representation
     * of this BigInteger that differ from its sign bit.  This method is
     * useful when implementing bit-vector style sets atop BigIntegers.
     *
     * @return number of bits in the two's complement representation
     *         of this BigInteger that differ from its sign bit.
     */
    public int bitCount() {
        int bc = bitCountPlusOne - 1;
        if (bc == -1) {  // bitCount not initialized yet
            bc = 0;      // offset by one to initialize
            // Count the bits in the magnitude
            for (int i=0; i < mag.length; i++)
                bc += Integer.bitCount(mag[i]);
            if (signum < 0) {
                // Count the trailing zeros in the magnitude
                int magTrailingZeroCount = 0, j;
                for (j=mag.length-1; mag[j] == 0; j--)
                    magTrailingZeroCount += 32;
                magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
                bc += magTrailingZeroCount - 1;
            }
            bitCountPlusOne = bc + 1;
        }
        return bc;
    }

    // Primality Testing

    /**
     * Returns {@code true} if this BigInteger is probably prime,
     * {@code false} if it's definitely composite.  If
     * {@code certainty} is &le; 0, {@code true} is
     * returned.
     *
     * @param  certainty a measure of the uncertainty that the caller is
     *         willing to tolerate: if the call returns {@code true}
     *         the probability that this BigInteger is prime exceeds
     *         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
     *         this method is proportional to the value of this parameter.
     * @return {@code true} if this BigInteger is probably prime,
     *         {@code false} if it's definitely composite.
     */
    public boolean isProbablePrime(int certainty) {
        if (certainty <= 0)
            return true;
        BigInteger w = this.abs();
        if (w.equals(TWO))
            return true;
        if (!w.testBit(0) || w.equals(ONE))
            return false;

        return w.primeToCertainty(certainty, null);
    }

    // Comparison Operations

    /**
     * Compares this BigInteger with the specified BigInteger.  This
     * method is provided in preference to individual methods for each
     * of the six boolean comparison operators ({@literal <}, ==,
     * {@literal >}, {@literal >=}, !=, {@literal <=}).  The suggested
     * idiom for performing these comparisons is: {@code
     * (x.compareTo(y)} &lt;<i>op</i>&gt; {@code 0)}, where
     * &lt;<i>op</i>&gt; is one of the six comparison operators.
     *
     * @param  val BigInteger to which this BigInteger is to be compared.
     * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
     *         to, or greater than {@code val}.
     */
    public int compareTo(BigInteger val) {
        if (signum == val.signum) {
            switch (signum) {
            case 1:
                return compareMagnitude(val);
            case -1:
                return val.compareMagnitude(this);
            default:
                return 0;
            }
        }
        return signum > val.signum ? 1 : -1;
    }

    /**
     * Compares the magnitude array of this BigInteger with the specified
     * BigInteger's. This is the version of compareTo ignoring sign.
     *
     * @param val BigInteger whose magnitude array to be compared.
     * @return -1, 0 or 1 as this magnitude array is less than, equal to or
     *         greater than the magnitude aray for the specified BigInteger's.
     */
    final int compareMagnitude(BigInteger val) {
        int[] m1 = mag;
        int len1 = m1.length;
        int[] m2 = val.mag;
        int len2 = m2.length;
        if (len1 < len2)
            return -1;
        if (len1 > len2)
            return 1;
        for (int i = 0; i < len1; i++) {
            int a = m1[i];
            int b = m2[i];
            if (a != b)
                return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
        }
        return 0;
    }

    /**
     * Version of compareMagnitude that compares magnitude with long value.
     * val can't be Long.MIN_VALUE.
     */
    final int compareMagnitude(long val) {
        assert val != Long.MIN_VALUE;
        int[] m1 = mag;
        int len = m1.length;
        if (len > 2) {
            return 1;
        }
        if (val < 0) {
            val = -val;
        }
        int highWord = (int)(val >>> 32);
        if (highWord == 0) {
            if (len < 1)
                return -1;
            if (len > 1)
                return 1;
            int a = m1[0];
            int b = (int)val;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            return 0;
        } else {
            if (len < 2)
                return -1;
            int a = m1[0];
            int b = highWord;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            a = m1[1];
            b = (int)val;
            if (a != b) {
                return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
            }
            return 0;
        }
    }

