Java example source code file (BigInteger.java)
The BigInteger.java Java example source code/*
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* 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
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/*
* 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 java.util.concurrent.ThreadLocalRandom;
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.
* 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.
*
* BigInteger must support values in the range
* -2<sup>{@code Integer.MAX_VALUE} (exclusive) to
* +2<sup>{@code Integer.MAX_VALUE} (exclusive)
* and may support values outside of that range.
*
* The range of probable prime values is limited and may be less than
* the full supported positive range of {@code BigInteger}.
* The range must be at least 1 to 2<sup>500000000.
*
* @implNote
* BigInteger constructors and operations throw {@code ArithmeticException} when
* the result is out of the supported range of
* -2<sup>{@code Integer.MAX_VALUE} (exclusive) to
* +2<sup>{@code Integer.MAX_VALUE} (exclusive).
*
* @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 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 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;
/**
* This constant limits {@code mag.length} of BigIntegers to the supported
* range.
*/
private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)
/**
* Bit lengths larger than this constant can cause overflow in searchLen
* calculation and in BitSieve.singleSearch method.
*/
private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;
/**
* The threshold value for using Karatsuba multiplication. If the number
* of ints in both mag arrays are greater than this number, then
* Karatsuba multiplication will be used. This value is found
* experimentally to work well.
*/
private static final int KARATSUBA_THRESHOLD = 80;
/**
* The threshold value for using 3-way Toom-Cook multiplication.
* If the number of ints in each mag array is greater than the
* Karatsuba threshold, and the number of ints in at least one of
* the mag arrays is greater than this threshold, then Toom-Cook
* multiplication will be used.
*/
private static final int TOOM_COOK_THRESHOLD = 240;
/**
* The threshold value for using Karatsuba squaring. If the number
* of ints in the number are larger than this value,
* Karatsuba squaring will be used. This value is found
* experimentally to work well.
*/
private static final int KARATSUBA_SQUARE_THRESHOLD = 128;
/**
* The threshold value for using Toom-Cook squaring. If the number
* of ints in the number are larger than this value,
* Toom-Cook squaring will be used. This value is found
* experimentally to work well.
*/
private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;
/**
* The threshold value for using Burnikel-Ziegler division. If the number
* of ints in the divisor are larger than this value, Burnikel-Ziegler
* division may be used. This value is found experimentally to work well.
*/
static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;
/**
* The offset value for using Burnikel-Ziegler division. If the number
* of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
* number of ints in the dividend is greater than the number of ints in the
* divisor plus this value, Burnikel-Ziegler division will be used. This
* value is found experimentally to work well.
*/
static final int BURNIKEL_ZIEGLER_OFFSET = 40;
/**
* The threshold value for using Schoenhage recursive base conversion. If
* the number of ints in the number are larger than this value,
* the Schoenhage algorithm will be used. In practice, it appears that the
* Schoenhage routine is faster for any threshold down to 2, and is
* relatively flat for thresholds between 2-25, so this choice may be
* varied within this range for very small effect.
*/
private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;
//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 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);
}
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* 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
* 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);
}
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* Translates the sign-magnitude representation of a BigInteger into a
* BigInteger. The sign is represented as an integer signum value: -1 for
* negative, 0 for zero, or 1 for positive. The magnitude is a byte array
* in <i>big-endian 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;
}
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* 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;
}
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* Translates the String representation of a BigInteger in the
* specified radix into a BigInteger. The String representation
* consists of an optional minus or plus sign followed by a
* sequence of one or more digits in the specified radix. The
* character-to-digit mapping is provided by {@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 >= 0) {
if (index1 != 0 || index2 >= 0) {
throw new NumberFormatException("Illegal embedded sign character");
}
sign = -1;
cursor = 1;
} else if (index2 >= 0) {
if (index2 != 0) {
throw new NumberFormatException("Illegal embedded sign character");
}
cursor = 1;
}
if (cursor == len)
throw new NumberFormatException("Zero length BigInteger");
// Skip leading zeros and compute number of digits in magnitude
while (cursor < len &&
Character.digit(val.charAt(cursor), radix) == 0) {
cursor++;
}
if (cursor == len) {
signum = 0;
mag = ZERO.mag;
return;
}
numDigits = len - cursor;
signum = sign;
// Pre-allocate array of expected size. May be too large but can
// never be too small. Typically exact.
