jdk-sandbox: comparison jdk/src/share/classes/java/math/BigInteger.java

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+/*
+* Portions Copyright 1996-2007 Sun Microsystems, Inc.  All Rights Reserved.
+* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
+*
+* This code is free software; you can redistribute it and/or modify it
+* under the terms of the GNU General Public License version 2 only, as
+* published by the Free Software Foundation.  Sun designates this
+* particular file as subject to the "Classpath" exception as provided
+* by Sun in the LICENSE file that accompanied this code.
+*
+* This code is distributed in the hope that it will be useful, but WITHOUT
+* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+* FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
+* version 2 for more details (a copy is included in the LICENSE file that
+* accompanied this code).
+*
+* You should have received a copy of the GNU General Public License version
+* 2 along with this work; if not, write to the Free Software Foundation,
+* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
+*
+* Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
+* CA 95054 USA or visit www.sun.com if you need additional information or
+* have any questions.
+*/
+/*
+* Portions Copyright (c) 1995  Colin Plumb.  All rights reserved.
+*/
+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 {@code ArithmeticException}, and
+* division of a negative by a positive yields a negative (or zero) remainder.
+* All of the details in the Spec concerning overflow are ignored, as
+* BigIntegers are made as large as necessary to accommodate the results of an
+* operation.
+*
+* <p>Semantics of shift operations extend those of Java's shift operators
+* to allow for negative shift distances.  A right-shift with a negative
+* shift distance results in a left shift, and vice-versa.  The unsigned
+* right shift operator ({@code >>>}) is omitted, as this operation makes
+* little sense in combination with the "infinite word size" abstraction
+* provided by this class.
+*
+* <p>Semantics of bitwise logical operations exactly mimic those of Java's
+* bitwise integer operators.  The binary operators ({@code and},
+* {@code or}, {@code xor}) implicitly perform sign extension on the shorter
+* of the two operands prior to performing the operation.
+*
+* <p>Comparison operations perform signed integer comparisons, analogous to
+* those performed by Java's relational and equality operators.
+*
+* <p>Modular arithmetic operations are provided to compute residues, perform
+* exponentiation, and compute multiplicative inverses.  These methods always
+* return a non-negative result, between {@code 0} and {@code (modulus - 1)},
+* inclusive.
+*
+* <p>Bit operations operate on a single bit of the two's-complement
+* representation of their operand.  If necessary, the operand is sign-
+* extended so that it contains the designated bit.  None of the single-bit
+* operations can produce a BigInteger with a different sign from the
+* BigInteger being operated on, as they affect only a single bit, and the
+* "infinite word size" abstraction provided by this class ensures that there
+* are infinitely many "virtual sign bits" preceding each BigInteger.
+*
+* <p>For the sake of brevity and clarity, pseudo-code is used throughout the
+* descriptions of BigInteger methods.  The pseudo-code expression
+* {@code (i + j)} is shorthand for "a BigInteger whose value is
+* that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
+* The pseudo-code expression {@code (i == j)} is shorthand for
+* "{@code true} if and only if the BigInteger {@code i} represents the same
+* value as the BigInteger {@code j}."  Other pseudo-code expressions are
+* interpreted similarly.
+*
+* <p>All methods and constructors in this class throw
+* {@code NullPointerException} when passed
+* a null object reference for any input parameter.
+*
+* @see     BigDecimal
+* @author  Josh Bloch
+* @author  Michael McCloskey
+* @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 ({@code mag[0]}) must be non-zero.  This is necessary to
+* ensure that there is exactly one representation for each BigInteger
+* value.  Note that this implies that the BigInteger zero has a
+* zero-length mag array.
+*/
+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 {@code val} is zero bytes long.
+*/
+public BigInteger(byte[] val) {
+if (val.length == 0)
+throw new NumberFormatException("Zero length BigInteger");
+if (val[0] < 0) {
+mag = makePositive(val);
+signum = -1;
+} else {
+mag = stripLeadingZeroBytes(val);
+signum = (mag.length == 0 ? 0 : 1);
+}
+}
+/**
+* This private constructor translates an int array containing the
+* two's-complement binary representation of a BigInteger into a
+* BigInteger. The input array is assumed to be in <i>big-endian</i>
+* int-order: the most significant int is in the zeroth element.