    /**
     * Compares this BigInteger with the specified Object for equality.
     *
     * @param  x Object to which this BigInteger is to be compared.
     * @return {@code true} if and only if the specified Object is a
     *         BigInteger whose value is numerically equal to this BigInteger.
     */
    public boolean equals(Object x) {
        // This test is just an optimization, which may or may not help
        if (x == this)
            return true;

        if (!(x instanceof BigInteger))
            return false;

        BigInteger xInt = (BigInteger) x;
        if (xInt.signum != signum)
            return false;

        int[] m = mag;
        int len = m.length;
        int[] xm = xInt.mag;
        if (len != xm.length)
            return false;

        for (int i = 0; i < len; i++)
            if (xm[i] != m[i])
                return false;

        return true;
    }

    /**
     * Returns the minimum of this BigInteger and {@code val}.
     *
     * @param  val value with which the minimum is to be computed.
     * @return the BigInteger whose value is the lesser of this BigInteger and
     *         {@code val}.  If they are equal, either may be returned.
     */
    public BigInteger min(BigInteger val) {
        return (compareTo(val) < 0 ? this : val);
    }

    /**
     * Returns the maximum of this BigInteger and {@code val}.
     *
     * @param  val value with which the maximum is to be computed.
     * @return the BigInteger whose value is the greater of this and
     *         {@code val}.  If they are equal, either may be returned.
     */
    public BigInteger max(BigInteger val) {
        return (compareTo(val) > 0 ? this : val);
    }


    // Hash Function

    /**
     * Returns the hash code for this BigInteger.
     *
     * @return hash code for this BigInteger.
     */
    public int hashCode() {
        int hashCode = 0;

        for (int i=0; i < mag.length; i++)
            hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));

        return hashCode * signum;
    }

    /**
     * Returns the String representation of this BigInteger in the
     * given radix.  If the radix is outside the range from {@link
     * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
     * it will default to 10 (as is the case for
     * {@code Integer.toString}).  The digit-to-character mapping
     * provided by {@code Character.forDigit} is used, and a minus
     * sign is prepended if appropriate.  (This representation is
     * compatible with the {@link #BigInteger(String, int) (String,
     * int)} constructor.)
     *
     * @param  radix  radix of the String representation.
     * @return String representation of this BigInteger in the given radix.
     * @see    Integer#toString
     * @see    Character#forDigit
     * @see    #BigInteger(java.lang.String, int)
     */
    public String toString(int radix) {
        if (signum == 0)
            return "0";
        if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
            radix = 10;

        // If it's small enough, use smallToString.
        if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
           return smallToString(radix);

        // Otherwise use recursive toString, which requires positive arguments.
        // The results will be concatenated into this StringBuilder
        StringBuilder sb = new StringBuilder();
        if (signum < 0) {
            toString(this.negate(), sb, radix, 0);
            sb.insert(0, '-');
        }
        else
            toString(this, sb, radix, 0);

        return sb.toString();
    }

    /** This method is used to perform toString when arguments are small. */
    private String smallToString(int radix) {
        if (signum == 0) {
            return "0";
        }

        // Compute upper bound on number of digit groups and allocate space
        int maxNumDigitGroups = (4*mag.length + 6)/7;
        String digitGroup[] = new String[maxNumDigitGroups];

        // Translate number to string, a digit group at a time
        BigInteger tmp = this.abs();
        int numGroups = 0;
        while (tmp.signum != 0) {
            BigInteger d = longRadix[radix];

            MutableBigInteger q = new MutableBigInteger(),
                              a = new MutableBigInteger(tmp.mag),
                              b = new MutableBigInteger(d.mag);
            MutableBigInteger r = a.divide(b, q);
            BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
            BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);

            digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
            tmp = q2;
        }

        // Put sign (if any) and first digit group into result buffer
        StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1);
        if (signum < 0) {
            buf.append('-');
        }
        buf.append(digitGroup[numGroups-1]);

        // Append remaining digit groups padded with leading zeros
        for (int i=numGroups-2; i >= 0; i--) {
            // Prepend (any) leading zeros for this digit group
            int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length();
            if (numLeadingZeros != 0) {
                buf.append(zeros[numLeadingZeros]);
            }
            buf.append(digitGroup[i]);
        }
        return buf.toString();
    }