long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
if (numBits + 31 >= (1L << 32)) {
reportOverflow();
}
int numWords = (int) (numBits + 31) >>> 5;
int[] magnitude = new int[numWords];
// Process first (potentially short) digit group
int firstGroupLen = numDigits % digitsPerInt[radix];
if (firstGroupLen == 0)
firstGroupLen = digitsPerInt[radix];
String group = val.substring(cursor, cursor += firstGroupLen);
magnitude[numWords - 1] = Integer.parseInt(group, radix);
if (magnitude[numWords - 1] < 0)
throw new NumberFormatException("Illegal digit");
// Process remaining digit groups
int superRadix = intRadix[radix];
int groupVal = 0;
while (cursor < len) {
group = val.substring(cursor, cursor += digitsPerInt[radix]);
groupVal = Integer.parseInt(group, radix);
if (groupVal < 0)
throw new NumberFormatException("Illegal digit");
destructiveMulAdd(magnitude, superRadix, groupVal);
}
// Required for cases where the array was overallocated.
mag = trustedStripLeadingZeroInts(magnitude);
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/*
* Constructs a new BigInteger using a char array with radix=10.
* Sign is precalculated outside and not allowed in the val.
*/
BigInteger(char[] val, int sign, int len) {
int cursor = 0, numDigits;
// Skip leading zeros and compute number of digits in magnitude
while (cursor < len && Character.digit(val[cursor], 10) == 0) {
cursor++;
}
if (cursor == len) {
signum = 0;
mag = ZERO.mag;
return;
}
numDigits = len - cursor;
signum = sign;
// Pre-allocate array of expected size
int numWords;
if (len < 10) {
numWords = 1;
} else {
long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
if (numBits + 31 >= (1L << 32)) {
reportOverflow();
}
numWords = (int) (numBits + 31) >>> 5;
}
int[] magnitude = new int[numWords];
// Process first (potentially short) digit group
int firstGroupLen = numDigits % digitsPerInt[10];
if (firstGroupLen == 0)
firstGroupLen = digitsPerInt[10];
magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);
// Process remaining digit groups
while (cursor < len) {
int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
destructiveMulAdd(magnitude, intRadix[10], groupVal);
}
mag = trustedStripLeadingZeroInts(magnitude);
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
// Create an integer with the digits between the two indexes
// Assumes start < end. The result may be negative, but it
// is to be treated as an unsigned value.
private int parseInt(char[] source, int start, int end) {
int result = Character.digit(source[start++], 10);
if (result == -1)
throw new NumberFormatException(new String(source));
for (int index = start; index < end; index++) {
int nextVal = Character.digit(source[index], 10);
if (nextVal == -1)
throw new NumberFormatException(new String(source));
result = 10*result + nextVal;
}
return result;
}
// bitsPerDigit in the given radix times 1024
// Rounded up to avoid underallocation.
private static long bitsPerDigit[] = { 0, 0,
1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
5253, 5295};
// Multiply x array times word y in place, and add word z
private static void destructiveMulAdd(int[] x, int y, int z) {
// Perform the multiplication word by word
long ylong = y & LONG_MASK;
long zlong = z & LONG_MASK;
int len = x.length;
long product = 0;
long carry = 0;
for (int i = len-1; i >= 0; i--) {
product = ylong * (x[i] & LONG_MASK) + carry;
x[i] = (int)product;
carry = product >>> 32;
}
// Perform the addition
long sum = (x[len-1] & LONG_MASK) + zlong;
x[len-1] = (int)sum;
carry = sum >>> 32;
for (int i = len-2; i >= 0; i--) {
sum = (x[i] & LONG_MASK) + carry;
x[i] = (int)sum;
carry = sum >>> 32;
}
}
/**
* Translates the decimal String representation of a BigInteger into a
* BigInteger. The String representation consists of an optional minus
* 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} - 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}). The execution time of
* this constructor is proportional to the value of this parameter.
* @param rnd source of random bits used to select candidates to be
* tested for primality.
* @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
* @see #bitLength()
*/
public BigInteger(int bitLength, int certainty, Random rnd) {
BigInteger prime;
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
prime = (bitLength < SMALL_PRIME_THRESHOLD
? smallPrime(bitLength, certainty, rnd)
: largePrime(bitLength, certainty, rnd));
signum = 1;
mag = prime.mag;
}
// Minimum size in bits that the requested prime number has
// before we use the large prime number generating algorithms.
