diff -r 000000000000 -r 724f3e1ea53e emul/compact/src/main/java/java/math/BigInteger.java --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/emul/compact/src/main/java/java/math/BigInteger.java Sat Sep 07 13:51:24 2013 +0200 @@ -0,0 +1,3186 @@ +/* + * Copyright (c) 1996, 2007, Oracle and/or its affiliates. All rights reserved. + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. + * + * This code is free software; you can redistribute it and/or modify it + * under the terms of the GNU General Public License version 2 only, as + * published by the Free Software Foundation. Oracle designates this + * particular file as subject to the "Classpath" exception as provided + * by Oracle in the LICENSE file that accompanied this code. + * + * This code is distributed in the hope that it will be useful, but WITHOUT + * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or + * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License + * version 2 for more details (a copy is included in the LICENSE file that + * accompanied this code). + * + * You should have received a copy of the GNU General Public License version + * 2 along with this work; if not, write to the Free Software Foundation, + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. + * + * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA + * or visit www.oracle.com if you need additional information or have any + * questions. + */ + +/* + * Portions Copyright (c) 1995 Colin Plumb. All rights reserved. + */ + +package java.math; + +import java.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. + * + *

Semantics of arithmetic operations exactly mimic those of Java's integer + * arithmetic operators, as defined in 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. + * + *

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. + * + *

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. + * + *

Comparison operations perform signed integer comparisons, analogous to + * those performed by Java's relational and equality operators. + * + *

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. + * + *

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. + * + *

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. + * + *

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 { + /** + * The signum of this BigInteger: -1 for negative, 0 for zero, or + * 1 for positive. Note that the BigInteger zero 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 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; + + //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 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); + } + } + + /** + * 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 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); + } + } + + /** + * 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 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; + } + } + + /** + * A constructor for internal use that translates the sign-magnitude + * representation of a BigInteger into a BigInteger. It checks the + * arguments and copies the magnitude so this constructor would be + * safe for external use. + */ + private BigInteger(int signum, int[] magnitude) { + this.mag = stripLeadingZeroInts(magnitude); + + if (signum < -1 || signum > 1) + throw(new NumberFormatException("Invalid signum value")); + + if (this.mag.length==0) { + this.signum = 0; + } else { + if (signum == 0) + throw(new NumberFormatException("signum-magnitude mismatch")); + this.signum = signum; + } + } + + /** + * Translates the String representation of a BigInteger in the + * specified radix into a BigInteger. The String representation + * consists of an optional minus or plus sign followed by a + * sequence of one or more digits in the specified radix. The + * character-to-digit mapping is provided by {@code + * Character.digit}. The String may not contain any extraneous + * characters (whitespace, for example). + * + * @param val String representation of BigInteger. + * @param radix radix to be used in interpreting {@code val}. + * @throws NumberFormatException {@code val} is not a valid representation + * of a BigInteger in the specified radix, or {@code radix} is + * outside the range from {@link Character#MIN_RADIX} to + * {@link Character#MAX_RADIX}, inclusive. + * @see Character#digit + */ + public BigInteger(String val, int radix) { + int