    /**
     * Converts the specified BigInteger to a string and appends to
     * {@code sb}.  This implements the recursive Schoenhage algorithm
     * for base conversions.
     * <p>
     * See Knuth, Donald,  _The Art of Computer Programming_, Vol. 2,
     * Answers to Exercises (4.4) Question 14.
     *
     * @param u      The number to convert to a string.
     * @param sb     The StringBuilder that will be appended to in place.
     * @param radix  The base to convert to.
     * @param digits The minimum number of digits to pad to.
     */
    private static void toString(BigInteger u, StringBuilder sb, int radix,
                                 int digits) {
        // If we're smaller than a certain threshold, use the smallToString
        // method, padding with leading zeroes when necessary.
        if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
            String s = u.smallToString(radix);

            // Pad with internal zeros if necessary.
            // Don't pad if we're at the beginning of the string.
            if ((s.length() < digits) && (sb.length() > 0)) {
                for (int i=s.length(); i < digits; i++) {
                    sb.append('0');
                }
            }

            sb.append(s);
            return;
        }

        int b, n;
        b = u.bitLength();

        // Calculate a value for n in the equation radix^(2^n) = u
        // and subtract 1 from that value.  This is used to find the
        // cache index that contains the best value to divide u.
        n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
        BigInteger v = getRadixConversionCache(radix, n);
        BigInteger[] results;
        results = u.divideAndRemainder(v);

        int expectedDigits = 1 << n;

        // Now recursively build the two halves of each number.
        toString(results[0], sb, radix, digits-expectedDigits);
        toString(results[1], sb, radix, expectedDigits);
    }

    /**
     * Returns the value radix^(2^exponent) from the cache.
     * If this value doesn't already exist in the cache, it is added.
     * <p>
     * This could be changed to a more complicated caching method using
     * {@code Future}.
     */
    private static BigInteger getRadixConversionCache(int radix, int exponent) {
        BigInteger[] cacheLine = powerCache[radix]; // volatile read
        if (exponent < cacheLine.length) {
            return cacheLine[exponent];
        }

        int oldLength = cacheLine.length;
        cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
        for (int i = oldLength; i <= exponent; i++) {
            cacheLine[i] = cacheLine[i - 1].pow(2);
        }

        BigInteger[][] pc = powerCache; // volatile read again
        if (exponent >= pc[radix].length) {
            pc = pc.clone();
            pc[radix] = cacheLine;
            powerCache = pc; // volatile write, publish
        }
        return cacheLine[exponent];
    }

    /* zero[i] is a string of i consecutive zeros. */
    private static String zeros[] = new String[64];
    static {
        zeros[63] =
            "000000000000000000000000000000000000000000000000000000000000000";
        for (int i=0; i < 63; i++)
            zeros[i] = zeros[63].substring(0, i);
    }

    /**
     * Returns the decimal String representation of this BigInteger.
     * The digit-to-character mapping provided by
     * {@code Character.forDigit} is used, and a minus sign is
     * prepended if appropriate.  (This representation is compatible
     * with the {@link #BigInteger(String) (String)} constructor, and
     * allows for String concatenation with Java's + operator.)
     *
     * @return decimal String representation of this BigInteger.
     * @see    Character#forDigit
     * @see    #BigInteger(java.lang.String)
     */
    public String toString() {
        return toString(10);
    }

    /**
     * Returns a byte array containing the two's-complement
     * representation of this BigInteger.  The byte array will be in
     * <i>big-endian</i> byte-order: the most significant byte is in
     * the zeroth element.  The array will contain the minimum number
     * of bytes required to represent this BigInteger, including at
     * least one sign bit, which is {@code (ceil((this.bitLength() +
     * 1)/8))}.  (This representation is compatible with the
     * {@link #BigInteger(byte[]) (byte[])} constructor.)
     *
     * @return a byte array containing the two's-complement representation of
     *         this BigInteger.
     * @see    #BigInteger(byte[])
     */
    public byte[] toByteArray() {
        int byteLen = bitLength()/8 + 1;
        byte[] byteArray = new byte[byteLen];

        for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i >= 0; i--) {
            if (bytesCopied == 4) {
                nextInt = getInt(intIndex++);
                bytesCopied = 1;
            } else {
                nextInt >>>= 8;
                bytesCopied++;
            }
            byteArray[i] = (byte)nextInt;
        }
        return byteArray;
    }