+*/
+private BigInteger(int[] val) {
+if (val.length == 0)
+throw new NumberFormatException("Zero length BigInteger");
+if (val[0] < 0) {
+mag = makePositive(val);
+signum = -1;
+} else {
+mag = trustedStripLeadingZeroInts(val);
+signum = (mag.length == 0 ? 0 : 1);
+}
+}
+/**
+* Translates the sign-magnitude representation of a BigInteger into a
+* BigInteger.  The sign is represented as an integer signum value: -1 for
+* negative, 0 for zero, or 1 for positive.  The magnitude is a byte array
+* in <i>big-endian</i> byte-order: the most significant byte is in the
+* zeroth element.  A zero-length magnitude array is permissible, and will
+* result in a BigInteger value of 0, whether signum is -1, 0 or 1.
+*
+* @param  signum signum of the number (-1 for negative, 0 for zero, 1
+*         for positive).
+* @param  magnitude big-endian binary representation of the magnitude of
+*         the number.
+* @throws NumberFormatException {@code signum} is not one of the three
+*         legal values (-1, 0, and 1), or {@code signum} is 0 and
+*         {@code magnitude} contains one or more non-zero bytes.
+*/
+public BigInteger(int signum, byte[] magnitude) {
+this.mag = stripLeadingZeroBytes(magnitude);
+if (signum < -1 || signum > 1)
+throw(new NumberFormatException("Invalid signum value"));
+if (this.mag.length==0) {
+this.signum = 0;
+} else {
+if (signum == 0)
+throw(new NumberFormatException("signum-magnitude mismatch"));
+this.signum = signum;
+}
+}
+/**
+* A constructor for internal use that translates the sign-magnitude
+* representation of a BigInteger into a BigInteger. It checks the
+* arguments and copies the magnitude so this constructor would be
+* safe for external use.
+*/
+private BigInteger(int signum, int[] magnitude) {
+this.mag = stripLeadingZeroInts(magnitude);
+if (signum < -1 || signum > 1)
+throw(new NumberFormatException("Invalid signum value"));
+if (this.mag.length==0) {
+this.signum = 0;
+} else {
+if (signum == 0)
+throw(new NumberFormatException("signum-magnitude mismatch"));
+this.signum = signum;
+}
+}
+/**
+* Translates the String representation of a BigInteger in the
+* specified radix into a BigInteger.  The String representation
+* consists of an optional minus or plus sign followed by a
+* sequence of one or more digits in the specified radix.  The
+* character-to-digit mapping is provided by {@code
+* Character.digit}.  The String may not contain any extraneous
+* characters (whitespace, for example).
+*
+* @param val String representation of BigInteger.
+* @param radix radix to be used in interpreting {@code val}.
+* @throws NumberFormatException {@code val} is not a valid representation
+*         of a BigInteger in the specified radix, or {@code radix} is
+*         outside the range from {@link Character#MIN_RADIX} to
+*         {@link Character#MAX_RADIX}, inclusive.
+* @see    Character#digit
+*/
+public BigInteger(String val, int radix) {
+int cursor = 0, numDigits;
+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 at most one leading sign
+signum = 1;
+int index1 = val.lastIndexOf('-');
+int index2 = val.lastIndexOf('+');
+if ((index1 + index2) <= -1) {
+// No leading sign character or at most one leading sign character
+if (index1 == 0 || index2 == 0) {
+cursor = 1;
+if (val.length() == 1)
+throw new NumberFormatException("Zero length BigInteger");
+}
+if (index1 == 0)
+signum = -1;
+} else
+throw new NumberFormatException("Illegal embedded sign character");
+// Skip leading zeros and compute number of digits in magnitude
+while (cursor < len &&
+Character.digit(val.charAt(cursor), radix) == 0)
+cursor++;
+if (cursor == len) {
+signum = 0;
+mag = ZERO.mag;
+return;
+} 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;
+} else if (val[0] == '+') {
+if (len == 1)
+throw new NumberFormatException("Zero length BigInteger");
+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 {@code Character.digit}.
+* The String may not contain any extraneous characters (whitespace, for
+* example).
+*
+* @param val decimal String representation of BigInteger.
+* @throws NumberFormatException {@code val} is not a valid representation
+*         of a BigInteger.
+* @see    Character#digit
+*/
+public BigInteger(String val) {
+this(val, 10);
+}
+/**
+* Constructs a randomly generated BigInteger, uniformly distributed over
+* the range {@code 0} to (2<sup>{@code numBits}</sup> - 1), inclusive.
