Back to index...

	/*
	* Copyright (c) 1996, 2021, 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.vm.annotation.ForceInline;
	import jdk.internal.vm.annotation.IntrinsicCandidate;
	import jdk.internal.vm.annotation.Stable;

	/**
	* 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.
	*
	* <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 since this operation
	* only makes sense for a fixed sized word and not for a
	* representation conceptually having an infinite number of leading
	* virtual sign bits.
	*
	* <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
	* arbitrarily large abstraction provided by this class ensures that conceptually
	* 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.
	*
	* An {@code ArithmeticException} is thrown when a BigInteger
	* constructor or method would generate a value outside of the
	* supported 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
	* In the reference implementation, 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");
	}
	Objects.checkFromIndexSize(off, len, val.length);

	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"));
	}
	Objects.checkFromIndexSize(off, len, magnitude.length);

	// 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 {@link
	* Character#digit(char, int) 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.
	*/
	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 or plus sign followed by a sequence of one or
	* more decimal digits. The character-to-digit mapping is
	* provided by {@link Character#digit(char, int)
	* 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.
	*/
	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) {
	byte[] magnitude = randomBits(numBits, rnd);

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

	if (this.mag.length == 0) {
	this.signum = 0;
	} else {
	this.signum = 1;
	}
	if (mag.length >= MAX_MAG_LENGTH) {
	checkRange();
	}
	} finally {
	Arrays.fill(magnitude, (byte)0);
	}
	}

	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(3L571113171923293137*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;
	@Stable
	private static final BigInteger[] posConst = new BigInteger[MAX_CONSTANT+1];
	@Stable
	private static final 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 {
	assert 0 < KARATSUBA_THRESHOLD
	&& KARATSUBA_THRESHOLD < TOOM_COOK_THRESHOLD
	&& TOOM_COOK_THRESHOLD < Integer.MAX_VALUE
	&& 0 < KARATSUBA_SQUARE_THRESHOLD
	&& KARATSUBA_SQUARE_THRESHOLD < TOOM_COOK_SQUARE_THRESHOLD
	&& TOOM_COOK_SQUARE_THRESHOLD < Integer.MAX_VALUE :
	"Algorithm thresholds are inconsistent";

	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 > 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 >= 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) {
	return multiply(val, false);
	}

	/**
	* Returns a BigInteger whose value is {@code (this * val)}. If
	* the invocation is recursive certain overflow checks are skipped.
	*
	* @param val value to be multiplied by this BigInteger.
	* @param isRecursion whether this is a recursive invocation
	* @return {@code this * val}
	*/
	private BigInteger multiply(BigInteger val, boolean isRecursion) {
	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 {
	//
	// In "Hacker's Delight" section 2-13, p.33, it is explained
	// that if x and y are unsigned 32-bit quantities and m and n
	// are their respective numbers of leading zeros within 32 bits,
	// then the number of leading zeros within their product as a
	// 64-bit unsigned quantity is either m + n or m + n + 1. If
	// their product is not to overflow, it cannot exceed 32 bits,
	// and so the number of leading zeros of the product within 64
	// bits must be at least 32, i.e., the leftmost set bit is at
	// zero-relative position 31 or less.
	//
	// From the above there are three cases:
	//
	// m + n leftmost set bit condition
	// ----- ---------------- ---------
	// >= 32 x <= 64 - 32 = 32 no overflow
	// == 31 x >= 64 - 32 = 32 possible overflow
	// <= 30 x >= 64 - 31 = 33 definite overflow
	//
	// The "possible overflow" condition cannot be detected by
	// examning data lengths alone and requires further calculation.
	//
	// By analogy, if 'this' and 'val' have m and n as their
	// respective numbers of leading zeros within 32*MAX_MAG_LENGTH
	// bits, then:
	//
	// m + n >= 32*MAX_MAG_LENGTH no overflow
	// m + n == 32*MAX_MAG_LENGTH - 1 possible overflow
	// m + n <= 32*MAX_MAG_LENGTH - 2 definite overflow
	//
	// Note however that if the number of ints in the result
	// were to be MAX_MAG_LENGTH and mag[0] < 0, then there would
	// be overflow. As a result the leftmost bit (of mag[0]) cannot
	// be used and the constraints must be adjusted by one bit to:
	//
	// m + n > 32*MAX_MAG_LENGTH no overflow
	// m + n == 32*MAX_MAG_LENGTH possible overflow
	// m + n < 32*MAX_MAG_LENGTH definite overflow
	//
	// The foregoing leading zero-based discussion is for clarity
	// only. The actual calculations use the estimated bit length
	// of the product as this is more natural to the internal
	// array representation of the magnitude which has no leading
	// zero elements.
	//
	if (!isRecursion) {
	// The bitLength() instance method is not used here as we
	// are only considering the magnitudes as non-negative. The
	// Toom-Cook multiplication algorithm determines the sign
	// at its end from the two signum values.
	if (bitLength(mag, mag.length) +
	bitLength(val.mag, val.mag.length) >
	32L*MAX_MAG_LENGTH) {
	reportOverflow();
	}
	}

