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Patched BigInteger using efficient algorithms for multiplication and division
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BigInteger.java.phase3
/*
* Copyright (c) 1996, 2013, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
* by Oracle in the LICENSE file that accompanied this code.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
/*
* Portions Copyright (c) 1995 Colin Plumb. All rights reserved.
*/
package java.math;
import java.io.IOException;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.io.ObjectStreamField;
import java.util.Arrays;
import java.util.Random;
import sun.misc.DoubleConsts;
import sun.misc.FloatConsts;
/**
* Immutable arbitrary-precision integers. All operations behave as if
* BigIntegers were represented in two's-complement notation (like Java's
* primitive integer types). BigInteger provides analogues to all of Java's
* primitive integer operators, and all relevant methods from java.lang.Math.
* Additionally, BigInteger provides operations for modular arithmetic, GCD
* calculation, primality testing, prime generation, bit manipulation,
* and a few other miscellaneous operations.
*
* <p>Semantics of arithmetic operations exactly mimic those of Java's integer
* arithmetic operators, as defined in <i>The Java Language Specification</i>.
* For example, division by zero throws an {@code ArithmeticException}, and
* division of a negative by a positive yields a negative (or zero) remainder.
* All of the details in the Spec concerning overflow are ignored, as
* BigIntegers are made as large as necessary to accommodate the results of an
* operation.
*
* <p>Semantics of shift operations extend those of Java's shift operators
* to allow for negative shift distances. A right-shift with a negative
* shift distance results in a left shift, and vice-versa. The unsigned
* right shift operator ({@code >>>}) is omitted, as this operation makes
* little sense in combination with the "infinite word size" abstraction
* provided by this class.
*
* <p>Semantics of bitwise logical operations exactly mimic those of Java's
* bitwise integer operators. The binary operators ({@code and},
* {@code or}, {@code xor}) implicitly perform sign extension on the shorter
* of the two operands prior to performing the operation.
*
* <p>Comparison operations perform signed integer comparisons, analogous to
* those performed by Java's relational and equality operators.
*
* <p>Modular arithmetic operations are provided to compute residues, perform
* exponentiation, and compute multiplicative inverses. These methods always
* return a non-negative result, between {@code 0} and {@code (modulus - 1)},
* inclusive.
*
* <p>Bit operations operate on a single bit of the two's-complement
* representation of their operand. If necessary, the operand is sign-
* extended so that it contains the designated bit. None of the single-bit
* operations can produce a BigInteger with a different sign from the
* BigInteger being operated on, as they affect only a single bit, and the
* "infinite word size" abstraction provided by this class ensures that there
* are infinitely many "virtual sign bits" preceding each BigInteger.
*
* <p>For the sake of brevity and clarity, pseudo-code is used throughout the
* descriptions of BigInteger methods. The pseudo-code expression
* {@code (i + j)} is shorthand for "a BigInteger whose value is
* that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
* The pseudo-code expression {@code (i == j)} is shorthand for
* "{@code true} if and only if the BigInteger {@code i} represents the same
* value as the BigInteger {@code j}." Other pseudo-code expressions are
* interpreted similarly.
*
* <p>All methods and constructors in this class throw
* {@code NullPointerException} when passed
* a null object reference for any input parameter.
*
* @see BigDecimal
* @author Josh Bloch
* @author Michael McCloskey
* @author Alan Eliasen
* @author Timothy Buktu
* @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
*/
final int signum;
/**
* The magnitude of this BigInteger, in <i>big-endian</i> order: the
* zeroth element of this array is the most-significant int of the
* magnitude. The magnitude must be "minimal" in that the most-significant
* int ({@code mag[0]}) must be non-zero. This is necessary to
* ensure that there is exactly one representation for each BigInteger
* value. Note that this implies that the BigInteger zero has a
* zero-length mag array.
*/
final int[] mag;
// These "redundant fields" are initialized with recognizable nonsense
// values, and cached the first time they are needed (or never, if they
// aren't needed).
/**
* One plus the bitCount of this BigInteger. Zeros means unitialized.
*
* @serial
* @see #bitCount
* @deprecated Deprecated since logical value is offset from stored
* value and correction factor is applied in accessor method.
*/
@Deprecated
private int bitCount;
/**
* One plus the bitLength of this BigInteger. Zeros means unitialized.
* (either value is acceptable).
*
* @serial
* @see #bitLength()
* @deprecated Deprecated since logical value is offset from stored
* value and correction factor is applied in accessor method.
*/
@Deprecated
private int bitLength;
/**
* Two plus the lowest set bit of this BigInteger, as returned by
* getLowestSetBit().
*
* @serial
* @see #getLowestSetBit
* @deprecated Deprecated since logical value is offset from stored
* value and correction factor is applied in accessor method.
*/
@Deprecated
private int lowestSetBit;
/**
* Two plus 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.
* @deprecated Deprecated since logical value is offset from stored
* value and correction factor is applied in accessor method.
*/
@Deprecated
private int firstNonzeroIntNum;
/**
* This mask is used to obtain the value of an int as if it were unsigned.
*/
final static long LONG_MASK = 0xffffffffL;
/**
* The threshold value for using Karatsuba multiplication. If the number
* of ints in both mag arrays are greater than this number, then
* Karatsuba multiplication will be used. This value is found
* experimentally to work well.
*/
private static final int KARATSUBA_THRESHOLD = 50;
/**
* The threshold value for using 3-way Toom-Cook multiplication.
* If the number of ints in each mag array is greater than the
* Karatsuba threshold, and the number of ints in at least one of
* the mag arrays is greater than this threshold, then Toom-Cook
* multiplication will be used.
*/
private static final int TOOM_COOK_THRESHOLD = 75;
/**
* The threshold value for using Karatsuba squaring. If the number
* of ints in the number are larger than this value,
* Karatsuba squaring will be used. This value is found
* experimentally to work well.
*/
private static final int KARATSUBA_SQUARE_THRESHOLD = 90;
/**
* The threshold value for using Toom-Cook squaring. If the number
* of ints in the number are larger than this value,
* Toom-Cook squaring will be used. This value is found
* experimentally to work well.
*/
private static final int TOOM_COOK_SQUARE_THRESHOLD = 140;
/**
* The threshold value for using Burnikel-Ziegler division. If the number
* of ints in the number are larger than this value,
* Burnikel-Ziegler division will be used. This value is found
* experimentally to work well.
*/
static final int BURNIKEL_ZIEGLER_THRESHOLD = 50;
/**
* The threshold value for using Schoenhage recursive base conversion. If
* the number of ints in the number are larger than this value,
* the Schoenhage algorithm will be used. In practice, it appears that the
* Schoenhage routine is faster for any threshold down to 2, and is
* relatively flat for thresholds between 2-25, so this choice may be
* varied within this range for very small effect.
*/
private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 8;
//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;
final int len = val.length();
if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
throw new NumberFormatException("Radix out of range");
if (len == 0)
throw new NumberFormatException("Zero length BigInteger");
// Check for at most one leading sign
int sign = 1;
int index1 = val.lastIndexOf('-');
int index2 = val.lastIndexOf('+');
if ((index1 + index2) <= -1) {
// No leading sign character or at most one leading sign character
if (index1 == 0 || index2 == 0) {
cursor = 1;
if (len == 1)
throw new NumberFormatException("Zero length BigInteger");
}
if (index1 == 0)
sign = -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;
}
numDigits = len - cursor;
signum = sign;
// 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) >>> 5;
int[] magnitude = new int[numWords];
// Process first (potentially short) digit group
int firstGroupLen = numDigits % digitsPerInt[radix];
if (firstGroupLen == 0)
firstGroupLen = digitsPerInt[radix];
String group = val.substring(cursor, cursor += firstGroupLen);
magnitude[numWords - 1] = Integer.parseInt(group, radix);
if (magnitude[numWords - 1] < 0)
throw new NumberFormatException("Illegal digit");
// Process remaining digit groups
int superRadix = intRadix[radix];
int groupVal = 0;
while (cursor < len) {
group = val.substring(cursor, cursor += digitsPerInt[radix]);
groupVal = Integer.parseInt(group, radix);
if (groupVal < 0)
throw new NumberFormatException("Illegal digit");
destructiveMulAdd(magnitude, superRadix, groupVal);
}
// Required for cases where the array was overallocated.
mag = trustedStripLeadingZeroInts(magnitude);
}
/*
* Constructs a new BigInteger using a char array with radix=10.
* Sign is precalculated outside and not allowed in the val.
*/
BigInteger(char[] val, int sign, int len) {
int cursor = 0, numDigits;
// Skip leading zeros and compute number of digits in magnitude
while (cursor < len && Character.digit(val[cursor], 10) == 0) {
cursor++;
}
if (cursor == len) {
signum = 0;
mag = ZERO.mag;
return;
}
numDigits = len - cursor;
signum = sign;
// Pre-allocate array of expected size
int numWords;
if (len < 10) {
numWords = 1;
} else {
int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1);
numWords = (numBits + 31) >>> 5;
}
int[] magnitude = new int[numWords];
// Process first (potentially short) digit group
int firstGroupLen = numDigits % digitsPerInt[10];
if (firstGroupLen == 0)
firstGroupLen = digitsPerInt[10];
magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
// Process remaining digit groups
while (cursor < len) {
int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
destructiveMulAdd(magnitude, intRadix[10], groupVal);
}
mag = trustedStripLeadingZeroInts(magnitude);
}
// 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 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");
prime = (bitLength < SMALL_PRIME_THRESHOLD
? smallPrime(bitLength, certainty, rnd)
: largePrime(bitLength, certainty, rnd));
signum = 1;
mag = prime.mag;
}
// Minimum size in bits that the requested prime number has
// before we use the large prime number generating algorithms.
// The cutoff of 95 was chosen empirically for best performance.
private static final int SMALL_PRIME_THRESHOLD = 95;
// Certainty required to meet the spec of probablePrime
private static final int DEFAULT_PRIME_CERTAINTY = 100;
/**
* Returns a positive BigInteger that is probably prime, with the
* specified bitLength. The probability that a BigInteger returned
* by this method is composite does not exceed 2<sup>-100</sup>.
*
* @param bitLength bitLength of the returned BigInteger.
* @param rnd source of random bits used to select candidates to be
* tested for primality.
* @return a BigInteger of {@code bitLength} bits that is probably prime
* @throws ArithmeticException {@code bitLength < 2}.
* @see #bitLength()
* @since 1.4
*/
public static BigInteger probablePrime(int bitLength, Random rnd) {
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
return (bitLength < SMALL_PRIME_THRESHOLD ?
smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
}
/**
* Find a random number of the specified bitLength that is probably prime.
* This method is used for smaller primes, its performance degrades on
* larger bitlengths.
*
* This method assumes bitLength > 1.
*/
private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
int magLen = (bitLength + 31) >>> 5;
int temp[] = new int[magLen];
int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
int highMask = (highBit << 1) - 1; // Bits to keep in high int
while (true) {
// Construct a candidate
for (int i=0; i < magLen; i++)
temp[i] = rnd.nextInt();
temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
if (bitLength > 2)
temp[magLen-1] |= 1; // Make odd if bitlen > 2
BigInteger p = new BigInteger(temp, 1);
// Do cheap "pre-test" if applicable
if (bitLength > 6) {
long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
continue; // Candidate is composite; try another
}
// All candidates of bitLength 2 and 3 are prime by this point
if (bitLength < 4)
return p;
// Do expensive test if we survive pre-test (or it's inapplicable)
if (p.primeToCertainty(certainty, rnd))
return p;
}
}
private static final BigInteger SMALL_PRIME_PRODUCT
= valueOf(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 = v2.subtract(n);
v2 = v2.shiftRight(1);
u = u2; v = v2;
if (k.testBit(i)) {
u2 = u.add(v).mod(n);
if (u2.testBit(0))
u2 = u2.subtract(n);
u2 = u2.shiftRight(1);
v2 = v.add(d.multiply(u)).mod(n);
if (v2.testBit(0))
v2 = v2.subtract(n);
v2 = v2.shiftRight(1);
u = u2; v = v2;
}
}
return u;
}
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 internal constructor differs from its public cousin
* with the arguments reversed in two ways: it assumes that its
* arguments are correct, and it doesn't copy the magnitude array.
*/
BigInteger(int[] magnitude, int signum) {
this.signum = (magnitude.length == 0 ? 0 : signum);
this.mag = magnitude;
}
/**
* 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);
}
//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) {
val = -val;
signum = -1;
} else {
signum = 1;
}
int highWord = (int)(val >>> 32);
if (highWord == 0) {
mag = new int[1];
mag[0] = (int)val;
} else {
mag = new int[2];
mag[0] = highWord;
mag[1] = (int)val;
}
}
/**
* Returns a BigInteger with the given two's complement representation.
* Assumes that the input array will not be modified (the returned
* BigInteger will reference the input array if feasible).
*/
private static BigInteger valueOf(int val[]) {
return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
}
// Constants
/**
* Initialize static constant array when class is loaded.
*/
private final static int MAX_CONSTANT = 16;
private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1];
private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1];
/**
* The cache of powers of each radix. This allows us to not have to
* recalculate powers of radix^(2^n) more than once. This speeds
* Schoenhage recursive base conversion significantly.
*/
private static volatile BigInteger[][] powerCache;
/** The cache of logarithms of radices for base conversion. */
private static final double[] logCache;
/** The natural log of 2. This is used in computing cache indices. */
private static final double LOG_TWO = Math.log(2.0);
static {
for (int i = 1; i <= MAX_CONSTANT; i++) {
int[] magnitude = new int[1];
magnitude[0] = i;
posConst[i] = new BigInteger(magnitude, 1);
negConst[i] = new BigInteger(magnitude, -1);
}
/*
* Initialize the cache of radix^(2^x) values used for base conversion
* with just the very first value. Additional values will be created
* on demand.
*/
powerCache = new BigInteger[Character.MAX_RADIX+1][];
logCache = new double[Character.MAX_RADIX+1];
for (int i=Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) };
logCache[i] = Math.log(i);
}
}
/**
* The BigInteger constant zero.
*
* @since 1.2
*/
public static final BigInteger ZERO = new BigInteger(new int[0], 0);
/**
* The BigInteger constant one.
*
* @since 1.2
*/
public static final BigInteger ONE = valueOf(1);
/**
* The BigInteger constant two. (Not exported.)
*/
private static final BigInteger TWO = valueOf(2);
/**
* The BigInteger constant -1. (Not exported.)
*/
private static final BigInteger NEGATIVE_ONE = valueOf(-1);
/**
* The BigInteger constant ten.
*
* @since 1.5
*/
public static final BigInteger TEN = valueOf(10);
// Arithmetic Operations
/**
* Returns a BigInteger whose value is {@code (this + val)}.
*
* @param val value to be added to this BigInteger.
* @return {@code this + val}
*/
public BigInteger add(BigInteger val) {
if (val.signum == 0)
return this;
if (signum == 0)
return val;
if (val.signum == signum)
return new BigInteger(add(mag, val.mag), signum);
int cmp = compareMagnitude(val);
if (cmp == 0)
return ZERO;
int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
: subtract(val.mag, mag));
resultMag = trustedStripLeadingZeroInts(resultMag);
return new BigInteger(resultMag, cmp == signum ? 1 : -1);
}
/**
* Package private methods used by BigDecimal code to add a BigInteger
* with a long. Assumes val is not equal to INFLATED.
*/
BigInteger add(long val) {
if (val == 0)
return this;
if (signum == 0)
return valueOf(val);
if (Long.signum(val) == signum)
return new BigInteger(add(mag, Math.abs(val)), signum);
int cmp = compareMagnitude(val);
if (cmp == 0)
return ZERO;
int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
resultMag = trustedStripLeadingZeroInts(resultMag);
return new BigInteger(resultMag, cmp == signum ? 1 : -1);
}
/**
* Adds the contents of the int array x and long value val. This
* method allocates a new int array to hold the answer and returns
* a reference to that array. Assumes x.length &gt; 0 and val is
* non-negative
*/
private static int[] add(int[] x, long val) {
int[] y;
long sum = 0;
int xIndex = x.length;
int[] result;
int highWord = (int)(val >>> 32);
if (highWord == 0) {
result = new int[xIndex];
sum = (x[--xIndex] & LONG_MASK) + val;
result[xIndex] = (int)sum;
} else {
if (xIndex == 1) {
result = new int[2];
sum = val + (x[0] & LONG_MASK);
result[1] = (int)sum;
result[0] = (int)(sum >>> 32);
return result;
} else {
result = new int[xIndex];
sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
result[xIndex] = (int)sum;
sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
result[xIndex] = (int)sum;
}
}
// Copy remainder of longer number while carry propagation is required
boolean carry = (sum >>> 32 != 0);
while (xIndex > 0 && carry)
carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
// Copy remainder of longer number
while (xIndex > 0)
result[--xIndex] = x[xIndex];
// Grow result if necessary
if (carry) {
int bigger[] = new int[result.length + 1];
System.arraycopy(result, 0, bigger, 1, result.length);
bigger[0] = 0x01;
return bigger;
}
return result;
}
/**
* Adds the contents of the int arrays x and y. This method allocates
* a new int array to hold the answer and returns a reference to that
* array.
*/
private static int[] add(int[] x, int[] y) {
// If x is shorter, swap the two arrays
if (x.length < y.length) {
int[] tmp = x;
x = y;
y = tmp;
}
int xIndex = x.length;
int yIndex = y.length;
int result[] = new int[xIndex];
long sum = 0;
if (yIndex == 1) {
sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
result[xIndex] = (int)sum;
} else {
// Add common parts of both numbers
while (yIndex > 0) {
sum = (x[--xIndex] & LONG_MASK) +
(y[--yIndex] & LONG_MASK) + (sum >>> 32);
result[xIndex] = (int)sum;
}
}
// Copy remainder of longer number while carry propagation is required
boolean carry = (sum >>> 32 != 0);
while (xIndex > 0 && carry)
carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
// Copy remainder of longer number
while (xIndex > 0)
result[--xIndex] = x[xIndex];
// Grow result if necessary
if (carry) {
int bigger[] = new int[result.length + 1];
System.arraycopy(result, 0, bigger, 1, result.length);
bigger[0] = 0x01;
return bigger;
}
return result;
}
private static int[] subtract(long val, int[] little) {
int highWord = (int)(val >>> 32);
if (highWord == 0) {
int result[] = new int[1];
result[0] = (int)(val - (little[0] & LONG_MASK));
return result;
} else {
int result[] = new int[2];
if (little.length == 1) {
long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
result[1] = (int)difference;
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
if (borrow) {
result[0] = highWord - 1;
} else { // Copy remainder of longer number
result[0] = highWord;
}
return result;
} else { // little.length == 2
long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
result[1] = (int)difference;
difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
result[0] = (int)difference;
return result;
}
}
}
/**
* Subtracts the contents of the second argument (val) from the
* first (big). The first int array (big) must represent a larger number
* than the second. This method allocates the space necessary to hold the
* answer.
* assumes val &gt;= 0
*/
private static int[] subtract(int[] big, long val) {
int highWord = (int)(val >>> 32);
int bigIndex = big.length;
int result[] = new int[bigIndex];
long difference = 0;
if (highWord == 0) {
difference = (big[--bigIndex] & LONG_MASK) - val;
result[bigIndex] = (int)difference;
} else {
difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
result[bigIndex] = (int)difference;
difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
result[bigIndex] = (int)difference;
}
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
while (bigIndex > 0 && borrow)
borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
// Copy remainder of longer number
while (bigIndex > 0)
result[--bigIndex] = big[bigIndex];
return result;
}
/**
* Returns a BigInteger whose value is {@code (this - val)}.
*
* @param val value to be subtracted from this BigInteger.
* @return {@code this - val}
*/
public BigInteger subtract(BigInteger val) {
if (val.signum == 0)
return this;
if (signum == 0)
return val.negate();
if (val.signum != signum)
return new BigInteger(add(mag, val.mag), signum);
int cmp = compareMagnitude(val);
if (cmp == 0)
return ZERO;
int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
: subtract(val.mag, mag));
resultMag = trustedStripLeadingZeroInts(resultMag);
return new BigInteger(resultMag, cmp == signum ? 1 : -1);
}
/**
* Subtracts the contents of the second int arrays (little) from the
* first (big). The first int array (big) must represent a larger number
* than the second. This method allocates the space necessary to hold the
* answer.
*/
private static int[] subtract(int[] big, int[] little) {
int bigIndex = big.length;
int result[] = new int[bigIndex];
int littleIndex = little.length;
long difference = 0;
// Subtract common parts of both numbers
while (littleIndex > 0) {
difference = (big[--bigIndex] & LONG_MASK) -
(little[--littleIndex] & LONG_MASK) +
(difference >> 32);
result[bigIndex] = (int)difference;
}
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
while (bigIndex > 0 && borrow)
borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
// Copy remainder of longer number
while (bigIndex > 0)
result[--bigIndex] = big[bigIndex];
return result;
}
/**
* Returns a BigInteger whose value is {@code (this * val)}.
