/*
 * Copyright 2004 Sun Microsystems, Inc. All rights reserved.
 * SUN PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
 */

/*
 * @(#)BigInteger.java	1.68 04/04/29
 */

package java.math;

import java.util.Random;
import java.io.*;

/**
 * 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 <tt>ArithmeticException</tt>, and
 * division of a negative by a positive yields a negative (or zero) remainder.
 * All of the details in the Spec concerning overflow are ignored, as
 * BigIntegers are made as large as necessary to accommodate the results of an
 * operation.
 * <p>
 * Semantics of shift operations extend those of Java's shift operators
 * to allow for negative shift distances.  A right-shift with a negative
 * shift distance results in a left shift, and vice-versa.  The unsigned
 * right shift operator (&gt;&gt;&gt;) is omitted, as this operation makes
 * little sense in combination with the "infinite word size" abstraction
 * provided by this class.
 * <p>
 * Semantics of bitwise logical operations exactly mimic those of Java's
 * bitwise integer operators.  The binary operators (<tt>and</tt>,
 * <tt>or</tt>, <tt>xor</tt>) 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 <tt>0</tt> and <tt>(modulus - 1)</tt>,
 * inclusive.
 * <p>
 * Bit operations operate on a single bit of the two's-complement
 * representation of their operand.  If necessary, the operand is sign-
 * extended so that it contains the designated bit.  None of the single-bit
 * operations can produce a BigInteger with a different sign from the
 * BigInteger being operated on, as they affect only a single bit, and the
 * "infinite word size" abstraction provided by this class ensures that there
 * are infinitely many "virtual sign bits" preceding each BigInteger.
 * <p>
 * For the sake of brevity and clarity, pseudo-code is used throughout the
 * descriptions of BigInteger methods.  The pseudo-code expression
 * <tt>(i + j)</tt> is shorthand for "a BigInteger whose value is
 * that of the BigInteger <tt>i</tt> plus that of the BigInteger <tt>j</tt>."
 * The pseudo-code expression <tt>(i == j)</tt> is shorthand for
 * "<tt>true</tt> if and only if the BigInteger <tt>i</tt> represents the same
 * value as the BigInteger <tt>j</tt>."  Other pseudo-code expressions are
 * interpreted similarly.
 * <p>
 * All methods and constructors in this class throw
 * <CODE>NullPointerException</CODE> when passed
 * a null object reference for any input parameter.
 *
 * @see     BigDecimal
 * @version 1.68, 04/29/04
 * @author  Josh Bloch
 * @author  Michael McCloskey
 * @since JDK1.1
 */

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

    // These "redundant fields" are initialized with recognizable nonsense
    // values, and cached the first time they are needed (or never, if they
    // aren't needed).

    /**
     * The bitCount of this BigInteger, as returned by bitCount(), or -1
     * (either value is acceptable).
     *
     * @serial
     * @see #bitCount
     */
    private int bitCount =  -1;

    /**
     * The bitLength of this BigInteger, as returned by bitLength(), or -1
     * (either value is acceptable).
     *
     * @serial
     * @see #bitLength
     */
    private int bitLength = -1;

    /**
     * The lowest set bit of this BigInteger, as returned by getLowestSetBit(),
     * or -2 (either value is acceptable).
     *
     * @serial
     * @see #getLowestSetBit
     */
    private int lowestSetBit = -2;

    /**
     * The index of the lowest-order byte in the magnitude of this BigInteger
     * that contains a nonzero byte, or -2 (either value is acceptable).  The
     * least significant byte has int-number 0, the next byte in order of
     * increasing significance has byte-number 1, and so forth.
     *
     * @serial
     */
    private int firstNonzeroByteNum = -2;

    /**
     * The index of the lowest-order int in the magnitude of this BigInteger
     * that contains a nonzero int, or -2 (either value is acceptable).  The
     * least significant int has int-number 0, the next int in order of
     * increasing significance has int-number 1, and so forth.
     */
    private int firstNonzeroIntNum = -2;

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

    //Constructors

    /**
     * Translates a byte array containing the two's-complement binary
     * representation of a BigInteger into a BigInteger.  The input array is
     * assumed to be in <i>big-endian</i> byte-order: the most significant
     * byte is in the zeroth element.
     *
     * @param  val big-endian two's-complement binary representation of
     *	       BigInteger.
     * @throws NumberFormatException <tt>val</tt> is zero bytes long.
     */
    public BigInteger(byte[] val) {
	if (val.length == 0)
	    throw new NumberFormatException("Zero length BigInteger");

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

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

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

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

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

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

    /**
     * A constructor for internal use that translates the sign-magnitude
     * representation of a BigInteger into a BigInteger. It checks the
     * arguments and copies the magnitude so this constructor would be
     * safe for external use.
     */
    private BigInteger(int signum, int[] magnitude) {
	this.mag = stripLeadingZeroInts(magnitude);

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

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

    /**
     * Translates the String representation of a BigInteger in the specified
     * radix into a BigInteger.  The String representation consists of an
     * optional minus sign followed by a sequence of one or more digits in the
     * specified radix.  The character-to-digit mapping is provided by
     * <tt>Character.digit</tt>.  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 <tt>val</tt>.
     * @throws NumberFormatException <tt>val</tt> is not a valid representation
     *	       of a BigInteger in the specified radix, or <tt>radix</tt> is
     *	       outside the range from {@link Character#MIN_RADIX} to
     *	       {@link Character#MAX_RADIX}, inclusive.
     * @see    Character#digit
     */
    public BigInteger(String val, int radix) {
	int cursor = 0, numDigits;
        int len = val.length();