// The cutoff of 95 was chosen empirically for best performance.
private static final int SMALL_PRIME_THRESHOLD = 95;
// Certainty required to meet the spec of probablePrime
private static final int DEFAULT_PRIME_CERTAINTY = 100;
/**
* Returns a positive BigInteger that is probably prime, with the
* specified bitLength. The probability that a BigInteger returned
* by this method is composite does not exceed 2<sup>-100.
*
* @param bitLength bitLength of the returned BigInteger.
* @param rnd source of random bits used to select candidates to be
* tested for primality.
* @return a BigInteger of {@code bitLength} bits that is probably prime
* @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
* @see #bitLength()
* @since 1.4
*/
public static BigInteger probablePrime(int bitLength, Random rnd) {
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
return (bitLength < SMALL_PRIME_THRESHOLD ?
smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
}
/**
* Find a random number of the specified bitLength that is probably prime.
* This method is used for smaller primes, its performance degrades on
* larger bitlengths.
*
* This method assumes bitLength > 1.
*/
private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
int magLen = (bitLength + 31) >>> 5;
int temp[] = new int[magLen];
int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
int highMask = (highBit << 1) - 1; // Bits to keep in high int
while (true) {
// Construct a candidate
for (int i=0; i < magLen; i++)
temp[i] = rnd.nextInt();
temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
if (bitLength > 2)
temp[magLen-1] |= 1; // Make odd if bitlen > 2
BigInteger p = new BigInteger(temp, 1);
// Do cheap "pre-test" if applicable
if (bitLength > 6) {
long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
continue; // Candidate is composite; try another
}
// All candidates of bitLength 2 and 3 are prime by this point
if (bitLength < 4)
return p;
// Do expensive test if we survive pre-test (or it's inapplicable)
if (p.primeToCertainty(certainty, rnd))
return p;
}
}
private static final BigInteger SMALL_PRIME_PRODUCT
= valueOf(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 = getPrimeSearchLen(bitLength);
BitSieve searchSieve = new BitSieve(p, searchLen);
BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
while ((candidate == null) || (candidate.bitLength() != bitLength)) {
p = p.add(BigInteger.valueOf(2*searchLen));
if (p.bitLength() != bitLength)
p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
p.mag[p.mag.length-1] &= 0xfffffffe;
searchSieve = new BitSieve(p, searchLen);
candidate = searchSieve.retrieve(p, certainty, rnd);
}
return candidate;
}
/**
* Returns the first integer greater than this {@code BigInteger} that
* is probably prime. The probability that the number returned by this
* method is composite does not exceed 2<sup>-100. This method will
* never skip over a prime when searching: if it returns {@code p}, there
* is no prime {@code q} such that {@code this < q < p}.
*
* @return the first integer greater than this {@code BigInteger} that
* is probably prime.
* @throws ArithmeticException {@code this < 0} or {@code this} is too large.
* @since 1.5
*/
public BigInteger nextProbablePrime() {
if (this.signum < 0)
throw new ArithmeticException("start < 0: " + this);
// Handle trivial cases
if ((this.signum == 0) || this.equals(ONE))
return TWO;
BigInteger result = this.add(ONE);
// Fastpath for small numbers
if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
// Ensure an odd number
if (!result.testBit(0))
result = result.add(ONE);
while (true) {
// Do cheap "pre-test" if applicable
if (result.bitLength() > 6) {
long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
result = result.add(TWO);
continue; // Candidate is composite; try another
}
}
// All candidates of bitLength 2 and 3 are prime by this point
if (result.bitLength() < 4)
return result;
// The expensive test
if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
return result;
result = result.add(TWO);
}
}
// Start at previous even number
if (result.testBit(0))
result = result.subtract(ONE);
// Looking for the next large prime
int searchLen = getPrimeSearchLen(result.bitLength());
while (true) {
BitSieve searchSieve = new BitSieve(result, searchLen);
BigInteger candidate = searchSieve.retrieve(result,
DEFAULT_PRIME_CERTAINTY, null);
if (candidate != null)
return candidate;
result = result.add(BigInteger.valueOf(2 * searchLen));
}
}
private static int getPrimeSearchLen(int bitLength) {
if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
throw new ArithmeticException("Prime search implementation restriction on bitLength");
}
return bitLength / 20 * 64;
}
/**
* Returns {@code true} if this BigInteger is probably prime,
* {@code false} if it's definitely composite.
*
* This method assumes bitLength > 2.
*
* @param certainty a measure of the uncertainty that the caller is
* willing to tolerate: if the call returns {@code true}
* the probability that this BigInteger is prime exceeds
* {@code (1 - 1/2<sup>certainty)}. The execution time of
* this method is proportional to the value of this parameter.