*
* @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 xlen = mag.length;
int ylen = val.mag.length;
if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
int resultSign = signum == val.signum ? 1 : -1;
if (val.mag.length == 1) {
return multiplyByInt(mag,val.mag[0], resultSign);
}
if (mag.length == 1) {
return multiplyByInt(val.mag,mag[0], resultSign);
}
int[] result = multiplyToLen(mag, xlen,
val.mag, ylen, null);
result = trustedStripLeadingZeroInts(result);
return new BigInteger(result, resultSign);
} else {
if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
return multiplyKaratsuba(this, val);
} else {
return multiplyToomCook3(this, val);
}
}
}
private static BigInteger multiplyByInt(int[] x, int y, int sign) {
if (Integer.bitCount(y) == 1) {
return new BigInteger(shiftLeft(x,Integer.numberOfTrailingZeros(y)), sign);
}
int xlen = x.length;
int[] rmag = new int[xlen + 1];
long carry = 0;
long yl = y & LONG_MASK;
int rstart = rmag.length - 1;
for (int i = xlen - 1; i >= 0; i--) {
long product = (x[i] & LONG_MASK) * yl + carry;
rmag[rstart--] = (int)product;
carry = product >>> 32;
}
if (carry == 0L) {
rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
} else {
rmag[rstart] = (int)carry;
}
return new BigInteger(rmag, sign);
}
/**
* Package private methods used by BigDecimal code to multiply a BigInteger
* with a long. Assumes v is not equal to INFLATED.
*/
BigInteger multiply(long v) {
if (v == 0 || signum == 0)
return ZERO;
if (v == BigDecimal.INFLATED)
return multiply(BigInteger.valueOf(v));
int rsign = (v > 0 ? signum : -signum);
if (v < 0)
v = -v;
long dh = v >>> 32; // higher order bits
long dl = v & LONG_MASK; // lower order bits
int xlen = mag.length;
int[] value = mag;
int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
long carry = 0;
int rstart = rmag.length - 1;
for (int i = xlen - 1; i >= 0; i--) {
long product = (value[i] & LONG_MASK) * dl + carry;
rmag[rstart--] = (int)product;
carry = product >>> 32;
}
rmag[rstart] = (int)carry;
if (dh != 0L) {
carry = 0;
rstart = rmag.length - 2;
for (int i = xlen - 1; i >= 0; i--) {
long product = (value[i] & LONG_MASK) * dh +
(rmag[rstart] & LONG_MASK) + carry;
rmag[rstart--] = (int)product;
carry = product >>> 32;
}
rmag[0] = (int)carry;
}
if (carry == 0L)
rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
return new BigInteger(rmag, rsign);
}
/**
* Multiplies int arrays x and y to the specified lengths and places
* the result into z. There will be no leading zeros in the resultant array.
*/
private 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;
}
/**
* Multiplies two BigIntegers using the Karatsuba multiplication
* algorithm. This is a recursive divide-and-conquer algorithm which is
* more efficient for large numbers than what is commonly called the
* "grade-school" algorithm used in multiplyToLen. If the numbers to be
* multiplied have length n, the "grade-school" algorithm has an
* asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm
* has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this
* increased performance by doing 3 multiplies instead of 4 when
* evaluating the product. As it has some overhead, should be used when
* both numbers are larger than a certain threshold (found
* experimentally).
*
* See: http://en.wikipedia.org/wiki/Karatsuba_algorithm
*/
private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
int xlen = x.mag.length;
int ylen = y.mag.length;
// The number of ints in each half of the number.
int half = (Math.max(xlen, ylen)+1) / 2;
// xl and yl are the lower halves of x and y respectively,
// xh and yh are the upper halves.
BigInteger xl = x.getLower(half);
BigInteger xh = x.getUpper(half);
BigInteger yl = y.getLower(half);
BigInteger yh = y.getUpper(half);
BigInteger p1 = xh.multiply(yh); // p1 = xh*yh
BigInteger p2 = xl.multiply(yl); // p2 = xl*yl
// p3=(xh+xl)*(yh+yl)
BigInteger p3 = xh.add(xl).multiply(yh.add(yl));
// result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
BigInteger result = p1.shiftLeft(32*half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32*half).add(p2);
if (x.signum != y.signum) {
return result.negate();
} else {
return result;
}
}
/**
* Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
* algorithm. This is a recursive divide-and-conquer algorithm which is
* more efficient for large numbers than what is commonly called the
* "grade-school" algorithm used in multiplyToLen. If the numbers to be
* multiplied have length n, the "grade-school" algorithm has an
* asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a
* complexity of about O(n^1.465). It achieves this increased asymptotic
* performance by breaking each number into three parts and by doing 5
* multiplies instead of 9 when evaluating the product. Due to overhead
* (additions, shifts, and one division) in the Toom-Cook algorithm, it
* should only be used when both numbers are larger than a certain
* threshold (found experimentally). This threshold is generally larger
* than that for Karatsuba multiplication, so this algorithm is generally
* only used when numbers become significantly larger.
*
* The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
* by Marco Bodrato.
*
* See: http://bodrato.it/toom-cook/
* http://bodrato.it/papers/#WAIFI2007
*
* "Towards Optimal Toom-Cook Multiplication for Univariate and
* Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
* In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
* LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
*
*/
private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
int alen = a.mag.length;
int blen = b.mag.length;
int largest = Math.max(alen, blen);
// k is the size (in ints) of the lower-order slices.
int k = (largest+2)/3; // Equal to ceil(largest/3)
// r is the size (in ints) of the highest-order slice.
int r = largest - 2*k;
// Obtain slices of the numbers. a2 and b2 are the most significant
// bits of the numbers a and b, and a0 and b0 the least significant.
BigInteger a0, a1, a2, b0, b1, b2;
a2 = a.getToomSlice(k, r, 0, largest);
a1 = a.getToomSlice(k, r, 1, largest);
a0 = a.getToomSlice(k, r, 2, largest);
b2 = b.getToomSlice(k, r, 0, largest);
b1 = b.getToomSlice(k, r, 1, largest);
b0 = b.getToomSlice(k, r, 2, largest);
BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;
v0 = a0.multiply(b0);
da1 = a2.add(a0);
db1 = b2.add(b0);
vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
da1 = da1.add(a1);
db1 = db1.add(b1);
v1 = da1.multiply(db1);
v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
db1.add(b2).shiftLeft(1).subtract(b0));
vinf = a2.multiply(b2);
// The algorithm requires two divisions by 2 and one by 3.
// All divisions are known to be exact, that is, they do not produce
// remainders, and all results are positive. The divisions by 2 are
// implemented as right shifts which are relatively efficient, leaving
// only an exact division by 3, which is done by a specialized
// linear-time algorithm.
t2 = v2.subtract(vm1).exactDivideBy3();
tm1 = v1.subtract(vm1).shiftRight(1);
t1 = v1.subtract(v0);
t2 = t2.subtract(t1).shiftRight(1);
t1 = t1.subtract(tm1).subtract(vinf);
t2 = t2.subtract(vinf.shiftLeft(1));
tm1 = tm1.subtract(t2);
// Number of bits to shift left.
int ss = k*32;
BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
if (a.signum != b.signum) {
return result.negate();
} else {
return result;
}
}
/**
* Returns a slice of a BigInteger for use in Toom-Cook multiplication.
*
* @param lowerSize The size of the lower-order bit slices.
* @param upperSize The size of the higher-order bit slices.
* @param slice The index of which slice is requested, which must be a
* number from 0 to size-1. Slice 0 is the highest-order bits, and slice
* size-1 are the lowest-order bits. Slice 0 may be of different size than
* the other slices.
* @param fullsize The size of the larger integer array, used to align
* slices to the appropriate position when multiplying different-sized
* numbers.
*/
private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
int fullsize) {
int start, end, sliceSize, len, offset;
len = mag.length;
offset = fullsize - len;
if (slice == 0) {
start = 0 - offset;
end = upperSize - 1 - offset;
} else {
start = upperSize + (slice-1)*lowerSize - offset;
end = start + lowerSize - 1;
}
if (start < 0) {
start = 0;
}
if (end < 0) {
return ZERO;
}
sliceSize = (end-start) + 1;
if (sliceSize <= 0) {
return ZERO;
}
// While performing Toom-Cook, all slices are positive and
// the sign is adjusted when the final number is composed.
if (start == 0 && sliceSize >= len) {
return this.abs();
}
int intSlice[] = new int[sliceSize];
System.arraycopy(mag, start, intSlice, 0, sliceSize);
return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
}
/**
* Does an exact division (that is, the remainder is known to be zero)
* of the specified number by 3. This is used in Toom-Cook
* multiplication. This is an efficient algorithm that runs in linear
* time. If the argument is not exactly divisible by 3, results are
* undefined. Note that this is expected to be called with positive
* arguments only.
*/
private BigInteger exactDivideBy3() {
int len = mag.length;
int[] result = new int[len];
long x, w, q, borrow;
borrow = 0L;
for (int i=len-1; i >= 0; i--) {
x = (mag[i] & LONG_MASK);
w = x - borrow;
if (borrow > x) { // Did we make the number go negative?
borrow = 1L;
} else {
borrow = 0L;
}
// 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus,
// the effect of this is to divide by 3 (mod 2^32).
// This is much faster than division on most architectures.
q = (w * 0xAAAAAAABL) & LONG_MASK;
result[i] = (int) q;
// Now check the borrow. The second check can of course be
// eliminated if the first fails.
if (q >= 0x55555556L) {
borrow++;
if (q >= 0xAAAAAAABL)
borrow++;
}
}
result = trustedStripLeadingZeroInts(result);
return new BigInteger(result, signum);
}
/**
* Returns a new BigInteger representing n lower ints of the number.
* This is used by Karatsuba multiplication and Karatsuba squaring.
*/
private BigInteger getLower(int n) {
int len = mag.length;
if (len <= n) {
return this;
}
int lowerInts[] = new int[n];
System.arraycopy(mag, len-n, lowerInts, 0, n);
return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
}
/**
* Returns a new BigInteger representing mag.length-n upper
* ints of the number. This is used by Karatsuba multiplication and
* Karatsuba squaring.
*/
private BigInteger getUpper(int n) {
int len = mag.length;
if (len <= n) {
return ZERO;
}
int upperLen = len - n;
int upperInts[] = new int[upperLen];
System.arraycopy(mag, 0, upperInts, 0, upperLen);
return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
}
// Squaring
/**
* Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
*
* @return {@code this<sup>2</sup>}
*/
private BigInteger square() {
if (signum == 0) {
return ZERO;
}
int len = mag.length;
if (len < KARATSUBA_SQUARE_THRESHOLD) {
int[] z = squareToLen(mag, len, null);
return new BigInteger(trustedStripLeadingZeroInts(z), 1);
} else {
if (len < TOOM_COOK_SQUARE_THRESHOLD) {
return squareKaratsuba();
} else {
return squareToomCook3();
}
}
}
/**
* Squares the contents of the int array x. The result is placed into the
* int array z. The contents of x are not changed.
*/
private static final int[] squareToLen(int[] x, int len, int[] z) {
/*
* 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;
}
/**
* Squares a BigInteger using the Karatsuba squaring algorithm. It should
* be used when both numbers are larger than a certain threshold (found
* experimentally). It is a recursive divide-and-conquer algorithm that
* has better asymptotic performance than the algorithm used in
* squareToLen.
*/
private BigInteger squareKaratsuba() {
int half = (mag.length+1) / 2;
BigInteger xl = getLower(half);
BigInteger xh = getUpper(half);
BigInteger xhs = xh.square(); // xhs = xh^2
BigInteger xls = xl.square(); // xls = xl^2
// xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
return xhs.shiftLeft(half*32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half*32).add(xls);
}
/**
* Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It
* should be used when both numbers are larger than a certain threshold
* (found experimentally). It is a recursive divide-and-conquer algorithm
* that has better asymptotic performance than the algorithm used in
* squareToLen or squareKaratsuba.
*/
private BigInteger squareToomCook3() {
int len = mag.length;
// k is the size (in ints) of the lower-order slices.
int k = (len+2)/3; // Equal to ceil(largest/3)
// r is the size (in ints) of the highest-order slice.
int r = len - 2*k;
// Obtain slices of the numbers. a2 is the most significant
// bits of the number, and a0 the least significant.
BigInteger a0, a1, a2;
a2 = getToomSlice(k, r, 0, len);
a1 = getToomSlice(k, r, 1, len);
a0 = getToomSlice(k, r, 2, len);
BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;
v0 = a0.square();
da1 = a2.add(a0);
vm1 = da1.subtract(a1).square();
da1 = da1.add(a1);
v1 = da1.square();
vinf = a2.square();
v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();
// The algorithm requires two divisions by 2 and one by 3.
// All divisions are known to be exact, that is, they do not produce
// remainders, and all results are positive. The divisions by 2 are
// implemented as right shifts which are relatively efficient, leaving
// only a division by 3.
// The division by 3 is done by an optimized algorithm for this case.
t2 = v2.subtract(vm1).exactDivideBy3();
tm1 = v1.subtract(vm1).shiftRight(1);
t1 = v1.subtract(v0);
t2 = t2.subtract(t1).shiftRight(1);
t1 = t1.subtract(tm1).subtract(vinf);
t2 = t2.subtract(vinf.shiftLeft(1));
tm1 = tm1.subtract(t2);
// Number of bits to shift left.
int ss = k*32;
return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0);
}
// Division
/**
* Returns a BigInteger whose value is {@code (this / val)}.
*
* @param val value by which this BigInteger is to be divided.
* @return {@code this / val}
* @throws ArithmeticException if {@code val} is zero.
*/
public BigInteger divide(BigInteger val) {
if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
return divideKnuth(val);
} else {
return divideBurnikelZiegler(val);
}
}
/**
* Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
*
* @param val value by which this BigInteger is to be divided.
* @return {@code this / val}
* @throws ArithmeticException if {@code val} is zero.
* @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
*/
private BigInteger divideKnuth(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
a.divideKnuth(b, q, false);
return q.toBigInteger(this.signum * val.signum);
}
/**
* Returns an array of two BigIntegers containing {@code (this / val)}
* followed by {@code (this % val)}.
*
* @param val value by which this BigInteger is to be divided, and the
* remainder computed.
* @return an array of two BigIntegers: the quotient {@code (this / val)}
* is the initial element, and the remainder {@code (this % val)}
* is the final element.
* @throws ArithmeticException if {@code val} is zero.
*/
public BigInteger[] divideAndRemainder(BigInteger val) {
if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
return divideAndRemainderKnuth(val);
} else {
return divideAndRemainderBurnikelZiegler(val);
}
}
/** Long division */
private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
BigInteger[] result = new BigInteger[2];
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
MutableBigInteger r = a.divideKnuth(b, q);
result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
result[1] = r.toBigInteger(this.signum);
return result;
}
/**
* Returns a BigInteger whose value is {@code (this % val)}.
*
* @param val value by which this BigInteger is to be divided, and the
* remainder computed.
* @return {@code this % val}
* @throws ArithmeticException if {@code val} is zero.
*/
public BigInteger remainder(BigInteger val) {
if (mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD) {
return remainderKnuth(val);
} else {
return remainderBurnikelZiegler(val);
}
}
/** Long division */
private BigInteger remainderKnuth(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
return a.divideKnuth(b, q).toBigInteger(this.signum);
}
/**
* Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
* @param val the divisor
* @return {@code this / val}
*/
private BigInteger divideBurnikelZiegler(BigInteger val) {
return divideAndRemainderBurnikelZiegler(val)[0];
}
/**
* Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
* @param val the divisor
* @return {@code this % val}
*/
private BigInteger remainderBurnikelZiegler(BigInteger val) {
return divideAndRemainderBurnikelZiegler(val)[1];
}
/**
* Computes {@code this / val} and {@code this % val} using the
* Burnikel-Ziegler algorithm.
* @param val the divisor
* @return an array containing the quotient and remainder
*/
private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
MutableBigInteger q = new MutableBigInteger();
MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
return new BigInteger[] {qBigInt, rBigInt};
}
/**
* Returns a BigInteger whose value is <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);
}
BigInteger partToSquare = this.abs();
// Factor out powers of two from the base, as the exponentiation of
// these can be done by left shifts only.
// The remaining part can then be exponentiated faster. The
// powers of two will be multiplied back at the end.
int powersOfTwo = partToSquare.getLowestSetBit();
int remainingBits;
// Factor the powers of two out quickly by shifting right, if needed.
if (powersOfTwo > 0) {
partToSquare = partToSquare.shiftRight(powersOfTwo);
remainingBits = partToSquare.bitLength();
if (remainingBits == 1) { // Nothing left but +/- 1?
if (signum < 0 && (exponent&1) == 1) {
return NEGATIVE_ONE.shiftLeft(powersOfTwo*exponent);
} else {
return ONE.shiftLeft(powersOfTwo*exponent);
}
}
} else {
remainingBits = partToSquare.bitLength();
if (remainingBits == 1) { // Nothing left but +/- 1?
if (signum < 0 && (exponent&1) == 1) {
return NEGATIVE_ONE;
} else {
return ONE;
}
}
}
// This is a quick way to approximate the size of the result,
// similar to doing log2[n] * exponent. This will give an upper bound
// of how big the result can be, and which algorithm to use.
int scaleFactor = remainingBits * exponent;
// Use slightly different algorithms for small and large operands.
// See if the result will safely fit into a long. (Largest 2^63-1)
if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
// Small number algorithm. Everything fits into a long.
int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
long result = 1;
long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
int workingExponent = exponent;
// Perform exponentiation using repeated squaring trick
while (workingExponent != 0) {
if ((workingExponent & 1) == 1) {
result = result * baseToPow2;
}
if ((workingExponent >>>= 1) != 0) {
baseToPow2 = baseToPow2 * baseToPow2;
}
}
// Multiply back the powers of two (quickly, by shifting left)
if (powersOfTwo > 0) {
int bitsToShift = powersOfTwo*exponent;
if (bitsToShift + scaleFactor <= 62) { // Fits in long?
return valueOf((result << bitsToShift) * newSign);
} else {
return valueOf(result*newSign).shiftLeft(bitsToShift);
}
}
else {
return valueOf(result*newSign);
}
} else {
// Large number algorithm. This is basically identical to
// the algorithm above, but calls multiply() and square()
// which may use more efficient algorithms for large numbers.
BigInteger answer = ONE;
int workingExponent = exponent;
// Perform exponentiation using repeated squaring trick
while (workingExponent != 0) {
if ((workingExponent & 1) == 1) {
answer = answer.multiply(partToSquare);
}
if ((workingExponent >>>= 1) != 0) {
partToSquare = partToSquare.square();
}
}
// Multiply back the (exponentiated) powers of two (quickly,
// by shifting left)
if (powersOfTwo > 0) {
answer = answer.shiftLeft(powersOfTwo*exponent);
}
if (signum < 0 && (exponent&1) == 1) {
return answer.negate();
} else {
return answer;
}
}
}
/**
* Returns a BigInteger whose value is the greatest common divisor of
* {@code abs(this)} and {@code abs(val)}. Returns 0 if
* {@code this == 0 && val == 0}.
*
* @param val value with which the GCD is to be computed.
* @return {@code GCD(abs(this), abs(val))}
*/
public BigInteger gcd(BigInteger val) {
if (val.signum == 0)
return this.abs();
else if (this.signum == 0)
return val.abs();
MutableBigInteger a = new MutableBigInteger(this);
MutableBigInteger b = new MutableBigInteger(val);
MutableBigInteger result = a.hybridGCD(b);
return result.toBigInteger(1);
}
/**
* Package private method to return bit length for an integer.
*/
static int bitLengthForInt(int n) {
return 32 - Integer.numberOfLeadingZeros(n);
}
/**
* Left shift int array a up to len by n bits. Returns the array that
* results from the shift since space may have to be reallocated.
*/
private static int[] leftShift(int[] a, int len, int n) {
int nInts = n >>> 5;
int nBits = n&0x1F;
int bitsInHighWord = bitLengthForInt(a[0]);
// If shift can be done without recopy, do so
if (n <= (32-bitsInHighWord)) {
primitiveLeftShift(a, len, nBits);
return a;
} else { // Array must be resized
if (nBits <= (32-bitsInHighWord)) {
int result[] = new int[nInts+len];
System.arraycopy(a, 0, result, 0, len);
primitiveLeftShift(result, result.length, nBits);
return result;
} else {
int result[] = new int[nInts+len+1];
System.arraycopy(a, 0, result, 0, len);
primitiveRightShift(result, result.length, 32 - nBits);
return result;
}
}
}
// shifts a up to len right n bits assumes no leading zeros, 0<n<32
static void primitiveRightShift(int[] a, int len, int n) {
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) + bitLengthForInt(val[0]);
}
/**
* Returns a BigInteger whose value is the absolute value of this
* BigInteger.
*
* @return {@code abs(this)}
*/
public BigInteger abs() {
return (signum >= 0 ? this : this.negate());
}
/**
* Returns a BigInteger whose value is {@code (-this)}.
*
* @return {@code -this}
*/
public BigInteger negate() {
return new BigInteger(this.mag, -this.signum);
}
/**
* Returns the signum function of this BigInteger.
*
* @return -1, 0 or 1 as the value of this BigInteger is negative, zero or
* positive.
*/
public int signum() {
return this.signum;
}
// Modular Arithmetic Operations
/**
* Returns a BigInteger whose value is {@code (this mod m}). This method
* differs from {@code remainder} in that it always returns a
* <i>non-negative</i> BigInteger.
*
* @param m the modulus.
* @return {@code this mod m}
* @throws ArithmeticException {@code m} &le; 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} &le; 0 or the exponent is
* negative and this BigInteger is not <i>relatively
* prime</i> to {@code m}.