+* The uniformity of the distribution assumes that a fair source of random
+* bits is provided in {@code rnd}.  Note that this constructor always
+* constructs a non-negative BigInteger.
+*
+* @param  numBits maximum bitLength of the new BigInteger.
+* @param  rnd source of randomness to be used in computing the new
+*         BigInteger.
+* @throws IllegalArgumentException {@code numBits} is negative.
+* @see #bitLength()
+*/
+public BigInteger(int numBits, Random rnd) {
+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 = (int)(((long)numBits+7)/8); // avoid overflow
+byte[] randomBits = new byte[numBytes];
+// Generate random bytes and mask out any excess bits
+if (numBytes > 0) {
+rnd.nextBytes(randomBits);
+int excessBits = 8*numBytes - numBits;
+randomBits[0] &= (1 << (8-excessBits)) - 1;
+}
+return randomBits;
+}
+/**
+* Constructs a randomly generated positive BigInteger that is probably
+* prime, with the specified bitLength.
+*
+* <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
+*         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
+*         this constructor is proportional to the value of this parameter.
+* @param  rnd source of random bits used to select candidates to be
+*         tested for primality.
+* @throws ArithmeticException {@code bitLength < 2}.
+* @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 {@code bitLength} bits that is probably prime
+* @throws ArithmeticException {@code bitLength < 2}.
+* @see    #bitLength()
+* @since 1.4
+*/
+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, rnd))
+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, rnd);
+while ((candidate == null) || (candidate.bitLength() != bitLength)) {
+p = p.add(BigInteger.valueOf(2*searchLen));
+if (p.bitLength() != bitLength)
+p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
+p.mag[p.mag.length-1] &= 0xfffffffe;
+searchSieve = new BitSieve(p, searchLen);
+candidate = searchSieve.retrieve(p, certainty, rnd);
+}
+return candidate;
+}
+/**
+* Returns the first integer greater than this {@code BigInteger} that
+* is probably prime.  The probability that the number returned by this
+* method is composite does not exceed 2<sup>-100</sup>. This method will
+* never skip over a prime when searching: if it returns {@code p}, there
+* is no prime {@code q} such that {@code this < q < p}.
+*
+* @return the first integer greater than this {@code BigInteger} that
+*         is probably prime.
+* @throws ArithmeticException {@code this < 0}.
+* @since 1.5
+*/
+public BigInteger nextProbablePrime() {
+if (this.signum < 0)
+throw new ArithmeticException("start < 0: " + this);
+// Handle trivial cases
+if ((this.signum == 0) || this.equals(ONE))
+return TWO;
+BigInteger result = this.add(ONE);
+// Fastpath for small numbers
+if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
+// Ensure an odd number
+if (!result.testBit(0))
+result = result.add(ONE);
+while(true) {
+// Do cheap "pre-test" if applicable
+if (result.bitLength() > 6) {
+long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
+if ((r%3==0)  || (r%5==0)  || (r%7==0)  || (r%11==0) ||
+(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
+(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
+result = result.add(TWO);
+continue; // Candidate is composite; try another
+}
+}
+// All candidates of bitLength 2 and 3 are prime by this point
+if (result.bitLength() < 4)
+return result;
+// The expensive test
+if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
+return result;
+result = result.add(TWO);
+}
+}
+// Start at previous even number
+if (result.testBit(0))
+result = result.subtract(ONE);
+// Looking for the next large prime
+int searchLen = (result.bitLength() / 20) * 64;
+while(true) {
+BitSieve searchSieve = new BitSieve(result, searchLen);
+BigInteger candidate = searchSieve.retrieve(result,
+DEFAULT_PRIME_CERTAINTY, null);
+if (candidate != null)
+return candidate;
+result = result.add(BigInteger.valueOf(2 * searchLen));
+}
+}
+/**
+* Returns {@code true} if this BigInteger is probably prime,
+* {@code false} if it's definitely composite.
+*
+* This method assumes bitLength > 2.
+*
+* @param  certainty a measure of the uncertainty that the caller is
+*         willing to tolerate: if the call returns {@code true}
+*         the probability that this BigInteger is prime exceeds
+*         {@code (1 - 1/2<sup>certainty</sup>)}.  The execution time of
+*         this method is proportional to the value of this parameter.
+* @return {@code true} if this BigInteger is probably prime,
+*         {@code false} if it's definitely composite.
