/* * 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. */ /* * %W% %E% */ 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