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

	// Check for minus sign
	signum = 1;
        int index = val.lastIndexOf("-");
        if (index != -1) {
            if (index == 0) {
                if (val.length() == 1)
                    throw new NumberFormatException("Zero length BigInteger");
                signum = -1;
                cursor = 1;
            } else {
                throw new NumberFormatException("Illegal embedded minus sign");
            }
        }

        // 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;
	} else {
	    numDigits = len - cursor;
	}

        // Pre-allocate array of expected size. May be too large but can
        // never be too small. Typically exact.
        int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1);
        int numWords = (numBits + 31) /32;
        mag = 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);
        mag[mag.length - 1] = Integer.parseInt(group, radix);
	if (mag[mag.length - 1] < 0)
	    throw new NumberFormatException("Illegal digit");
        
	// Process remaining digit groups
        int superRadix = intRadix[radix];
        int groupVal = 0;
	while (cursor < val.length()) {
	    group = val.substring(cursor, cursor += digitsPerInt[radix]);
	    groupVal = Integer.parseInt(group, radix);
	    if (groupVal < 0)
		throw new NumberFormatException("Illegal digit");
            destructiveMulAdd(mag, superRadix, groupVal);
	}
        // Required for cases where the array was overallocated.
        mag = trustedStripLeadingZeroInts(mag);
    }

    // Constructs a new BigInteger using a char array with radix=10
    BigInteger(char[] val) {
        int cursor = 0, numDigits;
        int len = val.length;

	// Check for leading minus sign
	signum = 1;
	if (val[0] == '-') {
	    if (len == 1)
		throw new NumberFormatException("Zero length BigInteger");
	    signum = -1;
	    cursor = 1;
	}

        // 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;
	} else {
	    numDigits = len - cursor;
	}

        // Pre-allocate array of expected size
        int numWords;
        if (len < 10) {
            numWords = 1;
        } else {    
            int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);
            numWords = (numBits + 31) /32;
        }
        mag = new int[numWords];
 
	// Process first (potentially short) digit group
	int firstGroupLen = numDigits % digitsPerInt[10];
	if (firstGroupLen == 0)
	    firstGroupLen = digitsPerInt[10];
        mag[mag.length-1] = parseInt(val, cursor,  cursor += firstGroupLen);
        
	// Process remaining digit groups
	while (cursor < len) {
	    int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
            destructiveMulAdd(mag, intRadix[10], groupVal);
	}
        mag = trustedStripLeadingZeroInts(mag);
    }

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

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

        return result;
    }

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

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

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

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

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

    /**
     * Constructs a randomly generated BigInteger, uniformly distributed over
     * the range <tt>0</tt> to <tt>(2<sup>numBits</sup> - 1)</tt>, inclusive.
     * The uniformity of the distribution assumes that a fair source of random
     * bits is provided in <tt>rnd</tt>.  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 <tt>numBits</tt> is negative.
     * @see #bitLength
     */
    public BigInteger(int numBits, Random rnd) {
	this(1, randomBits(numBits, rnd));
    }

    private static byte[] randomBits(int numBits, Random rnd) {
	if (numBits < 0)
	    throw new IllegalArgumentException("numBits must be non-negative");
	int numBytes = (numBits+7)/8;
	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.<p>
     *
     * 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
     *	       <tt>(1 - 1/2<sup>certainty</sup></tt>).  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 <tt>bitLength &lt; 2</tt>.
     * @see    #bitLength
     */
    public BigInteger(int bitLength, int certainty, Random rnd) {
        BigInteger prime;

	if (bitLength < 2)
	    throw new ArithmeticException("bitLength < 2");
        // The cutoff of 95 was chosen empirically for best performance
        prime = (bitLength < 95 ? 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
    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 <tt>bitLength</tt> bits that is probably prime
     * @throws ArithmeticException <tt>bitLength &lt; 2</tt>.
     * @see    #bitLength
     */
    public static BigInteger probablePrime(int bitLength, Random rnd) {
	if (bitLength < 2)
	    throw new ArithmeticException("bitLength < 2");

        // The cutoff of 95 was chosen empirically for best performance
        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))
                return p;
        }
    }

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

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

        // Use a sieve length likely to contain the next prime number
        int searchLen = (bitLength / 20) * 64;
        BitSieve searchSieve = new BitSieve(p, searchLen);
        BigInteger candidate = searchSieve.retrieve(p, certainty);

        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);
        }
        return candidate;
    }

   /**
    * Returns the first integer greater than this <code>BigInteger</code> 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 <tt>p</tt>, there
    * is no prime <tt>q</tt> such that <tt>this &lt; q &lt; p</tt>.
    *
    * @return the first integer greater than this <code>BigInteger</code> that
    *         is probably prime.
    * @throws ArithmeticException <tt>this &lt; 0</tt>.
    * @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))
                    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 = (result.bitLength() / 20) * 64;

