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* Copyright (c) 2006, <a href="http://lib.csdn.net/base/oracle" class='replace_word' title="Oracle知识库" target='_blank' style='color:#df3434; font-weight:bold;'>Oracle</a> and/or its affiliates. All rights reserved.
* ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
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/*
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package <a href="http://lib.csdn.net/base/javaee" class='replace_word' title="Java EE知识库" target='_blank' style='color:#df3434; font-weight:bold;'>Java</a>.math;
import java.util.Random;
import java.io.*;
/**
* Immutable arbitrary-precision integers. All operations behave as if
* BigIntegers were represented in two's-complement notation (like Java's
* primitive integer types). BigInteger provides analogues to all of Java's
* primitive integer operators, and all relevant methods from java.lang.Math.
* Additionally, BigInteger provides operations for modular arithmetic, GCD
* calculation, primality testing, prime generation, bit manipulation,
* and a few other miscellaneous operations.
* <p>
* Semantics of arithmetic operations exactly mimic those of Java's integer
* arithmetic operators, as defined in <i>The <a href="http://lib.csdn.net/base/java" class='replace_word' title="Java 知识库" target='_blank' style='color:#df3434; font-weight:bold;'>Java </a>Language Specification</i>.
* For example, division by zero throws an {@code ArithmeticException}, and
* division of a negative by a positive yields a negative (or zero) remainder.
* All of the details in the Spec concerning overflow are ignored, as
* BigIntegers are made as large as necessary to accommodate the results of an
* operation.
* <p>
* Semantics of shift operations extend those of Java's shift operators
* to allow for negative shift distances. A right-shift with a negative
* shift distance results in a left shift, and vice-versa. The unsigned
* right shift operator ({@code >>>}) is omitted, as this operation makes
* little sense in combination with the "infinite word size" abstraction
* provided by this class.
* <p>
* Semantics of bitwise logical operations exactly mimic those of Java's
* bitwise integer operators. The binary operators ({@code and},
* {@code or}, {@code xor}) implicitly perform sign extension on the shorter
* of the two operands prior to performing the operation.
* <p>
* Comparison operations perform signed integer comparisons, analogous to
* those performed by Java's relational and equality operators.
* <p>
* Modular arithmetic operations are provided to compute residues, perform
* exponentiation, and compute multiplicative inverses. These methods always
* return a non-negative result, between {@code 0} and {@code (modulus - 1)},
* inclusive.
* <p>
* Bit operations operate on a single bit of the two's-complement
* representation of their operand. If necessary, the operand is sign-
* extended so that it contains the designated bit. None of the single-bit
* operations can produce a BigInteger with a different sign from the
* BigInteger being operated on, as they affect only a single bit, and the
* "infinite word size" abstraction provided by this class ensures that there
* are infinitely many "virtual sign bits" preceding each BigInteger.
* <p>
* For the sake of brevity and clarity, pseudo-code is used throughout the
* descriptions of BigInteger methods. The pseudo-code expression
* {@code (i + j)} is shorthand for "a BigInteger whose value is
* that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
* The pseudo-code expression {@code (i == j)} is shorthand for
* "{@code true} if and only if the BigInteger {@code i} represents the same
* value as the BigInteger {@code j}." Other pseudo-code expressions are
* interpreted similarly.
* <p>
* All methods and constructors in this class throw
* {@code NullPointerException} when passed
* a null object reference for any input parameter.
*
* @see BigDecimal
* @author Josh Bloch
* @author Michael McCloskey
* @since JDK1.1
*/
public class BigInteger extends Number implements Comparable<BigInteger> {
/**
* The signum of this BigInteger: -1 for negative, 0 for zero, or
* 1 for positive. Note that the BigInteger zero <i>must</i> have
* a signum of 0. This is necessary to ensures that there is exactly one
* representation for each BigInteger value.
*
* @serial
*/
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;
//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 sign followed by a sequence of one or more digits in the
* specified radix. The character-to-digit mapping is provided by
* {@code Character.digit}. The String may not contain any extraneous
* characters (whitespace, for example).
*
* @param val String representation of BigInteger.
* @param radix radix to be used in interpreting {@code val}.