cursor = 0, numDigits; + final int len = val.length(); + + if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) + throw new NumberFormatException("Radix out of range"); + if (len == 0) + throw new NumberFormatException("Zero length BigInteger"); + + // Check for at most one leading sign + int sign = 1; + int index1 = val.lastIndexOf('-'); + int index2 = val.lastIndexOf('+'); + if ((index1 + index2) <= -1) { + // No leading sign character or at most one leading sign character + if (index1 == 0 || index2 == 0) { + cursor = 1; + if (len == 1) + throw new NumberFormatException("Zero length BigInteger"); + } + if (index1 == 0) + sign = -1; + } else + throw new NumberFormatException("Illegal embedded sign character"); + + // Skip leading zeros and compute number of digits in magnitude + while (cursor < len && + Character.digit(val.charAt(cursor), radix) == 0) + cursor++; + if (cursor == len) { + signum = 0; + mag = ZERO.mag; + return; + } + + numDigits = len - cursor; + signum = sign; + + // Pre-allocate array of expected size. May be too large but can + // never be too small. Typically exact. + int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1); + int numWords = (numBits + 31) >>> 5; + int[] magnitude = new int[numWords]; + + // Process first (potentially short) digit group + int firstGroupLen = numDigits % digitsPerInt[radix]; + if (firstGroupLen == 0) + firstGroupLen = digitsPerInt[radix]; + String group = val.substring(cursor, cursor += firstGroupLen); + magnitude[numWords - 1] = Integer.parseInt(group, radix); + if (magnitude[numWords - 1] < 0) + throw new NumberFormatException("Illegal digit"); + + // Process remaining digit groups + int superRadix = intRadix[radix]; + int groupVal = 0; + while (cursor < len) { + group = val.substring(cursor, cursor += digitsPerInt[radix]); + groupVal = Integer.parseInt(group, radix); + if (groupVal < 0) + throw new NumberFormatException("Illegal digit"); + destructiveMulAdd(magnitude, superRadix, groupVal); + } + // Required for cases where the array was overallocated. + mag = trustedStripLeadingZeroInts(magnitude); + } + + // Constructs a new BigInteger using a char array with radix=10 + 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; + } else if (val[0] == '+') { + if (len == 1) + throw new NumberFormatException("Zero length BigInteger"); + cursor = 1; + } + + // Skip leading zeros and compute number of digits in magnitude + while (cursor < len && Character.digit(val[cursor], 10) == 0) + cursor++; + if (cursor == len) { + signum = 0; + mag = ZERO.mag; + return; + } + + 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= 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^{{@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. + * + *

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^{{@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}. + * @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^-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}. + * @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 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^-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}. + * @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^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; + } + + 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= 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[] result = multiplyToLen(mag, mag.length, + val.mag, val.mag.length, null); + result = trustedStripLeadingZeroInts(result); + return new BigInteger(result, 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; + 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; + } + + /** + * Returns a BigInteger whose value is {@code (this²)}. + * + * @return {@code this²} + */ + 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>> 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 if {@code val} is zero. + */ + 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 ? 1 : -1); + } + + /** + * 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) { + 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 ? 