* @return {@code true} if this BigInteger is probably prime,
* {@code false} if it's definitely composite.
*/
boolean primeToCertainty(int certainty, Random random) {
int rounds = 0;
int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
// The relationship between the certainty and the number of rounds
// we perform is given in the draft standard ANSI X9.80, "PRIME
// NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
int sizeInBits = this.bitLength();
if (sizeInBits < 100) {
rounds = 50;
rounds = n < rounds ? n : rounds;
return passesMillerRabin(rounds, random);
}
if (sizeInBits < 256) {
rounds = 27;
} else if (sizeInBits < 512) {
rounds = 15;
} else if (sizeInBits < 768) {
rounds = 8;
} else if (sizeInBits < 1024) {
rounds = 4;
} else {
rounds = 2;
}
rounds = n < rounds ? n : rounds;
return passesMillerRabin(rounds, random) && passesLucasLehmer();
}
/**
* Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
*
* The following assumptions are made:
* This BigInteger is a positive, odd number.
*/
private boolean passesLucasLehmer() {
BigInteger thisPlusOne = this.add(ONE);
// Step 1
int d = 5;
while (jacobiSymbol(d, this) != -1) {
// 5, -7, 9, -11, ...
d = (d < 0) ? Math.abs(d)+2 : -(d+2);
}
// Step 2
BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
// Step 3
return u.mod(this).equals(ZERO);
}
/**
* Computes Jacobi(p,n).
* Assumes n positive, odd, n>=3.
*/
private static int jacobiSymbol(int p, BigInteger n) {
if (p == 0)
return 0;
// Algorithm and comments adapted from Colin Plumb's C library.
int j = 1;
int u = n.mag[n.mag.length-1];
// Make p positive
if (p < 0) {
p = -p;
int n8 = u & 7;
if ((n8 == 3) || (n8 == 7))
j = -j; // 3 (011) or 7 (111) mod 8
}
// Get rid of factors of 2 in p
while ((p & 3) == 0)
p >>= 2;
if ((p & 1) == 0) {
p >>= 1;
if (((u ^ (u>>1)) & 2) != 0)
j = -j; // 3 (011) or 5 (101) mod 8
}
if (p == 1)
return j;
// Then, apply quadratic reciprocity
if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
j = -j;
// And reduce u mod p
u = n.mod(BigInteger.valueOf(p)).intValue();
// Now compute Jacobi(u,p), u < p
while (u != 0) {
while ((u & 3) == 0)
u >>= 2;
if ((u & 1) == 0) {
u >>= 1;
if (((p ^ (p>>1)) & 2) != 0)
j = -j; // 3 (011) or 5 (101) mod 8
}
if (u == 1)
return j;
// Now both u and p are odd, so use quadratic reciprocity
assert (u < p);
int t = u; u = p; p = t;
if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
j = -j;
// Now u >= p, so it can be reduced
u %= p;
}
return 0;
}
private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
BigInteger d = BigInteger.valueOf(z);
BigInteger u = ONE; BigInteger u2;
BigInteger v = ONE; BigInteger v2;
for (int i=k.bitLength()-2; i >= 0; i--) {
u2 = u.multiply(v).mod(n);
v2 = v.square().add(d.multiply(u.square())).mod(n);
if (v2.testBit(0))
v2 = v2.subtract(n);
v2 = v2.shiftRight(1);
u = u2; v = v2;
if (k.testBit(i)) {
u2 = u.add(v).mod(n);
if (u2.testBit(0))
u2 = u2.subtract(n);
u2 = u2.shiftRight(1);
v2 = v.add(d.multiply(u)).mod(n);
if (v2.testBit(0))
v2 = v2.subtract(n);
v2 = v2.shiftRight(1);
u = u2; v = v2;
}
}
return u;
}
/**
* Returns true iff this BigInteger passes the specified number of
* Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
* 186-2).
*
* The following assumptions are made:
* This BigInteger is a positive, odd number greater than 2.
* iterations<=50.