* @throws NumberFormatException {@code val} is not a valid representation
* of a BigInteger in the specified radix, or {@code radix} is
* outside the range from {@link Character#MIN_RADIX} to
* {@link Character#MAX_RADIX}, inclusive.
* @see Character#digit
*/
public BigInteger(String val, int radix) {
int cursor = 0, numDigits;
int len = val.length();
if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
throw new NumberFormatException("Radix out of range");
if (val.length() == 0)
throw new NumberFormatException("Zero length BigInteger");
// Check for minus sign
int sign = 1;
int index = val.lastIndexOf("-");
if (index != -1) {
if (index == 0) {
if (val.length() == 1)
throw new NumberFormatException("Zero length BigInteger");
sign = -1;
cursor = 1;
} else {
throw new NumberFormatException("Illegal embedded minus sign");
}
}
// Skip leading zeros and compute number of digits in magnitude
while (cursor < len &&
Character.digit(val.charAt(cursor), radix) == 0)
cursor++;
if (cursor == len) {
mag = ZERO.mag;
signum = 0;
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[magnitude.length - 1] = Integer.parseInt(group, radix);
if (magnitude[magnitude.length - 1] < 0)
throw new NumberFormatException("Illegal digit");
// Process remaining digit groups
int superRadix = intRadix[radix];
int groupVal = 0;
while (cursor < val.length()) {
group = val.substring(cursor, cursor += digitsPerInt[radix]);
groupVal = Integer.parseInt(group, radix);
if (groupVal < 0)
throw new NumberFormatException("Illegal digit");
destructiveMulAdd(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
BigInteger(char[] val) {
int cursor = 0, numDigits;
int len = val.length;
// Check for leading minus sign
int sign = 1;
if (val[0] == '-') {
if (len == 1)
throw new NumberFormatException("Zero length BigInteger");
sign = -1;
cursor = 1;
}
// Skip leading zeros and compute number of digits in magnitude
while (cursor < len && Character.digit(val[cursor], 10) == 0)
cursor++;
if (cursor == len) {
signum = 0;
mag = ZERO.mag;
return;
}
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 {@code 0} to (2<sup>{@code numBits}</sup> - 1), inclusive.
* The uniformity of the distribution assumes that a fair source of random
* bits is provided in {@code rnd}. Note that this constructor always
* constructs a non-negative BigInteger.
*
* @param numBits maximum bitLength of the new BigInteger.
* @param rnd source of randomness to be used in computing the new
* BigInteger.
* @throws IllegalArgumentException {@code numBits} is negative.
* @see #bitLength()
*/
public BigInteger(int numBits, Random rnd) {
this(1, randomBits(numBits, rnd));
}
private static byte[] randomBits(int numBits, Random rnd) {
if (numBits < 0)
throw new IllegalArgumentException("numBits must be non-negative");
int numBytes = (int)(((long)numBits+7)/8); // avoid overflow
byte[] randomBits = new byte[numBytes];
// Generate random bytes and mask out any excess bits
if (numBytes > 0) {
rnd.nextBytes(randomBits);
int excessBits = 8*numBytes - numBits;
randomBits[0] &= (1 << (8-excessBits)) - 1;
}
return randomBits;
}
/**
* Constructs a randomly generated positive BigInteger that is probably
* prime, with the specified bitLength.<p>
*
* It is recommended that the {@link #probablePrime probablePrime}
* method be used in preference to this constructor unless there
* is a compelling need to specify a certainty.
*
* @param bitLength bitLength of the returned BigInteger.
* @param certainty a measure of the uncertainty that the caller is
* willing to tolerate. The probability that the new BigInteger
* represents a prime number will exceed
* (1 - 1/2<sup>{@code certainty}</sup>). The execution time of
* this constructor is proportional to the value of this parameter.
* @param rnd source of random bits used to select candidates to be
* tested for primality.
* @throws ArithmeticException {@code bitLength < 2}.