    /**
     * Converts this BigInteger to an {@code int}.  This
     * conversion is analogous to a
     * <i>narrowing primitive conversion</i> from {@code long} to
     * {@code int} as defined in
     * <cite>The Java&trade; Language Specification</cite>:
     * if this BigInteger is too big to fit in an
     * {@code int}, only the low-order 32 bits are returned.
     * Note that this conversion can lose information about the
     * overall magnitude of the BigInteger value as well as return a
     * result with the opposite sign.
     *
     * @return this BigInteger converted to an {@code int}.
     * @see #intValueExact()
     * @jls 5.1.3 Narrowing Primitive Conversion
     */
    public int intValue() {
        int result = 0;
        result = getInt(0);
        return result;
    }

    /**
     * Converts this BigInteger to a {@code long}.  This
     * conversion is analogous to a
     * <i>narrowing primitive conversion</i> from {@code long} to
     * {@code int} as defined in
     * <cite>The Java&trade; Language Specification</cite>:
     * if this BigInteger is too big to fit in a
     * {@code long}, only the low-order 64 bits are returned.
     * Note that this conversion can lose information about the
     * overall magnitude of the BigInteger value as well as return a
     * result with the opposite sign.
     *
     * @return this BigInteger converted to a {@code long}.
     * @see #longValueExact()
     * @jls 5.1.3 Narrowing Primitive Conversion
     */
    public long longValue() {
        long result = 0;

        for (int i=1; i >= 0; i--)
            result = (result << 32) + (getInt(i) & LONG_MASK);
        return result;
    }

    /**
     * Converts this BigInteger to a {@code float}.  This
     * conversion is similar to the
     * <i>narrowing primitive conversion</i> from {@code double} to
     * {@code float} as defined in
     * <cite>The Java&trade; Language Specification</cite>:
     * if this BigInteger has too great a magnitude
     * to represent as a {@code float}, it will be converted to
     * {@link Float#NEGATIVE_INFINITY} or {@link
     * Float#POSITIVE_INFINITY} as appropriate.  Note that even when
     * the return value is finite, this conversion can lose
     * information about the precision of the BigInteger value.
     *
     * @return this BigInteger converted to a {@code float}.
     * @jls 5.1.3 Narrowing Primitive Conversion
     */
    public float floatValue() {
        if (signum == 0) {
            return 0.0f;
        }

        int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

        // exponent == floor(log2(abs(this)))
        if (exponent < Long.SIZE - 1) {
            return longValue();
        } else if (exponent > Float.MAX_EXPONENT) {
            return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
        }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
        int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;

        int twiceSignifFloor;
        // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
        // We do the shift into an int directly to improve performance.

        int nBits = shift & 0x1f;
        int nBits2 = 32 - nBits;

        if (nBits == 0) {
            twiceSignifFloor = mag[0];
        } else {
            twiceSignifFloor = mag[0] >>> nBits;
            if (twiceSignifFloor == 0) {
                twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
            }
        }

        int signifFloor = twiceSignifFloor >> 1;
        signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
        boolean increment = (twiceSignifFloor & 1) != 0
                && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
        int signifRounded = increment ? signifFloor + 1 : signifFloor;
        int bits = ((exponent + FloatConsts.EXP_BIAS))
                << (FloatConsts.SIGNIFICAND_WIDTH - 1);
        bits += signifRounded;
        /*
         * If signifRounded == 2^24, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Float.MAX_EXPONENT, we round up (correctly) to
         * Float.POSITIVE_INFINITY.
         */
        bits |= signum & FloatConsts.SIGN_BIT_MASK;
        return Float.intBitsToFloat(bits);
    }

    /**
     * Converts this BigInteger to a {@code double}.  This
     * conversion is similar to the
     * <i>narrowing primitive conversion</i> from {@code double} to
     * {@code float} as defined in
     * <cite>The Java&trade; Language Specification</cite>:
     * if this BigInteger has too great a magnitude
     * to represent as a {@code double}, it will be converted to
     * {@link Double#NEGATIVE_INFINITY} or {@link
     * Double#POSITIVE_INFINITY} as appropriate.  Note that even when
     * the return value is finite, this conversion can lose
     * information about the precision of the BigInteger value.
     *
     * @return this BigInteger converted to a {@code double}.
     * @jls 5.1.3 Narrowing Primitive Conversion
     */
    public double doubleValue() {
        if (signum == 0) {
            return 0.0;
        }

        int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

        // exponent == floor(log2(abs(this))Double)
        if (exponent < Long.SIZE - 1) {
            return longValue();
        } else if (exponent > Double.MAX_EXPONENT) {
            return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
        }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
        int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;

        long twiceSignifFloor;
        // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
        // We do the shift into a long directly to improve performance.

