/*
 * @(#)BigInteger.java	1.43 01/02/16
 *
 * Copyright 1996-2001 Sun Microsystems, Inc. All Rights Reserved.
 * 
 * This software is the proprietary information of Sun Microsystems, Inc.  
 * Use is subject to license terms.
 * 
 */

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 (>>>) 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 the BigInteger <tt>j</tt>."  Other pseudo-code expressions are
 * interpreted similarly.
 *
 * @see     BigDecimal
 * @version 1.43, 02/16/01
 * @author  Josh Bloch
 * @author  Michael McCloskey
 * @since JDK1.1
 */

public class BigInteger extends Number implements Comparable {
    /**
     * 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.
     */
    transient int[] mag;

    /**
     * This field is required for historical reasons. The magnitude of a
     * BigInteger used to be in a byte representation, and is still serialized
     * that way. The mag field is used in all real computations but the
     * magnitude field is required for storage.
     *
     * @serial
     */
    private byte[] magnitude;

    // 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 transient 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 in in a BigInteger value of 0, whether signum is -1, 0 or 1.
     *
     * @param  signum signum of the number (-1 for negative, 0 for zero, 1
     * 	       for positive).
     * @param  magnitude big-endian binary representation of the magnitude of
     * 	       the number.
     * @throws NumberFormatException <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 <tt>Character.MIN_RADIX</tt> (2) to
     *	       <tt>Character.MAX_RADIX</tt> (36), 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 leading minus sign
	signum = 1;
	if (val.charAt(0) == '-') {
	    if (val.length() == 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.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);
        
	// 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);
            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 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 < 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;
    }

    /**
     * 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.
     * @throws ArithmeticException <tt>bitLength < 2</tt>.
     * @see    #bitLength
     */
    private 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 < 95 ? smallPrime(bitLength, 100, rnd)
                               : largePrime(bitLength, 100, 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(3*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 <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 = (certainty+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;
        int sign = 1;
        while (jacobiSymbol(Math.abs(d), this) != -1) {
            d += 2;
            sign = -sign;
        }
        d = d * sign;
        
        // Step 2
        BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);

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

    /**
     * Computes Jacobi(p,n).
     * Assumes n is positive, odd.
     */
    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];

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

	return new BigInteger(val);
    }

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

    // 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);
	    if ((exponent >>>= 1) != 0)
                baseToPow2 = squareToLen(baseToPow2, baseToPow2.length, null);
	}
        result = trustedStripLeadingZeroInts(result);
	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 && val==0</tt>.
     *
     * @param  val value with with 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 <= 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 <= 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 t