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) { + 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 (this^exponent). + * Note that {@code exponent} is an integer rather than a BigInteger. + * + * @param exponent exponent to which this BigInteger is to be raised. + * @return this^exponent + * @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; i0; 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>> 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 + * 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 + * (this^exponent mod m). (Unlike {@code pow}, this + * method permits negative exponents.) + * + * @param exponent the exponent. + * @param m the modulus. + * @return this^exponent mod m + * @throws ArithmeticException {@code m} ≤ 0 or the exponent is + * negative and this BigInteger is not relatively + * prime to {@code m}. + * @see #modInverse + */ + public BigInteger modPow(BigInteger exponent, BigInteger m) { + if (m.signum <= 0) + throw new ArithmeticException("BigInteger: modulus not positive"); + + // Trivial cases + if (exponent.signum == 0) + return (m.equals(ONE) ? ZERO : ONE); + + if (this.equals(ONE)) + return (m.equals(ONE) ? ZERO : ONE); + + if (this.equals(ZERO) && exponent.signum >= 0) + return ZERO; + + if (this.equals(negConst[1]) && (!exponent.testBit(0))) + return (m.equals(ONE) ? ZERO : ONE); + + boolean invertResult; + if ((invertResult = (exponent.signum < 0))) + exponent = exponent.negate(); + + BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0 + ? this.mod(m) : this); + BigInteger result; + if (m.testBit(0)) { // odd modulus + result = base.oddModPow(exponent, m); + } else { + /* + * Even modulus. Tear it into an "odd part" (m1) and power of two + * (m2), exponentiate mod m1, manually exponentiate mod m2, and + * use Chinese Remainder Theorem to combine results. + */ + + // Tear m apart into odd part (m1) and power of 2 (m2) + int p = m.getLowestSetBit(); // Max pow of 2 that divides m + + BigInteger m1 = m.shiftRight(p); // m/2**p + BigInteger m2 = ONE.shiftLeft(p); // 2**p + + // Calculate new base from m1 + BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0 + ? this.mod(m1) : this); + + // Caculate (base ** exponent) mod m1. + BigInteger a1 = (m1.equals(ONE) ? ZERO : + base2.oddModPow(exponent, m1)); + + // Calculate (this ** exponent) mod m2 + BigInteger a2 = base.modPow2(exponent, p); + + // Combine results using Chinese Remainder Theorem + BigInteger y1 = m2.modInverse(m1); + BigInteger y2 = m1.modInverse(m2); + + result = a1.multiply(m2).multiply(y1).add + (a2.multiply(m1).multiply(y2)).mod(m); + } + + return (invertResult ? result.modInverse(m) : result); + } + + static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793, + Integer.MAX_VALUE}; // Sentinel + + /** + * Returns a BigInteger whose value is x to the power of y mod z. + * Assumes: z is odd && x < z. + */ + private BigInteger oddModPow(BigInteger y, BigInteger z) { + /* + * The algorithm is adapted from Colin Plumb's C library. + * + * The window algorithm: + * The idea is to keep a running product of b1 = n^(high-order bits of exp) + * and then keep appending exponent bits to it. The following patterns + * apply to a 3-bit window (k = 3): + * To append 0: square + * To append 1: square, multiply by n^1 + * To append 10: square, multiply by n^1, square + * To append 11: square, square, multiply by n^3 + * To append 100: square, multiply by n^1, square, square + * To append 101: square, square, square, multiply by n^5 + * To append 110: square, square, multiply by n^3, square + * To append 111: square, square, square, multiply by n^7 + * + * Since each pattern involves only one multiply, the longer the pattern + * the better, except that a 0 (no multiplies) can be appended directly. + * We precompute a table of odd powers of n, up to 2^k, and can then + * multiply k bits of exponent at a time. Actually, assuming random + * exponents, there is on average one zero bit between needs to + * multiply (1/2 of the time there's none, 1/4 of the time there's 1, + * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so + * you have to do one multiply per k+1 bits of exponent. + * + * The loop walks down the exponent, squaring the result buffer as + * it goes. There is a wbits+1 bit lookahead buffer, buf, that is + * filled with the upcoming exponent bits. (What is read after the + * end of the exponent is unimportant, but it is filled with zero here.) + * When the most-significant bit of this buffer becomes set, i.e. + * (buf & tblmask) != 0, we have to decide what pattern to multiply + * by, and when to do it. We decide, remember to do it in future + * after a suitable number of squarings