* @see #bitLength()
*/
public BigInteger(int bitLength, int certainty, Random rnd) {
BigInteger prime;
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
// The cutoff of 95 was chosen empirically for best performance
prime = (bitLength < 95 ? smallPrime(bitLength, certainty, rnd)
: largePrime(bitLength, certainty, rnd));
signum = 1;
mag = prime.mag;
}
// Minimum size in bits that the requested prime number has
// before we use the large prime number generating algorithms
private static final int SMALL_PRIME_THRESHOLD = 95;
// Certainty required to meet the spec of probablePrime
private static final int DEFAULT_PRIME_CERTAINTY = 100;
/**
* Returns a positive BigInteger that is probably prime, with the
* specified bitLength. The probability that a BigInteger returned
* by this method is composite does not exceed 2<sup>-100</sup>.
*
* @param bitLength bitLength of the returned BigInteger.
* @param rnd source of random bits used to select candidates to be
* tested for primality.
* @return a BigInteger of {@code bitLength} bits that is probably prime
* @throws ArithmeticException {@code bitLength < 2}.
* @see #bitLength()
* @since 1.4
*/
public static BigInteger probablePrime(int bitLength, Random rnd) {
if (bitLength < 2)
throw new ArithmeticException("bitLength < 2");
// The cutoff of 95 was chosen empirically for best performance
return (bitLength < SMALL_PRIME_THRESHOLD ?
smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
}
/**
* Find a random number of the specified bitLength that is probably prime.
* This method is used for smaller primes, its performance degrades on
* larger bitlengths.
*
* This method assumes bitLength > 1.
*/
private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
int magLen = (bitLength + 31) >>> 5;
int temp[] = new int[magLen];
int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int
int highMask = (highBit << 1) - 1; // Bits to keep in high int
while(true) {
// Construct a candidate
for (int i=0; i<magLen; i++)
temp[i] = rnd.nextInt();
temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length
if (bitLength > 2)
temp[magLen-1] |= 1; // Make odd if bitlen > 2
BigInteger p = new BigInteger(temp, 1);
// Do cheap "pre-test" if applicable
if (bitLength > 6) {
long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0))
continue; // Candidate is composite; try another
}
// All candidates of bitLength 2 and 3 are prime by this point
if (bitLength < 4)
return p;
// Do expensive test if we survive pre-test (or it's inapplicable)
if (p.primeToCertainty(certainty, rnd))
return p;
}
}
private static final BigInteger SMALL_PRIME_PRODUCT
= valueOf(3L*5*7*11*13*17*19*23*29*31*37*41);
/**
* Find a random number of the specified bitLength that is probably prime.
* This method is more appropriate for larger bitlengths since it uses
* a sieve to eliminate most composites before using a more expensive
* test.
*/
private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
BigInteger p;
p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
p.mag[p.mag.length-1] &= 0xfffffffe;
// Use a sieve length likely to contain the next prime number
int searchLen = (bitLength / 20) * 64;
BitSieve searchSieve = new BitSieve(p, searchLen);
BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);
while ((candidate == null) || (candidate.bitLength() != bitLength)) {
p = p.add(BigInteger.valueOf(2*searchLen));
if (p.bitLength() != bitLength)
p = new BigInteger(bitLength, rnd).setBit(bitLength-1);
p.mag[p.mag.length-1] &= 0xfffffffe;
searchSieve = new BitSieve(p, searchLen);
candidate = searchSieve.retrieve(p, certainty, rnd);
}
return candidate;
}
/**
* Returns the first integer greater than this {@code BigInteger} that
* is probably prime. The probability that the number returned by this
* method is composite does not exceed 2<sup>-100</sup>. This method will
* never skip over a prime when searching: if it returns {@code p}, there
* is no prime {@code q} such that {@code this < q < p}.
*
* @return the first integer greater than this {@code BigInteger} that
* is probably prime.
* @throws ArithmeticException {@code this < 0}.