        int nBits = shift & 0x1f;
        int nBits2 = 32 - nBits;

        int highBits;
        int lowBits;
        if (nBits == 0) {
            highBits = mag[0];
            lowBits = mag[1];
        } else {
            highBits = mag[0] >>> nBits;
            lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
            if (highBits == 0) {
                highBits = lowBits;
                lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
            }
        }

        twiceSignifFloor = ((highBits & LONG_MASK) << 32)
                | (lowBits & LONG_MASK);

        long signifFloor = twiceSignifFloor >> 1;
        signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
        boolean increment = (twiceSignifFloor & 1) != 0
                && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
        long signifRounded = increment ? signifFloor + 1 : signifFloor;
        long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
                << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
        bits += signifRounded;
        /*
         * If signifRounded == 2^53, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Double.MAX_EXPONENT, we round up (correctly) to
         * Double.POSITIVE_INFINITY.
         */
        bits |= signum & DoubleConsts.SIGN_BIT_MASK;
        return Double.longBitsToDouble(bits);
    }

    /**
     * Returns a copy of the input array stripped of any leading zero bytes.
     */
    private static int[] stripLeadingZeroInts(int val[]) {
        int vlen = val.length;
        int keep;

        // Find first nonzero byte
        for (keep = 0; keep < vlen && val[keep] == 0; keep++)
            ;
        return java.util.Arrays.copyOfRange(val, keep, vlen);
    }

    /**
     * Returns the input array stripped of any leading zero bytes.
     * Since the source is trusted the copying may be skipped.
     */
    private static int[] trustedStripLeadingZeroInts(int val[]) {
        int vlen = val.length;
        int keep;

        // Find first nonzero byte
        for (keep = 0; keep < vlen && val[keep] == 0; keep++)
            ;
        return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
    }

    /**
     * Returns a copy of the input array stripped of any leading zero bytes.
     */
    private static int[] stripLeadingZeroBytes(byte a[], int off, int len) {
        int indexBound = off + len;
        int keep;

        // Find first nonzero byte
        for (keep = off; keep < indexBound && a[keep] == 0; keep++)
            ;

        // Allocate new array and copy relevant part of input array
        int intLength = ((indexBound - keep) + 3) >>> 2;
        int[] result = new int[intLength];
        int b = indexBound - 1;
        for (int i = intLength-1; i >= 0; i--) {
            result[i] = a[b--] & 0xff;
            int bytesRemaining = b - keep + 1;
            int bytesToTransfer = Math.min(3, bytesRemaining);
            for (int j=8; j <= (bytesToTransfer << 3); j += 8)
                result[i] |= ((a[b--] & 0xff) << j);
        }
        return result;
    }

    /**
     * Takes an array a representing a negative 2's-complement number and
     * returns the minimal (no leading zero bytes) unsigned whose value is -a.
     */
    private static int[] makePositive(byte a[], int off, int len) {
        int keep, k;
        int indexBound = off + len;

        // Find first non-sign (0xff) byte of input
        for (keep=off; keep < indexBound && a[keep] == -1; keep++)
            ;


        /* Allocate output array.  If all non-sign bytes are 0x00, we must
         * allocate space for one extra output byte. */
        for (k=keep; k < indexBound && a[k] == 0; k++)
            ;

        int extraByte = (k == indexBound) ? 1 : 0;
        int intLength = ((indexBound - keep + extraByte) + 3) >>> 2;
        int result[] = new int[intLength];

        /* Copy one's complement of input into output, leaving extra
         * byte (if it exists) == 0x00 */
        int b = indexBound - 1;
        for (int i = intLength-1; i >= 0; i--) {
            result[i] = a[b--] & 0xff;
            int numBytesToTransfer = Math.min(3, b-keep+1);
            if (numBytesToTransfer < 0)
                numBytesToTransfer = 0;
            for (int j=8; j <= 8*numBytesToTransfer; j += 8)
                result[i] |= ((a[b--] & 0xff) << j);