have passed (e.g. a pattern + * of "100" in the buffer requires that we multiply by n^1 immediately; + * a pattern of "110" calls for multiplying by n^3 after one more + * squaring), clear the buffer, and continue. + * + * When we start, there is one more optimization: the result buffer + * is implcitly one, so squaring it or multiplying by it can be + * optimized away. Further, if we start with a pattern like "100" + * in the lookahead window, rather than placing n into the buffer + * and then starting to square it, we have already computed n^2 + * to compute the odd-powers table, so we can place that into + * the buffer and save a squaring. + * + * This means that if you have a k-bit window, to compute n^z, + * where z is the high k bits of the exponent, 1/2 of the time + * it requires no squarings. 1/4 of the time, it requires 1 + * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings. + * And the remaining 1/2^(k-1) of the time, the top k bits are a + * 1 followed by k-1 0 bits, so it again only requires k-2 + * squarings, not k-1. The average of these is 1. Add that + * to the one squaring we have to do to compute the table, + * and you'll see that a k-bit window saves k-2 squarings + * as well as reducing the multiplies. (It actually doesn't + * hurt in the case k = 1, either.) + */ + // Special case for exponent of one + if (y.equals(ONE)) + return this; + + // Special case for base of zero + if (signum==0) + return ZERO; + + int[] base = mag.clone(); + int[] exp = y.mag; + int[] mod = z.mag; + int modLen = mod.length; + + // Select an appropriate window size + int wbits = 0; + int ebits = bitLength(exp, exp.length); + // if exponent is 65537 (0x10001), use minimum window size + if ((ebits != 17) || (exp[0] != 65537)) { + while (ebits > bnExpModThreshTable[wbits]) { + wbits++; + } + } + + // Calculate appropriate table size + int tblmask = 1 << wbits; + + // Allocate table for precomputed odd powers of base in Montgomery form + int[][] table = new int[tblmask][]; + for (int i=0; i>>= 1; + if (bitpos == 0) { + eIndex++; + bitpos = 1 << (32-1); + elen--; + } + } + + int multpos = ebits; + + // The first iteration, which is hoisted out of the main loop + ebits--; + boolean isone = true; + + multpos = ebits - wbits; + while ((buf & 1) == 0) { + buf >>>= 1; + multpos++; + } + + int[] mult = table[buf >>> 1]; + + buf = 0; + if (multpos == ebits) + isone = false; + + // The main loop + while(true) { + ebits--; + // Advance the window + buf <<= 1; + + if (elen != 0) { + buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0; + bitpos >>>= 1; + if (bitpos == 0) { + eIndex++; + bitpos = 1 << (32-1); + elen--; + } + } + + // Examine the window for pending multiplies + if ((buf & tblmask) != 0) { + multpos = ebits - wbits; + while ((buf & 1) == 0) { + buf >>>= 1; + multpos++; + } + mult = table[buf >>> 1]; + buf = 0; + } + + // Perform multiply + if (ebits == multpos) { + if (isone) { + b = mult.clone(); + isone = false; + } else { + t = b; + a = multiplyToLen(t, modLen, mult, modLen, a); + a = montReduce(a, mod, modLen, inv); + t = a; a = b; b = t; + } + } + + // Check if done + if (ebits == 0) + break; + + // Square the input + if (!isone) { + t = b; + a = squareToLen(t, modLen, a); + a = montReduce(a, mod, modLen, inv); + t = a; a = b; b = t; + } + } + + // Convert result out of Montgomery form and return + int[] t2 = new int[2*modLen]; + for(int i=0; i 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 b2) + return 1; + } + return 0; + } + + /** + * Subtracts two numbers of same length, returning borrow. + */ + private static int subN(int[] a, int[] b, int len) { + long sum = 0; + + while(--len >= 0) { + sum = (a[len] & LONG_MASK) - + (b[len] & LONG_MASK) + (sum >> 32); + a[len] = (int)sum; + } + + return (int)(sum >> 32); + } + + /** + * Multiply an array by one word k and add to result, return the carry + */ + static int mulAdd(int[] out, int[] in, int offset, int len, int k) { + long kLong = k & LONG_MASK; + long carry = 0; + + offset = out.length-offset - 1; + for (int j=len-1; j >= 0; j--) { + long product = (in[j] & LONG_MASK) * kLong + + (out[offset] & LONG_MASK) + carry; + out[offset--] = (int)product; + carry = product >>> 32; + } + return (int)carry; + } + + /** + * Add one word to the number a mlen words into a. Return the resulting + * carry. + */ + static int addOne(int[] a, int offset, int