* @since 1.5
*/
public BigInteger nextProbablePrime() {
if (this.signum < 0)
throw new ArithmeticException("start < 0: " + this);
// Handle trivial cases
if ((this.signum == 0) || this.equals(ONE))
return TWO;
BigInteger result = this.add(ONE);
// Fastpath for small numbers
if (result.bitLength() < SMALL_PRIME_THRESHOLD) {
// Ensure an odd number
if (!result.testBit(0))
result = result.add(ONE);
while(true) {
// Do cheap "pre-test" if applicable
if (result.bitLength() > 6) {
long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) ||
(r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) ||
(r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) {
result = result.add(TWO);
continue; // Candidate is composite; try another
}
}
// All candidates of bitLength 2 and 3 are prime by this point
if (result.bitLength() < 4)
return result;
// The expensive test
if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null))
return result;
result = result.add(TWO);
}
}
// Start at previous even number
if (result.testBit(0))
result = result.subtract(ONE);
// Looking for the next large prime
int searchLen = (result.bitLength() / 20) * 64;
while(true) {
BitSieve searchSieve = new BitSieve(result, searchLen);
BigInteger candidate = searchSieve.retrieve(result,
DEFAULT_PRIME_CERTAINTY, null);
if (candidate != null)
return candidate;
result = result.add(BigInteger.valueOf(2 * searchLen));
}
}
/**
* Returns {@code true} if this BigInteger is probably prime,
* {@code false} if it's definitely composite.
*
* This method assumes bitLength > 2.
*
* @param certainty a measure of the uncertainty that the caller is
* willing to tolerate: if the call returns {@code true}
* the probability that this BigInteger is prime exceeds
* {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of
* this method is proportional to the value of this parameter.
* @return {@code true} if this BigInteger is probably prime,
* {@code false} if it's definitely composite.
*/
boolean primeToCertainty(int certainty, Random random) {
int rounds = 0;
int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2;
// The relationship between the certainty and the number of rounds
// we perform is given in the draft standard ANSI X9.80, "PRIME
// NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
int sizeInBits = this.bitLength();
if (sizeInBits < 100) {
rounds = 50;
rounds = n < rounds ? n : rounds;
return passesMillerRabin(rounds, random);
}
if (sizeInBits < 256) {
rounds = 27;
} else if (sizeInBits < 512) {
rounds = 15;
} else if (sizeInBits < 768) {
rounds = 8;
} else if (sizeInBits < 1024) {
rounds = 4;
} else {
rounds = 2;
}
rounds = n < rounds ? n : rounds;
return passesMillerRabin(rounds, random) && passesLucasLehmer();
}
/**
* Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
*
* The following assumptions are made:
* This BigInteger is a positive, odd number.
*/
private boolean passesLucasLehmer() {
BigInteger thisPlusOne = this.add(ONE);
// Step 1
int d = 5;
while (jacobiSymbol(d, this) != -1) {
// 5, -7, 9, -11, ...
d = (d<0) ? Math.abs(d)+2 : -(d+2);
}
// Step 2
BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);
// Step 3
return u.mod(this).equals(ZERO);
}
/**
* Computes Jacobi(p,n).
* Assumes n positive, odd, n>=3.
*/
private static int jacobiSymbol(int p, BigInteger n) {
if (p == 0)
return 0;
// Algorithm and comments adapted from Colin Plumb's C library.
int j = 1;
int u = n.mag[n.mag.length-1];
// Make p positive
if (p < 0) {
p = -p;
int n8 = u & 7;
if ((n8 == 3) || (n8 == 7))
j = -j; // 3 (011) or 7 (111) mod 8
}
// Get rid of factors of 2 in p
while ((p & 3) == 0)
p >>= 2;
if ((p & 1) == 0) {
p >>= 1;
if (((u ^ (u>>1)) & 2) != 0)
j = -j; // 3 (011) or 5 (101) mod 8
}
if (p == 1)
return j;
// Then, apply quadratic reciprocity
if ((p & u & 2) != 0) // p = u = 3 (mod 4)?
j = -j;
// And reduce u mod p
u = n.mod(BigInteger.valueOf(p)).intValue();
// Now compute Jacobi(u,p), u < p
while (u != 0) {
while ((u & 3) == 0)
u >>= 2;
if ((u & 1) == 0) {
u >>= 1;
if (((p ^ (p>>1)) & 2) != 0)
j = -j; // 3 (011) or 5 (101) mod 8
}
if (u == 1)
return j;
// Now both u and p are odd, so use quadratic reciprocity
assert (u < p);
int t = u; u = p; p = t;
if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
j = -j;
// Now u >= p, so it can be reduced
u %= p;
}
return 0;
}
private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
BigInteger d = BigInteger.valueOf(z);
BigInteger u = ONE; BigInteger u2;
BigInteger v = ONE; BigInteger v2;
for (int i=k.bitLength()-2; i>=0; i--) {
u2 = u.multiply(v).mod(n);
v2 = v.square().add(d.multiply(u.square())).mod(n);
if (v2.testBit(0))
v2 = 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];
static {
for (int i = 1; i <= MAX_CONSTANT; i++) {
int[] magnitude = new int[1];
magnitude[0] = i;
posConst[i] = new BigInteger(magnitude, 1);
negConst[i] = new BigInteger(magnitude, -1);
}
}
/**
* The BigInteger constant zero.