* @see #modInverse
*/
public BigInteger modPow(BigInteger exponent, BigInteger m) {
if (m.signum <= 0)
throw new ArithmeticException("BigInteger: modulus not positive");
// Trivial cases
if (exponent.signum == 0)
return (m.equals(ONE) ? ZERO : ONE);
if (this.equals(ONE))
return (m.equals(ONE) ? ZERO : ONE);
if (this.equals(ZERO) && exponent.signum >= 0)
return ZERO;
if (this.equals(negConst[1]) && (!exponent.testBit(0)))
return (m.equals(ONE) ? ZERO : ONE);
boolean invertResult;
if ((invertResult = (exponent.signum < 0)))
exponent = exponent.negate();
BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
? this.mod(m) : this);
BigInteger result;
if (m.testBit(0)) { // odd modulus
result = base.oddModPow(exponent, m);
} else {
/*
* Even modulus. Tear it into an "odd part" (m1) and power of two
* (m2), exponentiate mod m1, manually exponentiate mod m2, and
* use Chinese Remainder Theorem to combine results.
*/
// Tear m apart into odd part (m1) and power of 2 (m2)
int p = m.getLowestSetBit(); // Max pow of 2 that divides m
BigInteger m1 = m.shiftRight(p); // m/2**p
BigInteger m2 = ONE.shiftLeft(p); // 2**p
// Calculate new base from m1
BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
? this.mod(m1) : this);
// 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(),
a2 = new MutableBigInteger(a),
b2 = new MutableBigInteger(mod);
MutableBigInteger r= a2.divide(b2, q);
table[0] = r.toIntArray();
// Pad table[0] with leading zeros so its length is at least modLen
if (table[0].length < modLen) {
int offset = modLen - table[0].length;
int[] t2 = new int[modLen];
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 = Arrays.copyOf(b, modLen);
// 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];
System.arraycopy(b, 0, t2, modLen, modLen);
b = montReduce(t2, mod, modLen, inv);
t2 = Arrays.copyOf(b, modLen);
return new BigInteger(1, t2);
}
/**
* Montgomery reduce n, modulo mod. This reduces modulo mod and divides
* by 2^(32*mlen). Adapted from Colin Plumb's C library.
*/
private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
int c=0;
int len = mlen;
int offset=0;
do {
int nEnd = n[n.length-1-offset];
int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
c += addOne(n, offset, mlen, carry);
offset++;
} while (--len > 0);
while (c > 0)
c += subN(n, mod, mlen);
while (intArrayCmpToLen(n, mod, mlen) >= 0)
subN(n, mod, mlen);
return n;
}
/*
* Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
* equal to, or greater than arg2 up to length len.
*/
private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
for (int i=0; i < len; i++) {
long b1 = arg1[i] & LONG_MASK;
long b2 = arg2[i] & LONG_MASK;
if (b1 < b2)
return -1;
if (b1 > b2)
return 1;
}
return 0;
}
/**
* Subtracts two numbers of same length, returning borrow.
*/
private static int subN(int[] a, int[] b, int len) {
long sum = 0;
while (--len >= 0) {
sum = (a[len] & LONG_MASK) -
(b[len] & LONG_MASK) + (sum >> 32);
a[len] = (int)sum;
}
return (int)(sum >> 32);
}
/**
* Multiply an array by one word k and add to result, return the carry
*/
static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
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 = ONE;
BigInteger baseToPow2 = this.mod2(p);
int expOffset = 0;
int limit = exponent.bitLength();
if (this.testBit(0))
limit = (p-1) < limit ? (p-1) : limit;
while (expOffset < limit) {
if (exponent.testBit(expOffset))
result = result.multiply(baseToPow2).mod2(p);
expOffset++;
if (expOffset < limit)
baseToPow2 = baseToPow2.square().mod2(p);
}
return result;
}
/**
* Returns a BigInteger whose value is this mod(2**p).
* Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
*/
private BigInteger mod2(int p) {
if (bitLength() <= p)
return this;
// Copy remaining ints of mag
int numInts = (p + 31) >>> 5;
int[] mag = new int[numInts];
System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);
// Mask out any excess bits
int excessBits = (numInts << 5) - p;
mag[0] &= (1L << (32-excessBits)) - 1;
return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
}
/**
* Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
*
* @param m the modulus.
* @return {@code this}<sup>-1</sup> {@code mod m}.
* @throws ArithmeticException {@code m} &le; 0, or this BigInteger
* has no multiplicative inverse mod m (that is, this BigInteger
* is not <i>relatively prime</i> to m).
*/
public BigInteger modInverse(BigInteger m) {
if (m.signum != 1)
throw new ArithmeticException("BigInteger: modulus not positive");
if (m.equals(ONE))
return ZERO;
// Calculate (this mod m)
BigInteger modVal = this;
if (signum < 0 || (this.compareMagnitude(m) >= 0))
modVal = this.mod(m);
if (modVal.equals(ONE))
return ONE;
MutableBigInteger a = new MutableBigInteger(modVal);
MutableBigInteger b = new MutableBigInteger(m);
MutableBigInteger result = a.mutableModInverse(b);
return result.toBigInteger(1);
}
// Shift Operations
/**
* Returns a BigInteger whose value is {@code (this << n)}.
* The shift distance, {@code n}, may be negative, in which case
* this method performs a right shift.
* (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
*
* @param n shift distance, in bits.
* @return {@code this << n}
* @throws ArithmeticException if the shift distance is {@code
* Integer.MIN_VALUE}.
* @see #shiftRight
*/
public BigInteger shiftLeft(int n) {
if (signum == 0)
return ZERO;
if (n == 0)
return this;
if (n < 0) {
if (n == Integer.MIN_VALUE) {
throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
} else {
return shiftRight(-n);
}
}
int[] newMag = shiftLeft(mag, n);
return new BigInteger(newMag, signum);
}
private static int[] shiftLeft(int[] mag, int n) {
int nInts = n >>> 5;
int nBits = n & 0x1f;
int magLen = mag.length;
int newMag[] = null;
if (nBits == 0) {
newMag = new int[magLen + nInts];
System.arraycopy(mag, 0, newMag, 0, magLen);
} else {
int i = 0;
int nBits2 = 32 - nBits;
int highBits = mag[0] >>> nBits2;
if (highBits != 0) {
newMag = new int[magLen + nInts + 1];
newMag[i++] = highBits;
} else {
newMag = new int[magLen + nInts];
}
int j=0;
while (j < magLen-1)
newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
newMag[i] = mag[j] << nBits;
}
return newMag;
}
/**
* 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}
* @throws ArithmeticException if the shift distance is {@code
* Integer.MIN_VALUE}.
* @see #shiftLeft
*/
public BigInteger shiftRight(int n) {
if (n == 0)
return this;
if (n < 0) {
if (n == Integer.MIN_VALUE) {
throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported.");
} else {
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 = Arrays.copyOf(mag, newMagLen);
} else {
int i = 0;
int highBits = mag[0] >>> nBits;
if (highBits != 0) {
newMag = new int[magLen - nInts];
newMag[i++] = highBits;
} else {
newMag = new int[magLen - nInts -1];
}
int 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 >>> 5) & (1 << (n & 31))) != 0;
}
/**
* Returns a BigInteger whose value is equivalent to this BigInteger
* with the designated bit set. (Computes {@code (this | (1<<n))}.)
*
* @param n index of bit to set.
* @return {@code this | (1<<n)}
* @throws ArithmeticException {@code n} is negative.
*/
public BigInteger setBit(int n) {
if (n < 0)
throw new ArithmeticException("Negative bit address");
int intNum = n >>> 5;
int[] result = new int[Math.max(intLength(), intNum+2)];
for (int i=0; i < result.length; i++)
result[result.length-i-1] = getInt(i);
result[result.length-intNum-1] |= (1 << (n & 31));
return valueOf(result);
}
/**
* Returns a BigInteger whose value is equivalent to this BigInteger
* with the designated bit cleared.
* (Computes {@code (this & ~(1<<n))}.)
*
* @param n index of bit to clear.
* @return {@code this & ~(1<<n)}
* @throws ArithmeticException {@code n} is negative.
*/
public BigInteger clearBit(int n) {
if (n < 0)
throw new ArithmeticException("Negative bit address");
int intNum = n >>> 5;
int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];
for (int i=0; i < result.length; i++)
result[result.length-i-1] = getInt(i);
result[result.length-intNum-1] &= ~(1 << (n & 31));
return valueOf(result);
}
/**
* Returns a BigInteger whose value is equivalent to this BigInteger
* with the designated bit flipped.
* (Computes {@code (this ^ (1<<n))}.)
*
* @param n index of bit to flip.
* @return {@code this ^ (1<<n)}
* @throws ArithmeticException {@code n} is negative.
*/
public BigInteger flipBit(int n) {
if (n < 0)
throw new ArithmeticException("Negative bit address");
int intNum = n >>> 5;
int[] result = new int[Math.max(intLength(), intNum+2)];
for (int i=0; i < result.length; i++)
result[result.length-i-1] = getInt(i);
result[result.length-intNum-1] ^= (1 << (n & 31));
return valueOf(result);
}
/**
* Returns the index of the rightmost (lowest-order) one bit in this
* BigInteger (the number of zero bits to the right of the rightmost
* one bit). Returns -1 if this BigInteger contains no one bits.
* (Computes {@code (this == 0? -1 : log2(this & -this))}.)
*
* @return index of the rightmost one bit in this BigInteger.
*/
public int getLowestSetBit() {
@SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
if (lsb == -2) { // lowestSetBit not initialized yet
lsb = 0;
if (signum == 0) {
lsb -= 1;
} else {
// Search for lowest order nonzero int
int i,b;
for (i=0; (b = getInt(i)) == 0; i++)
;
lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
}
lowestSetBit = lsb + 2;
}
return lsb;
}
// 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() {
@SuppressWarnings("deprecation") int n = bitLength - 1;
if (n == -1) { // bitLength not initialized yet
int[] m = mag;
int len = m.length;
if (len == 0) {
n = 0; // offset by one to initialize
} else {
// Calculate the bit length of the magnitude
int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
if (signum < 0) {
// Check if magnitude is a power of two
boolean pow2 = (Integer.bitCount(mag[0]) == 1);
for (int i=1; i< len && pow2; i++)
pow2 = (mag[i] == 0);
n = (pow2 ? magBitLength -1 : magBitLength);
} else {
n = magBitLength;
}
}
bitLength = n + 1;
}
return n;
}
/**
* Returns the number of bits in the two's complement representation
* of this BigInteger that differ from its sign bit. This method is
* useful when implementing bit-vector style sets atop BigIntegers.
*
* @return number of bits in the two's complement representation
* of this BigInteger that differ from its sign bit.
*/
public int bitCount() {
@SuppressWarnings("deprecation") int bc = bitCount - 1;
if (bc == -1) { // bitCount not initialized yet
bc = 0; // offset by one to initialize
// Count the bits in the magnitude
for (int i=0; i < mag.length; i++)
bc += Integer.bitCount(mag[i]);
if (signum < 0) {
// Count the trailing zeros in the magnitude
int magTrailingZeroCount = 0, j;
for (j=mag.length-1; mag[j] == 0; j--)
magTrailingZeroCount += 32;
magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
bc += magTrailingZeroCount - 1;
}
bitCount = bc + 1;
}
return bc;
}
// Primality Testing
/**
* Returns {@code true} if this BigInteger is probably prime,
* {@code false} if it's definitely composite. If
* {@code certainty} is &le; 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) {
if (signum == val.signum) {
switch (signum) {
case 1:
return compareMagnitude(val);
case -1:
return val.compareMagnitude(this);
default:
return 0;
}
}
return signum > val.signum ? 1 : -1;
}
/**
* Compares the magnitude array of this BigInteger with the specified
* BigInteger's. This is the version of compareTo ignoring sign.
*
* @param val BigInteger whose magnitude array to be compared.
* @return -1, 0 or 1 as this magnitude array is less than, equal to or
* greater than the magnitude aray for the specified BigInteger's.
*/
final int compareMagnitude(BigInteger val) {
int[] m1 = mag;
int len1 = m1.length;
int[] m2 = val.mag;
int len2 = m2.length;
if (len1 < len2)
return -1;
if (len1 > len2)
return 1;
for (int i = 0; i < len1; i++) {
int a = m1[i];
int b = m2[i];
if (a != b)
return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
}
return 0;
}
/**
* Version of compareMagnitude that compares magnitude with long value.
* val can't be Long.MIN_VALUE.
*/
final int compareMagnitude(long val) {
assert val != Long.MIN_VALUE;
int[] m1 = mag;
int len = m1.length;
if (len > 2) {
return 1;
}
if (val < 0) {
val = -val;
}
int highWord = (int)(val >>> 32);
if (highWord == 0) {
if (len < 1)
return -1;
if (len > 1)
return 1;
int a = m1[0];
int b = (int)val;
if (a != b) {
return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
}
return 0;
} else {
if (len < 2)
return -1;
int a = m1[0];
int b = highWord;
if (a != b) {
return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
}
a = m1[1];
b = (int)val;
if (a != b) {
return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1;
}
return 0;
}
}
/**
* Compares this BigInteger with the specified Object for equality.
*
* @param x Object to which this BigInteger is to be compared.
* @return {@code true} if and only if the specified Object is a
* BigInteger whose value is numerically equal to this BigInteger.
*/
public boolean equals(Object x) {
// This test is just an optimization, which may or may not help
if (x == this)
return true;
if (!(x instanceof BigInteger))
return false;
BigInteger xInt = (BigInteger) x;
if (xInt.signum != signum)
return false;
int[] m = mag;
int len = m.length;
int[] xm = xInt.mag;
if (len != xm.length)
return false;
for (int i = 0; i < len; i++)
if (xm[i] != m[i])
return false;
return true;
}
/**
* Returns the minimum of this BigInteger and {@code val}.
*
* @param val value with which the minimum is to be computed.
* @return the BigInteger whose value is the lesser of this BigInteger and
* {@code val}. If they are equal, either may be returned.
*/
public BigInteger min(BigInteger val) {
return (compareTo(val) < 0 ? this : val);
}
/**
* Returns the maximum of this BigInteger and {@code val}.
*
* @param val value with which the maximum is to be computed.
* @return the BigInteger whose value is the greater of this and
* {@code val}. If they are equal, either may be returned.
*/
public BigInteger max(BigInteger val) {
return (compareTo(val) > 0 ? this : val);
}
// Hash Function
/**
* Returns the hash code for this BigInteger.
*
* @return hash code for this BigInteger.
*/
public int hashCode() {
int hashCode = 0;
for (int i=0; i < mag.length; i++)
hashCode = (int)(31*hashCode + (mag[i] & LONG_MASK));
return hashCode * signum;
}
/**
* Returns the String representation of this BigInteger in the
* given radix. If the radix is outside the range from {@link
* Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
* it will default to 10 (as is the case for
* {@code Integer.toString}). The digit-to-character mapping
* provided by {@code Character.forDigit} is used, and a minus
* sign is prepended if appropriate. (This representation is
* compatible with the {@link #BigInteger(String, int) (String,
* int)} constructor.)
*
* @param radix radix of the String representation.
* @return String representation of this BigInteger in the given radix.
* @see Integer#toString
* @see Character#forDigit
* @see #BigInteger(java.lang.String, int)
*/
public String toString(int radix) {
if (signum == 0)
return "0";
if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
radix = 10;
// If it's small enough, use smallToString.
if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD)
return smallToString(radix);
// Otherwise use recursive toString, which requires positive arguments.
// The results will be concatenated into this StringBuilder
StringBuilder sb = new StringBuilder();
if (signum < 0) {
toString(this.negate(), sb, radix, 0);
sb.insert(0, '-');
}
else
toString(this, sb, radix, 0);
return sb.toString();
}
/** This method is used to perform toString when arguments are small. */
private String smallToString(int radix) {
if (signum == 0) {
return "0";
}
// 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(),
a = new MutableBigInteger(tmp.mag),
b = new MutableBigInteger(d.mag);
MutableBigInteger r = a.divide(b, q);
BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);
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();
}
/**
* Converts the specified BigInteger to a string and appends to
* {@code sb}. This implements the recursive Schoenhage algorithm
* for base conversions.
* <p/>
* See Knuth, Donald, _The Art of Computer Programming_, Vol. 2,
* Answers to Exercises (4.4) Question 14.
*
* @param u The number to convert to a string.
* @param sb The StringBuilder that will be appended to in place.
* @param radix The base to convert to.
* @param digits The minimum number of digits to pad to.
*/
private static void toString(BigInteger u, StringBuilder sb, int radix,
int digits) {
/* If we're smaller than a certain threshold, use the smallToString
method, padding with leading zeroes when necessary. */
if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
String s = u.smallToString(radix);
// Pad with internal zeros if necessary.
// Don't pad if we're at the beginning of the string.
if ((s.length() < digits) && (sb.length() > 0)) {
for (int i=s.length(); i < digits; i++) { // May be a faster way to
sb.append('0'); // do this?
}
}
sb.append(s);
return;
}
int b, n;
b = u.bitLength();
// Calculate a value for n in the equation radix^(2^n) = u
// and subtract 1 from that value. This is used to find the
// cache index that contains the best value to divide u.
n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
BigInteger v = getRadixConversionCache(radix, n);
BigInteger[] results;
results = u.divideAndRemainder(v);
int expectedDigits = 1 << n;
// Now recursively build the two halves of each number.
toString(results[0], sb, radix, digits-expectedDigits);
toString(results[1], sb, radix, expectedDigits);
}
/**
* Returns the value radix^(2^exponent) from the cache.
* If this value doesn't already exist in the cache, it is added.
* <p/>
* This could be changed to a more complicated caching method using
* {@code Future}.
*/
private static BigInteger getRadixConversionCache(int radix, int exponent) {
BigInteger[] cacheLine = powerCache[radix]; // volatile read
if (exponent < cacheLine.length) {
return cacheLine[exponent];
}
int oldLength = cacheLine.length;
cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
for (int i = oldLength; i <= exponent; i++) {
cacheLine[i] = cacheLine[i - 1].pow(2);
}
BigInteger[][] pc = powerCache; // volatile read again
if (exponent >= pc[radix].length) {
pc = pc.clone();
pc[radix] = cacheLine;
powerCache = pc; // volatile write, publish
}
return cacheLine[exponent];
}
/* 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
* <i>narrowing primitive conversion</i> from {@code long} to
* {@code int} as defined in section 5.1.3 of
* <cite>The Java&trade; Language Specification</cite>:
* if this BigInteger is too big to fit in an
* {@code int}, only the low-order 32 bits are returned.
* Note that this conversion can lose information about the
* overall magnitude of the BigInteger value as well as return a
* result with the opposite sign.
*
* @return this BigInteger converted to an {@code int}.
* @see #intValueExact()
*/
public int intValue() {
int result = 0;
result = getInt(0);
return result;
}
/**
* Converts this BigInteger to a {@code long}. This
* conversion is analogous to a
* <i>narrowing primitive conversion</i> from {@code long} to
* {@code int} as defined in section 5.1.3 of
* <cite>The Java&trade; Language Specification</cite>:
* if this BigInteger is too big to fit in a
* {@code long}, only the low-order 64 bits are returned.
* Note that this conversion can lose information about the
* overall magnitude of the BigInteger value as well as return a
* result with the opposite sign.
*
* @return this BigInteger converted to a {@code long}.
* @see #longValueExact()
*/
public long longValue() {
long result = 0;
for (int i=1; i >= 0; i--)
result = (result << 32) + (getInt(i) & LONG_MASK);
return result;
}
/**
* Converts this BigInteger to a {@code float}. This
* conversion is similar to the
* <i>narrowing primitive conversion</i> from {@code double} to
* {@code float} as defined in section 5.1.3 of
* <cite>The Java&trade; Language Specification</cite>:
* if this BigInteger has too great a magnitude
* to represent as a {@code float}, it will be converted to
* {@link Float#NEGATIVE_INFINITY} or {@link
* Float#POSITIVE_INFINITY} as appropriate. Note that even when
* the return value is finite, this conversion can lose
* information about the precision of the BigInteger value.
*
* @return this BigInteger converted to a {@code float}.
*/
public float floatValue() {
if (signum == 0) {
return 0.0f;
}
int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
// exponent == floor(log2(abs(this)))
if (exponent < Long.SIZE - 1) {
return longValue();
} else if (exponent > Float.MAX_EXPONENT) {
return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
}
/*
* We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
* one bit. To make rounding easier, we pick out the top
* SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
* down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
* bits, and signifFloor the top SIGNIFICAND_WIDTH.
*
* It helps to consider the real number signif = abs(this) *
* 2^(SIGNIFICAND_WIDTH - 1 - exponent).
*/
int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;
int twiceSignifFloor;
// twiceSignifFloor will be == abs().shiftRight(shift).intValue()
// We do the shift into an int directly to improve performance.
int nBits = shift & 0x1f;
int nBits2 = 32 - nBits;
if (nBits == 0) {
twiceSignifFloor = mag[0];
} else {
twiceSignifFloor = mag[0] >>> nBits;
if (twiceSignifFloor == 0) {
twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
}
}
int signifFloor = twiceSignifFloor >> 1;
signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit
/*
* We round up if either the fractional part of signif is strictly
* greater than 0.5 (which is true if the 0.5 bit is set and any lower
* bit is set), or if the fractional part of signif is >= 0.5 and
* signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
* are set). This is equivalent to the desired HALF_EVEN rounding.
*/
boolean increment = (twiceSignifFloor & 1) != 0
&& ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
int signifRounded = increment ? signifFloor + 1 : signifFloor;
int bits = ((exponent + FloatConsts.EXP_BIAS))
<< (FloatConsts.SIGNIFICAND_WIDTH - 1);
bits += signifRounded;
/*
* If signifRounded == 2^24, we'd need to set all of the significand
* bits to zero and add 1 to the exponent. This is exactly the behavior
* we get from just adding signifRounded to bits directly. If the
* exponent is Float.MAX_EXPONENT, we round up (correctly) to
* Float.POSITIVE_INFINITY.