+*/
+boolean primeToCertainty(int certainty, Random random) {
+int rounds = 0;
+int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
+// The relationship between the certainty and the number of rounds
+// we perform is given in the draft standard ANSI X9.80, "PRIME
+// NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
+int sizeInBits = this.bitLength();
+if (sizeInBits < 100) {
+rounds = 50;
+rounds = n < rounds ? n : rounds;
+return passesMillerRabin(rounds, random);
+}
+if (sizeInBits < 256) {
+rounds = 27;
+} else if (sizeInBits < 512) {
+rounds = 15;
+} else if (sizeInBits < 768) {
+rounds = 8;
+} else if (sizeInBits < 1024) {
+rounds = 4;
+} else {
+rounds = 2;
+}
+rounds = n < rounds ? n : rounds;
+return passesMillerRabin(rounds, random) && passesLucasLehmer();
+}
+/**
+* Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
+*
+* The following assumptions are made:
+* This BigInteger is a positive, odd number.
+*/
+private boolean passesLucasLehmer() {
+BigInteger thisPlusOne = this.add(ONE);
+// Step 1
+int d = 5;
+while (jacobiSymbol(d, this) != -1) {
+// 5, -7, 9, -11, ...
+d = (d<0) ? Math.abs(d)+2 : -(d+2);
+}
+// Step 2
+BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
+// Step 3
+return u.mod(this).equals(ZERO);
+}
+/**
+* Computes Jacobi(p,n).
+* Assumes n positive, odd, n>=3.
+*/
+private static int jacobiSymbol(int p, BigInteger n) {
+if (p == 0)
+return 0;
+// Algorithm and comments adapted from Colin Plumb's C library.
+int j = 1;
+int u = n.mag[n.mag.length-1];
+// Make p positive
+if (p < 0) {
+p = -p;
+int n8 = u & 7;
+if ((n8 == 3) || (n8 == 7))
+j = -j; // 3 (011) or 7 (111) mod 8
+}
+// Get rid of factors of 2 in p
+while ((p & 3) == 0)
+p >>= 2;
+if ((p & 1) == 0) {
+p >>= 1;
+if (((u ^ (u>>1)) & 2) != 0)
+j = -j; // 3 (011) or 5 (101) mod 8
+}
+if (p == 1)
+return j;
+// Then, apply quadratic reciprocity
+if ((p & u & 2) != 0)   // p = u = 3 (mod 4)?
+j = -j;
+// And reduce u mod p
+u = n.mod(BigInteger.valueOf(p)).intValue();
+// Now compute Jacobi(u,p), u < p
+while (u != 0) {
+while ((u & 3) == 0)
+u >>= 2;
+if ((u & 1) == 0) {
+u >>= 1;
+if (((p ^ (p>>1)) & 2) != 0)
+j = -j;     // 3 (011) or 5 (101) mod 8
+}
+if (u == 1)
+return j;
+// Now both u and p are odd, so use quadratic reciprocity
+assert (u < p);
+int t = u; u = p; p = t;
+if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
+j = -j;
+// Now u >= p, so it can be reduced
+u %= p;
+}
+return 0;
+}
+private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
+BigInteger d = BigInteger.valueOf(z);
+BigInteger u = ONE; BigInteger u2;
+BigInteger v = ONE; BigInteger v2;
+for (int i=k.bitLength()-2; i>=0; i--) {
+u2 = u.multiply(v).mod(n);
+v2 = v.square().add(d.multiply(u.square())).mod(n);
+if (v2.testBit(0)) {
+v2 = 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;
+}
+private static volatile Random staticRandom;
+private static Random getSecureRandom() {
+if (staticRandom == null) {
+staticRandom = new java.security.SecureRandom();
+}
+return staticRandom;
+}
+/**
+* Returns true iff this BigInteger passes the specified number of
+* Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
+* 186-2).
+*
+* The following assumptions are made:
+* This BigInteger is a positive, odd number greater than 2.
+* iterations<=50.
+*/
+private boolean passesMillerRabin(int iterations, Random rnd) {
+// Find a and m such that m is odd and this == 1 + 2**a * m
+BigInteger thisMinusOne = this.subtract(ONE);
+BigInteger m = thisMinusOne;
+int a = m.getLowestSetBit();
+m = m.shiftRight(a);
+// Do the tests
+if (rnd == null) {
+rnd = getSecureRandom();
+}
+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}.  This "static factory method" is
+* provided in preference to a ({@code long}) constructor
+* because it allows for reuse of frequently used BigIntegers.