*
* @since 1.2
*/
public static final BigInteger ZERO = new BigInteger(new int[0], 0);
/**
* The BigInteger constant one.
*
* @since 1.2
*/
public static final BigInteger ONE = valueOf(1);
/**
* The BigInteger constant two. (Not exported.)
*/
private static final BigInteger TWO = valueOf(2);
/**
* The BigInteger constant ten.
*
* @since 1.5
*/
public static final BigInteger TEN = valueOf(10);
// Arithmetic Operations
/**
* Returns a BigInteger whose value is {@code (this + val)}.
*
* @param val value to be added to this BigInteger.
* @return {@code this + val}
*/
public BigInteger add(BigInteger val) {
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);
}
/**
* Adds the contents of the int arrays x and y. This method allocates
* a new int array to hold the answer and returns a reference to that
* array.
*/
private static int[] add(int[] x, int[] y) {
// If x is shorter, swap the two arrays
if (x.length < y.length) {
int[] tmp = x;
x = y;
y = tmp;
}
int xIndex = x.length;
int yIndex = y.length;
int result[] = new int[xIndex];
long sum = 0;
// Add common parts of both numbers
while(yIndex > 0) {
sum = (x[--xIndex] & LONG_MASK) +
(y[--yIndex] & LONG_MASK) + (sum >>> 32);
result[xIndex] = (int)sum;
}
// Copy remainder of longer number while carry propagation is required
boolean carry = (sum >>> 32 != 0);
while (xIndex > 0 && carry)
carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
// Copy remainder of longer number
while (xIndex > 0)
result[--xIndex] = x[xIndex];
// Grow result if necessary
if (carry) {
int bigger[] = new int[result.length + 1];
System.arraycopy(result, 0, bigger, 1, result.length);
bigger[0] = 0x01;
return bigger;
}
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[] res = multiplyToLen(mag, mag.length, val.mag, val.mag.length, null);
res = trustedStripLeadingZeroInts(res);
return new BigInteger(res, signum == val.signum ? 1 : -1);
}
/**
* 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;
assert v != BigDecimal.INFLATED;
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;
}
/**
* Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
*
* @return {@code this<sup>2</sup>}
*/
private BigInteger square() {
if (signum == 0)
return ZERO;
int[] z = squareToLen(mag, mag.length, null);
return new BigInteger(trustedStripLeadingZeroInts(z), 1);
}
/**
* Squares the contents of the int array x. The result is placed into the
* int array z. The contents of x are not changed.
*/
private static final int[] squareToLen(int[] x, int len, int[] z) {
/*
* The algorithm used here is adapted from Colin Plumb's C library.
* Technique: Consider the partial products in the multiplication
* of "abcde" by itself:
*
* a b c d e
* * a b c d e
* ==================
* ae be ce de ee
* ad bd cd dd de
* ac bc cc cd ce
* ab bb bc bd be
* aa ab ac ad ae
*
* Note that everything above the main diagonal:
* ae be ce de = (abcd) * e
* ad bd cd = (abc) * d
* ac bc = (ab) * c
* ab = (a) * b
*
* is a copy of everything below the main diagonal:
* de
* cd ce
* bc bd be
* ab ac ad ae
*
* Thus, the sum is 2 * (off the diagonal) + diagonal.
*
* This is accumulated beginning with the diagonal (which
* consist of the squares of the digits of the input), which is then
* divided by two, the off-diagonal added, and multiplied by two
* again. The low bit is simply a copy of the low bit of the
* input, so it doesn't need special care.