*/
private boolean passesMillerRabin(int iterations, Random rnd) {
// Find a and m such that m is odd and this == 1 + 2**a * m
BigInteger thisMinusOne = this.subtract(ONE);
BigInteger m = thisMinusOne;
int a = m.getLowestSetBit();
m = m.shiftRight(a);
// Do the tests
if (rnd == null) {
rnd = ThreadLocalRandom.current();
}
for (int i=0; i < iterations; i++) {
// Generate a uniform random on (1, this)
BigInteger b;
do {
b = new BigInteger(this.bitLength(), rnd);
} while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);
int j = 0;
BigInteger z = b.modPow(m, this);
while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
if (j > 0 && z.equals(ONE) || ++j == a)
return false;
z = z.modPow(TWO, this);
}
}
return true;
}
/**
* This internal constructor differs from its public cousin
* with the arguments reversed in two ways: it assumes that its
* arguments are correct, and it doesn't copy the magnitude array.
*/
BigInteger(int[] magnitude, int signum) {
this.signum = (magnitude.length == 0 ? 0 : signum);
this.mag = magnitude;
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* This private constructor is for internal use and assumes that its
* arguments are correct.
*/
private BigInteger(byte[] magnitude, int signum) {
this.signum = (magnitude.length == 0 ? 0 : signum);
this.mag = stripLeadingZeroBytes(magnitude);
if (mag.length >= MAX_MAG_LENGTH) {
checkRange();
}
}
/**
* Throws an {@code ArithmeticException} if the {@code BigInteger} would be
* out of the supported range.
*
* @throws ArithmeticException if {@code this} exceeds the supported range.
*/
private void checkRange() {
if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
reportOverflow();
}
}
private static void reportOverflow() {
throw new ArithmeticException("BigInteger would overflow supported range");
}
//Static Factory Methods
/**
* Returns a BigInteger whose value is equal to that of the
* specified {@code long}. 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 > 0 and val is
* non-negative
*/
private static int[] add(int[] x, long val) {
int[] y;
long sum = 0;
int xIndex = x.length;
int[] result;
int highWord = (int)(val >>> 32);
if (highWord == 0) {
result = new int[xIndex];
sum = (x[--xIndex] & LONG_MASK) + val;
result[xIndex] = (int)sum;
} else {
if (xIndex == 1) {
result = new int[2];
sum = val + (x[0] & LONG_MASK);
result[1] = (int)sum;
result[0] = (int)(sum >>> 32);
return result;
} else {
result = new int[xIndex];
sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
result[xIndex] = (int)sum;
sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
result[xIndex] = (int)sum;
}
}
// Copy remainder of longer number while carry propagation is required
boolean carry = (sum >>> 32 != 0);
while (xIndex > 0 && carry)
carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
// Copy remainder of longer number
while (xIndex > 0)
result[--xIndex] = x[xIndex];
// Grow result if necessary
if (carry) {
int bigger[] = new int[result.length + 1];
System.arraycopy(result, 0, bigger, 1, result.length);
bigger[0] = 0x01;
return bigger;
}
return result;
}
/**
* Adds the contents of the int arrays x and y. This method allocates
* a new int array to hold the answer and returns a reference to that
* array.
*/
private static int[] add(int[] x, int[] y) {
// If x is shorter, swap the two arrays
if (x.length < y.length) {
int[] tmp = x;
x = y;
y = tmp;
}
int xIndex = x.length;
int yIndex = y.length;
int result[] = new int[xIndex];
long sum = 0;
if (yIndex == 1) {
sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK) ;
result[xIndex] = (int)sum;
} else {
// Add common parts of both numbers
while (yIndex > 0) {
sum = (x[--xIndex] & LONG_MASK) +
(y[--yIndex] & LONG_MASK) + (sum >>> 32);
result[xIndex] = (int)sum;
}
}
// Copy remainder of longer number while carry propagation is required
boolean carry = (sum >>> 32 != 0);
while (xIndex > 0 && carry)
carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
// Copy remainder of longer number
while (xIndex > 0)
result[--xIndex] = x[xIndex];
// Grow result if necessary
if (carry) {
int bigger[] = new int[result.length + 1];
System.arraycopy(result, 0, bigger, 1, result.length);
bigger[0] = 0x01;
return bigger;
}
return result;
}
private static int[] subtract(long val, int[] little) {
int highWord = (int)(val >>> 32);
if (highWord == 0) {
int result[] = new int[1];
result[0] = (int)(val - (little[0] & LONG_MASK));
return result;
} else {
int result[] = new int[2];
if (little.length == 1) {
long difference = ((int)val & LONG_MASK) - (little[0] & LONG_MASK);
result[1] = (int)difference;
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
if (borrow) {
result[0] = highWord - 1;
} else { // Copy remainder of longer number
result[0] = highWord;
}
return result;
} else { // little.length == 2
long difference = ((int)val & LONG_MASK) - (little[1] & LONG_MASK);
result[1] = (int)difference;
difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
result[0] = (int)difference;
return result;
}
}
}
/**
* Subtracts the contents of the second argument (val) from the
* first (big). The first int array (big) must represent a larger number
* than the second. This method allocates the space necessary to hold the
* answer.