+*
+* @param  val value of the BigInteger to return.
+* @return a BigInteger with the specified value.
+*/
+public static BigInteger valueOf(long val) {
+// If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
+if (val == 0)
+return ZERO;
+if (val > 0 && val <= MAX_CONSTANT)
+return posConst[(int) val];
+else if (val < 0 && val >= -MAX_CONSTANT)
+return negConst[(int) -val];
+return new BigInteger(val);
+}
+/**
+* Constructs a BigInteger with the specified value, which may not be zero.
+*/
+private BigInteger(long val) {
+if (val < 0) {
+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] = 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 {@code (this + val)}.
+*
+* @param  val value to be added to this BigInteger.
+* @return {@code this + val}
+*/
+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 {@code (this - val)}.
+*
+* @param  val value to be subtracted from this BigInteger.
+* @return {@code this - val}
+*/
+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 {@code (this * val)}.
+*
+* @param  val value to be multiplied by this BigInteger.
+* @return {@code this * val}
+*/
+public BigInteger multiply(BigInteger val) {
+if (val.signum == 0 || 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 {@code (this<sup>2</sup>)}.
+*
+* @return {@code this<sup>2</sup>}
+*/
+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 {@code (this / val)}.
+*
+* @param  val value by which this BigInteger is to be divided.
+* @return {@code this / val}
+* @throws ArithmeticException {@code val==0}
+*/
+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 {@code (this / val)}
+* followed by {@code (this % val)}.
+*
+* @param  val value by which this BigInteger is to be divided, and the
+*         remainder computed.
+* @return an array of two BigIntegers: the quotient {@code (this / val)}
+*         is the initial element, and the remainder {@code (this % val)}
+*         is the final element.
+* @throws ArithmeticException {@code val==0}
+*/
+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 {@code (this % val)}.
+*
+* @param  val value by which this BigInteger is to be divided, and the
+*         remainder computed.
+* @return {@code this % val}
+* @throws ArithmeticException {@code val==0}
+*/
+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 {@code exponent} 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 {@code exponent} is negative.  (This would
+*         cause the operation to yield a non-integer value.)
+*/
+public BigInteger pow(int exponent) {
+if (exponent < 0)
+throw new ArithmeticException("Negative exponent");
+if (signum==0)
+return (exponent==0 ? ONE : this);
+// 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
+* {@code abs(this)} and {@code abs(val)}.  Returns 0 if
+* {@code this==0 && val==0}.
+*
+* @param  val value with which the GCD is to be computed.
+* @return {@code GCD(abs(this), abs(val))}
+*/
+public BigInteger gcd(BigInteger val) {
+if (val.signum == 0)
+return this.abs();
+else if (this.signum == 0)
+return val.abs();
+MutableBigInteger a = new MutableBigInteger(this);
+MutableBigInteger b = new MutableBigInteger(val);
+MutableBigInteger result = a.hybridGCD(b);
+return 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 {@code abs(this)}
+*/
+public BigInteger abs() {
+return (signum >= 0 ? this : this.negate());
+}
+/**
+* Returns a BigInteger whose value is {@code (-this)}.
+*
+* @return {@code -this}
+*/
+public BigInteger negate() {
+return new BigInteger(this.mag, -this.signum);
+}
+/**
+* Returns the signum function of this BigInteger.
+*
+* @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
+*         positive.
+*/
+public int signum() {
+return this.signum;
+}
+// Modular Arithmetic Operations
+/**
+* Returns a BigInteger whose value is {@code (this mod m}).  This method
+* differs from {@code remainder} in that it always returns a
+* <i>non-negative</i> BigInteger.
+*
+* @param  m the modulus.
+* @return {@code this mod m}
+* @throws ArithmeticException {@code m <= 0}
+* @see    #remainder
+*/
+public BigInteger mod(BigInteger m) {
+if (m.signum <= 0)
+throw new ArithmeticException("BigInteger: modulus not positive");
+BigInteger result = this.remainder(m);
+return (result.signum >= 0 ? result : result.add(m));
+}
+/**
+* Returns a BigInteger whose value is
+* <tt>(this<sup>exponent</sup> mod m)</tt>.  (Unlike {@code pow}, this
+* method permits negative exponents.)
+*
+* @param  exponent the exponent.
+* @param  m the modulus.