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

    /**
     * Returns <tt>true</tt> if this BigInteger is probably prime,
     * <tt>false</tt> 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 <tt>true</tt>
     *	       the probability that this BigInteger is prime exceeds
     *	       <tt>(1 - 1/2<sup>certainty</sup>)</tt>.  The execution time of
     * 	       this method is proportional to the value of this parameter.
     * @return <tt>true</tt> if this BigInteger is probably prime,
     * 	       <tt>false</tt> if it's definitely composite.
     */
    boolean primeToCertainty(int certainty) {
        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);
        }

        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) && 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 = n.subtract(v2);
                v2.signum = - v2.signum;
            }
            v2 = v2.shiftRight(1);

            u = u2; v = v2;
            if (k.testBit(i)) {
                u2 = u.add(v).mod(n);
                if (u2.testBit(0)) {
                    u2 = n.subtract(u2);
                    u2.signum = - u2.signum;
                }
                u2 = u2.shiftRight(1);
                
                v2 = v.add(d.multiply(u)).mod(n);
                if (v2.testBit(0)) {
                    v2 = n.subtract(v2);
                    v2.signum = - v2.signum;
                }
                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) {
	// 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
        Random rnd = new Random();
	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 private 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.
     */
    private BigInteger(int[] magnitude, int signum) {
	this.signum = (magnitude.length==0 ? 0 : signum);
	this.mag = magnitude;
    }

    /**
     * This private constructor is for internal use and assumes that its
     * arguments are correct.
     */
    private BigInteger(byte[] magnitude, int signum) {
	this.signum = (magnitude.length==0 ? 0 : signum);
        this.mag = stripLeadingZeroBytes(magnitude);
    }

    /**
     * This private constructor is for internal use in converting
     * from a MutableBigInteger object into a BigInteger.
     */
    BigInteger(MutableBigInteger val, int sign) {
        if (val.offset > 0 || val.value.length != val.intLen) {
            mag = new int[val.intLen];
            for(int i=0; i<val.intLen; i++)
                mag[i] = val.value[val.offset+i];
        } else {
            mag = val.value;
        }

	this.signum = (val.intLen == 0) ? 0 : sign;
    }

    //Static Factory Methods

    /**
     * Returns a BigInteger whose value is equal to that of the
     * specified <code>long</code>.  This "static factory method" is
     * provided in preference to a (<code>long</code>) 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) {
            signum = -1;
            val = -val;
        } 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 final static int MAX_CONSTANT = 16;
    private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
    private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
    static {
	for (int i = 1; i <= MAX_CONSTANT; i++) {
	    int[] magnitude = new int[1];
	    magnitude[0] = (int) i;
	    posConst[i] = new BigInteger(magnitude,  1);
	    negConst[i] = new BigInteger(magnitude, -1);
	}
    }

    /**
     * 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.  (Not exported.)
     */
    private static final BigInteger TWO = valueOf(2);

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

    // Arithmetic Operations

    /**
     * Returns a BigInteger whose value is <tt>(this + val)</tt>.
     *
     * @param  val value to be added to this BigInteger.
     * @return <tt>this + val</tt>
     */
    public BigInteger add(BigInteger val) {
        int[] resultMag;
	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 = intArrayCmp(mag, val.mag);
        if (cmp==0)
            return ZERO;
        resultMag = (cmp>0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);

        return new BigInteger(resultMag, cmp*signum);
    }

    /**
     * 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;

        // 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 newLen = result.length + 1;
            int temp[] = new int[newLen];
            for (int i = 1; i<newLen; i++)
                temp[i] = result[i-1];
            temp[0] = 0x01;
            result = temp;
        }
        return result;
    }

    /**
     * Returns a BigInteger whose value is <tt>(this - val)</tt>.
     *
     * @param  val value to be subtracted from this BigInteger.
     * @return <tt>this - val</tt>
     */
    public BigInteger subtract(BigInteger val) {
        int[] resultMag;
	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 = intArrayCmp(mag, val.mag);
        if (cmp==0)
            return ZERO;
        resultMag = (cmp>0 ? subtract(mag, val.mag)
                           : subtract(val.mag, mag));
        resultMag = trustedStripLeadingZeroInts(resultMag);
        return new BigInteger(resultMag, cmp*signum);
    }

    /**
     * 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 <tt>(this * val)</tt>.
     *
     * @param  val value to be multiplied by this BigInteger.
     * @return <tt>this * val</tt>
     */
    public BigInteger multiply(BigInteger val) {
        if (signum == 0 || val.signum==0)
	    return ZERO;
        
        int[] result = multiplyToLen(mag, mag.length, 
                                     val.mag, val.mag.length, null);
        result = trustedStripLeadingZeroInts(result);
        return new BigInteger(result, signum*val.signum);
    }

    /**
     * Multiplies int arrays x and y to the specified lengths and places
     * the result into z.
     */
    private int[] multiplyToLen(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;
    }

    /**
     * Returns a BigInteger whose value is <tt>(this<sup>2</sup>)</tt>.
     *
     * @return <tt>this<sup>2</sup></tt>
     */
    private BigInteger square() {
        if (signum == 0)
	    return ZERO;
        int[] z = squareToLen(mag, mag.length, null);
        return new BigInteger(trustedStripLeadingZeroInts(z), 1);
    }