            // Mask indicates which bits must be complemented
            int mask = -1 >>> (8*(3-numBytesToTransfer));
            result[i] = ~result[i] & mask;
        }

        // Add one to one's complement to generate two's complement
        for (int i=result.length-1; i >= 0; i--) {
            result[i] = (int)((result[i] & LONG_MASK) + 1);
            if (result[i] != 0)
                break;
        }

        return result;
    }

    /**
     * Takes an array a representing a negative 2's-complement number and
     * returns the minimal (no leading zero ints) unsigned whose value is -a.
     */
    private static int[] makePositive(int a[]) {
        int keep, j;

        // Find first non-sign (0xffffffff) int of input
        for (keep=0; keep < a.length && a[keep] == -1; keep++)
            ;

        /* Allocate output array.  If all non-sign ints are 0x00, we must
         * allocate space for one extra output int. */
        for (j=keep; j < a.length && a[j] == 0; j++)
            ;
        int extraInt = (j == a.length ? 1 : 0);
        int result[] = new int[a.length - keep + extraInt];

        /* Copy one's complement of input into output, leaving extra
         * int (if it exists) == 0x00 */
        for (int i = keep; i < a.length; i++)
            result[i - keep + extraInt] = ~a[i];

        // Add one to one's complement to generate two's complement
        for (int i=result.length-1; ++result[i] == 0; i--)
            ;

        return result;
    }

    /*
     * The following two arrays are used for fast String conversions.  Both
     * are indexed by radix.  The first is the number of digits of the given
     * radix that can fit in a Java long without "going negative", i.e., the
     * highest integer n such that radix**n < 2**63.  The second is the
     * "long radix" that tears each number into "long digits", each of which
     * consists of the number of digits in the corresponding element in
     * digitsPerLong (longRadix[i] = i**digitPerLong[i]).  Both arrays have
     * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
     * used.
     */
    private static int digitsPerLong[] = {0, 0,
        62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
        14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};

    private static BigInteger longRadix[] = {null, null,
        valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
        valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
        valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
        valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
        valueOf( 0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
        valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
        valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
        valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
        valueOf(0x5da0e1e53c5c8000L), valueOf( 0xb16a458ef403f19L),
        valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
        valueOf(0x5658597bcaa24000L), valueOf( 0x6feb266931a75b7L),
        valueOf( 0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
        valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
        valueOf(0x5a3c23e39c000000L), valueOf( 0x4e900abb53e6b71L),
        valueOf( 0x7600ec618141000L), valueOf( 0xaee5720ee830681L),
        valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
        valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
        valueOf(0x41c21cb8e1000000L)};

    /*
     * These two arrays are the integer analogue of above.
     */
    private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
        11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
        6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};

    private static int intRadix[] = {0, 0,
        0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
        0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
        0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f,  0x10000000,
        0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
        0x6c20a40,  0x8d2d931,  0xb640000,  0xe8d4a51,  0x1269ae40,
        0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
        0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
    };

    /**
     * These routines provide access to the two's complement representation
     * of BigIntegers.
     */

    /**
     * Returns the length of the two's complement representation in ints,
     * including space for at least one sign bit.
     */
    private int intLength() {
        return (bitLength() >>> 5) + 1;
    }

    /* Returns sign bit */
    private int signBit() {
        return signum < 0 ? 1 : 0;
    }

    /* Returns an int of sign bits */
    private int signInt() {
        return signum < 0 ? -1 : 0;
    }

    /**
     * Returns the specified int of the little-endian two's complement
     * representation (int 0 is the least significant).  The int number can
     * be arbitrarily high (values are logically preceded by infinitely many
     * sign ints).
     */
    private int getInt(int n) {
        if (n < 0)
            return 0;
        if (n >= mag.length)
            return signInt();

        int magInt = mag[mag.length-n-1];

        return (signum >= 0 ? magInt :
                (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
    }