* assumes val >= 0
*/
private static int[] subtract(int[] big, long val) {
int highWord = (int)(val >>> 32);
int bigIndex = big.length;
int result[] = new int[bigIndex];
long difference = 0;
if (highWord == 0) {
difference = (big[--bigIndex] & LONG_MASK) - val;
result[bigIndex] = (int)difference;
} else {
difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
result[bigIndex] = (int)difference;
difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
result[bigIndex] = (int)difference;
}
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
while (bigIndex > 0 && borrow)
borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
// Copy remainder of longer number
while (bigIndex > 0)
result[--bigIndex] = big[bigIndex];
return result;
}
/**
* Returns a BigInteger whose value is {@code (this - val)}.
*
* @param val value to be subtracted from this BigInteger.
* @return {@code this - val}
*/
public BigInteger subtract(BigInteger val) {
if (val.signum == 0)
return this;
if (signum == 0)
return val.negate();
if (val.signum != signum)
return new BigInteger(add(mag, val.mag), signum);
int cmp = compareMagnitude(val);
if (cmp == 0)
return ZERO;
int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
: subtract(val.mag, mag));
resultMag = trustedStripLeadingZeroInts(resultMag);
return new BigInteger(resultMag, cmp == signum ? 1 : -1);
}
/**
* Subtracts the contents of the second int arrays (little) from the
* first (big). The first int array (big) must represent a larger number
* than the second. This method allocates the space necessary to hold the
* answer.
*/
private static int[] subtract(int[] big, int[] little) {
int bigIndex = big.length;
int result[] = new int[bigIndex];
int littleIndex = little.length;
long difference = 0;
// Subtract common parts of both numbers
while (littleIndex > 0) {
difference = (big[--bigIndex] & LONG_MASK) -
(little[--littleIndex] & LONG_MASK) +
(difference >> 32);
result[bigIndex] = (int)difference;
}
// Subtract remainder of longer number while borrow propagates
boolean borrow = (difference >> 32 != 0);
while (bigIndex > 0 && borrow)
borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
// Copy remainder of longer number
while (bigIndex > 0)
result[--bigIndex] = big[bigIndex];
return result;
}
/**
* Returns a BigInteger whose value is {@code (this * val)}.
*
* @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 abs();
}
int lowerInts[] = new int[n];
System.arraycopy(mag, len-n, lowerInts, 0, n);
return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
}
/**
* Returns a new BigInteger representing mag.length-n upper
* ints of the number. This is used by Karatsuba multiplication and
* Karatsuba squaring.
*/
private BigInteger getUpper(int n) {
int len = mag.length;
if (len <= n) {
return ZERO;
}
int upperLen = len - n;
int upperInts[] = new int[upperLen];
System.arraycopy(mag, 0, upperInts, 0, upperLen);
return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
}
// Squaring
/**
* Returns a BigInteger whose value is {@code (this<sup>2)}.
*
* @return {@code this<sup>2}
*/
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 (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
return divideKnuth(val);
} else {
return divideBurnikelZiegler(val);
}
}
/**
* Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
*
* @param val value by which this BigInteger is to be divided.
* @return {@code this / val}
* @throws ArithmeticException if {@code val} is zero.
* @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
*/
private BigInteger divideKnuth(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
a.divideKnuth(b, q, false);
return q.toBigInteger(this.signum * val.signum);
}
/**
* Returns an array of two BigIntegers containing {@code (this / val)}
* followed by {@code (this % val)}.
*
* @param val value by which this BigInteger is to be divided, and the
* remainder computed.
* @return an array of two BigIntegers: the quotient {@code (this / val)}
* is the initial element, and the remainder {@code (this % val)}
* is the final element.
* @throws ArithmeticException if {@code val} is zero.
*/
public BigInteger[] divideAndRemainder(BigInteger val) {
if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
return divideAndRemainderKnuth(val);
} else {
return divideAndRemainderBurnikelZiegler(val);
}
}
/** Long division */
private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
BigInteger[] result = new BigInteger[2];
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
MutableBigInteger r = a.divideKnuth(b, q);
result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
result[1] = r.toBigInteger(this.signum);
return result;
}
/**
* Returns a BigInteger whose value is {@code (this % val)}.