    /**
     * 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) {
        /*
         * 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.
         */
        int zlen = len << 1;
        if (z == null || z.length < zlen)
            z = new int[zlen];
        
        // 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;
    }

    /**
     * Returns a BigInteger whose value is <tt>(this / val)</tt>.
     *
     * @param  val value by which this BigInteger is to be divided.
     * @return <tt>this / val</tt>
     * @throws ArithmeticException <tt>val==0</tt>
     */
    public BigInteger divide(BigInteger val) {
        MutableBigInteger q = new MutableBigInteger(),
                          r = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);

        a.divide(b, q, r);
        return new BigInteger(q, this.signum * val.signum);
    }

    /**
     * Returns an array of two BigIntegers containing <tt>(this / val)</tt>
     * followed by <tt>(this % val)</tt>.
     *
     * @param  val value by which this BigInteger is to be divided, and the
     *	       remainder computed.
     * @return an array of two BigIntegers: the quotient <tt>(this / val)</tt>
     *	       is the initial element, and the remainder <tt>(this % val)</tt>
     *	       is the final element.
     * @throws ArithmeticException <tt>val==0</tt>
     */
    public BigInteger[] divideAndRemainder(BigInteger val) {
        BigInteger[] result = new BigInteger[2];
        MutableBigInteger q = new MutableBigInteger(),
                          r = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);
        a.divide(b, q, r);
        result[0] = new BigInteger(q, this.signum * val.signum);
        result[1] = new BigInteger(r, this.signum);
        return result;
    }

    /**
     * Returns a BigInteger whose value is <tt>(this % val)</tt>.
     *
     * @param  val value by which this BigInteger is to be divided, and the
     *	       remainder computed.
     * @return <tt>this % val</tt>
     * @throws ArithmeticException <tt>val==0</tt>
     */
    public BigInteger remainder(BigInteger val) {
        MutableBigInteger q = new MutableBigInteger(),
                          r = new MutableBigInteger(),
                          a = new MutableBigInteger(this.mag),
                          b = new MutableBigInteger(val.mag);

        a.divide(b, q, r);
        return new BigInteger(r, this.signum);
    }

    /**
     * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
     * Note that <tt>exponent</tt> is an integer rather than a BigInteger.
     *
     * @param  exponent exponent to which this BigInteger is to be raised.
     * @return <tt>this<sup>exponent</sup></tt>
     * @throws ArithmeticException <tt>exponent</tt> 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);

	// Perform exponentiation using repeated squaring trick
        int newSign = (signum<0 && (exponent&1)==1 ? -1 : 1);
	int[] baseToPow2 = this.mag;
        int[] result = {1};

	while (exponent != 0) {
	    if ((exponent & 1)==1) {
		result = multiplyToLen(result, result.length, 
                                       baseToPow2, baseToPow2.length, null);
		result = trustedStripLeadingZeroInts(result);
	    }
	    if ((exponent >>>= 1) != 0) {
                baseToPow2 = squareToLen(baseToPow2, baseToPow2.length, null);
		baseToPow2 = trustedStripLeadingZeroInts(baseToPow2);
	    }
	}
	return new BigInteger(result, newSign);
    }

    /**
     * Returns a BigInteger whose value is the greatest common divisor of
     * <tt>abs(this)</tt> and <tt>abs(val)</tt>.  Returns 0 if
     * <tt>this==0 &amp;&amp; val==0</tt>.
     *
     * @param  val value with which the GCD is to be computed.
     * @return <tt>GCD(abs(this), abs(val))</tt>
     */
    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 new BigInteger(result, 1);
    }

    /**
     * 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 = bitLen(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];
                for (int i=0; i<len; i++)
                    result[i] = a[i];
                primitiveLeftShift(result, result.length, nBits);
                return result;
            } else {
                int result[] = new int[nInts+len+1];
                for (int i=0; i<len; i++)
                    result[i] = a[i];
                primitiveRightShift(result, result.length, 32 - nBits);
                return result;
            }
        }
    }

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

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

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

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

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

    /**
     * Returns a BigInteger whose value is <tt>(-this)</tt>.
     *
     * @return <tt>-this</tt>
     */
    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 <tt>(this mod m</tt>).  This method
     * differs from <tt>remainder</tt> in that it always returns a
     * <i>non-negative</i> BigInteger.
     *
     * @param  m the modulus.
     * @return <tt>this mod m</tt>
     * @throws ArithmeticException <tt>m &lt;= 0</tt>
     * @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
     * <tt>(this<sup>exponent</sup> mod m)</tt>.  (Unlike <tt>pow</tt>, this
     * method permits negative exponents.)
     *
     * @param  exponent the exponent.
     * @param  m the modulus.
     * @return <tt>this<sup>exponent</sup> mod m</tt>
     * @throws ArithmeticException <tt>m &lt;= 0</tt>
     * @see    #modInverse
     */
    public BigInteger modPow(BigInteger exponent, BigInteger m) {
	if (m.signum <= 0)
	    throw new ArithmeticException("BigInteger: modulus not positive");

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

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

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

        if (this.equals(negConst[1]) && (!exponent.testBit(0)))
            return (m.equals(ONE) ? ZERO : ONE);
            
	boolean invertResult;
	if ((invertResult = (exponent.signum < 0)))
	    exponent = exponent.negate();

