/*

 * Copyright (c) 1996, 2013, 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.Random;

import sun.misc.DoubleConsts;

import sun.misc.FloatConsts;

/**

 * 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.

 *

 * @see     BigDecimal

 * @author  Josh Bloch

 * @author  Michael McCloskey

 * @author  Alan Eliasen

 * @author  Timothy Buktu

 * @since JDK1.1

 */

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 <i>must</i> have

     * a signum of 0.  This is necessary to ensures that there is exactly one

     * representation for each BigInteger value.

     *

     * @serial

     */

    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;

    // These "redundant fields" are initialized with recognizable nonsense

    // values, and cached the first time they are needed (or never, if they

    // aren't needed).

     /**

     * One plus the bitCount of this BigInteger. Zeros means unitialized.

     *

     * @serial

     * @see #bitCount

     * @deprecated Deprecated since logical value is offset from stored

     * value and correction factor is applied in accessor method.

     */

    @Deprecated

    private int bitCount;

    /**

     * One plus the bitLength of this BigInteger. Zeros means unitialized.

     * (either value is acceptable).

     *

     * @serial

     * @see #bitLength()

     * @deprecated Deprecated since logical value is offset from stored

     * value and correction factor is applied in accessor method.

     */

    @Deprecated

    private int bitLength;

    /**

     * Two plus the lowest set bit of this BigInteger, as returned by

     * getLowestSetBit().

     *

     * @serial

     * @see #getLowestSetBit

     * @deprecated Deprecated since logical value is offset from stored

     * value and correction factor is applied in accessor method.

     */

    @Deprecated

    private int lowestSetBit;

    /**

     * Two plus the index of the lowest-order int in the magnitude of this

     * BigInteger that contains a nonzero int, or -2 (either value is acceptable).

     * The least significant int has int-number 0, the next int in order of

     * increasing significance has int-number 1, and so forth.

     * @deprecated Deprecated since logical value is offset from stored

     * value and correction factor is applied in accessor method.

     */

    @Deprecated

    private int firstNonzeroIntNum;

    /**

     * This mask is used to obtain the value of an int as if it were unsigned.

     */

    final static long LONG_MASK = 0xffffffffL;

    /**

     * 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 = 50;

    /**

     * 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 = 75;

    /**

     * 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 = 90;

    /**

     * 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 = 140;

    /**

     * The threshold value for using Burnikel-Ziegler division.  If the number

     * of ints in the number are larger than this value,

     * Burnikel-Ziegler division will be used.   This value is found

     * experimentally to work well.

     */

    static final int BURNIKEL_ZIEGLER_THRESHOLD = 50;

    /**

     * 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 = 8;

    //Constructors

    /**

     * 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.

     *

     * @param  val big-endian two's-complement binary representation of

     *         BigInteger.

     * @throws NumberFormatException {@code val} is zero bytes long.

     */

    public BigInteger(byte[] val) {

        if (val.length == 0)

            throw new NumberFormatException("Zero length BigInteger");

        if (val[0] < 0) {

            mag = makePositive(val);

            signum = -1;

        } else {

            mag = stripLeadingZeroBytes(val);

            signum = (mag.length == 0 ? 0 : 1);

        }

    }

    /**

     * 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.

     */

    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);

        }

    }

    /**

     * 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 in 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.

     *

     * @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.mag = stripLeadingZeroBytes(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;

        }

    }

    /**

     * 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.

     */

    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;

        }

    }

    /**

     * 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 + index2) <= -1) {

            // No leading sign character or at most one leading sign character

            if (index1 == 0 || index2 == 0) {

                cursor = 1;

                if (len == 1)

                    throw new NumberFormatException("Zero length BigInteger");

            }

            if (index1 == 0)

                sign = -1;

        } else

            throw new NumberFormatException("Illegal embedded sign character");

        // 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.

        int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1);

        int numWords = (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);

    }

    /*

     * Constructs a new BigInteger using a char array with radix=10.

     * Sign is precalculated outside and not allowed in the val.

     */

    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 {

            int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);

            numWords = (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);

    }

    // 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;

        }

    }

    /**