mlen, int carry) { + offset = a.length-1-mlen-offset; + long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK); + + a[offset] = (int)t; + if ((t >>> 32) == 0) + return 0; + while (--mlen >= 0) { + if (--offset < 0) { // Carry out of number + return 1; + } else { + a[offset]++; + if (a[offset] != 0) + return 0; + } + } + return 1; + } + + /** + * Returns a BigInteger whose value is (this ** exponent) mod (2**p) + */ + private BigInteger modPow2(BigInteger exponent, int p) { + /* + * Perform exponentiation using repeated squaring trick, chopping off + * high order bits as indicated by modulus. + */ + BigInteger result = valueOf(1); + BigInteger baseToPow2 = this.mod2(p); + int expOffset = 0; + + int limit = exponent.bitLength(); + + if (this.testBit(0)) + limit = (p-1) < limit ? (p-1) : limit; + + while (expOffset < limit) { + if (exponent.testBit(expOffset)) + result = result.multiply(baseToPow2).mod2(p); + expOffset++; + if (expOffset < limit) + baseToPow2 = baseToPow2.square().mod2(p); + } + + return result; + } + + /** + * Returns a BigInteger whose value is this mod(2**p). + * Assumes that this {@code BigInteger >= 0} and {@code p > 0}. + */ + private BigInteger mod2(int p) { + if (bitLength() <= p) + return this; + + // Copy remaining ints of mag + int numInts = (p + 31) >>> 5; + int[] mag = new int[numInts]; + for (int i=0; i-1 {@code mod m)}. + * + * @param m the modulus. + * @return {@code this}^-1 {@code mod m}. + * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger + * has no multiplicative inverse mod m (that is, this BigInteger + * is not 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 floor(this * 2ⁿ).) + * + * @param n shift distance, in bits. + * @return {@code this << n} + * @throws ArithmeticException if the shift distance is {@code + * Integer.MIN_VALUE}. + * @see #shiftRight + */ + public BigInteger shiftLeft(int n) { + if (signum == 0) + return ZERO; + if (n==0) + return this; + if (n<0) { + if (n == Integer.MIN_VALUE) { + throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported."); + } else { + return shiftRight(-n); + } + } + + int nInts = n >>> 5; + int nBits = n & 0x1f; + int magLen = mag.length; + int newMag[] = null; + + if (nBits == 0) { + newMag = new int[magLen + nInts]; + for (int i=0; i>> nBits2; + if (highBits != 0) { + newMag = new int[magLen + nInts + 1]; + newMag[i++] = highBits; + } else { + newMag = new int[magLen + nInts]; + } + int j=0; + while (j < magLen-1) + newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2; + newMag[i] = mag[j] << nBits; + } + + return new BigInteger(newMag, signum); + } + + /** + * Returns a BigInteger whose value is {@code (this >> n)}. Sign + * extension is performed. The shift distance, {@code n}, may be + * negative, in which case this method performs a left shift. + * (Computes floor(this / 2ⁿ).) + * + * @param n shift distance, in bits. + * @return {@code this >> n} + * @throws ArithmeticException if the shift distance is {@code + * Integer.MIN_VALUE}. + * @see #shiftLeft + */ + public BigInteger shiftRight(int n) { + if (n==0) + return this; + if (n<0) { + if (n == Integer.MIN_VALUE) { + throw new ArithmeticException("Shift distance of Integer.MIN_VALUE not supported."); + } else { + return shiftLeft(-n); + } + } + + int nInts = n >>> 5; + int nBits = n & 0x1f; + int magLen = mag.length; + int newMag[] = null; + + // Special case: entire contents shifted off the end + if (nInts >= magLen) + return (signum >= 0 ? ZERO : negConst[1]); + + if (nBits == 0) { + int newMagLen = magLen - nInts; + newMag = new int[newMagLen]; + for (int i=0; i>> 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>> 5) & (1 << (n & 31))) != 0; + } + + /** + * Returns a BigInteger whose value is equivalent to this BigInteger + * with the designated bit set. (Computes {@code (this | (1<>> 5; + int[] result = new int[Math.max(intLength(), intNum+2)]; + + for (int i=0; i>> 5; + int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)]; + + for (int i=0; i>> 5; + int[] result = new int[Math.max(intLength(), intNum+2)]; + + for (int i=0; iexcluding a sign bit. + * For positive BigIntegers, this is equivalent to the number of bits in + * the ordinary binary representation. (Computes + * {@code (ceil(log2(this < 0 ? -this : this+1)))}.) + * + * @return number of bits in the minimal two's-complement + * representation of this BigInteger, excluding a