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

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

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

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

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

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

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

	    result = a1.multiply(m2).multiply(y1).add
		     (a2.multiply(m1).multiply(y2)).mod(m);
	}

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

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

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

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

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

        // 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
        int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]);

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

        MutableBigInteger q = new MutableBigInteger(),
                          r = new MutableBigInteger(),
                          a2 = new MutableBigInteger(a),
                          b2 = new MutableBigInteger(mod);

        a2.divide(b2, q, r);
        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];
           for (int i=0; i<table[0].length; i++)
               t2[i+offset] = table[0][i];
           table[0] = t2;
        }

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

        // Set t to high half of b
        int[] t = new int[modLen];
        for(int i=0; i<modLen; i++)
            t[i] = b[i];

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

        // 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 = (int[])mult.clone();
                    isone = false;
                } else {
                    t = b;
                    a = multiplyToLen(t, modLen, mult, modLen, a);
                    a = montReduce(a, mod, modLen, inv);
                    t = a; a = b; b = t;
                }
            }

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

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

        // Convert result out of Montgomery form and return
        int[] t2 = new int[2*modLen];
        for(int i=0; i<modLen; i++)
            t2[i+modLen] = b[i];

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

        t2 = new int[modLen];
        for(int i=0; i<modLen; i++)
            t2[i] = b[i];
           
        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) {
        long kLong = k & LONG_MASK;
        long carry = 0;

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

    /**
     * Add one word to the number a mlen words into a. Return the resulting
     * carry.
     */
    static int addOne(int[] a, int offset, int mlen, int carry) {
        offset = a.length-1-mlen-offset;
        long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);
        
        a[offset] = (int)t;
        if ((t >>> 32) == 0)
            return 0;
        while (--mlen >= 0) {
            if (--offset < 0) { // Carry out of number
                return 1;
            } else {
                a[offset]++;
                if (a[offset] != 0)
                    return 0;
            }
        }
        return 1;
    }

    /**
     * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
     */
    private BigInteger modPow2(BigInteger exponent, int p) {
	/*
	 * Perform exponentiation using repeated squaring trick, chopping off
	 * high order bits as indicated by modulus.
	 */
	BigInteger result = valueOf(1);
	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 BigInteger &gt;= 0 and p &gt; 0.
     */
    private BigInteger mod2(int p) {
	if (bitLength() <= p)
	    return this;

	// Copy remaining ints of mag
	int numInts = (p+31)/32;
	int[] mag = new int[numInts];
	for (int i=0; i<numInts; i++)
	    mag[i] = this.mag[i + (this.mag.length - 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 <tt>(this<sup>-1</sup> mod m)</tt>.
     *
     * @param  m the modulus.
     * @return <tt>this<sup>-1</sup> mod m</tt>.
     * @throws ArithmeticException <tt> m &lt;= 0</tt>, 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 || (intArrayCmp(mag, m.mag) >= 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 new BigInteger(result, 1);
    }

    // Shift Operations

    /**
     * Returns a BigInteger whose value is <tt>(this &lt;&lt; n)</tt>.
     * The shift distance, <tt>n</tt>, may be negative, in which case
     * this method performs a right shift.
     * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
     *
     * @param  n shift distance, in bits.
     * @return <tt>this &lt;&lt; n</tt>
     * @see #shiftRight
     */
    public BigInteger shiftLeft(int n) {
        if (signum == 0)
            return ZERO;
        if (n==0)
            return this;
        if (n<0)
            return shiftRight(-n);

        int nInts = n >>> 5;
        int nBits = n & 0x1f;
        int magLen = mag.length;
        int newMag[] = null;

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

        return new BigInteger(newMag, signum);
    }

    /**
     * Returns a BigInteger whose value is <tt>(this &gt;&gt; n)</tt>.  Sign
     * extension is performed.  The shift distance, <tt>n</tt>, may be
     * negative, in which case this method performs a left shift.
     * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.) 
     *
     * @param  n shift distance, in bits.
     * @return <tt>this &gt;&gt; n</tt>
     * @see #shiftLeft
     */
    public BigInteger shiftRight(int n) {
        if (n==0)
            return this;
        if (n<0)
            return shiftLeft(-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 = new int[newMagLen];
            for (int i=0; i<newMagLen; i++)
                newMag[i] = mag[i];
        } else {
            int i = 0;
            int highBits = mag[0] >>> nBits;
            if (highBits != 0) {
                newMag = new int[magLen - nInts];
                newMag[i++] = highBits;
            } else {
                newMag = new int[magLen - nInts -1];
            }