    /**
    * Returns the index of the int that contains the first nonzero int in the
    * little-endian binary representation of the magnitude (int 0 is the
    * least significant). If the magnitude is zero, return value is undefined.
    *
    * <p>Note: never used for a BigInteger with a magnitude of zero.
    * @see #getInt.
    */
    private int firstNonzeroIntNum() {
        int fn = firstNonzeroIntNumPlusTwo - 2;
        if (fn == -2) { // firstNonzeroIntNum not initialized yet
            // Search for the first nonzero int
            int i;
            int mlen = mag.length;
            for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
                ;
            fn = mlen - i - 1;
            firstNonzeroIntNumPlusTwo = fn + 2; // offset by two to initialize
        }
        return fn;
    }

    /** use serialVersionUID from JDK 1.1. for interoperability */
    private static final long serialVersionUID = -8287574255936472291L;

    /**
     * Serializable fields for BigInteger.
     *
     * @serialField signum  int
     *              signum of this BigInteger
     * @serialField magnitude byte[]
     *              magnitude array of this BigInteger
     * @serialField bitCount  int
     *              appears in the serialized form for backward compatibility
     * @serialField bitLength int
     *              appears in the serialized form for backward compatibility
     * @serialField firstNonzeroByteNum int
     *              appears in the serialized form for backward compatibility
     * @serialField lowestSetBit int
     *              appears in the serialized form for backward compatibility
     */
    private static final ObjectStreamField[] serialPersistentFields = {
        new ObjectStreamField("signum", Integer.TYPE),
        new ObjectStreamField("magnitude", byte[].class),
        new ObjectStreamField("bitCount", Integer.TYPE),
        new ObjectStreamField("bitLength", Integer.TYPE),
        new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
        new ObjectStreamField("lowestSetBit", Integer.TYPE)
        };

    /**
     * Reconstitute the {@code BigInteger} instance from a stream (that is,
     * deserialize it). The magnitude is read in as an array of bytes
     * for historical reasons, but it is converted to an array of ints
     * and the byte array is discarded.
     * Note:
     * The current convention is to initialize the cache fields, bitCountPlusOne,
     * bitLengthPlusOne and lowestSetBitPlusTwo, to 0 rather than some other
     * marker value. Therefore, no explicit action to set these fields needs to
     * be taken in readObject because those fields already have a 0 value by
     * default since defaultReadObject is not being used.
     */
    private void readObject(java.io.ObjectInputStream s)
        throws java.io.IOException, ClassNotFoundException {
        // prepare to read the alternate persistent fields
        ObjectInputStream.GetField fields = s.readFields();

        // Read the alternate persistent fields that we care about
        int sign = fields.get("signum", -2);
        byte[] magnitude = (byte[])fields.get("magnitude", null);

        // Validate signum
        if (sign < -1 || sign > 1) {
            String message = "BigInteger: Invalid signum value";
            if (fields.defaulted("signum"))
                message = "BigInteger: Signum not present in stream";
            throw new java.io.StreamCorruptedException(message);
        }
        int[] mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length);
        if ((mag.length == 0) != (sign == 0)) {
            String message = "BigInteger: signum-magnitude mismatch";
            if (fields.defaulted("magnitude"))
                message = "BigInteger: Magnitude not present in stream";
            throw new java.io.StreamCorruptedException(message);
        }

        // Commit final fields via Unsafe
        UnsafeHolder.putSign(this, sign);

        // Calculate mag field from magnitude and discard magnitude
        UnsafeHolder.putMag(this, mag);
        if (mag.length >= MAX_MAG_LENGTH) {
            try {
                checkRange();
            } catch (ArithmeticException e) {
                throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
            }
        }
    }

    // Support for resetting final fields while deserializing
    private static class UnsafeHolder {
        private static final jdk.internal.misc.Unsafe unsafe;
        private static final long signumOffset;
        private static final long magOffset;
        static {
            try {
                unsafe = jdk.internal.misc.Unsafe.getUnsafe();
                signumOffset = unsafe.objectFieldOffset
                    (BigInteger.class.getDeclaredField("signum"));
                magOffset = unsafe.objectFieldOffset
                    (BigInteger.class.getDeclaredField("mag"));
            } catch (Exception ex) {
                throw new ExceptionInInitializerError(ex);
            }
        }

        static void putSign(BigInteger bi, int sign) {
            unsafe.putInt(bi, signumOffset, sign);
        }

        static void putMag(BigInteger bi, int[] magnitude) {
            unsafe.putObject(bi, magOffset, magnitude);
        }
    }