*/
int zlen = len << 1;
if (z == null || z.length < zlen)
z = new int[zlen];
// Store the squares, right shifted one bit (i.e., divided by 2)
int lastProductLowWord = 0;
for (int j=0, i=0; j<len; j++) {
long piece = (x[j] & LONG_MASK);
long product = piece * piece;
z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33);
z[i++] = (int)(product >>> 1);
lastProductLowWord = (int)product;
}
// Add in off-diagonal sums
for (int i=len, offset=1; i>0; i--, offset+=2) {
int t = x[i-1];
t = mulAdd(z, x, offset, i-1, t);
addOne(z, offset-1, i, t);
}
// Shift back up and set low bit
primitiveLeftShift(z, zlen, 1);
z[zlen-1] |= x[len-1] & 1;
return z;
}
/**
* Returns a BigInteger whose value is {@code (this / val)}.
*
* @param val value by which this BigInteger is to be divided.
* @return {@code this / val}
* @throws ArithmeticException {@code val==0}
*/
public BigInteger divide(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
a.divide(b, q);
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 {@code val==0}
*/
public BigInteger[] divideAndRemainder(BigInteger val) {
BigInteger[] result = new BigInteger[2];
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
MutableBigInteger r = a.divide(b, q);
result[0] = q.toBigInteger(this.signum * val.signum);
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 {@code val==0}
*/
public BigInteger remainder(BigInteger val) {
MutableBigInteger q = new MutableBigInteger(),
a = new MutableBigInteger(this.mag),
b = new MutableBigInteger(val.mag);
return a.divide(b, q).toBigInteger(this.signum);
}
/**
* Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
* Note that {@code exponent} is an integer rather than a BigInteger.
*
* @param exponent exponent to which this BigInteger is to be raised.
* @return <tt>this<sup>exponent</sup></tt>
* @throws ArithmeticException {@code exponent} is negative. (This would
* cause the operation to yield a non-integer value.)
*/
public BigInteger pow(int exponent) {
if (exponent < 0)
throw new ArithmeticException("Negative exponent");
if (signum==0)
return (exponent==0 ? ONE : this);
// Perform exponentiation using repeated squaring trick
int newSign = (signum<0 && (exponent&1)==1 ? -1 : 1);
int[] baseToPow2 = this.mag;
int[] result = {1};
while (exponent != 0) {
if ((exponent & 1)==1) {
result = multiplyToLen(result, result.length,
baseToPow2, baseToPow2.length, null);
result = trustedStripLeadingZeroInts(result);
}
if ((exponent >>>= 1) != 0) {
baseToPow2 = squareToLen(baseToPow2, baseToPow2.length, null);
baseToPow2 = trustedStripLeadingZeroInts(baseToPow2);
}
}
return new BigInteger(result, newSign);
}
/**
* Returns a BigInteger whose value is the greatest common divisor of
* {@code abs(this)} and {@code abs(val)}. Returns 0 if
* {@code this==0 && val==0}.
*
* @param val value with which the GCD is to be computed.
* @return {@code GCD(abs(this), abs(val))}
*/
public BigInteger gcd(BigInteger val) {
if (val.signum == 0)
return this.abs();
else if (this.signum == 0)
return val.abs();
MutableBigInteger a = new MutableBigInteger(this);
MutableBigInteger b = new MutableBigInteger(val);
MutableBigInteger result = a.hybridGCD(b);
return 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];
for (int i=0; i<len; i++)
result[i] = a[i];
primitiveLeftShift(result, result.length, nBits);
return result;
} else {
int result[] = new int[nInts+len+1];
for (int i=0; i<len; i++)
result[i] = a[i];
primitiveRightShift(result, result.length, 32 - nBits);
return result;
}
}
}
// shifts a up to len right n bits assumes no leading zeros, 0<n<32
static void primitiveRightShift(int[] a, int len, int n) {
int n2 = 32 - n;
for (int i=len-1, c=a[i]; i>0; i--) {
int b = c;
c = a[i-1];
a[i] = (c << n2) | (b >>> n);
}
a[0] >>>= n;
}
// shifts a up to len left n bits assumes no leading zeros, 0<=n<32
static void primitiveLeftShift(int[] a, int len, int n) {
if (len == 0 || n == 0)
return;
int n2 = 32 - n;
for (int i=0, c=a[i], m=i+len-1; i<m; i++) {
int b = c;
c = a[i+1];
a[i] = (b << n) | (c >>> n2);
}
a[len-1] <<= n;
}
/**
* Calculate bitlength
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