	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);
	}

	@IntrinsicCandidate
	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^(322half) + (p3 - p1 - p2) * 2^(32*half) + p2
	BigInteger result = p1.shiftLeft(32half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32half).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, true);
	da1 = a2.add(a0);
	db1 = b2.add(b0);
	vm1 = da1.subtract(a1).multiply(db1.subtract(b1), true);
	da1 = da1.add(a1);
	db1 = db1.add(b1);
	v1 = da1.multiply(db1, true);
	v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
	db1.add(b2).shiftLeft(1).subtract(b0), true);
	vinf = a2.multiply(b2, true);

	// 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() {
	return square(false);
	}

	/**
	* Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. If
	* the invocation is recursive certain overflow checks are skipped.
	*
	* @param isRecursion whether this is a recursive invocation
	* @return {@code this<sup>2</sup>}
	*/
	private BigInteger square(boolean isRecursion) {
	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 {
	//
	// For a discussion of overflow detection see multiply()
	//
	if (!isRecursion) {
	if (bitLength(mag, mag.length) > 16L*MAX_MAG_LENGTH) {
	reportOverflow();
	}
	}

	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.
	*/
	@IntrinsicCandidate
	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(half32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half32).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(true);
	da1 = a2.add(a0);
	vm1 = da1.subtract(a1).square(true);
	da1 = da1.add(a1);
	v1 = da1.square(true);
	vinf = a2.square(true);
	v2 = da1.add(a2).shiftLeft(1).subtract(a0).square(true);

	// 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 bitsToShiftLong = (long)powersOfTwo * exponent;
	if (bitsToShiftLong > Integer.MAX_VALUE) {
	reportOverflow();
	}
	int bitsToShift = (int)bitsToShiftLong;

	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(bitsToShift);
	} else {
	return ONE.shiftLeft(bitsToShift);
	}
	}
	} 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(bitsToShift);
	}
	} else {
	return valueOf(result*newSign);
	}
	} else {
	if ((long)bitLength() * exponent / Integer.SIZE > MAX_MAG_LENGTH) {
	reportOverflow();
	}

	// 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(bitsToShift);
	}

	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) {
	Objects.checkFromToIndex(0, len, a.length);
	shiftRightImplWorker(a, a, 1, n, len-1);
	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;
	Objects.checkFromToIndex(0, len, a.length);
	shiftLeftImplWorker(a, a, 0, n, len-1);
	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} ≤ 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} ≤ 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);

	// Calculate (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 replaced by virtual machine
	// intrinsics.
	@IntrinsicCandidate
	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);
	}
	@IntrinsicCandidate
	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 requires 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.
	*/
	@IntrinsicCandidate
	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 (2p)
	*/
	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} ≤ 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 numIter = magLen - 1;
	Objects.checkFromToIndex(0, numIter + 1, mag.length);
	Objects.checkFromToIndex(i, numIter + i + 1, newMag.length);
	shiftLeftImplWorker(newMag, mag, i, nBits, numIter);
	newMag[numIter + i] = mag[numIter] << nBits;
	}
	return newMag;
	}

	@ForceInline
	@IntrinsicCandidate
	private static void shiftLeftImplWorker(int[] newArr, int[] oldArr, int newIdx, int shiftCount, int numIter) {
	int shiftCountRight = 32 - shiftCount;
	int oldIdx = 0;
	while (oldIdx < numIter) {
	newArr[newIdx++] = (oldArr[oldIdx++] << shiftCount) \| (oldArr[oldIdx] >>> shiftCountRight);
	}
	}

	/**
	* 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 numIter = magLen - nInts - 1;
	Objects.checkFromToIndex(0, numIter + 1, mag.length);
	Objects.checkFromToIndex(i, numIter + i, newMag.length);
	shiftRightImplWorker(newMag, mag, i, nBits, numIter);
	}