    /**
     * Save the {@code BigInteger} instance to a stream.  The magnitude of a
     * {@code BigInteger} is serialized as a byte array for historical reasons.
     * To maintain compatibility with older implementations, the integers
     * -1, -1, -2, and -2 are written as the values of the obsolete fields
     * {@code bitCount}, {@code bitLength}, {@code lowestSetBit}, and
     * {@code firstNonzeroByteNum}, respectively.  These values are compatible
     * with older implementations, but will be ignored by current
     * implementations.
     */
    private void writeObject(ObjectOutputStream s) throws IOException {
        // set the values of the Serializable fields
        ObjectOutputStream.PutField fields = s.putFields();
        fields.put("signum", signum);
        fields.put("magnitude", magSerializedForm());
        // The values written for cached fields are compatible with older
        // versions, but are ignored in readObject so don't otherwise matter.
        fields.put("bitCount", -1);
        fields.put("bitLength", -1);
        fields.put("lowestSetBit", -2);
        fields.put("firstNonzeroByteNum", -2);

        // save them
        s.writeFields();
    }

    /**
     * Returns the mag array as an array of bytes.
     */
    private byte[] magSerializedForm() {
        int len = mag.length;

        int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
        int byteLen = (bitLen + 7) >>> 3;
        byte[] result = new byte[byteLen];

        for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
             i >= 0; i--) {
            if (bytesCopied == 4) {
                nextInt = mag[intIndex--];
                bytesCopied = 1;
            } else {
                nextInt >>>= 8;
                bytesCopied++;
            }
            result[i] = (byte)nextInt;
        }
        return result;
    }

    /**
     * Converts this {@code BigInteger} to a {@code long}, checking
     * for lost information.  If the value of this {@code BigInteger}
     * is out of the range of the {@code long} type, then an
     * {@code ArithmeticException} is thrown.
     *
     * @return this {@code BigInteger} converted to a {@code long}.
     * @throws ArithmeticException if the value of {@code this} will
     * not exactly fit in a {@code long}.
     * @see BigInteger#longValue
     * @since  1.8
     */
    public long longValueExact() {
        if (mag.length <= 2 && bitLength() <= 63)
            return longValue();
        else
            throw new ArithmeticException("BigInteger out of long range");
    }

    /**
     * Converts this {@code BigInteger} to an {@code int}, checking
     * for lost information.  If the value of this {@code BigInteger}
     * is out of the range of the {@code int} type, then an
     * {@code ArithmeticException} is thrown.
     *
     * @return this {@code BigInteger} converted to an {@code int}.
     * @throws ArithmeticException if the value of {@code this} will
     * not exactly fit in an {@code int}.
     * @see BigInteger#intValue
     * @since  1.8
     */
    public int intValueExact() {
        if (mag.length <= 1 && bitLength() <= 31)
            return intValue();
        else
            throw new ArithmeticException("BigInteger out of int range");
    }

    /**
     * Converts this {@code BigInteger} to a {@code short}, checking
     * for lost information.  If the value of this {@code BigInteger}
     * is out of the range of the {@code short} type, then an
     * {@code ArithmeticException} is thrown.
     *
     * @return this {@code BigInteger} converted to a {@code short}.
     * @throws ArithmeticException if the value of {@code this} will
     * not exactly fit in a {@code short}.
     * @see BigInteger#shortValue
     * @since  1.8
     */
    public short shortValueExact() {
        if (mag.length <= 1 && bitLength() <= 31) {
            int value = intValue();
            if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
                return shortValue();
        }
        throw new ArithmeticException("BigInteger out of short range");
    }

    /**
     * Converts this {@code BigInteger} to a {@code byte}, checking
     * for lost information.  If the value of this {@code BigInteger}
     * is out of the range of the {@code byte} type, then an
     * {@code ArithmeticException} is thrown.
     *
     * @return this {@code BigInteger} converted to a {@code byte}.
     * @throws ArithmeticException if the value of {@code this} will
     * not exactly fit in a {@code byte}.
     * @see BigInteger#byteValue
     * @since  1.8
     */
    public byte byteValueExact() {
        if (mag.length <= 1 && bitLength() <= 31) {
            int value = intValue();
            if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
                return byteValue();
        }
        throw new ArithmeticException("BigInteger out of byte range");
    }
}