*/
bits |= signum & FloatConsts.SIGN_BIT_MASK;
return Float.intBitsToFloat(bits);
}
/**
* Converts this BigInteger to a {@code double}. This
* conversion is similar to the
* <i>narrowing primitive conversion</i> from {@code double} to
* {@code float} as defined in section 5.1.3 of
* <cite>The Java&trade; Language Specification</cite>:
* if this BigInteger has too great a magnitude
* to represent as a {@code double}, it will be converted to
* {@link Double#NEGATIVE_INFINITY} or {@link
* Double#POSITIVE_INFINITY} as appropriate. Note that even when
* the return value is finite, this conversion can lose
* information about the precision of the BigInteger value.
*
* @return this BigInteger converted to a {@code double}.
*/
public double doubleValue() {
if (signum == 0) {
return 0.0;
}
int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;
// exponent == floor(log2(abs(this))Double)
if (exponent < Long.SIZE - 1) {
return longValue();
} else if (exponent > Double.MAX_EXPONENT) {
return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
}
/*
* We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
* one bit. To make rounding easier, we pick out the top
* SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
* down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
* bits, and signifFloor the top SIGNIFICAND_WIDTH.
*
* It helps to consider the real number signif = abs(this) *
* 2^(SIGNIFICAND_WIDTH - 1 - exponent).
*/
int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;
long twiceSignifFloor;
// twiceSignifFloor will be == abs().shiftRight(shift).longValue()
// We do the shift into a long directly to improve performance.
int nBits = shift & 0x1f;
int nBits2 = 32 - nBits;
int highBits;
int lowBits;
if (nBits == 0) {
highBits = mag[0];
lowBits = mag[1];
} else {
highBits = mag[0] >>> nBits;
lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
if (highBits == 0) {
highBits = lowBits;
lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
}
}
twiceSignifFloor = ((highBits & LONG_MASK) << 32)
| (lowBits & LONG_MASK);
long signifFloor = twiceSignifFloor >> 1;
signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit
/*
* We round up if either the fractional part of signif is strictly
* greater than 0.5 (which is true if the 0.5 bit is set and any lower
* bit is set), or if the fractional part of signif is >= 0.5 and
* signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
* are set). This is equivalent to the desired HALF_EVEN rounding.
*/
boolean increment = (twiceSignifFloor & 1) != 0
&& ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
long signifRounded = increment ? signifFloor + 1 : signifFloor;
long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
<< (DoubleConsts.SIGNIFICAND_WIDTH - 1);
bits += signifRounded;
/*
* If signifRounded == 2^53, we'd need to set all of the significand
* bits to zero and add 1 to the exponent. This is exactly the behavior
* we get from just adding signifRounded to bits directly. If the
* exponent is Double.MAX_EXPONENT, we round up (correctly) to
* Double.POSITIVE_INFINITY.
*/
bits |= signum & DoubleConsts.SIGN_BIT_MASK;
return Double.longBitsToDouble(bits);
}
/**
* Returns a copy of the input array stripped of any leading zero bytes.
*/
private static int[] stripLeadingZeroInts(int val[]) {
int vlen = val.length;
int keep;
// Find first nonzero byte
for (keep = 0; keep < vlen && val[keep] == 0; keep++)
;
return java.util.Arrays.copyOfRange(val, keep, vlen);
}
/**
* Returns the input array stripped of any leading zero bytes.
* Since the source is trusted the copying may be skipped.
*/
private static int[] trustedStripLeadingZeroInts(int val[]) {
int vlen = val.length;
int keep;
// Find first nonzero byte
for (keep = 0; keep < vlen && val[keep] == 0; keep++)
;
return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
}
/**
* Returns a copy of the input array stripped of any leading zero bytes.
*/
private static int[] stripLeadingZeroBytes(byte a[]) {
int byteLength = a.length;
int keep;
// Find first nonzero byte
for (keep = 0; keep < byteLength && a[keep] == 0; keep++)
;
// Allocate new array and copy relevant part of input array
int intLength = ((byteLength - keep) + 3) >>> 2;
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 <= (bytesToTransfer << 3); j += 8)
result[i] |= ((a[b--] & 0xff) << j);
}
return result;
}
/**
* Takes an array a representing a negative 2's-complement number and
* returns the minimal (no leading zero bytes) unsigned whose value is -a.
*/
private static int[] makePositive(byte a[]) {
int 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() >>> 5) + 1;
}
/* Returns sign bit */
private int signBit() {
return signum < 0 ? 1 : 0;
}
/* Returns an int of sign bits */
private int signInt() {
return signum < 0 ? -1 : 0;
}
/**
* Returns the specified int of the little-endian two's complement
* representation (int 0 is the least significant). The int number can
* be arbitrarily high (values are logically preceded by infinitely many
* sign ints).
*/
private int getInt(int n) {
if (n < 0)
return 0;
if (n >= mag.length)
return signInt();
int magInt = mag[mag.length-n-1];
return (signum >= 0 ? magInt :
(n <= firstNonzeroIntNum() ? -magInt : ~magInt));
}
/**
* Returns the index of the int that contains the first nonzero int in the
* little-endian binary representation of the magnitude (int 0 is the
* least significant). If the magnitude is zero, return value is undefined.
*/
private int firstNonzeroIntNum() {
int fn = firstNonzeroIntNum - 2;
if (fn == -2) { // firstNonzeroIntNum not initialized yet
fn = 0;
// Search for the first nonzero int
int i;
int mlen = mag.length;
for (i = mlen - 1; i >= 0 && mag[i] == 0; i--)
;
fn = mlen - i - 1;
firstNonzeroIntNum = fn + 2; // offset by two to initialize
}
return fn;
}
/** 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.
* Note:
* The current convention is to initialize the cache fields, bitCount,
* bitLength and lowestSetBit, to 0 rather than some other marker value.
* Therefore, no explicit action to set these fields needs to be taken in
* readObject because those fields already have a 0 value be default since
* defaultReadObject is not being used.
*/
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
int sign = fields.get("signum", -2);
byte[] magnitude = (byte[])fields.get("magnitude", null);
// Validate signum
if (sign < -1 || sign > 1) {
String message = "BigInteger: Invalid signum value";
if (fields.defaulted("signum"))
message = "BigInteger: Signum not present in stream";
throw new java.io.StreamCorruptedException(message);
}
if ((magnitude.length == 0) != (sign == 0)) {
String message = "BigInteger: signum-magnitude mismatch";
if (fields.defaulted("magnitude"))
message = "BigInteger: Magnitude not present in stream";
throw new java.io.StreamCorruptedException(message);
}
// Commit final fields via Unsafe
UnsafeHolder.putSign(this, sign);
// Calculate mag field from magnitude and discard magnitude
UnsafeHolder.putMag(this, stripLeadingZeroBytes(magnitude));
}
// Support for resetting final fields while deserializing
private static class UnsafeHolder {
private static final sun.misc.Unsafe unsafe;
private static final long signumOffset;
private static final long magOffset;
static {
try {
unsafe = sun.misc.Unsafe.getUnsafe();
signumOffset = unsafe.objectFieldOffset
(BigInteger.class.getDeclaredField("signum"));
magOffset = unsafe.objectFieldOffset
(BigInteger.class.getDeclaredField("mag"));
} catch (Exception ex) {
throw new ExceptionInInitializerError(ex);
}
}
static void putSign(BigInteger bi, int sign) {
unsafe.putIntVolatile(bi, signumOffset, sign);
}
static void putMag(BigInteger bi, int[] magnitude) {
unsafe.putObjectVolatile(bi, magOffset, 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());
// The values written for cached fields are compatible with older
// versions, but are ignored in readObject so don't otherwise matter.
fields.put("bitCount", -1);
fields.put("bitLength", -1);
fields.put("lowestSetBit", -2);
fields.put("firstNonzeroByteNum", -2);
// save them
s.writeFields();
}
/**
* Returns the mag array as an array of bytes.
*/
private byte[] magSerializedForm() {
int len = mag.length;
int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
int byteLen = (bitLen + 7) >>> 3;
byte[] result = new byte[byteLen];
for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
i >= 0; i--) {
if (bytesCopied == 4) {
nextInt = mag[intIndex--];
bytesCopied = 1;
} else {
nextInt >>>= 8;
bytesCopied++;
}
result[i] = (byte)nextInt;
}
return result;
}
/**
* Converts this {@code BigInteger} to a {@code long}, checking
* for lost information. If the value of this {@code BigInteger}
* is out of the range of the {@code long} type, then an
* {@code ArithmeticException} is thrown.
*
* @return this {@code BigInteger} converted to a {@code long}.
* @throws ArithmeticException if the value of {@code this} will
* not exactly fit in a {@code long}.
* @see BigInteger#longValue
* @since 1.8
*/
public long longValueExact() {
if (mag.length <= 2 && bitLength() <= 63)
return longValue();
else
throw new ArithmeticException("BigInteger out of long range");
}
/**
* Converts this {@code BigInteger} to an {@code int}, checking
* for lost information. If the value of this {@code BigInteger}
* is out of the range of the {@code int} type, then an
* {@code ArithmeticException} is thrown.
*
* @return this {@code BigInteger} converted to an {@code int}.
* @throws ArithmeticException if the value of {@code this} will
* not exactly fit in a {@code int}.
* @see BigInteger#intValue
* @since 1.8
*/
public int intValueExact() {
if (mag.length <= 1 && bitLength() <= 31)
return intValue();
else
throw new ArithmeticException("BigInteger out of int range");
}
/**
* Converts this {@code BigInteger} to a {@code short}, checking
* for lost information. If the value of this {@code BigInteger}
* is out of the range of the {@code short} type, then an
* {@code ArithmeticException} is thrown.
*
* @return this {@code BigInteger} converted to a {@code short}.
* @throws ArithmeticException if the value of {@code this} will
* not exactly fit in a {@code short}.
* @see BigInteger#shortValue
* @since 1.8
*/
public short shortValueExact() {
if (mag.length <= 1 && bitLength() <= 31) {
int value = intValue();
if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE)
return shortValue();
}
throw new ArithmeticException("BigInteger out of short range");
}
/**
* Converts this {@code BigInteger} to a {@code byte}, checking
* for lost information. If the value of this {@code BigInteger}
* is out of the range of the {@code byte} type, then an
* {@code ArithmeticException} is thrown.
*
* @return this {@code BigInteger} converted to a {@code byte}.
* @throws ArithmeticException if the value of {@code this} will
* not exactly fit in a {@code byte}.
* @see BigInteger#byteValue
* @since 1.8
*/
public byte byteValueExact() {
if (mag.length <= 1 && bitLength() <= 31) {
int value = intValue();
if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE)
return byteValue();
}
throw new ArithmeticException("BigInteger out of byte range");
}
}
Raw
BigIntegerTest.java.phase3
/*
* Copyright (c) 1998, 2013, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
/*
* @test
* @bug 4181191 4161971 4227146 4194389 4823171 4624738 4812225 4837946
* @summary tests methods in BigInteger
* @run main/timeout=400 BigIntegerTest
* @author madbot
*/
import java.io.File;
import java.io.FileInputStream;
import java.io.FileOutputStream;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.math.BigInteger;
import java.util.Random;
/**
* This is a simple test class created to ensure that the results
* generated by BigInteger adhere to certain identities. Passing
* this test is a strong assurance that the BigInteger operations
* are working correctly.
*
* Four arguments may be specified which give the number of
* decimal digits you desire in the four batches of test numbers.
*
* The tests are performed on arrays of random numbers which are
* generated by a Random class as well as special cases which
* throw in boundary numbers such as 0, 1, maximum sized, etc.
*
*/
public class BigIntegerTest {
//
// Bit large number thresholds based on the int thresholds
// defined in BigInteger itself:
//
// KARATSUBA_THRESHOLD = 50 ints = 1600 bits
// TOOM_COOK_THRESHOLD = 75 ints = 2400 bits
// KARATSUBA_SQUARE_THRESHOLD = 90 ints = 2880 bits
// TOOM_COOK_SQUARE_THRESHOLD = 140 ints = 4480 bits
//
// SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 8 ints = 256 bits
//
// BURNIKEL_ZIEGLER_THRESHOLD = 50 ints = 1600 bits
//
static final int BITS_KARATSUBA = 1600;
static final int BITS_TOOM_COOK = 2400;
static final int BITS_KARATSUBA_SQUARE = 2880;
static final int BITS_TOOM_COOK_SQUARE = 4480;
static final int BITS_SCHOENHAGE_BASE = 256;
static final int BITS_BURNIKEL_ZIEGLER = 1600;
static final int ORDER_SMALL = 60;
static final int ORDER_MEDIUM = 100;
// #bits for testing Karatsuba
static final int ORDER_KARATSUBA = 1800;
// #bits for testing Toom-Cook and Burnikel-Ziegler
static final int ORDER_TOOM_COOK = 4000;
// #bits for testing Karatsuba squaring
static final int ORDER_KARATSUBA_SQUARE = 3200;
// #bits for testing Toom-Cook squaring
static final int ORDER_TOOM_COOK_SQUARE = 4600;
static final int SIZE = 1000; // numbers per batch
static Random rnd = new Random();
static boolean failure = false;
public static void pow(int order) {
int failCount1 = 0;
for (int i=0; i<SIZE; i++) {
// Test identity x^power == x*x*x ... *x
int power = rnd.nextInt(6) + 2;
BigInteger x = fetchNumber(order);
BigInteger y = x.pow(power);
BigInteger z = x;
for (int j=1; j<power; j++)
z = z.multiply(x);
if (!y.equals(z))
failCount1++;
}
report("pow for " + order + " bits", failCount1);
}
public static void square(int order) {
int failCount1 = 0;
for (int i=0; i<SIZE; i++) {
// Test identity x^2 == x*x
BigInteger x = fetchNumber(order);
BigInteger xx = x.multiply(x);
BigInteger x2 = x.pow(2);
if (!x2.equals(xx))
failCount1++;
}
report("square for " + order + " bits", failCount1);
}
public static void arithmetic(int order) {
int failCount = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order);
while(x.compareTo(BigInteger.ZERO) != 1)
x = fetchNumber(order);
BigInteger y = fetchNumber(order/2);
while(x.compareTo(y) == -1)
y = fetchNumber(order/2);
if (y.equals(BigInteger.ZERO))
y = y.add(BigInteger.ONE);
// Test identity ((x/y))*y + x%y - x == 0
// using separate divide() and remainder()
BigInteger baz = x.divide(y);
baz = baz.multiply(y);
baz = baz.add(x.remainder(y));
baz = baz.subtract(x);
if (!baz.equals(BigInteger.ZERO))
failCount++;
}
report("Arithmetic I for " + order + " bits", failCount);
failCount = 0;
for (int i=0; i<100; i++) {
BigInteger x = fetchNumber(order);
while(x.compareTo(BigInteger.ZERO) != 1)
x = fetchNumber(order);
BigInteger y = fetchNumber(order/2);
while(x.compareTo(y) == -1)
y = fetchNumber(order/2);
if (y.equals(BigInteger.ZERO))
y = y.add(BigInteger.ONE);
// Test identity ((x/y))*y + x%y - x == 0
// using divideAndRemainder()
BigInteger baz[] = x.divideAndRemainder(y);
baz[0] = baz[0].multiply(y);
baz[0] = baz[0].add(baz[1]);
baz[0] = baz[0].subtract(x);
if (!baz[0].equals(BigInteger.ZERO))
failCount++;
}
report("Arithmetic II for " + order + " bits", failCount);
}
/**
* Sanity test for Karatsuba and 3-way Toom-Cook multiplication.
* For each of the Karatsuba and 3-way Toom-Cook multiplication thresholds,
* construct two factors each with a mag array one element shorter than the
* threshold, and with the most significant bit set and the rest of the bits
* random. Each of these numbers will therefore be below the threshold but
* if shifted left be above the threshold. Call the numbers 'u' and 'v' and
* define random shifts 'a' and 'b' in the range [1,32]. Then we have the
* identity
* <pre>
* (u << a)*(v << b) = (u*v) << (a + b)
* </pre>
* For Karatsuba multiplication, the right hand expression will be evaluated
* using the standard naive algorithm, and the left hand expression using
* the Karatsuba algorithm. For 3-way Toom-Cook multiplication, the right
* hand expression will be evaluated using Karatsuba multiplication, and the
* left hand expression using 3-way Toom-Cook multiplication.
*/
public static void multiplyLarge() {
int failCount = 0;
BigInteger base = BigInteger.ONE.shiftLeft(BITS_KARATSUBA - 32 - 1);
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(BITS_KARATSUBA - 32 - 1);
BigInteger u = base.add(x);
int a = 1 + rnd.nextInt(31);
BigInteger w = u.shiftLeft(a);
BigInteger y = fetchNumber(BITS_KARATSUBA - 32 - 1);
BigInteger v = base.add(y);
int b = 1 + rnd.nextInt(32);
BigInteger z = v.shiftLeft(b);
BigInteger multiplyResult = u.multiply(v).shiftLeft(a + b);
BigInteger karatsubaMultiplyResult = w.multiply(z);
if (!multiplyResult.equals(karatsubaMultiplyResult)) {
failCount++;
}
}
report("multiplyLarge Karatsuba", failCount);
failCount = 0;
base = base.shiftLeft(BITS_TOOM_COOK - BITS_KARATSUBA);
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(BITS_TOOM_COOK - 32 - 1);
BigInteger u = base.add(x);
BigInteger u2 = u.shiftLeft(1);
BigInteger y = fetchNumber(BITS_TOOM_COOK - 32 - 1);
BigInteger v = base.add(y);
BigInteger v2 = v.shiftLeft(1);
BigInteger multiplyResult = u.multiply(v).shiftLeft(2);
BigInteger toomCookMultiplyResult = u2.multiply(v2);
if (!multiplyResult.equals(toomCookMultiplyResult)) {
failCount++;
}
}
report("multiplyLarge Toom-Cook", failCount);
}
/**
* Sanity test for Karatsuba and 3-way Toom-Cook squaring.
* This test is analogous to {@link AbstractMethodError#multiplyLarge}
* with both factors being equal. The squaring methods will not be tested
* unless the <code>bigInteger.multiply(bigInteger)</code> tests whether
* the parameter is the same instance on which the method is being invoked
* and calls <code>square()</code> accordingly.
*/
public static void squareLarge() {
int failCount = 0;
BigInteger base = BigInteger.ONE.shiftLeft(BITS_KARATSUBA_SQUARE - 32 - 1);
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(BITS_KARATSUBA_SQUARE - 32 - 1);
BigInteger u = base.add(x);
int a = 1 + rnd.nextInt(31);
BigInteger w = u.shiftLeft(a);
BigInteger squareResult = u.multiply(u).shiftLeft(2*a);
BigInteger karatsubaSquareResult = w.multiply(w);
if (!squareResult.equals(karatsubaSquareResult)) {
failCount++;
}
}
report("squareLarge Karatsuba", failCount);
failCount = 0;
base = base.shiftLeft(BITS_TOOM_COOK_SQUARE - BITS_KARATSUBA_SQUARE);
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(BITS_TOOM_COOK_SQUARE - 32 - 1);
BigInteger u = base.add(x);
int a = 1 + rnd.nextInt(31);
BigInteger w = u.shiftLeft(a);
BigInteger squareResult = u.multiply(u).shiftLeft(2*a);
BigInteger toomCookSquareResult = w.multiply(w);
if (!squareResult.equals(toomCookSquareResult)) {
failCount++;
}
}
report("squareLarge Toom-Cook", failCount);
}
/**
* Sanity test for Burnikel-Ziegler division. The Burnikel-Ziegler division
* algorithm is used when each of the dividend and the divisor has at least
* a specified number of ints in its representation. This test is based on
* the observation that if {@code w = u*pow(2,a)} and {@code z = v*pow(2,b)}
* where {@code abs(u) > abs(v)} and {@code a > b && b > 0}, then if
* {@code w/z = q1*z + r1} and {@code u/v = q2*v + r2}, then
* {@code q1 = q2*pow(2,a-b)} and {@code r1 = r2*pow(2,b)}. The test
* ensures that {@code v} is just under the B-Z threshold and that {@code w}
* and {@code z} are both over the threshold. This implies that {@code u/v}
* uses the standard division algorithm and {@code w/z} uses the B-Z
* algorithm. The results of the two algorithms are then compared using the
* observation described in the foregoing and if they are not equal a
* failure is logged.