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

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

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

        return new BigInteger(newMag, signum);
    }

    int[] javaIncrement(int[] val) {
        boolean done = false;
        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 <tt>(this &amp; val)</tt>.  (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 <tt>this &amp; val</tt>
     */
    public BigInteger and(BigInteger val) {
	int[] result = new int[Math.max(intLength(), val.intLength())];
	for (int i=0; i<result.length; i++)
	    result[i] = (int) (getInt(result.length-i-1)
				& val.getInt(result.length-i-1));

	return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is <tt>(this | val)</tt>.  (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 <tt>this | val</tt>
     */
    public BigInteger or(BigInteger val) {
	int[] result = new int[Math.max(intLength(), val.intLength())];
	for (int i=0; i<result.length; i++)
	    result[i] = (int) (getInt(result.length-i-1)
				| val.getInt(result.length-i-1));

	return valueOf(result);
    }

    /**
     * Returns a BigInteger whose value is <tt>(this ^ val)</tt>.  (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 <tt>this ^ val</tt>
     */
    public BigInteger xor(BigInteger val) {
	int[] result = new int[Math.max(intLength(), val.intLength())];
	for (int i=0; i<result.length; i++)
	    result[i] = (int) (getInt(result.length-i-1)
				^ val.getInt(result.length-i-1));

	return valueOf(result);
    }

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

	return valueOf(result);
    }

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

	return valueOf(result);
    }


    // Single Bit Operations

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

	return (getInt(n/32) & (1 << (n%32))) != 0;
    }

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

	int intNum = n/32;
	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%32));

	return valueOf(result);
    }

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

	int intNum = n/32;
	int[] result = new int[Math.max(intLength(), (n+1)/32+1)];

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

	result[result.length-intNum-1] &= ~(1 << (n%32));

	return valueOf(result);
    }

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

	int intNum = n/32;
	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%32));

	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 <tt>(this==0? -1 : log<sub>2</sub>(this &amp; -this))</tt>.)
     *
     * @return index of the rightmost one bit in this BigInteger.
     */
    public int getLowestSetBit() {
	/*
	 * Initialize lowestSetBit field the first time this method is
	 * executed. This method depends on the atomicity of int modifies;
	 * without this guarantee, it would have to be synchronized.
	 */
	if (lowestSetBit == -2) {
	    if (signum == 0) {
		lowestSetBit = -1;
	    } else {
		// Search for lowest order nonzero int
		int i,b;
		for (i=0; (b = getInt(i))==0; i++)
		    ;
		lowestSetBit = (i << 5) + trailingZeroCnt(b);
	    }
	}
	return lowestSetBit;
    }


    // Miscellaneous Bit Operations

    /**
     * Returns the number of bits in the minimal two's-complement
     * representation of this BigInteger, <i>excluding</i> a sign bit.
     * For positive BigIntegers, this is equivalent to the number of bits in
     * the ordinary binary representation.  (Computes
     * <tt>(ceil(log<sub>2</sub>(this &lt; 0 ? -this : this+1)))</tt>.)
     *
     * @return number of bits in the minimal two's-complement
     *         representation of this BigInteger, <i>excluding</i> a sign bit.
     */
    public int bitLength() {
	/*
	 * Initialize bitLength field the first time this method is executed.
	 * This method depends on the atomicity of int modifies; without
	 * this guarantee, it would have to be synchronized.
	 */
	if (bitLength == -1) {
	    if (signum == 0) {
		bitLength = 0;
	    } else {
		// Calculate the bit length of the magnitude
		int magBitLength = ((mag.length-1) << 5) + bitLen(mag[0]);

		if (signum < 0) {
		    // Check if magnitude is a power of two
		    boolean pow2 = (bitCnt(mag[0]) == 1);
		    for(int i=1; i<mag.length && pow2; i++)
			pow2 = (mag[i]==0);

		    bitLength = (pow2 ? magBitLength-1 : magBitLength);
		} else {
		    bitLength = magBitLength;
		}
	    }
	}
	return bitLength;
    }

    /**
     * bitLen(val) is the number of bits in val.
     */
    static int bitLen(int w) {
        // Binary search - decision tree (5 tests, rarely 6)
        return
         (w < 1<<15 ?
          (w < 1<<7 ?
           (w < 1<<3 ?
            (w < 1<<1 ? (w < 1<<0 ? (w<0 ? 32 : 0) : 1) : (w < 1<<2 ? 2 : 3)) :
            (w < 1<<5 ? (w < 1<<4 ? 4 : 5) : (w < 1<<6 ? 6 : 7))) :
           (w < 1<<11 ?
            (w < 1<<9 ? (w < 1<<8 ? 8 : 9) : (w < 1<<10 ? 10 : 11)) :
            (w < 1<<13 ? (w < 1<<12 ? 12 : 13) : (w < 1<<14 ? 14 : 15)))) :
          (w < 1<<23 ?
           (w < 1<<19 ?
            (w < 1<<17 ? (w < 1<<16 ? 16 : 17) : (w < 1<<18 ? 18 : 19)) :
            (w < 1<<21 ? (w < 1<<20 ? 20 : 21) : (w < 1<<22 ? 22 : 23))) :
           (w < 1<<27 ?
            (w < 1<<25 ? (w < 1<<24 ? 24 : 25) : (w < 1<<26 ? 26 : 27)) :
            (w < 1<<29 ? (w < 1<<28 ? 28 : 29) : (w < 1<<30 ? 30 : 31)))));
    }

    /*
     * trailingZeroTable[i] is the number of trailing zero bits in the binary
     * representation of i.
     */
    final static byte trailingZeroTable[] = {
      -25, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
	4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0};