+* @return <tt>this<sup>exponent</sup> mod m</tt>
+* @throws ArithmeticException {@code m <= 0}
+* @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 = 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 = 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 {@code BigInteger >= 0} and {@code p > 0}.
+*/
+private BigInteger mod2(int p) {
+if (bitLength() <= p)
+return this;
+// Copy remaining ints of mag
+int numInts = (p+31)/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 {@code (this}<sup>-1</sup> {@code mod m)}.
+*
+* @param  m the modulus.
+* @return {@code this}<sup>-1</sup> {@code mod m}.
+* @throws ArithmeticException {@code  m <= 0}, or this BigInteger
+*         has no multiplicative inverse mod m (that is, this BigInteger
+*         is not <i>relatively prime</i> to m).
+*/
+public BigInteger modInverse(BigInteger m) {
+if (m.signum != 1)
+throw new ArithmeticException("BigInteger: modulus not positive");
+if (m.equals(ONE))
+return ZERO;
+// Calculate (this mod m)
+BigInteger modVal = this;
+if (signum < 0 || (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 {@code (this << n)}.
+* The shift distance, {@code n}, 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 {@code this << n}
+* @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 {@code (this >> n)}.  Sign
+* extension is performed.  The shift distance, {@code n}, may be
+* negative, in which case this method performs a left shift.
+* (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
+*
+* @param  n shift distance, in bits.
+* @return {@code this >> n}
+* @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) {
+int lastSum = 0;
+for (int i=val.length-1;  i >= 0 && lastSum == 0; i--)
+lastSum = (val[i] += 1);
+if (lastSum == 0) {
+val = new int[val.length+1];
+val[0] = 1;
+}
+return val;
+}
+// Bitwise Operations
+/**
+* Returns a BigInteger whose value is {@code (this & val)}.  (This
+* method returns a negative BigInteger if and only if this and val are
+* both negative.)
+*
+* @param val value to be AND'ed with this BigInteger.
+* @return {@code this & val}
+*/
+public BigInteger and(BigInteger val) {
+int[] result = new int[Math.max(intLength(), val.intLength())];
+for (int i=0; i<result.length; i++)
+result[i] = (getInt(result.length-i-1)
+& val.getInt(result.length-i-1));
+return valueOf(result);
+}
+/**
+* Returns a BigInteger whose value is {@code (this | val)}.  (This method
+* returns a negative BigInteger if and only if either this or val is
+* negative.)
+*
+* @param val value to be OR'ed with this BigInteger.
+* @return {@code this | val}
+*/
+public BigInteger or(BigInteger val) {
+int[] result = new int[Math.max(intLength(), val.intLength())];
+for (int i=0; i<result.length; i++)
+result[i] = (getInt(result.length-i-1)
+| val.getInt(result.length-i-1));
+return valueOf(result);
+}
+/**
+* Returns a BigInteger whose value is {@code (this ^ val)}.  (This method
+* returns a negative BigInteger if and only if exactly one of this and
+* val are negative.)
+*
+* @param val value to be XOR'ed with this BigInteger.
+* @return {@code this ^ val}
+*/
+public BigInteger xor(BigInteger val) {
+int[] result = new int[Math.max(intLength(), val.intLength())];
+for (int i=0; i<result.length; i++)
+result[i] = (getInt(result.length-i-1)
+^ val.getInt(result.length-i-1));
+return valueOf(result);
+}
+/**
+* Returns a BigInteger whose value is {@code (~this)}.  (This method
+* returns a negative value if and only if this BigInteger is
+* non-negative.)
+*
+* @return {@code ~this}
+*/
+public BigInteger not() {
+int[] result = new int[intLength()];
+for (int i=0; i<result.length; i++)
+result[i] = ~getInt(result.length-i-1);
+return valueOf(result);
+}
+/**
+* Returns a BigInteger whose value is {@code (this & ~val)}.  This
+* method, which is equivalent to {@code and(val.not())}, is provided as
+* a convenience for masking operations.  (This method returns a negative
+* BigInteger if and only if {@code this} is negative and {@code val} is
+* positive.)
+*
+* @param val value to be complemented and AND'ed with this BigInteger.
+* @return {@code this & ~val}
+*/
+public BigInteger andNot(BigInteger val) {
+int[] result = new int[Math.max(intLength(), val.intLength())];
+for (int i=0; i<result.length; i++)
+result[i] = (getInt(result.length-i-1)
+& ~val.getInt(result.length-i-1));
+return valueOf(result);
+}
+// Single Bit Operations
+/**
+* Returns {@code true} if and only if the designated bit is set.
