/*

 * Copyright (c) 1996, 2014, 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 java.util.concurrent.ThreadLocalRandom;

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.

 *

 * 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

 * @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 <i>must</i> 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.

     */

    final static 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;

    // 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 1.9

     */

    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 1.9

     */

    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;