*
* @param val value by which this BigInteger is to be divided, and the
* remainder computed.
* @return {@code this % val}
* @throws ArithmeticException if {@code val} is zero.
*/
public BigInteger remainder(BigInteger val) {
if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
return remainderKnuth(val);
} else {
return remainderBurnikelZiegler(val);
}
}
/** Long division */
private BigInteger remainderKnuth(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
return a.divideKnuth(b, q).toBigInteger(this.signum);
}
/**
* Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
* @param val the divisor
* @return {@code this / val}
*/
private BigInteger divideBurnikelZiegler(BigInteger val) {
return divideAndRemainderBurnikelZiegler(val)[0];
}
/**
* Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
* @param val the divisor
* @return {@code this % val}
*/
private BigInteger remainderBurnikelZiegler(BigInteger val) {
return divideAndRemainderBurnikelZiegler(val)[1];
}
/**
* Computes {@code this / val} and {@code this % val} using the
* Burnikel-Ziegler algorithm.
* @param val the divisor
* @return an array containing the quotient and remainder
*/
private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
MutableBigInteger q = new MutableBigInteger();
MutableBigInteger r = new MutableBigInteger(this).divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum*val.signum);
BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
return new BigInteger[] {qBigInt, rBigInt};
}
/**
* Returns a BigInteger whose value is <tt>(thisexponent).
* Note that {@code exponent} is an integer rather than a BigInteger.
*
* @param exponent exponent to which this BigInteger is to be raised.
* @return <tt>thisexponent
* @throws ArithmeticException {@code exponent} is negative. (This would
* cause the operation to yield a non-integer value.)
*/
public BigInteger pow(int exponent) {
if (exponent < 0) {
throw new ArithmeticException("Negative exponent");
}
if (signum == 0) {
return (exponent == 0 ? ONE : this);
}
BigInteger partToSquare = this.abs();
// Factor out powers of two from the base, as the exponentiation of
// these can be done by left shifts only.
// The remaining part can then be exponentiated faster. The
// powers of two will be multiplied back at the end.
int powersOfTwo = partToSquare.getLowestSetBit();
long bitsToShift = (long)powersOfTwo * exponent;
if (bitsToShift > Integer.MAX_VALUE) {
reportOverflow();
}
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.
long scaleFactor = (long)remainingBits * exponent;
// Use slightly different algorithms for small and large operands.
// See if the result will safely fit into a long. (Largest 2^63-1)
if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
// Small number algorithm. Everything fits into a long.
int newSign = (signum <0 && (exponent&1) == 1 ? -1 : 1);
long result = 1;
long baseToPow2 = partToSquare.mag[0] & LONG_MASK;
int workingExponent = exponent;
// Perform exponentiation using repeated squaring trick
while (workingExponent != 0) {
if ((workingExponent & 1) == 1) {
result = result * baseToPow2;
}
if ((workingExponent >>>= 1) != 0) {
baseToPow2 = baseToPow2 * baseToPow2;
}
}
// Multiply back the powers of two (quickly, by shifting left)
if (powersOfTwo > 0) {
if (bitsToShift + scaleFactor <= 62) { // Fits in long?
return valueOf((result << bitsToShift) * newSign);
} else {
return valueOf(result*newSign).shiftLeft((int) 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 BigInteger.
*
* @param m the modulus.
* @return {@code this mod m}
* @throws ArithmeticException {@code m} ? 0
* @see #remainder
*/
public BigInteger mod(BigInteger m) {
if (m.signum <= 0)
throw new ArithmeticException("BigInteger: modulus not positive");
BigInteger result = this.remainder(m);
return (result.signum >= 0 ? result : result.add(m));
}
/**
* Returns a BigInteger whose value is
* <tt>(thisexponent mod m). (Unlike {@code pow}, this
* method permits negative exponents.)
*
* @param exponent the exponent.
* @param m the modulus.
* @return <tt>thisexponent mod m
* @throws ArithmeticException {@code m} ? 0 or the exponent is
* negative and this BigInteger is not <i>relatively
* prime</i> to {@code m}.