*/
public static void divideLarge() {
int failCount = 0;
BigInteger base = BigInteger.ONE.shiftLeft(BITS_BURNIKEL_ZIEGLER - 33);
for (int i=0; i<SIZE; i++) {
BigInteger addend = new BigInteger(BITS_BURNIKEL_ZIEGLER - 34, rnd);
BigInteger v = base.add(addend);
BigInteger u = v.multiply(BigInteger.valueOf(2 + rnd.nextInt(Short.MAX_VALUE - 1)));
if(rnd.nextBoolean()) {
u = u.negate();
}
if(rnd.nextBoolean()) {
v = v.negate();
}
int a = 17 + rnd.nextInt(16);
int b = 1 + rnd.nextInt(16);
BigInteger w = u.multiply(BigInteger.valueOf(1L << a));
BigInteger z = v.multiply(BigInteger.valueOf(1L << b));
BigInteger[] divideResult = u.divideAndRemainder(v);
divideResult[0] = divideResult[0].multiply(BigInteger.valueOf(1L << (a - b)));
divideResult[1] = divideResult[1].multiply(BigInteger.valueOf(1L << b));
BigInteger[] bzResult = w.divideAndRemainder(z);
if (divideResult[0].compareTo(bzResult[0]) != 0 ||
divideResult[1].compareTo(bzResult[1]) != 0) {
failCount++;
}
}
report("divideLarge", failCount);
}
public static void bitCount() {
int failCount = 0;
for (int i=0; i<SIZE*10; i++) {
int x = rnd.nextInt();
BigInteger bigX = BigInteger.valueOf((long)x);
int bit = (x < 0 ? 0 : 1);
int tmp = x, bitCount = 0;
for (int j=0; j<32; j++) {
bitCount += ((tmp & 1) == bit ? 1 : 0);
tmp >>= 1;
}
if (bigX.bitCount() != bitCount) {
//System.err.println(x+": "+bitCount+", "+bigX.bitCount());
failCount++;
}
}
report("Bit Count", failCount);
}
public static void bitLength() {
int failCount = 0;
for (int i=0; i<SIZE*10; i++) {
int x = rnd.nextInt();
BigInteger bigX = BigInteger.valueOf((long)x);
int signBit = (x < 0 ? 0x80000000 : 0);
int tmp = x, bitLength, j;
for (j=0; j<32 && (tmp & 0x80000000)==signBit; j++)
tmp <<= 1;
bitLength = 32 - j;
if (bigX.bitLength() != bitLength) {
//System.err.println(x+": "+bitLength+", "+bigX.bitLength());
failCount++;
}
}
report("BitLength", failCount);
}
public static void bitOps(int order) {
int failCount1 = 0, failCount2 = 0, failCount3 = 0;
for (int i=0; i<SIZE*5; i++) {
BigInteger x = fetchNumber(order);
BigInteger y;
// Test setBit and clearBit (and testBit)
if (x.signum() < 0) {
y = BigInteger.valueOf(-1);
for (int j=0; j<x.bitLength(); j++)
if (!x.testBit(j))
y = y.clearBit(j);
} else {
y = BigInteger.ZERO;
for (int j=0; j<x.bitLength(); j++)
if (x.testBit(j))
y = y.setBit(j);
}
if (!x.equals(y))
failCount1++;
// Test flipBit (and testBit)
y = BigInteger.valueOf(x.signum()<0 ? -1 : 0);
for (int j=0; j<x.bitLength(); j++)
if (x.signum()<0 ^ x.testBit(j))
y = y.flipBit(j);
if (!x.equals(y))
failCount2++;
}
report("clearBit/testBit for " + order + " bits", failCount1);
report("flipBit/testBit for " + order + " bits", failCount2);
for (int i=0; i<SIZE*5; i++) {
BigInteger x = fetchNumber(order);
// Test getLowestSetBit()
int k = x.getLowestSetBit();
if (x.signum() == 0) {
if (k != -1)
failCount3++;
} else {
BigInteger z = x.and(x.negate());
int j;
for (j=0; j<z.bitLength() && !z.testBit(j); j++)
;
if (k != j)
failCount3++;
}
}
report("getLowestSetBit for " + order + " bits", failCount3);
}
public static void bitwise(int order) {
// Test identity x^y == x|y &~ x&y
int failCount = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order);
BigInteger y = fetchNumber(order);
BigInteger z = x.xor(y);
BigInteger w = x.or(y).andNot(x.and(y));
if (!z.equals(w))
failCount++;
}
report("Logic (^ | & ~) for " + order + " bits", failCount);
// Test identity x &~ y == ~(~x | y)
failCount = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order);
BigInteger y = fetchNumber(order);
BigInteger z = x.andNot(y);
BigInteger w = x.not().or(y).not();
if (!z.equals(w))
failCount++;
}
report("Logic (&~ | ~) for " + order + " bits", failCount);
}
public static void shift(int order) {
int failCount1 = 0;
int failCount2 = 0;
int failCount3 = 0;
for (int i=0; i<100; i++) {
BigInteger x = fetchNumber(order);
int n = Math.abs(rnd.nextInt()%200);
if (!x.shiftLeft(n).equals
(x.multiply(BigInteger.valueOf(2L).pow(n))))
failCount1++;
BigInteger y[] =x.divideAndRemainder(BigInteger.valueOf(2L).pow(n));
BigInteger z = (x.signum()<0 && y[1].signum()!=0
? y[0].subtract(BigInteger.ONE)
: y[0]);
BigInteger b = x.shiftRight(n);
if (!b.equals(z)) {
System.err.println("Input is "+x.toString(2));
System.err.println("shift is "+n);
System.err.println("Divided "+z.toString(2));
System.err.println("Shifted is "+b.toString(2));
if (b.toString().equals(z.toString()))
System.err.println("Houston, we have a problem.");
failCount2++;
}
if (!x.shiftLeft(n).shiftRight(n).equals(x))
failCount3++;
}
report("baz shiftLeft for " + order + " bits", failCount1);
report("baz shiftRight for " + order + " bits", failCount2);
report("baz shiftLeft/Right for " + order + " bits", failCount3);
}
public static void divideAndRemainder(int order) {
int failCount1 = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order).abs();
while(x.compareTo(BigInteger.valueOf(3L)) != 1)
x = fetchNumber(order).abs();
BigInteger z = x.divide(BigInteger.valueOf(2L));
BigInteger y[] = x.divideAndRemainder(x);
if (!y[0].equals(BigInteger.ONE)) {
failCount1++;
System.err.println("fail1 x :"+x);
System.err.println(" y :"+y);
}
else if (!y[1].equals(BigInteger.ZERO)) {
failCount1++;
System.err.println("fail2 x :"+x);
System.err.println(" y :"+y);
}
y = x.divideAndRemainder(z);
if (!y[0].equals(BigInteger.valueOf(2))) {
failCount1++;
System.err.println("fail3 x :"+x);
System.err.println(" y :"+y);
}
}
report("divideAndRemainder for " + order + " bits", failCount1);
}
public static void stringConv() {
int failCount = 0;
// Generic string conversion.
for (int i=0; i<100; i++) {
byte xBytes[] = new byte[Math.abs(rnd.nextInt())%100+1];
rnd.nextBytes(xBytes);
BigInteger x = new BigInteger(xBytes);
for (int radix=Character.MIN_RADIX; radix < Character.MAX_RADIX; radix++) {
String result = x.toString(radix);
BigInteger test = new BigInteger(result, radix);
if (!test.equals(x)) {
failCount++;
System.err.println("BigInteger toString: "+x);
System.err.println("Test: "+test);
System.err.println(radix);
}
}
}
// String conversion straddling the Schoenhage algorithm crossover
// threshold, and at twice and four times the threshold.
for (int k = 0; k <= 2; k++) {
int factor = 1 << k;
int upper = factor * BITS_SCHOENHAGE_BASE + 33;
int lower = upper - 35;
for (int bits = upper; bits >= lower; bits--) {
for (int i = 0; i < 50; i++) {
BigInteger x = BigInteger.ONE.shiftLeft(bits - 1).or(new BigInteger(bits - 2, rnd));
for (int radix = Character.MIN_RADIX; radix < Character.MAX_RADIX; radix++) {
String result = x.toString(radix);
BigInteger test = new BigInteger(result, radix);
if (!test.equals(x)) {
failCount++;
System.err.println("BigInteger toString: " + x);
System.err.println("Test: " + test);
System.err.println(radix);
}
}
}
}
}
report("String Conversion", failCount);
}
public static void byteArrayConv(int order) {
int failCount = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order);
while (x.equals(BigInteger.ZERO))
x = fetchNumber(order);
BigInteger y = new BigInteger(x.toByteArray());
if (!x.equals(y)) {
failCount++;
System.err.println("orig is "+x);
System.err.println("new is "+y);
}
}
report("Array Conversion for " + order + " bits", failCount);
}
public static void modInv(int order) {
int failCount = 0, successCount = 0, nonInvCount = 0;
for (int i=0; i<SIZE; i++) {
BigInteger x = fetchNumber(order);
while(x.equals(BigInteger.ZERO))
x = fetchNumber(order);
BigInteger m = fetchNumber(order).abs();
while(m.compareTo(BigInteger.ONE) != 1)
m = fetchNumber(order).abs();
try {
BigInteger inv = x.modInverse(m);
BigInteger prod = inv.multiply(x).remainder(m);
if (prod.signum() == -1)
prod = prod.add(m);
if (prod.equals(BigInteger.ONE))
successCount++;
else
failCount++;
} catch(ArithmeticException e) {
nonInvCount++;
}
}
report("Modular Inverse for " + order + " bits", failCount);
}
public static void modExp(int order1, int order2) {
int failCount = 0;
for (int i=0; i<SIZE/10; i++) {
BigInteger m = fetchNumber(order1).abs();
while(m.compareTo(BigInteger.ONE) != 1)
m = fetchNumber(order1).abs();
BigInteger base = fetchNumber(order2);
BigInteger exp = fetchNumber(8).abs();
BigInteger z = base.modPow(exp, m);
BigInteger w = base.pow(exp.intValue()).mod(m);
if (!z.equals(w)) {
System.err.println("z is "+z);
System.err.println("w is "+w);
System.err.println("mod is "+m);
System.err.println("base is "+base);
System.err.println("exp is "+exp);
failCount++;
}
}
report("Exponentiation I for " + order1 + " and " +
order2 + " bits", failCount);
}
// This test is based on Fermat's theorem
// which is not ideal because base must not be multiple of modulus
// and modulus must be a prime or pseudoprime (Carmichael number)
public static void modExp2(int order) {
int failCount = 0;
for (int i=0; i<10; i++) {
BigInteger m = new BigInteger(100, 5, rnd);
while(m.compareTo(BigInteger.ONE) != 1)
m = new BigInteger(100, 5, rnd);
BigInteger exp = m.subtract(BigInteger.ONE);
BigInteger base = fetchNumber(order).abs();
while(base.compareTo(m) != -1)
base = fetchNumber(order).abs();
while(base.equals(BigInteger.ZERO))
base = fetchNumber(order).abs();
BigInteger one = base.modPow(exp, m);
if (!one.equals(BigInteger.ONE)) {
System.err.println("m is "+m);
System.err.println("base is "+base);
System.err.println("exp is "+exp);
failCount++;
}
}
report("Exponentiation II for " + order + " bits", failCount);
}
private static final int[] mersenne_powers = {
521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
1257787, 1398269, 2976221, 3021377, 6972593, 13466917 };
private static final long[] carmichaels = {
561,1105,1729,2465,2821,6601,8911,10585,15841,29341,41041,46657,52633,
62745,63973,75361,101101,115921,126217,162401,172081,188461,252601,
278545,294409,314821,334153,340561,399001,410041,449065,488881,512461,
225593397919L };
// Note: testing the larger ones takes too long.
private static final int NUM_MERSENNES_TO_TEST = 7;
// Note: this constant used for computed Carmichaels, not the array above
private static final int NUM_CARMICHAELS_TO_TEST = 5;
private static final String[] customer_primes = {
"120000000000000000000000000000000019",
"633825300114114700748351603131",
"1461501637330902918203684832716283019651637554291",
"779626057591079617852292862756047675913380626199",
"857591696176672809403750477631580323575362410491",
"910409242326391377348778281801166102059139832131",
"929857869954035706722619989283358182285540127919",
"961301750640481375785983980066592002055764391999",
"1267617700951005189537696547196156120148404630231",
"1326015641149969955786344600146607663033642528339" };
private static final BigInteger ZERO = BigInteger.ZERO;
private static final BigInteger ONE = BigInteger.ONE;
private static final BigInteger TWO = new BigInteger("2");
private static final BigInteger SIX = new BigInteger("6");
private static final BigInteger TWELVE = new BigInteger("12");
private static final BigInteger EIGHTEEN = new BigInteger("18");
public static void prime() {
BigInteger p1, p2, c1;
int failCount = 0;
// Test consistency
for(int i=0; i<10; i++) {
p1 = BigInteger.probablePrime(100, rnd);
if (!p1.isProbablePrime(100)) {
System.err.println("Consistency "+p1.toString(16));
failCount++;
}
}
// Test some known Mersenne primes (2^n)-1
// The array holds the exponents, not the numbers being tested
for (int i=0; i<NUM_MERSENNES_TO_TEST; i++) {
p1 = new BigInteger("2");
p1 = p1.pow(mersenne_powers[i]);
p1 = p1.subtract(BigInteger.ONE);
if (!p1.isProbablePrime(100)) {
System.err.println("Mersenne prime "+i+ " failed.");
failCount++;
}
}
// Test some primes reported by customers as failing in the past
for (int i=0; i<customer_primes.length; i++) {
p1 = new BigInteger(customer_primes[i]);
if (!p1.isProbablePrime(100)) {
System.err.println("Customer prime "+i+ " failed.");
failCount++;
}
}
// Test some known Carmichael numbers.
for (int i=0; i<carmichaels.length; i++) {
c1 = BigInteger.valueOf(carmichaels[i]);
if(c1.isProbablePrime(100)) {
System.err.println("Carmichael "+i+ " reported as prime.");
failCount++;
}
}
// Test some computed Carmichael numbers.
// Numbers of the form (6k+1)(12k+1)(18k+1) are Carmichael numbers if
// each of the factors is prime
int found = 0;
BigInteger f1 = new BigInteger(40, 100, rnd);
while (found < NUM_CARMICHAELS_TO_TEST) {
BigInteger k = null;
BigInteger f2, f3;
f1 = f1.nextProbablePrime();
BigInteger[] result = f1.subtract(ONE).divideAndRemainder(SIX);
if (result[1].equals(ZERO)) {
k = result[0];
f2 = k.multiply(TWELVE).add(ONE);
if (f2.isProbablePrime(100)) {
f3 = k.multiply(EIGHTEEN).add(ONE);
if (f3.isProbablePrime(100)) {
c1 = f1.multiply(f2).multiply(f3);
if (c1.isProbablePrime(100)) {
System.err.println("Computed Carmichael "
+c1.toString(16));
failCount++;
}
found++;
}
}
}
f1 = f1.add(TWO);
}
// Test some composites that are products of 2 primes
for (int i=0; i<50; i++) {
p1 = BigInteger.probablePrime(100, rnd);
p2 = BigInteger.probablePrime(100, rnd);
c1 = p1.multiply(p2);
if (c1.isProbablePrime(100)) {
System.err.println("Composite failed "+c1.toString(16));
failCount++;
}
}
for (int i=0; i<4; i++) {
p1 = BigInteger.probablePrime(600, rnd);
p2 = BigInteger.probablePrime(600, rnd);
c1 = p1.multiply(p2);
if (c1.isProbablePrime(100)) {
System.err.println("Composite failed "+c1.toString(16));
failCount++;
}
}
report("Prime", failCount);
}
private static final long[] primesTo100 = {
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
};
private static final long[] aPrimeSequence = {
1999999003L, 1999999013L, 1999999049L, 1999999061L, 1999999081L,
1999999087L, 1999999093L, 1999999097L, 1999999117L, 1999999121L,
1999999151L, 1999999171L, 1999999207L, 1999999219L, 1999999271L,
1999999321L, 1999999373L, 1999999423L, 1999999439L, 1999999499L,
1999999553L, 1999999559L, 1999999571L, 1999999609L, 1999999613L,
1999999621L, 1999999643L, 1999999649L, 1999999657L, 1999999747L,
1999999763L, 1999999777L, 1999999811L, 1999999817L, 1999999829L,
1999999853L, 1999999861L, 1999999871L, 1999999873
};
public static void nextProbablePrime() throws Exception {
int failCount = 0;
BigInteger p1, p2, p3;
p1 = p2 = p3 = ZERO;
// First test nextProbablePrime on the low range starting at zero
for (int i=0; i<primesTo100.length; i++) {
p1 = p1.nextProbablePrime();
if (p1.longValue() != primesTo100[i]) {
System.err.println("low range primes failed");
System.err.println("p1 is "+p1);
System.err.println("expected "+primesTo100[i]);
failCount++;
}
}
// Test nextProbablePrime on a relatively small, known prime sequence
p1 = BigInteger.valueOf(aPrimeSequence[0]);
for (int i=1; i<aPrimeSequence.length; i++) {
p1 = p1.nextProbablePrime();
if (p1.longValue() != aPrimeSequence[i]) {
System.err.println("prime sequence failed");
failCount++;
}
}
// Next, pick some large primes, use nextProbablePrime to find the
// next one, and make sure there are no primes in between
for (int i=0; i<100; i+=10) {
p1 = BigInteger.probablePrime(50 + i, rnd);
p2 = p1.add(ONE);
p3 = p1.nextProbablePrime();
while(p2.compareTo(p3) < 0) {
if (p2.isProbablePrime(100)){
System.err.println("nextProbablePrime failed");
System.err.println("along range "+p1.toString(16));
System.err.println("to "+p3.toString(16));
failCount++;
break;
}
p2 = p2.add(ONE);
}
}
report("nextProbablePrime", failCount);
}
public static void serialize() throws Exception {
int failCount = 0;
String bitPatterns[] = {
"ffffffff00000000ffffffff00000000ffffffff00000000",
"ffffffffffffffffffffffff000000000000000000000000",
"ffffffff0000000000000000000000000000000000000000",
"10000000ffffffffffffffffffffffffffffffffffffffff",
"100000000000000000000000000000000000000000000000",
"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
"-ffffffff00000000ffffffff00000000ffffffff00000000",
"-ffffffffffffffffffffffff000000000000000000000000",
"-ffffffff0000000000000000000000000000000000000000",
"-10000000ffffffffffffffffffffffffffffffffffffffff",
"-100000000000000000000000000000000000000000000000",
"-aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"
};
for(int i = 0; i < bitPatterns.length; i++) {
BigInteger b1 = new BigInteger(bitPatterns[i], 16);
BigInteger b2 = null;
File f = new File("serialtest");
try (FileOutputStream fos = new FileOutputStream(f)) {
try (ObjectOutputStream oos = new ObjectOutputStream(fos)) {
oos.writeObject(b1);
oos.flush();
}
try (FileInputStream fis = new FileInputStream(f);
ObjectInputStream ois = new ObjectInputStream(fis))
{
b2 = (BigInteger)ois.readObject();
}
if (!b1.equals(b2) ||
!b1.equals(b1.or(b2))) {
failCount++;
System.err.println("Serialized failed for hex " +
b1.toString(16));
}
}
f.delete();
}
for(int i=0; i<10; i++) {
BigInteger b1 = fetchNumber(rnd.nextInt(100));
BigInteger b2 = null;
File f = new File("serialtest");
try (FileOutputStream fos = new FileOutputStream(f)) {
try (ObjectOutputStream oos = new ObjectOutputStream(fos)) {
oos.writeObject(b1);
oos.flush();
}
try (FileInputStream fis = new FileInputStream(f);
ObjectInputStream ois = new ObjectInputStream(fis))
{
b2 = (BigInteger)ois.readObject();
}
}
if (!b1.equals(b2) ||
!b1.equals(b1.or(b2)))
failCount++;
f.delete();
}
report("Serialize", failCount);
}
/**
* Main to interpret arguments and run several tests.
*
* Up to three arguments may be given to specify the SIZE of BigIntegers
* used for call parameters 1, 2, and 3. The SIZE is interpreted as
* the maximum number of decimal digits that the parameters will have.
*
*/
public static void main(String[] args) throws Exception {
// Some variables for sizing test numbers in bits
int order1 = ORDER_MEDIUM;
int order2 = ORDER_SMALL;
int order3 = ORDER_KARATSUBA;
int order4 = ORDER_TOOM_COOK;
if (args.length >0)
order1 = (int)((Integer.parseInt(args[0]))* 3.333);
if (args.length >1)
order2 = (int)((Integer.parseInt(args[1]))* 3.333);
if (args.length >2)
order3 = (int)((Integer.parseInt(args[2]))* 3.333);
if (args.length >3)
order4 = (int)((Integer.parseInt(args[3]))* 3.333);
prime();
nextProbablePrime();
arithmetic(order1); // small numbers
arithmetic(order3); // Karatsuba range
arithmetic(order4); // Toom-Cook / Burnikel-Ziegler range
divideAndRemainder(order1); // small numbers
divideAndRemainder(order3); // Karatsuba range
divideAndRemainder(order4); // Toom-Cook / Burnikel-Ziegler range
pow(order1);
pow(order3);
pow(order4);
square(ORDER_MEDIUM);
square(ORDER_KARATSUBA_SQUARE);
square(ORDER_TOOM_COOK_SQUARE);
bitCount();
bitLength();
bitOps(order1);
bitwise(order1);
shift(order1);
byteArrayConv(order1);
modInv(order1); // small numbers
modInv(order3); // Karatsuba range
modInv(order4); // Toom-Cook / Burnikel-Ziegler range
modExp(order1, order2);
modExp2(order1);
stringConv();
serialize();
multiplyLarge();
squareLarge();
divideLarge();
if (failure)
throw new RuntimeException("Failure in BigIntegerTest.");
}
/*
* Get a random or boundary-case number. This is designed to provide
* a lot of numbers that will find failure points, such as max sized
* numbers, empty BigIntegers, etc.
*
* If order is less than 2, order is changed to 2.