+* (Computes {@code ((this & (1<<n)) != 0)}.)
+*
+* @param  n index of bit to test.
+* @return {@code true} if and only if the designated bit is set.
+* @throws ArithmeticException {@code n} is negative.
+*/
+public boolean testBit(int n) {
+if (n<0)
+throw new ArithmeticException("Negative bit address");
+return (getInt(n/32) & (1 << (n%32))) != 0;
+}
+/**
+* Returns a BigInteger whose value is equivalent to this BigInteger
+* with the designated bit set.  (Computes {@code (this | (1<<n))}.)
+*
+* @param  n index of bit to set.
+* @return {@code this | (1<<n)}
+* @throws ArithmeticException {@code n} is negative.
+*/
+public BigInteger setBit(int n) {
+if (n<0)
+throw new ArithmeticException("Negative bit address");
+int intNum = n/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 {@code (this & ~(1<<n))}.)
+*
+* @param  n index of bit to clear.
+* @return {@code this & ~(1<<n)}
+* @throws ArithmeticException {@code n} is negative.
+*/
+public BigInteger clearBit(int n) {
+if (n<0)
+throw new ArithmeticException("Negative bit address");
+int intNum = n/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 {@code (this ^ (1<<n))}.)
+*
+* @param  n index of bit to flip.
+* @return {@code this ^ (1<<n)}
+* @throws ArithmeticException {@code n} is negative.
+*/
+public BigInteger flipBit(int n) {
+if (n<0)
+throw new ArithmeticException("Negative bit address");
+int intNum = n/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 {@code (this==0? -1 : log2(this & -this))}.)
+*
+* @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
+* {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
+*
+* @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 {@code true} if this BigInteger is probably prime,
+* {@code false} if it's definitely composite.  If
+* {@code certainty} is {@code  <= 0}, {@code true} is
+* returned.
+*
+* @param  certainty a measure of the uncertainty that the caller is
+*         willing to tolerate: if the call returns {@code true}
+*         the probability that this BigInteger is prime exceeds
+*         (1 - 1/2<sup>{@code certainty}</sup>).  The execution time of
+*         this method is proportional to the value of this parameter.
+* @return {@code true} if this BigInteger is probably prime,
+*         {@code false} if it's definitely composite.
+*/
+public boolean isProbablePrime(int certainty) {
+if (certainty <= 0)
+return true;
+BigInteger w = this.abs();
+if (w.equals(TWO))
+return true;
+if (!w.testBit(0) || w.equals(ONE))
+return false;
+return w.primeToCertainty(certainty, null);
+}
+// Comparison Operations
+/**
+* Compares this BigInteger with the specified BigInteger.  This
+* method is provided in preference to individual methods for each
+* of the six boolean comparison operators ({@literal <}, ==,
+* {@literal >}, {@literal >=}, !=, {@literal <=}).  The suggested
+* idiom for performing these comparisons is: {@code
+* (x.compareTo(y)} &lt;<i>op</i>&gt; {@code 0)}, 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 {@code val}.
+*/
+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 {@code true} if and only if the specified Object is a
+*         BigInteger whose value is numerically equal to this BigInteger.
+*/
+public boolean equals(Object x) {
+// This test is just an optimization, which may or may not help
+if (x == this)
+return true;
+if (!(x instanceof BigInteger))
+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 {@code val}.
+*
+* @param  val value with which the minimum is to be computed.
+* @return the BigInteger whose value is the lesser of this BigInteger and
+*         {@code val}.  If they are equal, either may be returned.
+*/
+public BigInteger min(BigInteger val) {
+return (compareTo(val)<0 ? this : val);
+}
+/**
+* Returns the maximum of this BigInteger and {@code val}.
+*
+* @param  val value with which the maximum is to be computed.
+* @return the BigInteger whose value is the greater of this and
+*         {@code val}.  If they are equal, either may be returned.
+*/
+public BigInteger max(BigInteger val) {
+return (compareTo(val)>0 ? this : val);
+}
+// Hash Function
+/**
+* Returns the hash code for this BigInteger.
+*
+* @return hash code for this BigInteger.