    /**
     * 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() {
	/*
	 * Initialize bitCount field the first time this method is executed.
	 * This method depends on the atomicity of int modifies; without
	 * this guarantee, it would have to be synchronized.
	 */
	if (bitCount == -1) {
	    // Count the bits in the magnitude
	    int magBitCount = 0;
	    for (int i=0; i<mag.length; i++)
		magBitCount += bitCnt(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 +=
                            trailingZeroCnt(mag[j]);

		bitCount = magBitCount + magTrailingZeroCount - 1;
	    } else {
		bitCount = magBitCount;
	    }
	}
	return bitCount;
    }

    static int bitCnt(int val) {
        val -= (0xaaaaaaaa & val) >>> 1;
        val = (val & 0x33333333) + ((val >>> 2) & 0x33333333);
        val = val + (val >>> 4) & 0x0f0f0f0f;
        val += val >>> 8;
        val += val >>> 16;
        return val & 0xff;
    }

    static int trailingZeroCnt(int val) {
        // Loop unrolled for performance
        int byteVal = val & 0xff;
        if (byteVal != 0)
            return trailingZeroTable[byteVal];

        byteVal = (val >>> 8) & 0xff;
        if (byteVal != 0)
            return trailingZeroTable[byteVal] + 8;

        byteVal = (val >>> 16) & 0xff;
        if (byteVal != 0)
            return trailingZeroTable[byteVal] + 16;

        byteVal = (val >>> 24) & 0xff;
        return trailingZeroTable[byteVal] + 24;
    }

    // Primality Testing

    /**
     * Returns <tt>true</tt> if this BigInteger is probably prime,
     * <tt>false</tt> if it's definitely composite.  If
     * <tt>certainty</tt> is <tt> &lt;= 0</tt>, <tt>true</tt> is
     * returned.
     *
     * @param  certainty a measure of the uncertainty that the caller is
     *	       willing to tolerate: if the call returns <tt>true</tt>
     *	       the probability that this BigInteger is prime exceeds
     *	       <tt>(1 - 1/2<sup>certainty</sup>)</tt>.  The execution time of
     * 	       this method is proportional to the value of this parameter.
     * @return <tt>true</tt> if this BigInteger is probably prime,
     * 	       <tt>false</tt> 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);
    }

    // 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 (&lt;, ==, &gt;, &gt;=, !=, &lt;=).  The
     * suggested idiom for performing these comparisons is:
     * <tt>(x.compareTo(y)</tt> &lt;<i>op</i>&gt; <tt>0)</tt>,
     * where &lt;<i>op</i>&gt; is one of the six comparison operators.
     *
     * @param  val BigInteger to which this BigInteger is to be compared.
     * @return -1, 0 or 1 as this BigInteger is numerically less than, equal
     *         to, or greater than <tt>val</tt>.
     */
    public int compareTo(BigInteger val) {
	return (signum==val.signum
		? signum*intArrayCmp(mag, val.mag)
		: (signum>val.signum ? 1 : -1));
    }

    /*
     * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is
     * less than, equal to, or greater than arg2.
     */
    private static int intArrayCmp(int[] arg1, int[] arg2) {
	if (arg1.length < arg2.length)
	    return -1;
	if (arg1.length > arg2.length)
	    return 1;

	// Argument lengths are equal; compare the values
	for (int i=0; i<arg1.length; 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;
    }

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

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

	if (xInt.signum != signum || xInt.mag.length != mag.length)
	    return false;

	for (int i=0; i<mag.length; i++)
	    if (xInt.mag[i] != mag[i])
		return false;

	return true;
    }

    /**
     * Returns the minimum of this BigInteger and <tt>val</tt>.
     *
     * @param  val value with which the minimum is to be computed.
     * @return the BigInteger whose value is the lesser of this BigInteger and 
     *	       <tt>val</tt>.  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 <tt>val</tt>.
     *
     * @param  val value with which the maximum is to be computed.
     * @return the BigInteger whose value is the greater of this and
     *         <tt>val</tt>.  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
     * <tt>Integer.toString</tt>).  The digit-to-character mapping
     * provided by <tt>Character.forDigit</tt> is used, and a minus
     * sign is prepended if appropriate.  (This representation is
     * compatible with the {@link #BigInteger(String, int) (String,
     * <code>int</code>)} 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;

	// Compute upper bound on number of digit groups and allocate space
	int maxNumDigitGroups = (4*mag.length + 6)/7;
	String digitGroup[] = new String[maxNumDigitGroups];
        
	// Translate number to string, a digit group at a time
	BigInteger tmp = this.abs();
	int numGroups = 0;
	while (tmp.signum != 0) {
            BigInteger d = longRadix[radix];

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

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

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

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

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

    /**
     * Returns the decimal String representation of this BigInteger.
     * The digit-to-character mapping provided by
     * <tt>Character.forDigit</tt> 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 <tt>(ceil((this.bitLength() +
     * 1)/8))</tt>.  (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</code>.  This
     * conversion is analogous to a <a
     * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing
     * primitive conversion</i></a> from <code>long</code> to
     * <code>int</code> as defined in the <a
     * href="http://java.sun.com/docs/books/jls/html/">Java Language
     * Specification</a>: if this BigInteger is too big to fit in an
     * <code>int</code>, 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</code>.
     */
    public int intValue() {
	int result = 0;
	result = getInt(0);
	return result;
    }