*/
private static BigInteger fetchNumber(int order) {
boolean negative = rnd.nextBoolean();
int numType = rnd.nextInt(7);
BigInteger result = null;
if (order < 2) order = 2;
switch (numType) {
case 0: // Empty
result = BigInteger.ZERO;
break;
case 1: // One
result = BigInteger.ONE;
break;
case 2: // All bits set in number
int numBytes = (order+7)/8;
byte[] fullBits = new byte[numBytes];
for(int i=0; i<numBytes; i++)
fullBits[i] = (byte)0xff;
int excessBits = 8*numBytes - order;
fullBits[0] &= (1 << (8-excessBits)) - 1;
result = new BigInteger(1, fullBits);
break;
case 3: // One bit in number
result = BigInteger.ONE.shiftLeft(rnd.nextInt(order));
break;
case 4: // Random bit density
byte[] val = new byte[(order+7)/8];
int iterations = rnd.nextInt(order);
for (int i=0; i<iterations; i++) {
int bitIdx = rnd.nextInt(order);
val[bitIdx/8] |= 1 << (bitIdx%8);
}
result = new BigInteger(1, val);
break;
case 5: // Runs of consecutive ones and zeros
result = ZERO;
int remaining = order;
int bit = rnd.nextInt(2);
while (remaining > 0) {
int runLength = Math.min(remaining, rnd.nextInt(order));
result = result.shiftLeft(runLength);
if (bit > 0)
result = result.add(ONE.shiftLeft(runLength).subtract(ONE));
remaining -= runLength;
bit = 1 - bit;
}
break;
default: // random bits
result = new BigInteger(order, rnd);
}
if (negative)
result = result.negate();
return result;
}
static void report(String testName, int failCount) {
System.err.println(testName+": " +
(failCount==0 ? "Passed":"Failed("+failCount+")"));
if (failCount > 0)
failure = true;
}
}
Raw
MutableBigInteger.java.phase3
/*
* Copyright (c) 1999, 2013, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
* by Oracle in the LICENSE file that accompanied this code.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
package java.math;
/**
* A class used to represent multiprecision integers that makes efficient
* use of allocated space by allowing a number to occupy only part of
* an array so that the arrays do not have to be reallocated as often.
* When performing an operation with many iterations the array used to
* hold a number is only reallocated when necessary and does not have to
* be the same size as the number it represents. A mutable number allows
* calculations to occur on the same number without having to create
* a new number for every step of the calculation as occurs with
* BigIntegers.
*
* @see BigInteger
* @author Michael McCloskey
* @author Timothy Buktu
* @since 1.3
*/
import static java.math.BigDecimal.INFLATED;
import static java.math.BigInteger.LONG_MASK;
import java.util.Arrays;
class MutableBigInteger {
/**
* Holds the magnitude of this MutableBigInteger in big endian order.
* The magnitude may start at an offset into the value array, and it may
* end before the length of the value array.
*/
int[] value;
/**
* The number of ints of the value array that are currently used
* to hold the magnitude of this MutableBigInteger. The magnitude starts
* at an offset and offset + intLen may be less than value.length.
*/
int intLen;
/**
* The offset into the value array where the magnitude of this
* MutableBigInteger begins.
*/
int offset = 0;
// Constants
/**
* MutableBigInteger with one element value array with the value 1. Used by
* BigDecimal divideAndRound to increment the quotient. Use this constant
* only when the method is not going to modify this object.
*/
static final MutableBigInteger ONE = new MutableBigInteger(1);
/**
* The minimum {@code intLen} for cancelling powers of two before
* dividing.
* If the number of ints is less than this threshold,
* {@code divideKnuth} does not eliminate common powers of two from
* the dividend and divisor.
*/
static final int KNUTH_POW2_THRESH_LEN = 6;
/**
* The minimum number of trailing zero ints for cancelling powers of two
* before dividing.
* If the dividend and divisor don't share at least this many zero ints
* at the end, {@code divideKnuth} does not eliminate common powers
* of two from the dividend and divisor.
*/
static final int KNUTH_POW2_THRESH_ZEROS = 3;
// Constructors
/**
* The default constructor. An empty MutableBigInteger is created with
* a one word capacity.
*/
MutableBigInteger() {
value = new int[1];
intLen = 0;
}
/**
* Construct a new MutableBigInteger with a magnitude specified by
* the int val.
*/
MutableBigInteger(int val) {
value = new int[1];
intLen = 1;
value[0] = val;
}
/**
* Construct a new MutableBigInteger with the specified value array
* up to the length of the array supplied.
*/
MutableBigInteger(int[] val) {
value = val;
intLen = val.length;
}
/**
* Construct a new MutableBigInteger with a magnitude equal to the
* specified BigInteger.
*/
MutableBigInteger(BigInteger b) {
intLen = b.mag.length;
value = Arrays.copyOf(b.mag, intLen);
}
/**
* Construct a new MutableBigInteger with a magnitude equal to the
* specified MutableBigInteger.
*/
MutableBigInteger(MutableBigInteger val) {
intLen = val.intLen;
value = Arrays.copyOfRange(val.value, val.offset, val.offset + intLen);
}
/**
* Makes this number an {@code n}-int number all of whose bits are ones.
* Used by Burnikel-Ziegler division.
* @param n number of ints in the {@code value} array
* @return a number equal to {@code ((1<<(32*n)))-1}
*/
private void ones(int n) {
if (n > value.length)
value = new int[n];
Arrays.fill(value, -1);
offset = 0;
intLen = n;
}
/**
* Internal helper method to return the magnitude array. The caller is not
* supposed to modify the returned array.
*/
private int[] getMagnitudeArray() {
if (offset > 0 || value.length != intLen)
return Arrays.copyOfRange(value, offset, offset + intLen);
return value;
}
/**
* Convert this MutableBigInteger to a long value. The caller has to make
* sure this MutableBigInteger can be fit into long.
*/
private long toLong() {
assert (intLen <= 2) : "this MutableBigInteger exceeds the range of long";
if (intLen == 0)
return 0;
long d = value[offset] & LONG_MASK;
return (intLen == 2) ? d << 32 | (value[offset + 1] & LONG_MASK) : d;
}
/**
* Convert this MutableBigInteger to a BigInteger object.
*/
BigInteger toBigInteger(int sign) {
if (intLen == 0 || sign == 0)
return BigInteger.ZERO;
return new BigInteger(getMagnitudeArray(), sign);
}
/**
* Converts this number to a nonnegative {@code BigInteger}.
*/
BigInteger toBigInteger() {
normalize();
return toBigInteger(isZero() ? 0 : 1);
}
/**
* Convert this MutableBigInteger to BigDecimal object with the specified sign
* and scale.
*/
BigDecimal toBigDecimal(int sign, int scale) {
if (intLen == 0 || sign == 0)
return BigDecimal.zeroValueOf(scale);
int[] mag = getMagnitudeArray();
int len = mag.length;
int d = mag[0];
// If this MutableBigInteger can't be fit into long, we need to
// make a BigInteger object for the resultant BigDecimal object.
if (len > 2 || (d < 0 && len == 2))
return new BigDecimal(new BigInteger(mag, sign), INFLATED, scale, 0);
long v = (len == 2) ?
((mag[1] & LONG_MASK) | (d & LONG_MASK) << 32) :
d & LONG_MASK;
return BigDecimal.valueOf(sign == -1 ? -v : v, scale);
}
/**
* This is for internal use in converting from a MutableBigInteger
* object into a long value given a specified sign.
* returns INFLATED if value is not fit into long
*/
long toCompactValue(int sign) {
if (intLen == 0 || sign == 0)
return 0L;
int[] mag = getMagnitudeArray();
int len = mag.length;
int d = mag[0];
// If this MutableBigInteger can not be fitted into long, we need to
// make a BigInteger object for the resultant BigDecimal object.
if (len > 2 || (d < 0 && len == 2))
return INFLATED;
long v = (len == 2) ?
((mag[1] & LONG_MASK) | (d & LONG_MASK) << 32) :
d & LONG_MASK;
return sign == -1 ? -v : v;
}
/**
* Clear out a MutableBigInteger for reuse.
*/
void clear() {
offset = intLen = 0;
for (int index=0, n=value.length; index < n; index++)
value[index] = 0;
}
/**
* Set a MutableBigInteger to zero, removing its offset.
*/
void reset() {
offset = intLen = 0;
}
/**
* Compare the magnitude of two MutableBigIntegers. Returns -1, 0 or 1
* as this MutableBigInteger is numerically less than, equal to, or
* greater than <tt>b</tt>.
*/
final int compare(MutableBigInteger b) {
int blen = b.intLen;
if (intLen < blen)
return -1;
if (intLen > blen)
return 1;
// Add Integer.MIN_VALUE to make the comparison act as unsigned integer
// comparison.
int[] bval = b.value;
for (int i = offset, j = b.offset; i < intLen + offset; i++, j++) {
int b1 = value[i] + 0x80000000;
int b2 = bval[j] + 0x80000000;
if (b1 < b2)
return -1;
if (b1 > b2)
return 1;
}
return 0;
}
/**
* Returns a value equal to what {@code b.leftShift(32*ints); return compare(b);}
* would return, but doesn't change the value of {@code b}.
*/
private int compareShifted(MutableBigInteger b, int ints) {
int blen = b.intLen;
int alen = intLen - ints;
if (alen < blen)
return -1;
if (alen > blen)
return 1;
// Add Integer.MIN_VALUE to make the comparison act as unsigned integer
// comparison.
int[] bval = b.value;
for (int i = offset, j = b.offset; i < alen + offset; i++, j++) {
int b1 = value[i] + 0x80000000;
int b2 = bval[j] + 0x80000000;
if (b1 < b2)
return -1;
if (b1 > b2)
return 1;
}
return 0;
}
/**
* Compare this against half of a MutableBigInteger object (Needed for
* remainder tests).
* Assumes no leading unnecessary zeros, which holds for results
* from divide().
*/
final int compareHalf(MutableBigInteger b) {
int blen = b.intLen;
int len = intLen;
if (len <= 0)
return blen <= 0 ? 0 : -1;
if (len > blen)
return 1;
if (len < blen - 1)
return -1;
int[] bval = b.value;
int bstart = 0;
int carry = 0;
// Only 2 cases left:len == blen or len == blen - 1
if (len != blen) { // len == blen - 1
if (bval[bstart] == 1) {
++bstart;
carry = 0x80000000;
} else
return -1;
}
// compare values with right-shifted values of b,
// carrying shifted-out bits across words
int[] val = value;
for (int i = offset, j = bstart; i < len + offset;) {
int bv = bval[j++];
long hb = ((bv >>> 1) + carry) & LONG_MASK;
long v = val[i++] & LONG_MASK;
if (v != hb)
return v < hb ? -1 : 1;
carry = (bv & 1) << 31; // carray will be either 0x80000000 or 0
}
return carry == 0 ? 0 : -1;
}
/**
* Return the index of the lowest set bit in this MutableBigInteger. If the
* magnitude of this MutableBigInteger is zero, -1 is returned.
*/
private final int getLowestSetBit() {
if (intLen == 0)
return -1;
int j, b;
for (j=intLen-1; (j > 0) && (value[j+offset] == 0); j--)
;
b = value[j+offset];
if (b == 0)
return -1;
return ((intLen-1-j)<<5) + Integer.numberOfTrailingZeros(b);
}
/**
* Return the int in use in this MutableBigInteger at the specified
* index. This method is not used because it is not inlined on all
* platforms.
*/
private final int getInt(int index) {
return value[offset+index];
}
/**
* Return a long which is equal to the unsigned value of the int in
* use in this MutableBigInteger at the specified index. This method is
* not used because it is not inlined on all platforms.
*/
private final long getLong(int index) {
return value[offset+index] & LONG_MASK;
}
/**
* Ensure that the MutableBigInteger is in normal form, specifically
* making sure that there are no leading zeros, and that if the
* magnitude is zero, then intLen is zero.
*/
final void normalize() {
if (intLen == 0) {
offset = 0;
return;
}
int index = offset;
if (value[index] != 0)
return;
int indexBound = index+intLen;
do {
index++;
} while(index < indexBound && value[index] == 0);
int numZeros = index - offset;
intLen -= numZeros;
offset = (intLen == 0 ? 0 : offset+numZeros);
}
/**
* If this MutableBigInteger cannot hold len words, increase the size
* of the value array to len words.
*/
private final void ensureCapacity(int len) {
if (value.length < len) {
value = new int[len];
offset = 0;
intLen = len;
}
}
/**
* Convert this MutableBigInteger into an int array with no leading
* zeros, of a length that is equal to this MutableBigInteger's intLen.
*/
int[] toIntArray() {
int[] result = new int[intLen];
for(int i=0; i < intLen; i++)
result[i] = value[offset+i];
return result;
}
/**
* Sets the int at index+offset in this MutableBigInteger to val.
* This does not get inlined on all platforms so it is not used
* as often as originally intended.
*/
void setInt(int index, int val) {
value[offset + index] = val;
}
/**
* Sets this MutableBigInteger's value array to the specified array.
* The intLen is set to the specified length.
*/
void setValue(int[] val, int length) {
value = val;
intLen = length;
offset = 0;
}
/**
* Sets this MutableBigInteger's value array to a copy of the specified
* array. The intLen is set to the length of the new array.
*/
void copyValue(MutableBigInteger src) {
int len = src.intLen;
if (value.length < len)
value = new int[len];
System.arraycopy(src.value, src.offset, value, 0, len);
intLen = len;
offset = 0;
}
/**
* Sets this MutableBigInteger's value array to a copy of the specified
* array. The intLen is set to the length of the specified array.
*/
void copyValue(int[] val) {
int len = val.length;
if (value.length < len)
value = new int[len];
System.arraycopy(val, 0, value, 0, len);
intLen = len;
offset = 0;
}
/**
* Returns true iff this MutableBigInteger has a value of one.
*/
boolean isOne() {
return (intLen == 1) && (value[offset] == 1);
}
/**
* Returns true iff this MutableBigInteger has a value of zero.
*/
boolean isZero() {
return (intLen == 0);
}
/**
* Returns true iff this MutableBigInteger is even.
*/
boolean isEven() {
return (intLen == 0) || ((value[offset + intLen - 1] & 1) == 0);
}
/**
* Returns true iff this MutableBigInteger is odd.
*/
boolean isOdd() {
return isZero() ? false : ((value[offset + intLen - 1] & 1) == 1);
}
/**
* Returns true iff this MutableBigInteger is in normal form. A
* MutableBigInteger is in normal form if it has no leading zeros
* after the offset, and intLen + offset <= value.length.
*/
boolean isNormal() {
if (intLen + offset > value.length)
return false;
if (intLen == 0)
return true;
return (value[offset] != 0);
}
/**
* Returns a String representation of this MutableBigInteger in radix 10.
*/
public String toString() {
BigInteger b = toBigInteger(1);
return b.toString();
}
/**
* Like {@link #rightShift(int)} but {@code n} can be greater than the length of the number.
*/
void safeRightShift(int n) {
if (n/32 >= intLen) {
reset();
} else {
rightShift(n);
}
}
/**
* Right shift this MutableBigInteger n bits. The MutableBigInteger is left
* in normal form.
*/
void rightShift(int n) {
if (intLen == 0)
return;
int nInts = n >>> 5;
int nBits = n & 0x1F;
this.intLen -= nInts;
if (nBits == 0)
return;
int bitsInHighWord = BigInteger.bitLengthForInt(value[offset]);
if (nBits >= bitsInHighWord) {
this.primitiveLeftShift(32 - nBits);
this.intLen--;
} else {
primitiveRightShift(nBits);
}
}
/**
* Like {@link #leftShift(int)} but {@code n} can be zero.
*/
void safeLeftShift(int n) {
if (n > 0) {
leftShift(n);
}
}
/**
* Left shift this MutableBigInteger n bits.
*/
void leftShift(int n) {
/*
* If there is enough storage space in this MutableBigInteger already
* the available space will be used. Space to the right of the used
* ints in the value array is faster to utilize, so the extra space
* will be taken from the right if possible.
*/
if (intLen == 0)
return;
int nInts = n >>> 5;
int nBits = n&0x1F;
int bitsInHighWord = BigInteger.bitLengthForInt(value[offset]);
// If shift can be done without moving words, do so
if (n <= (32-bitsInHighWord)) {
primitiveLeftShift(nBits);
return;
}
int newLen = intLen + nInts +1;
if (nBits <= (32-bitsInHighWord))
newLen--;
if (value.length < newLen) {
// The array must grow
int[] result = new int[newLen];
for (int i=0; i < intLen; i++)
result[i] = value[offset+i];
setValue(result, newLen);
} else if (value.length - offset >= newLen) {
// Use space on right
for(int i=0; i < newLen - intLen; i++)
value[offset+intLen+i] = 0;
} else {
// Must use space on left
for (int i=0; i < intLen; i++)
value[i] = value[offset+i];
for (int i=intLen; i < newLen; i++)
value[i] = 0;
offset = 0;
}
intLen = newLen;
if (nBits == 0)
return;
if (nBits <= (32-bitsInHighWord))
primitiveLeftShift(nBits);
else
primitiveRightShift(32 -nBits);
}
/**
* A primitive used for division. This method adds in one multiple of the
* divisor a back to the dividend result at a specified offset. It is used
* when qhat was estimated too large, and must be adjusted.
*/
private int divadd(int[] a, int[] result, int offset) {
long carry = 0;
for (int j=a.length-1; j >= 0; j--) {
long sum = (a[j] & LONG_MASK) +
(result[j+offset] & LONG_MASK) + carry;
result[j+offset] = (int)sum;
carry = sum >>> 32;
}
return (int)carry;
}
/**
* This method is used for division. It multiplies an n word input a by one
* word input x, and subtracts the n word product from q. This is needed
* when subtracting qhat*divisor from dividend.
*/
private int mulsub(int[] q, int[] a, int x, int len, int offset) {
long xLong = x & LONG_MASK;
long carry = 0;
offset += len;
for (int j=len-1; j >= 0; j--) {
long product = (a[j] & LONG_MASK) * xLong + carry;
long difference = q[offset] - product;
q[offset--] = (int)difference;
carry = (product >>> 32)
+ (((difference & LONG_MASK) >
(((~(int)product) & LONG_MASK))) ? 1:0);
}
return (int)carry;
}
/**
* The method is the same as mulsun, except the fact that q array is not
* updated, the only result of the method is borrow flag.
*/
private int mulsubBorrow(int[] q, int[] a, int x, int len, int offset) {
long xLong = x & LONG_MASK;
long carry = 0;
offset += len;
for (int j=len-1; j >= 0; j--) {
long product = (a[j] & LONG_MASK) * xLong + carry;
long difference = q[offset--] - product;
carry = (product >>> 32)
+ (((difference & LONG_MASK) >
(((~(int)product) & LONG_MASK))) ? 1:0);
}
return (int)carry;
}
/**
* Right shift this MutableBigInteger n bits, where n is
* less than 32.
* Assumes that intLen > 0, n > 0 for speed
*/
private final void primitiveRightShift(int n) {
int[] val = value;
int n2 = 32 - n;
for (int i=offset+intLen-1, c=val[i]; i > offset; i--) {
int b = c;
c = val[i-1];
val[i] = (c << n2) | (b >>> n);
}
val[offset] >>>= n;
}
/**
* Left shift this MutableBigInteger n bits, where n is
* less than 32.
* Assumes that intLen > 0, n > 0 for speed
*/
private final void primitiveLeftShift(int n) {
int[] val = value;
int n2 = 32 - n;
for (int i=offset, c=val[i], m=i+intLen-1; i < m; i++) {
int b = c;
c = val[i+1];
val[i] = (b << n) | (c >>> n2);
}
val[offset+intLen-1] <<= n;
}
/**
* Returns a {@code BigInteger} equal to the {@code n}
* low ints of this number.
*/
private BigInteger getLower(int n) {
if (isZero()) {
return BigInteger.ZERO;
} else if (intLen < n) {
return toBigInteger(1);
} else {
// strip zeros
int len = n;
while (len > 0 && value[offset+intLen-len] == 0)
len--;
int sign = len > 0 ? 1 : 0;
return new BigInteger(Arrays.copyOfRange(value, offset+intLen-len, offset+intLen), sign);
}
}
/**
* Discards all ints whose index is greater than {@code n}.
*/
private void keepLower(int n) {
if (intLen >= n) {
offset += intLen - n;
intLen = n;
}
}
/**
* Adds the contents of two MutableBigInteger objects.The result
* is placed within this MutableBigInteger.
* The contents of the addend are not changed.
*/
void add(MutableBigInteger addend) {
int x = intLen;
int y = addend.intLen;
int resultLen = (intLen > addend.intLen ? intLen : addend.intLen);
int[] result = (value.length < resultLen ? new int[resultLen] : value);
int rstart = result.length-1;
long sum;
long carry = 0;
// Add common parts of both numbers
while(x > 0 && y > 0) {
x--; y--;
sum = (value[x+offset] & LONG_MASK) +
(addend.value[y+addend.offset] & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
// Add remainder of the longer number
while(x > 0) {
x--;
if (carry == 0 && result == value && rstart == (x + offset))
return;
sum = (value[x+offset] & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
while(y > 0) {
y--;
sum = (addend.value[y+addend.offset] & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
if (carry > 0) { // Result must grow in length
resultLen++;
if (result.length < resultLen) {
int temp[] = new int[resultLen];
// Result one word longer from carry-out; copy low-order
// bits into new result.
System.arraycopy(result, 0, temp, 1, result.length);
temp[0] = 1;
result = temp;
} else {
result[rstart--] = 1;
}
}
value = result;
intLen = resultLen;
offset = result.length - resultLen;
}
/**
* Adds the value of {@code addend} shifted {@code n} ints to the left.
* Has the same effect as {@code addend.leftShift(32*ints); add(addend);}
* but doesn't change the value of {@code addend}.