+*/
+public int hashCode() {
+int hashCode = 0;
+for (int i=0; i<mag.length; i++)
+hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
+return hashCode * signum;
+}
+/**
+* Returns the String representation of this BigInteger in the
+* given radix.  If the radix is outside the range from {@link
+* Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
+* it will default to 10 (as is the case for
+* {@code Integer.toString}).  The digit-to-character mapping
+* provided by {@code Character.forDigit} is used, and a minus
+* sign is prepended if appropriate.  (This representation is
+* compatible with the {@link #BigInteger(String, int) (String,
+* int)} constructor.)
+*
+* @param  radix  radix of the String representation.
+* @return String representation of this BigInteger in the given radix.
+* @see    Integer#toString
+* @see    Character#forDigit
+* @see    #BigInteger(java.lang.String, int)
+*/
+public String toString(int radix) {
+if (signum == 0)
+return "0";
+if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
+radix = 10;
+// 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
+* {@code Character.forDigit} is used, and a minus sign is
+* prepended if appropriate.  (This representation is compatible
+* with the {@link #BigInteger(String) (String)} constructor, and
+* allows for String concatenation with Java's + operator.)
+*
+* @return decimal String representation of this BigInteger.
+* @see    Character#forDigit
+* @see    #BigInteger(java.lang.String)
+*/
+public String toString() {
+return toString(10);
+}
+/**
+* Returns a byte array containing the two's-complement
+* representation of this BigInteger.  The byte array will be in
+* <i>big-endian</i> byte-order: the most significant byte is in
+* the zeroth element.  The array will contain the minimum number
+* of bytes required to represent this BigInteger, including at
+* least one sign bit, which is {@code (ceil((this.bitLength() +
+* 1)/8))}.  (This representation is compatible with the
+* {@link #BigInteger(byte[]) (byte[])} constructor.)
+*
+* @return a byte array containing the two's-complement representation of
+*         this BigInteger.
+* @see    #BigInteger(byte[])
+*/
+public byte[] toByteArray() {
+int byteLen = bitLength()/8 + 1;
+byte[] byteArray = new byte[byteLen];
+for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i>=0; i--) {
+if (bytesCopied == 4) {
+nextInt = getInt(intIndex++);
+bytesCopied = 1;
+} else {
+nextInt >>>= 8;
+bytesCopied++;
+}
+byteArray[i] = (byte)nextInt;
+}
+return byteArray;
+}
+/**
+* Converts this BigInteger to an {@code int}.  This
+* conversion is analogous to a <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} to
+* {@code int} 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}, 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}.
+*/
+public int intValue() {
+int result = 0;
+result = getInt(0);
+return result;
+}
+/**
+* Converts this BigInteger to a {@code long}.  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} to
+* {@code int} 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}, 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}.
+*/
+public long longValue() {
+long result = 0;
+for (int i=1; i>=0; i--)
+result = (result << 32) + (getInt(i) & LONG_MASK);
+return result;
+}
+/**
+* Converts this BigInteger to a {@code float}.  This
+* conversion is similar to the <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} to
+* {@code float} 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}, 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}.
+*/
+public float floatValue() {
+// Somewhat inefficient, but guaranteed to work.
+return Float.parseFloat(this.toString());
+}
+/**
+* Converts this BigInteger to a {@code double}.  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} to
+* {@code float} 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}, 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}.
+*/
+public double doubleValue() {
+// Somewhat inefficient, but guaranteed to work.
+return Double.parseDouble(this.toString());
+}
+/**
+* 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 signum < 0 ? -1 : 0;
+}
+/**
+* Returns the specified int of the little-endian two's complement
+* representation (int 0 is the least significant).  The int number can
+* be arbitrarily high (values are logically preceded by infinitely many
+* sign ints).
+*/
+private int getInt(int n) {
+if (n < 0)
+return 0;
+if (n >= mag.length)
+return signInt();
+int magInt = mag[mag.length-n-1];
+return (signum >= 0 ? magInt :
+(n <= firstNonzeroIntNum() ? -magInt : ~magInt));
+}
+/**
+* Returns the index of the int that contains the first nonzero int in the
+* little-endian binary representation of the magnitude (int 0 is the
+* least significant). If the magnitude is zero, return value is undefined.
+*/
+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 {@code BigInteger} instance from a stream (that is,
+* deserialize it). The magnitude is read in as an array of bytes
+* for historical reasons, but it is converted to an array of ints
+* and the byte array is discarded.
+*/
+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 = 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 {@code BigInteger} 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;
+}
+}

changeset 2	90ce3da70b43
child 2922	dd6d609861f0