    /**
     * Converts this BigInteger to a <code>long</code>.  This
     * conversion is analogous to a <a
     * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing
     * primitive conversion</i></a> from <code>long</code> to
     * <code>int</code> as defined in the <a
     * href="http://java.sun.com/docs/books/jls/html/">Java Language
     * Specification</a>: if this BigInteger is too big to fit in a
     * <code>long</code>, 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</code>.
     */
    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</code>.  This
     * conversion is similar to the <a
     * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing
     * primitive conversion</i></a> from <code>double</code> to
     * <code>float</code> defined in the <a
     * href="http://java.sun.com/docs/books/jls/html/">Java Language
     * Specification</a>: if this BigInteger has too great a magnitude
     * to represent as a <code>float</code>, 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</code>.
     */
    public float floatValue() {
	// Somewhat inefficient, but guaranteed to work.
	return Float.valueOf(this.toString()).floatValue();
    }

    /**
     * Converts this BigInteger to a <code>double</code>.  This
     * conversion is similar to the <a
     * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing
     * primitive conversion</i></a> from <code>double</code> to
     * <code>float</code> defined in the <a
     * href="http://java.sun.com/docs/books/jls/html/">Java Language
     * Specification</a>: if this BigInteger has too great a magnitude
     * to represent as a <code>double</code>, 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</code>.
     */
    public double doubleValue() {
	// Somewhat inefficient, but guaranteed to work.
	return Double.valueOf(this.toString()).doubleValue();
    }

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

	// Find first nonzero byte
        for (keep=0; keep<val.length && val[keep]==0; keep++)
            ;

        int result[] = new int[val.length - keep];
        for(int i=0; i<val.length - keep; i++)
            result[i] = val[keep+i];

        return result;
    }

    /**
     * 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 byteLength = val.length;
	int keep;

	// Find first nonzero byte
        for (keep=0; keep<val.length && val[keep]==0; keep++)
            ;

        // Only perform copy if necessary
        if (keep > 0) {
            int result[] = new int[val.length - keep];
            for(int i=0; i<val.length - keep; i++)
               result[i] = val[keep+i];
            return result; 
        }
        return val;
    }

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

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

	// Allocate new array and copy relevant part of input array
        int intLength = ((byteLength - keep) + 3)/4;
	int[] result = new int[intLength];
        int b = byteLength - 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 <= 8*bytesToTransfer; 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 keep, k;
        int byteLength = a.length;

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

	int extraByte = (k==byteLength) ? 1 : 0;
        int intLength = ((byteLength - keep + extraByte) + 3)/4;
	int result[] = new int[intLength];

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

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

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

	return result;
    }

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

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

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

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

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

	return result;
    }

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

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

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

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

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

    /**
     * Returns the length of the two's complement representation in ints,
     * including space for at least one sign bit.
     */
    private int intLength() {
	return bitLength()/32 + 1;
    }

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

    /* Returns an int of sign bits */
    private int signInt() {
	return (int) (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 (int) (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.
     */
     private int firstNonzeroIntNum() {
	/*
	 * Initialize firstNonzeroIntNum field the first time this method is
	 * executed. This method depends on the atomicity of int modifies;
	 * without this guarantee, it would have to be synchronized.
	 */
	if (firstNonzeroIntNum == -2) {
	    // Search for the first nonzero int
	    int i;
	    for (i=mag.length-1; i>=0 && mag[i]==0; i--)
		;
	    firstNonzeroIntNum = mag.length-i-1;
	}
	return firstNonzeroIntNum;
    }

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

    /**
     * Serializable fields for BigInteger.
     * 
     * @serialField signum  int
     *              signum of this BigInteger.
     * @serialField magnitude int[]
     *              magnitude array of this BigInteger.
     * @serialField bitCount  int
     *              number of bits in this BigInteger
     * @serialField bitLength int
     *              the number of bits in the minimal two's-complement
     *              representation of this BigInteger
     * @serialField lowestSetBit int
     *              lowest set bit in the twos complement representation
     */
    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 <tt>BigInteger</tt> 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.
     */
    private void readObject(java.io.ObjectInputStream s)
        throws java.io.IOException, ClassNotFoundException {
        /*
         * In order to maintain compatibility with previous serialized forms,
         * the magnitude of a BigInteger is serialized as an array of bytes.
         * The magnitude field is used as a temporary store for the byte array
         * that is deserialized. The cached computation fields should be
         * transient but are serialized for compatibility reasons.
         */

        // prepare to read the alternate persistent fields
        ObjectInputStream.GetField fields = s.readFields();
            
        // Read the alternate persistent fields that we care about
        signum = (int)fields.get("signum", -2);
        byte[] magnitude = (byte[])fields.get("magnitude", null);

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

        // Set "cached computation" fields to their initial values
        bitCount = bitLength = -1;
        lowestSetBit = firstNonzeroByteNum = firstNonzeroIntNum = -2;

        // Calculate mag field from magnitude and discard magnitude
	mag = stripLeadingZeroBytes(magnitude);
    }

    /**
     * Save the <tt>BigInteger</tt> instance to a stream.
     * The magnitude of a BigInteger is serialized as a byte array for
     * historical reasons.
     * 
     * @serialData two necessary fields are written as well as obsolete
     *             fields for compatibility with older versions.
     */
    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());
        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 bitLen = (mag.length == 0 ? 0 :
                      ((mag.length - 1) << 5) + bitLen(mag[0]));
        int byteLen = (bitLen + 7)/8;
        byte[] result = new byte[byteLen];

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