*/
void addShifted(MutableBigInteger addend, int n) {
if (addend.isZero()) {
return;
}
int x = intLen;
int y = addend.intLen + n;
int resultLen = (intLen > y ? intLen : y);
int[] result = (value.length < resultLen ? new int[resultLen] : value);
int rstart = result.length-1;
long sum;
long carry = 0;
// Add common parts of both numbers
while (x > 0 && y > 0) {
x--; y--;
int bval = y+addend.offset < addend.value.length ? addend.value[y+addend.offset] : 0;
sum = (value[x+offset] & LONG_MASK) +
(bval & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
// Add remainder of the longer number
while (x > 0) {
x--;
if (carry == 0 && result == value && rstart == (x + offset)) {
return;
}
sum = (value[x+offset] & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
while (y > 0) {
y--;
int bval = y+addend.offset < addend.value.length ? addend.value[y+addend.offset] : 0;
sum = (bval & LONG_MASK) + carry;
result[rstart--] = (int)sum;
carry = sum >>> 32;
}
if (carry > 0) { // Result must grow in length
resultLen++;
if (result.length < resultLen) {
int temp[] = new int[resultLen];
// Result one word longer from carry-out; copy low-order
// bits into new result.
System.arraycopy(result, 0, temp, 1, result.length);
temp[0] = 1;
result = temp;
} else {
result[rstart--] = 1;
}
}
value = result;
intLen = resultLen;
offset = result.length - resultLen;
}
/**
* Like {@link #addShifted(MutableBigInteger, int)} but {@code this.intLen} must
* not be greater than {@code n}. In other words, concatenates {@code this}
* and {@code addend}.
*/
void addDisjoint(MutableBigInteger addend, int n) {
if (addend.isZero())
return;
int x = intLen;
int y = addend.intLen + n;
int resultLen = (intLen > y ? intLen : y);
int[] result;
if (value.length < resultLen)
result = new int[resultLen];
else {
result = value;
Arrays.fill(value, offset+intLen, value.length, 0);
}
int rstart = result.length-1;
// copy from this if needed
System.arraycopy(value, offset, result, rstart+1-x, x);
y -= x;
rstart -= x;
int len = Math.min(y, addend.value.length-addend.offset);
System.arraycopy(addend.value, addend.offset, result, rstart+1-y, len);
// zero the gap
for (int i=rstart+1-y+len; i < rstart+1; i++)
result[i] = 0;
value = result;
intLen = resultLen;
offset = result.length - resultLen;
}
/**
* Adds the low {@code n} ints of {@code addend}.
*/
void addLower(MutableBigInteger addend, int n) {
MutableBigInteger a = new MutableBigInteger(addend);
if (a.offset + a.intLen >= n) {
a.offset = a.offset + a.intLen - n;
a.intLen = n;
}
a.normalize();
add(a);
}
/**
* Subtracts the smaller of this and b from the larger and places the
* result into this MutableBigInteger.
*/
int subtract(MutableBigInteger b) {
MutableBigInteger a = this;
int[] result = value;
int sign = a.compare(b);
if (sign == 0) {
reset();
return 0;
}
if (sign < 0) {
MutableBigInteger tmp = a;
a = b;
b = tmp;
}
int resultLen = a.intLen;
if (result.length < resultLen)
result = new int[resultLen];
long diff = 0;
int x = a.intLen;
int y = b.intLen;
int rstart = result.length - 1;
// Subtract common parts of both numbers
while (y > 0) {
x--; y--;
diff = (a.value[x+a.offset] & LONG_MASK) -
(b.value[y+b.offset] & LONG_MASK) - ((int)-(diff>>32));
result[rstart--] = (int)diff;
}
// Subtract remainder of longer number
while (x > 0) {
x--;
diff = (a.value[x+a.offset] & LONG_MASK) - ((int)-(diff>>32));
result[rstart--] = (int)diff;
}
value = result;
intLen = resultLen;
offset = value.length - resultLen;
normalize();
return sign;
}
/**
* Subtracts the smaller of a and b from the larger and places the result
* into the larger. Returns 1 if the answer is in a, -1 if in b, 0 if no
* operation was performed.
*/
private int difference(MutableBigInteger b) {
MutableBigInteger a = this;
int sign = a.compare(b);
if (sign == 0)
return 0;
if (sign < 0) {
MutableBigInteger tmp = a;
a = b;
b = tmp;
}
long diff = 0;
int x = a.intLen;
int y = b.intLen;
// Subtract common parts of both numbers
while (y > 0) {
x--; y--;
diff = (a.value[a.offset+ x] & LONG_MASK) -
(b.value[b.offset+ y] & LONG_MASK) - ((int)-(diff>>32));
a.value[a.offset+x] = (int)diff;
}
// Subtract remainder of longer number
while (x > 0) {
x--;
diff = (a.value[a.offset+ x] & LONG_MASK) - ((int)-(diff>>32));
a.value[a.offset+x] = (int)diff;
}
a.normalize();
return sign;
}
/**
* Multiply the contents of two MutableBigInteger objects. The result is
* placed into MutableBigInteger z. The contents of y are not changed.
*/
void multiply(MutableBigInteger y, MutableBigInteger z) {
int xLen = intLen;
int yLen = y.intLen;
int newLen = xLen + yLen;
// Put z into an appropriate state to receive product
if (z.value.length < newLen)
z.value = new int[newLen];
z.offset = 0;
z.intLen = newLen;
// The first iteration is hoisted out of the loop to avoid extra add
long carry = 0;
for (int j=yLen-1, k=yLen+xLen-1; j >= 0; j--, k--) {
long product = (y.value[j+y.offset] & LONG_MASK) *
(value[xLen-1+offset] & LONG_MASK) + carry;
z.value[k] = (int)product;
carry = product >>> 32;
}
z.value[xLen-1] = (int)carry;
// Perform the multiplication word by word
for (int i = xLen-2; i >= 0; i--) {
carry = 0;
for (int j=yLen-1, k=yLen+i; j >= 0; j--, k--) {
long product = (y.value[j+y.offset] & LONG_MASK) *
(value[i+offset] & LONG_MASK) +
(z.value[k] & LONG_MASK) + carry;
z.value[k] = (int)product;
carry = product >>> 32;
}
z.value[i] = (int)carry;
}
// Remove leading zeros from product
z.normalize();
}
/**
* Multiply the contents of this MutableBigInteger by the word y. The
* result is placed into z.
*/
void mul(int y, MutableBigInteger z) {
if (y == 1) {
z.copyValue(this);
return;
}
if (y == 0) {
z.clear();
return;
}
// Perform the multiplication word by word
long ylong = y & LONG_MASK;
int[] zval = (z.value.length < intLen+1 ? new int[intLen + 1]
: z.value);
long carry = 0;
for (int i = intLen-1; i >= 0; i--) {
long product = ylong * (value[i+offset] & LONG_MASK) + carry;
zval[i+1] = (int)product;
carry = product >>> 32;
}
if (carry == 0) {
z.offset = 1;
z.intLen = intLen;
} else {
z.offset = 0;
z.intLen = intLen + 1;
zval[0] = (int)carry;
}
z.value = zval;
}
/**
* This method is used for division of an n word dividend by a one word
* divisor. The quotient is placed into quotient. The one word divisor is
* specified by divisor.
*
* @return the remainder of the division is returned.
*
*/
int divideOneWord(int divisor, MutableBigInteger quotient) {
long divisorLong = divisor & LONG_MASK;
// Special case of one word dividend
if (intLen == 1) {
long dividendValue = value[offset] & LONG_MASK;
int q = (int) (dividendValue / divisorLong);
int r = (int) (dividendValue - q * divisorLong);
quotient.value[0] = q;
quotient.intLen = (q == 0) ? 0 : 1;
quotient.offset = 0;
return r;
}
if (quotient.value.length < intLen)
quotient.value = new int[intLen];
quotient.offset = 0;
quotient.intLen = intLen;
// Normalize the divisor
int shift = Integer.numberOfLeadingZeros(divisor);
int rem = value[offset];
long remLong = rem & LONG_MASK;
if (remLong < divisorLong) {
quotient.value[0] = 0;
} else {
quotient.value[0] = (int)(remLong / divisorLong);
rem = (int) (remLong - (quotient.value[0] * divisorLong));
remLong = rem & LONG_MASK;
}
int xlen = intLen;
while (--xlen > 0) {
long dividendEstimate = (remLong << 32) |
(value[offset + intLen - xlen] & LONG_MASK);
int q;
if (dividendEstimate >= 0) {
q = (int) (dividendEstimate / divisorLong);
rem = (int) (dividendEstimate - q * divisorLong);
} else {
long tmp = divWord(dividendEstimate, divisor);
q = (int) (tmp & LONG_MASK);
rem = (int) (tmp >>> 32);
}
quotient.value[intLen - xlen] = q;
remLong = rem & LONG_MASK;
}
quotient.normalize();
// Unnormalize
if (shift > 0)
return rem % divisor;
else
return rem;
}
/**
* Calculates the quotient of this div b and places the quotient in the
* provided MutableBigInteger objects and the remainder object is returned.
*
*/
MutableBigInteger divide(MutableBigInteger b, MutableBigInteger quotient) {
return divide(b,quotient,true);
}
MutableBigInteger divide(MutableBigInteger b, MutableBigInteger quotient, boolean needRemainder) {
if (intLen < BigInteger.BURNIKEL_ZIEGLER_THRESHOLD ||
b.intLen < BigInteger.BURNIKEL_ZIEGLER_THRESHOLD) {
return divideKnuth(b, quotient, needRemainder);
} else {
return divideAndRemainderBurnikelZiegler(b, quotient);
}
}
/**
* @see #divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
*/
MutableBigInteger divideKnuth(MutableBigInteger b, MutableBigInteger quotient) {
return divideKnuth(b,quotient,true);
}
/**
* Calculates the quotient of this div b and places the quotient in the
* provided MutableBigInteger objects and the remainder object is returned.
*
* Uses Algorithm D in Knuth section 4.3.1.
* Many optimizations to that algorithm have been adapted from the Colin
* Plumb C library.
* It special cases one word divisors for speed. The content of b is not
* changed.
*
*/
MutableBigInteger divideKnuth(MutableBigInteger b, MutableBigInteger quotient, boolean needRemainder) {
if (b.intLen == 0)
throw new ArithmeticException("BigInteger divide by zero");
// Dividend is zero
if (intLen == 0) {
quotient.intLen = quotient.offset = 0;
return needRemainder ? new MutableBigInteger() : null;
}
int cmp = compare(b);
// Dividend less than divisor
if (cmp < 0) {
quotient.intLen = quotient.offset = 0;
return needRemainder ? new MutableBigInteger(this) : null;
}
// Dividend equal to divisor
if (cmp == 0) {
quotient.value[0] = quotient.intLen = 1;
quotient.offset = 0;
return needRemainder ? new MutableBigInteger() : null;
}
quotient.clear();
// Special case one word divisor
if (b.intLen == 1) {
int r = divideOneWord(b.value[b.offset], quotient);
if(needRemainder) {
if (r == 0)
return new MutableBigInteger();
return new MutableBigInteger(r);
} else {
return null;
}
}
// Cancel common powers of two if we're above the KNUTH_POW2_* thresholds
if (intLen >= KNUTH_POW2_THRESH_LEN) {
int trailingZeroBits = Math.min(getLowestSetBit(), b.getLowestSetBit());
if (trailingZeroBits >= KNUTH_POW2_THRESH_ZEROS*32) {
MutableBigInteger a = new MutableBigInteger(this);
b = new MutableBigInteger(b);
a.rightShift(trailingZeroBits);
b.rightShift(trailingZeroBits);
MutableBigInteger r = a.divideKnuth(b, quotient);
r.leftShift(trailingZeroBits);
return r;
}
}
return divideMagnitude(b, quotient, needRemainder);
}
/**
* Computes {@code this/b} and {@code this%b} using the
* <a href="http://cr.yp.to/bib/1998/burnikel.ps"> Burnikel-Ziegler algorithm</a>.
* This method implements algorithm 3 from pg. 9 of the Burnikel-Ziegler paper.
* The parameter beta was chosen to b 2<sup>32</sup> so almost all shifts are
* multiples of 32 bits.<br/>
* {@code this} and {@code b} must be nonnegative.
* @param b the divisor
* @param quotient output parameter for {@code this/b}
* @return the remainder
*/
MutableBigInteger divideAndRemainderBurnikelZiegler(MutableBigInteger b, MutableBigInteger quotient) {
int r = intLen;
int s = b.intLen;
if (r < s) {
quotient.intLen = quotient.offset = 0;
return this;
} else {
// Unlike Knuth division, we don't check for common powers of two here because
// BZ already runs faster if both numbers contain powers of two and cancelling them has no
// additional benefit.
// step 1: let m = min{2^k | (2^k)*BURNIKEL_ZIEGLER_THRESHOLD > s}
int m = 1 << (32-Integer.numberOfLeadingZeros(s/BigInteger.BURNIKEL_ZIEGLER_THRESHOLD));
int j = (s+m-1) / m; // step 2a: j = ceil(s/m)
int n = j * m; // step 2b: block length in 32-bit units
int n32 = 32 * n; // block length in bits
int sigma = Math.max(0, n32 - b.bitLength()); // step 3: sigma = max{T | (2^T)*B < beta^n}
MutableBigInteger bShifted = new MutableBigInteger(b);
bShifted.safeLeftShift(sigma); // step 4a: shift b so its length is a multiple of n
safeLeftShift(sigma); // step 4b: shift this by the same amount
// step 5: t is the number of blocks needed to accommodate this plus one additional bit
int t = (bitLength()+n32) / n32;
if (t < 2) {
t = 2;
}
// step 6: conceptually split this into blocks a[t-1], ..., a[0]
MutableBigInteger a1 = getBlock(t-1, t, n); // the most significant block of this
// step 7: z[t-2] = [a[t-1], a[t-2]]
MutableBigInteger z = getBlock(t-2, t, n); // the second to most significant block
z.addDisjoint(a1, n); // z[t-2]
// do schoolbook division on blocks, dividing 2-block numbers by 1-block numbers
MutableBigInteger qi = new MutableBigInteger();
MutableBigInteger ri;
quotient.offset = quotient.intLen = 0;
for (int i=t-2; i > 0; i--) {
// step 8a: compute (qi,ri) such that z=b*qi+ri
ri = z.divide2n1n(bShifted, qi);
// step 8b: z = [ri, a[i-1]]
z = getBlock(i-1, t, n); // a[i-1]
z.addDisjoint(ri, n);
quotient.addShifted(qi, i*n); // update q (part of step 9)
}
// final iteration of step 8: do the loop one more time for i=0 but leave z unchanged
ri = z.divide2n1n(bShifted, qi);
quotient.add(qi);
ri.rightShift(sigma); // step 9: this and b were shifted, so shift back
return ri;
}
}
/**
* This method implements algorithm 1 from pg. 4 of the Burnikel-Ziegler paper.
* It divides a 2n-digit number by a n-digit number.<br/>
* The parameter beta is 2<sup>32</sup> so all shifts are multiples of 32 bits.
* <br/>
* {@code this} must be a nonnegative number such that {@code this.bitLength() <= 2*b.bitLength()}
* @param b a positive number such that {@code b.bitLength()} is even
* @param quotient output parameter for {@code this/b}
* @return {@code this%b}
*/
private MutableBigInteger divide2n1n(MutableBigInteger b, MutableBigInteger quotient) {
int n = b.intLen;
// step 1: base case
if (n%2 != 0 || n < BigInteger.BURNIKEL_ZIEGLER_THRESHOLD) {
return divideKnuth(b, quotient);
}
// step 2: view this as [a1,a2,a3,a4] where each ai is n/2 ints or less
MutableBigInteger aUpper = new MutableBigInteger(this);
aUpper.safeRightShift(32*(n/2)); // aUpper = [a1,a2,a3]
keepLower(n/2); // this = a4
// step 3: q1=aUpper/b, r1=aUpper%b
MutableBigInteger q1 = new MutableBigInteger();
MutableBigInteger r1 = aUpper.divide3n2n(b, q1);
// step 4: quotient=[r1,this]/b, r2=[r1,this]%b
addDisjoint(r1, n/2); // this = [r1,this]
MutableBigInteger r2 = divide3n2n(b, quotient);
// step 5: let quotient=[q1,quotient] and return r2
quotient.addDisjoint(q1, n/2);
return r2;
}
/**
* This method implements algorithm 2 from pg. 5 of the Burnikel-Ziegler paper.
* It divides a 3n-digit number by a 2n-digit number.<br/>
* The parameter beta is 2<sup>32</sup> so all shifts are multiples of 32 bits.<br/>
* <br/>
* {@code this} must be a nonnegative number such that {@code 2*this.bitLength() <= 3*b.bitLength()}
* @param quotient output parameter for {@code this/b}
* @return {@code this%b}
*/
private MutableBigInteger divide3n2n(MutableBigInteger b, MutableBigInteger quotient) {
int n = b.intLen / 2; // half the length of b in ints
// step 1: view this as [a1,a2,a3] where each ai is n ints or less; let a12=[a1,a2]
MutableBigInteger a12 = new MutableBigInteger(this);
a12.safeRightShift(32*n);
// step 2: view b as [b1,b2] where each bi is n ints or less
MutableBigInteger b1 = new MutableBigInteger(b);
b1.safeRightShift(n * 32);
BigInteger b2 = b.getLower(n);
MutableBigInteger r;
MutableBigInteger d;
if (compareShifted(b, n) < 0) {
// step 3a: if a1<b1, let quotient=a12/b1 and r=a12%b1
r = a12.divide2n1n(b1, quotient);
// step 4: d=quotient*b2
d = new MutableBigInteger(quotient.toBigInteger().multiply(b2));
} else {
// step 3b: if a1>=b1, let quotient=beta^n-1 and r=a12-b1*2^n+b1
quotient.ones(n);
a12.add(b1);
b1.leftShift(32*n);
a12.subtract(b1);
r = a12;
// step 4: d=quotient*b2=(b2 << 32*n) - b2
d = new MutableBigInteger(b2);
d.leftShift(32 * n);
d.subtract(new MutableBigInteger(b2));
}
// step 5: r = r*beta^n + a3 - d (paper says a4)
// However, don't subtract d until after the while loop so r doesn't become negative
r.leftShift(32 * n);
r.addLower(this, n);
// step 6: add b until r>=d
while (r.compare(d) < 0) {
r.add(b);
quotient.subtract(MutableBigInteger.ONE);
}
r.subtract(d);
return r;
}
/**
* Returns a {@code MutableBigInteger} containing {@code blockLength} ints from
* {@code this} number, starting at {@code index*blockLength}.<br/>
* Used by Burnikel-Ziegler division.
* @param index the block index
* @param numBlocks the total number of blocks in {@code this} number
* @param blockLength length of one block in units of 32 bits
* @return
*/
private MutableBigInteger getBlock(int index, int numBlocks, int blockLength) {
int blockStart = index * blockLength;
if (blockStart >= intLen) {
return new MutableBigInteger();
}
int blockEnd;
if (index == numBlocks-1) {
blockEnd = intLen;
} else {
blockEnd = (index+1) * blockLength;
}
if (blockEnd > intLen) {
return new MutableBigInteger();
}
int[] newVal = Arrays.copyOfRange(value, offset+intLen-blockEnd, offset+intLen-blockStart);
return new MutableBigInteger(newVal);
}
/** @see BigInteger#bitLength() */
int bitLength() {
if (intLen == 0)
return 0;
return intLen*32 - Integer.numberOfLeadingZeros(value[offset]);
}
/**
* Internally used to calculate the quotient of this div v and places the
* quotient in the provided MutableBigInteger object and the remainder is
* returned.
*
* @return the remainder of the division will be returned.
*/
long divide(long v, MutableBigInteger quotient) {
if (v == 0)
throw new ArithmeticException("BigInteger divide by zero");
// Dividend is zero
if (intLen == 0) {
quotient.intLen = quotient.offset = 0;
return 0;
}
if (v < 0)
v = -v;
int d = (int)(v >>> 32);
quotient.clear();
// Special case on word divisor
if (d == 0)
return divideOneWord((int)v, quotient) & LONG_MASK;
else {
return divideLongMagnitude(v, quotient).toLong();
}
}
private static void copyAndShift(int[] src, int srcFrom, int srcLen, int[] dst, int dstFrom, int shift) {
int n2 = 32 - shift;
int c=src[srcFrom];
for (int i=0; i < srcLen-1; i++) {
int b = c;
c = src[++srcFrom];
dst[dstFrom+i] = (b << shift) | (c >>> n2);
}
dst[dstFrom+srcLen-1] = c << shift;
}
/**
* Divide this MutableBigInteger by the divisor.
* The quotient will be placed into the provided quotient object &
* the remainder object is returned.