	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);
	}

	@ForceInline
	@IntrinsicCandidate
	private static void shiftRightImplWorker(int[] newArr, int[] oldArr, int newIdx, int shiftCount, int numIter) {
	int shiftCountLeft = 32 - shiftCount;
	int idx = numIter;
	int nidx = (newIdx == 0) ? numIter - 1 : numIter;
	while (nidx >= newIdx) {
	newArr[nidx--] = (oldArr[idx--] >>> shiftCount) \| (oldArr[idx] << shiftCountLeft);
	}
	}

	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. For zero this method returns
	* {@code 0}. (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 ≤ 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)} <<i>op</i>> {@code 0)}, where
	* <<i>op</i>> 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) {
	return switch (signum) {
	case 1 -> compareMagnitude(val);
	case -1 -> val.compareMagnitude(this);
	default -> 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 xInt))
	return false;

	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;

	BigInteger abs = this.abs();

	// Ensure buffer capacity sufficient to contain string representation
	// floor(bitLength*log(2)/log(radix)) + 1
	// plus an additional character for the sign if negative.
	int b = abs.bitLength();
	int numChars = (int)(Math.floor(b*LOG_TWO/logCache[radix]) + 1) +
	(signum < 0 ? 1 : 0);
	StringBuilder sb = new StringBuilder(numChars);

	if (signum < 0) {
	sb.append('-');
	}

	// Use recursive toString.
	toString(abs, sb, radix, 0);

	return sb.toString();
	}

	/**
	* If {@code numZeros > 0}, appends that many zeros to the
	* specified StringBuilder; otherwise, does nothing.
	*
	* @param buf The StringBuilder that will be appended to.
	* @param numZeros The number of zeros to append.
	*/
	private static void padWithZeros(StringBuilder buf, int numZeros) {
	while (numZeros >= NUM_ZEROS) {
	buf.append(ZEROS);
	numZeros -= NUM_ZEROS;
	}
	if (numZeros > 0) {
	buf.append(ZEROS, 0, numZeros);
	}
	}

	/**
	* This method is used to perform toString when arguments are small.
	* The value must be non-negative. If {@code digits <= 0} no padding
	* (pre-pending with zeros) will be effected.
	*
	* @param radix The base to convert to.
	* @param buf The StringBuilder that will be appended to in place.
	* @param digits The minimum number of digits to pad to.
	*/
	private void smallToString(int radix, StringBuilder buf, int digits) {
	assert signum >= 0;

	if (signum == 0) {
	padWithZeros(buf, digits);
	return;
	}

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

	// Translate number to string, a digit group at a time
	BigInteger tmp = this;
	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);

	digitGroups[numGroups++] = r2.longValue();
	tmp = q2;
	}

	// Get string version of first digit group
	String s = Long.toString(digitGroups[numGroups-1], radix);

	// Pad with internal zeros if necessary.
	padWithZeros(buf, digits - (s.length() +
	(numGroups - 1)*digitsPerLong[radix]));

	// Put first digit group into result buffer
	buf.append(s);

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

	/**
	* Converts the specified BigInteger to a string and appends to
	* {@code sb}. This implements the recursive Schoenhage algorithm
	* for base conversions. This method can only be called for non-negative
	* numbers.
	* <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) {
	assert u.signum() >= 0;

	// If we're smaller than a certain threshold, use the smallToString
	// method, padding with leading zeroes when necessary unless we're
	// at the beginning of the string or digits <= 0. As u.signum() >= 0,
	// smallToString() will not prepend a negative sign.
	if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
	u.smallToString(radix, sb, digits);
	return;
	}

	// 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.
	int b = u.bitLength();
	int 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];
	}

	/* Size of ZEROS string. */
	private static int NUM_ZEROS = 63;

	/* ZEROS is a string of NUM_ZEROS consecutive zeros. */
	private static final String ZEROS = "0".repeat(NUM_ZEROS);

	/**
	* 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 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 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 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 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 radixn < 263. 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 */
	@java.io.Serial
	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
	*/
	@java.io.Serial
	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.
	*
	* @param s the stream being read.
	* @throws IOException if an I/O error occurs
	* @throws ClassNotFoundException if a serialized class cannot be loaded
	*/
	@java.io.Serial
	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
	= jdk.internal.misc.Unsafe.getUnsafe();
	private static final long signumOffset
	= unsafe.objectFieldOffset(BigInteger.class, "signum");
	private static final long magOffset
	= unsafe.objectFieldOffset(BigInteger.class, "mag");

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

	static void putMag(BigInteger bi, int[] magnitude) {
	unsafe.putReference(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.
	*
	* @param s the stream to serialize to.
	* @throws IOException if an I/O error occurs
	*/
	@java.io.Serial
	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");
	}
	}

Back to index...