* @see #modInverse
*/
public BigInteger modPow(BigInteger exponent, BigInteger m) {
if (m.signum <= 0)
throw new ArithmeticException("BigInteger: modulus not positive");
// Trivial cases
if (exponent.signum == 0)
return (m.equals(ONE) ? ZERO : ONE);
if (this.equals(ONE))
return (m.equals(ONE) ? ZERO : ONE);
if (this.equals(ZERO) && exponent.signum >= 0)
return ZERO;
if (this.equals(negConst[1]) && (!exponent.testBit(0)))
return (m.equals(ONE) ? ZERO : ONE);
boolean invertResult;
if ((invertResult = (exponent.signum < 0)))
exponent = exponent.negate();
BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
? this.mod(m) : this);
BigInteger result;
if (m.testBit(0)) { // odd modulus
result = base.oddModPow(exponent, m);
} else {
/*
* Even modulus. Tear it into an "odd part" (m1) and power of two
* (m2), exponentiate mod m1, manually exponentiate mod m2, and
* use Chinese Remainder Theorem to combine results.
*/
// Tear m apart into odd part (m1) and power of 2 (m2)
int p = m.getLowestSetBit(); // Max pow of 2 that divides m
BigInteger m1 = m.shiftRight(p); // m/2**p
BigInteger m2 = ONE.shiftLeft(p); // 2**p
// Calculate new base from m1
BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
? this.mod(m1) : this);
// 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);
if (m.mag.length < MAX_MAG_LENGTH / 2) {
result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
} else {
MutableBigInteger t1 = new MutableBigInteger();
new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
MutableBigInteger t2 = new MutableBigInteger();
new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
t1.add(t2);
MutableBigInteger q = new MutableBigInteger();
result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
}
}
return (invertResult ? result.modInverse(m) : result);
}
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 {@code mod m)}.
*
* @param m the modulus.
* @return {@code this}<sup>-1 {@code mod m}.
* @throws ArithmeticException {@code m} ? 0, or this BigInteger
* has no multiplicative inverse mod m (that is, this BigInteger
* is not <i>relatively prime 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 * 2n).)
*
* @param n shift distance, in bits.
* @return {@code this << n}
* @see #shiftRight
*/
public BigInteger shiftLeft(int n) {
if (signum == 0)
return ZERO;
if (n > 0) {
return new BigInteger(shiftLeft(mag, n), signum);
} else if (n == 0) {
return this;
} else {
// Possible int overflow in (-n) is not a trouble,
// because shiftRightImpl considers its argument unsigned
return shiftRightImpl(-n);
}
}
/**
* Returns a magnitude array whose value is {@code (mag << n)}.
* The shift distance, {@code n}, is considered unnsigned.
* (Computes <tt>this * 2n.)
*
* @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
* @param n unsigned shift distance, in bits.
* @return {@code mag << n}
*/
private static int[] shiftLeft(int[] mag, int n) {
int nInts = n >>> 5;
int nBits = n & 0x1f;
int magLen = mag.length;
int newMag[] = null;
if (nBits == 0) {
newMag = new int[magLen + nInts];
System.arraycopy(mag, 0, newMag, 0, magLen);
} else {
int i = 0;
int nBits2 = 32 - nBits;
int highBits = mag[0] >>> nBits2;
if (highBits != 0) {
newMag = new int[magLen + nInts + 1];
newMag[i++] = highBits;
} else {
newMag = new int[magLen + nInts];
}
int 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 / 2n).)
*
* @param n shift distance, in bits.
* @return {@code this >> n}
* @see #shiftLeft
*/
public BigInteger shiftRight(int n) {
if (signum == 0)
return ZERO;
if (n > 0) {
return shiftRightImpl(n);
} else if (n == 0) {
return this;
} else {
// Possible int overflow in {@code -n} is not a trouble,
// because shiftLeft considers its argument unsigned
return new BigInteger(shiftLeft(mag, -n), signum);
}
}
/**
* Returns a BigInteger whose value is {@code (this >> n)}. The shift
* distance, {@code n}, is considered unsigned.
* (Computes <tt>floor(this * 2-n).)
*
* @param n unsigned shift distance, in bits.
* @return {@code this >> n}
*/
private BigInteger shiftRightImpl(int n) {
int nInts = n >>> 5;
int nBits = n & 0x1f;
int magLen = mag.length;
int newMag[] = null;
// Special case: entire contents shifted off the end
if (nInts >= magLen)
return (signum >= 0 ? ZERO : negConst[1]);
if (nBits == 0) {
int newMagLen = magLen - nInts;
newMag = Arrays.copyOf(mag, newMagLen);
} else {
int i = 0;
int highBits = mag[0] >>> nBits;
if (highBits != 0) {
newMag = new int[magLen - nInts];
newMag[i++] = highBits;
} else {
newMag = new int[magLen - nInts -1];
}
int 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<
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