*/
private MutableBigInteger divideMagnitude(MutableBigInteger div,
MutableBigInteger quotient,
boolean needRemainder ) {
// assert div.intLen > 1
// D1 normalize the divisor
int shift = Integer.numberOfLeadingZeros(div.value[div.offset]);
// Copy divisor value to protect divisor
final int dlen = div.intLen;
int[] divisor;
MutableBigInteger rem; // Remainder starts as dividend with space for a leading zero
if (shift > 0) {
divisor = new int[dlen];
copyAndShift(div.value,div.offset,dlen,divisor,0,shift);
if (Integer.numberOfLeadingZeros(value[offset]) >= shift) {
int[] remarr = new int[intLen + 1];
rem = new MutableBigInteger(remarr);
rem.intLen = intLen;
rem.offset = 1;
copyAndShift(value,offset,intLen,remarr,1,shift);
} else {
int[] remarr = new int[intLen + 2];
rem = new MutableBigInteger(remarr);
rem.intLen = intLen+1;
rem.offset = 1;
int rFrom = offset;
int c=0;
int n2 = 32 - shift;
for (int i=1; i < intLen+1; i++,rFrom++) {
int b = c;
c = value[rFrom];
remarr[i] = (b << shift) | (c >>> n2);
}
remarr[intLen+1] = c << shift;
}
} else {
divisor = Arrays.copyOfRange(div.value, div.offset, div.offset + div.intLen);
rem = new MutableBigInteger(new int[intLen + 1]);
System.arraycopy(value, offset, rem.value, 1, intLen);
rem.intLen = intLen;
rem.offset = 1;
}
int nlen = rem.intLen;
// Set the quotient size
final int limit = nlen - dlen + 1;
if (quotient.value.length < limit) {
quotient.value = new int[limit];
quotient.offset = 0;
}
quotient.intLen = limit;
int[] q = quotient.value;
// Must insert leading 0 in rem if its length did not change
if (rem.intLen == nlen) {
rem.offset = 0;
rem.value[0] = 0;
rem.intLen++;
}
int dh = divisor[0];
long dhLong = dh & LONG_MASK;
int dl = divisor[1];
// D2 Initialize j
for (int j=0; j < limit-1; j++) {
// D3 Calculate qhat
// estimate qhat
int qhat = 0;
int qrem = 0;
boolean skipCorrection = false;
int nh = rem.value[j+rem.offset];
int nh2 = nh + 0x80000000;
int nm = rem.value[j+1+rem.offset];
if (nh == dh) {
qhat = ~0;
qrem = nh + nm;
skipCorrection = qrem + 0x80000000 < nh2;
} else {
long nChunk = (((long)nh) << 32) | (nm & LONG_MASK);
if (nChunk >= 0) {
qhat = (int) (nChunk / dhLong);
qrem = (int) (nChunk - (qhat * dhLong));
} else {
long tmp = divWord(nChunk, dh);
qhat = (int) (tmp & LONG_MASK);
qrem = (int) (tmp >>> 32);
}
}
if (qhat == 0)
continue;
if (!skipCorrection) { // Correct qhat
long nl = rem.value[j+2+rem.offset] & LONG_MASK;
long rs = ((qrem & LONG_MASK) << 32) | nl;
long estProduct = (dl & LONG_MASK) * (qhat & LONG_MASK);
if (unsignedLongCompare(estProduct, rs)) {
qhat--;
qrem = (int)((qrem & LONG_MASK) + dhLong);
if ((qrem & LONG_MASK) >= dhLong) {
estProduct -= (dl & LONG_MASK);
rs = ((qrem & LONG_MASK) << 32) | nl;
if (unsignedLongCompare(estProduct, rs))
qhat--;
}
}
}
// D4 Multiply and subtract
rem.value[j+rem.offset] = 0;
int borrow = mulsub(rem.value, divisor, qhat, dlen, j+rem.offset);
// D5 Test remainder
if (borrow + 0x80000000 > nh2) {
// D6 Add back
divadd(divisor, rem.value, j+1+rem.offset);
qhat--;
}
// Store the quotient digit
q[j] = qhat;
} // D7 loop on j
// D3 Calculate qhat
// estimate qhat
int qhat = 0;
int qrem = 0;
boolean skipCorrection = false;
int nh = rem.value[limit - 1 + rem.offset];
int nh2 = nh + 0x80000000;
int nm = rem.value[limit + rem.offset];
if (nh == dh) {
qhat = ~0;
qrem = nh + nm;
skipCorrection = qrem + 0x80000000 < nh2;
} else {
long nChunk = (((long) nh) << 32) | (nm & LONG_MASK);
if (nChunk >= 0) {
qhat = (int) (nChunk / dhLong);
qrem = (int) (nChunk - (qhat * dhLong));
} else {
long tmp = divWord(nChunk, dh);
qhat = (int) (tmp & LONG_MASK);
qrem = (int) (tmp >>> 32);
}
}
if (qhat != 0) {
if (!skipCorrection) { // Correct qhat
long nl = rem.value[limit + 1 + rem.offset] & LONG_MASK;
long rs = ((qrem & LONG_MASK) << 32) | nl;
long estProduct = (dl & LONG_MASK) * (qhat & LONG_MASK);
if (unsignedLongCompare(estProduct, rs)) {
qhat--;
qrem = (int) ((qrem & LONG_MASK) + dhLong);
if ((qrem & LONG_MASK) >= dhLong) {
estProduct -= (dl & LONG_MASK);
rs = ((qrem & LONG_MASK) << 32) | nl;
if (unsignedLongCompare(estProduct, rs))
qhat--;
}
}
}
// D4 Multiply and subtract
int borrow;
rem.value[limit - 1 + rem.offset] = 0;
if(needRemainder)
borrow = mulsub(rem.value, divisor, qhat, dlen, limit - 1 + rem.offset);
else
borrow = mulsubBorrow(rem.value, divisor, qhat, dlen, limit - 1 + rem.offset);
// D5 Test remainder
if (borrow + 0x80000000 > nh2) {
// D6 Add back
if(needRemainder)
divadd(divisor, rem.value, limit - 1 + 1 + rem.offset);
qhat--;
}
// Store the quotient digit
q[(limit - 1)] = qhat;
}
if (needRemainder) {
// D8 Unnormalize
if (shift > 0)
rem.rightShift(shift);
rem.normalize();
}
quotient.normalize();
return needRemainder ? rem : null;
}
/**
* Divide this MutableBigInteger by the divisor represented by positive long
* value. The quotient will be placed into the provided quotient object &
* the remainder object is returned.
*/
private MutableBigInteger divideLongMagnitude(long ldivisor, MutableBigInteger quotient) {
// Remainder starts as dividend with space for a leading zero
MutableBigInteger rem = new MutableBigInteger(new int[intLen + 1]);
System.arraycopy(value, offset, rem.value, 1, intLen);
rem.intLen = intLen;
rem.offset = 1;
int nlen = rem.intLen;
int limit = nlen - 2 + 1;
if (quotient.value.length < limit) {
quotient.value = new int[limit];
quotient.offset = 0;
}
quotient.intLen = limit;
int[] q = quotient.value;
// D1 normalize the divisor
int shift = Long.numberOfLeadingZeros(ldivisor);
if (shift > 0) {
ldivisor<<=shift;
rem.leftShift(shift);
}
// Must insert leading 0 in rem if its length did not change
if (rem.intLen == nlen) {
rem.offset = 0;
rem.value[0] = 0;
rem.intLen++;
}
int dh = (int)(ldivisor >>> 32);
long dhLong = dh & LONG_MASK;
int dl = (int)(ldivisor & LONG_MASK);
// D2 Initialize j
for (int j = 0; j < limit; j++) {
// D3 Calculate qhat
// estimate qhat
int qhat = 0;
int qrem = 0;
boolean skipCorrection = false;
int nh = rem.value[j + rem.offset];
int nh2 = nh + 0x80000000;
int nm = rem.value[j + 1 + rem.offset];
if (nh == dh) {
qhat = ~0;
qrem = nh + nm;
skipCorrection = qrem + 0x80000000 < nh2;
} else {
long nChunk = (((long) nh) << 32) | (nm & LONG_MASK);
if (nChunk >= 0) {
qhat = (int) (nChunk / dhLong);
qrem = (int) (nChunk - (qhat * dhLong));
} else {
long tmp = divWord(nChunk, dh);
qhat =(int)(tmp & LONG_MASK);
qrem = (int)(tmp>>>32);
}
}
if (qhat == 0)
continue;
if (!skipCorrection) { // Correct qhat
long nl = rem.value[j + 2 + rem.offset] & LONG_MASK;
long rs = ((qrem & LONG_MASK) << 32) | nl;
long estProduct = (dl & LONG_MASK) * (qhat & LONG_MASK);
if (unsignedLongCompare(estProduct, rs)) {
qhat--;
qrem = (int) ((qrem & LONG_MASK) + dhLong);
if ((qrem & LONG_MASK) >= dhLong) {
estProduct -= (dl & LONG_MASK);
rs = ((qrem & LONG_MASK) << 32) | nl;
if (unsignedLongCompare(estProduct, rs))
qhat--;
}
}
}
// D4 Multiply and subtract
rem.value[j + rem.offset] = 0;
int borrow = mulsubLong(rem.value, dh, dl, qhat, j + rem.offset);
// D5 Test remainder
if (borrow + 0x80000000 > nh2) {
// D6 Add back
divaddLong(dh,dl, rem.value, j + 1 + rem.offset);
qhat--;
}
// Store the quotient digit
q[j] = qhat;
} // D7 loop on j
// D8 Unnormalize
if (shift > 0)
rem.rightShift(shift);
quotient.normalize();
rem.normalize();
return rem;
}
/**
* A primitive used for division by long.
* Specialized version of the method divadd.
* dh is a high part of the divisor, dl is a low part
*/
private int divaddLong(int dh, int dl, int[] result, int offset) {
long carry = 0;
long sum = (dl & LONG_MASK) + (result[1+offset] & LONG_MASK);
result[1+offset] = (int)sum;
sum = (dh & LONG_MASK) + (result[offset] & LONG_MASK) + carry;
result[offset] = (int)sum;
carry = sum >>> 32;
return (int)carry;
}
/**
* This method is used for division by long.
* Specialized version of the method sulsub.
* dh is a high part of the divisor, dl is a low part
*/
private int mulsubLong(int[] q, int dh, int dl, int x, int offset) {
long xLong = x & LONG_MASK;
offset += 2;
long product = (dl & LONG_MASK) * xLong;
long difference = q[offset] - product;
q[offset--] = (int)difference;
long carry = (product >>> 32)
+ (((difference & LONG_MASK) >
(((~(int)product) & LONG_MASK))) ? 1:0);
product = (dh & LONG_MASK) * xLong + carry;
difference = q[offset] - product;
q[offset--] = (int)difference;
carry = (product >>> 32)
+ (((difference & LONG_MASK) >
(((~(int)product) & LONG_MASK))) ? 1:0);
return (int)carry;
}
/**
* Compare two longs as if they were unsigned.
* Returns true iff one is bigger than two.
*/
private boolean unsignedLongCompare(long one, long two) {
return (one+Long.MIN_VALUE) > (two+Long.MIN_VALUE);
}
/**
* This method divides a long quantity by an int to estimate
* qhat for two multi precision numbers. It is used when
* the signed value of n is less than zero.
* Returns long value where high 32 bits contain remainder value and
* low 32 bits contain quotient value.
*/
static long divWord(long n, int d) {
long dLong = d & LONG_MASK;
long r;
long q;
if (dLong == 1) {
q = (int)n;
r = 0;
return (r << 32) | (q & LONG_MASK);
}
// Approximate the quotient and remainder
q = (n >>> 1) / (dLong >>> 1);
r = n - q*dLong;
// Correct the approximation
while (r < 0) {
r += dLong;
q--;
}
while (r >= dLong) {
r -= dLong;
q++;
}
// n - q*dlong == r && 0 <= r <dLong, hence we're done.
return (r << 32) | (q & LONG_MASK);
}
/**
* Calculate GCD of this and b. This and b are changed by the computation.
*/
MutableBigInteger hybridGCD(MutableBigInteger b) {
// Use Euclid's algorithm until the numbers are approximately the
// same length, then use the binary GCD algorithm to find the GCD.
MutableBigInteger a = this;
MutableBigInteger q = new MutableBigInteger();
while (b.intLen != 0) {
if (Math.abs(a.intLen - b.intLen) < 2)
return a.binaryGCD(b);
MutableBigInteger r = a.divide(b, q);
a = b;
b = r;
}
return a;
}
/**
* Calculate GCD of this and v.
* Assumes that this and v are not zero.
*/
private MutableBigInteger binaryGCD(MutableBigInteger v) {
// Algorithm B from Knuth section 4.5.2
MutableBigInteger u = this;
MutableBigInteger r = new MutableBigInteger();
// step B1
int s1 = u.getLowestSetBit();
int s2 = v.getLowestSetBit();
int k = (s1 < s2) ? s1 : s2;
if (k != 0) {
u.rightShift(k);
v.rightShift(k);
}
// step B2
boolean uOdd = (k == s1);
MutableBigInteger t = uOdd ? v: u;
int tsign = uOdd ? -1 : 1;
int lb;
while ((lb = t.getLowestSetBit()) >= 0) {
// steps B3 and B4
t.rightShift(lb);
// step B5
if (tsign > 0)
u = t;
else
v = t;
// Special case one word numbers
if (u.intLen < 2 && v.intLen < 2) {
int x = u.value[u.offset];
int y = v.value[v.offset];
x = binaryGcd(x, y);
r.value[0] = x;
r.intLen = 1;
r.offset = 0;
if (k > 0)
r.leftShift(k);
return r;
}
// step B6
if ((tsign = u.difference(v)) == 0)
break;
t = (tsign >= 0) ? u : v;
}
if (k > 0)
u.leftShift(k);
return u;
}
/**
* Calculate GCD of a and b interpreted as unsigned integers.
*/
static int binaryGcd(int a, int b) {
if (b == 0)
return a;
if (a == 0)
return b;
// Right shift a & b till their last bits equal to 1.
int aZeros = Integer.numberOfTrailingZeros(a);
int bZeros = Integer.numberOfTrailingZeros(b);
a >>>= aZeros;
b >>>= bZeros;
int t = (aZeros < bZeros ? aZeros : bZeros);
while (a != b) {
if ((a+0x80000000) > (b+0x80000000)) { // a > b as unsigned
a -= b;
a >>>= Integer.numberOfTrailingZeros(a);
} else {
b -= a;
b >>>= Integer.numberOfTrailingZeros(b);
}
}
return a<<t;
}
/**
* Returns the modInverse of this mod p. This and p are not affected by
* the operation.
*/
MutableBigInteger mutableModInverse(MutableBigInteger p) {
// Modulus is odd, use Schroeppel's algorithm
if (p.isOdd())
return modInverse(p);
// Base and modulus are even, throw exception
if (isEven())
throw new ArithmeticException("BigInteger not invertible.");
// Get even part of modulus expressed as a power of 2
int powersOf2 = p.getLowestSetBit();
// Construct odd part of modulus
MutableBigInteger oddMod = new MutableBigInteger(p);
oddMod.rightShift(powersOf2);
if (oddMod.isOne())
return modInverseMP2(powersOf2);
// Calculate 1/a mod oddMod
MutableBigInteger oddPart = modInverse(oddMod);
// Calculate 1/a mod evenMod
MutableBigInteger evenPart = modInverseMP2(powersOf2);
// Combine the results using Chinese Remainder Theorem
MutableBigInteger y1 = modInverseBP2(oddMod, powersOf2);
MutableBigInteger y2 = oddMod.modInverseMP2(powersOf2);
MutableBigInteger temp1 = new MutableBigInteger();
MutableBigInteger temp2 = new MutableBigInteger();
MutableBigInteger result = new MutableBigInteger();
oddPart.leftShift(powersOf2);
oddPart.multiply(y1, result);
evenPart.multiply(oddMod, temp1);
temp1.multiply(y2, temp2);
result.add(temp2);
return result.divide(p, temp1);
}
/*
* Calculate the multiplicative inverse of this mod 2^k.
*/
MutableBigInteger modInverseMP2(int k) {
if (isEven())
throw new ArithmeticException("Non-invertible. (GCD != 1)");
if (k > 64)
return euclidModInverse(k);
int t = inverseMod32(value[offset+intLen-1]);
if (k < 33) {
t = (k == 32 ? t : t & ((1 << k) - 1));
return new MutableBigInteger(t);
}
long pLong = (value[offset+intLen-1] & LONG_MASK);
if (intLen > 1)
pLong |= ((long)value[offset+intLen-2] << 32);
long tLong = t & LONG_MASK;
tLong = tLong * (2 - pLong * tLong); // 1 more Newton iter step
tLong = (k == 64 ? tLong : tLong & ((1L << k) - 1));
MutableBigInteger result = new MutableBigInteger(new int[2]);
result.value[0] = (int)(tLong >>> 32);
result.value[1] = (int)tLong;
result.intLen = 2;
result.normalize();
return result;
}
/**
* Returns the multiplicative inverse of val mod 2^32. Assumes val is odd.
*/
static int inverseMod32(int val) {
// Newton's iteration!
int t = val;
t *= 2 - val*t;
t *= 2 - val*t;
t *= 2 - val*t;
t *= 2 - val*t;
return t;
}
/**
* Calculate the multiplicative inverse of 2^k mod mod, where mod is odd.
*/
static MutableBigInteger modInverseBP2(MutableBigInteger mod, int k) {
// Copy the mod to protect original
return fixup(new MutableBigInteger(1), new MutableBigInteger(mod), k);
}
/**
* Calculate the multiplicative inverse of this mod mod, where mod is odd.
* This and mod are not changed by the calculation.
*
* This method implements an algorithm due to Richard Schroeppel, that uses
* the same intermediate representation as Montgomery Reduction
* ("Montgomery Form"). The algorithm is described in an unpublished
* manuscript entitled "Fast Modular Reciprocals."
*/
private MutableBigInteger modInverse(MutableBigInteger mod) {
MutableBigInteger p = new MutableBigInteger(mod);
MutableBigInteger f = new MutableBigInteger(this);
MutableBigInteger g = new MutableBigInteger(p);
SignedMutableBigInteger c = new SignedMutableBigInteger(1);
SignedMutableBigInteger d = new SignedMutableBigInteger();
MutableBigInteger temp = null;
SignedMutableBigInteger sTemp = null;
int k = 0;
// Right shift f k times until odd, left shift d k times
if (f.isEven()) {
int trailingZeros = f.getLowestSetBit();
f.rightShift(trailingZeros);
d.leftShift(trailingZeros);
k = trailingZeros;
}
// The Almost Inverse Algorithm
while (!f.isOne()) {
// If gcd(f, g) != 1, number is not invertible modulo mod
if (f.isZero())
throw new ArithmeticException("BigInteger not invertible.");
// If f < g exchange f, g and c, d
if (f.compare(g) < 0) {
temp = f; f = g; g = temp;
sTemp = d; d = c; c = sTemp;
}
// If f == g (mod 4)
if (((f.value[f.offset + f.intLen - 1] ^
g.value[g.offset + g.intLen - 1]) & 3) == 0) {
f.subtract(g);
c.signedSubtract(d);
} else { // If f != g (mod 4)
f.add(g);
c.signedAdd(d);
}
// Right shift f k times until odd, left shift d k times
int trailingZeros = f.getLowestSetBit();
f.rightShift(trailingZeros);
d.leftShift(trailingZeros);
k += trailingZeros;
}
while (c.sign < 0)
c.signedAdd(p);
return fixup(c, p, k);
}
/**
* The Fixup Algorithm
* Calculates X such that X = C * 2^(-k) (mod P)
* Assumes C<P and P is odd.
*/
static MutableBigInteger fixup(MutableBigInteger c, MutableBigInteger p,
int k) {
MutableBigInteger temp = new MutableBigInteger();
// Set r to the multiplicative inverse of p mod 2^32
int r = -inverseMod32(p.value[p.offset+p.intLen-1]);
for (int i=0, numWords = k >> 5; i < numWords; i++) {
// V = R * c (mod 2^j)
int v = r * c.value[c.offset + c.intLen-1];
// c = c + (v * p)
p.mul(v, temp);
c.add(temp);
// c = c / 2^j
c.intLen--;
}
int numBits = k & 0x1f;
if (numBits != 0) {
// V = R * c (mod 2^j)
int v = r * c.value[c.offset + c.intLen-1];
v &= ((1<<numBits) - 1);
// c = c + (v * p)
p.mul(v, temp);
c.add(temp);
// c = c / 2^j
c.rightShift(numBits);
}
// In theory, c may be greater than p at this point (Very rare!)
while (c.compare(p) >= 0)
c.subtract(p);
return c;
}
/**
* Uses the extended Euclidean algorithm to compute the modInverse of base
* mod a modulus that is a power of 2. The modulus is 2^k.
*/
MutableBigInteger euclidModInverse(int k) {
MutableBigInteger b = new MutableBigInteger(1);
b.leftShift(k);
MutableBigInteger mod = new MutableBigInteger(b);
MutableBigInteger a = new MutableBigInteger(this);
MutableBigInteger q = new MutableBigInteger();
MutableBigInteger r = b.divide(a, q);
MutableBigInteger swapper = b;
// swap b & r
b = r;
r = swapper;
MutableBigInteger t1 = new MutableBigInteger(q);
MutableBigInteger t0 = new MutableBigInteger(1);
MutableBigInteger temp = new MutableBigInteger();
while (!b.isOne()) {
r = a.divide(b, q);
if (r.intLen == 0)
throw new ArithmeticException("BigInteger not invertible.");
swapper = r;
a = swapper;
if (q.intLen == 1)
t1.mul(q.value[q.offset], temp);
else
q.multiply(t1, temp);
swapper = q;
q = temp;
temp = swapper;
t0.add(q);
if (a.isOne())
return t0;
r = b.divide(a, q);
if (r.intLen == 0)
throw new ArithmeticException("BigInteger not invertible.");
swapper = b;
b = r;
if (q.intLen == 1)
t0.mul(q.value[q.offset], temp);
else
q.multiply(t0, temp);
swapper = q; q = temp; temp = swapper;
t1.add(q);
}
mod.subtract(t1);
return mod;
}
}
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