sign bit. + */ + public int bitLength() { + @SuppressWarnings("deprecation") int n = bitLength - 1; + if (n == -1) { // bitLength not initialized yet + int[] m = mag; + int len = m.length; + if (len == 0) { + n = 0; // offset by one to initialize + } else { + // Calculate the bit length of the magnitude + int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]); + if (signum < 0) { + // Check if magnitude is a power of two + boolean pow2 = (Integer.bitCount(mag[0]) == 1); + for(int i=1; i< len && pow2; i++) + pow2 = (mag[i] == 0); + + n = (pow2 ? magBitLength -1 : magBitLength); + } else { + n = magBitLength; + } + } + bitLength = n + 1; + } + return n; + } + + /** + * Returns the number of bits in the two's complement representation + * of this BigInteger that differ from its sign bit. This method is + * useful when implementing bit-vector style sets atop BigIntegers. + * + * @return number of bits in the two's complement representation + * of this BigInteger that differ from its sign bit. + */ + public int bitCount() { + @SuppressWarnings("deprecation") int bc = bitCount - 1; + if (bc == -1) { // bitCount not initialized yet + bc = 0; // offset by one to initialize + // Count the bits in the magnitude + for (int i=0; i{@code 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. + */ + public boolean isProbablePrime(int certainty) { + if (certainty <= 0) + return true; + BigInteger w = this.abs(); + if (w.equals(TWO)) + return true; + if (!w.testBit(0) || w.equals(ONE)) + return false; + + return w.primeToCertainty(certainty, null); + } + + // Comparison Operations + + /** + * Compares this BigInteger with the specified BigInteger. This + * method is provided in preference to individual methods for each + * of the six boolean comparison operators ({@literal <}, ==, + * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested + * idiom for performing these comparisons is: {@code + * (x.compareTo(y)} <op> {@code 0)}, where + * <op> is one of the six comparison operators. + * + * @param val BigInteger to which this BigInteger is to be compared. + * @return -1, 0 or 1 as this BigInteger is numerically less than, equal + * to, or greater than {@code val}. + */ + public int compareTo(BigInteger val) { + if (signum == val.signum) { + switch (signum) { + case 1: + return compareMagnitude(val); + case -1: + return val.compareMagnitude(this); + default: + return 0; + } + } + return signum > val.signum ? 1 : -1; + } + + /** + * Compares the magnitude array of this BigInteger with the specified + * BigInteger's. This is the version of compareTo ignoring sign. + * + * @param val BigInteger whose magnitude array to be compared. + * @return -1, 0 or 1 as this magnitude array is less than, equal to or + * greater than the magnitude aray for the specified BigInteger's. + */ + final int compareMagnitude(BigInteger val) { + int[] m1 = mag; + int len1 = m1.length; + int[] m2 = val.mag; + int len2 = m2.length; + if (len1 < len2) + return -1; + if (len1 > len2) + return 1; + for (int i = 0; i < len1; i++) { + int a = m1[i]; + int b = m2[i]; + if (a != b) + return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; + } + return 0; + } + + /** + * Compares this BigInteger with the specified Object for equality. + * + * @param x Object to which this BigInteger is to be compared. + * @return {@code true} if and only if the specified Object is a + * BigInteger whose value is numerically equal to this BigInteger. + */ + public boolean equals(Object x) { + // This test is just an optimization, which may or may not help + if (x == this) + return true; + + if (!(x instanceof BigInteger)) + return false; + + BigInteger xInt = (BigInteger) x; + if (xInt.signum != signum) + return false; + + int[] m = mag; + int len = m.length; + int[] xm = xInt.mag; + if (len != xm.length) + return false; + + for (int i = 0; i < len; i++) + if (xm[i] != m[i]) + return false; + + return true; + } + + /** + * Returns the minimum of this BigInteger and {@code val}. + * + * @param val value with which the minimum is to be computed. + * @return the BigInteger whose value is the lesser of this BigInteger and + * {@code val}. If they are equal, either may be returned. + */ + public BigInteger min(BigInteger val) { + return (compareTo(val)<0 ? this : val); + } + + /** + * Returns the maximum of this BigInteger and {@code val}. + * + * @param val value with which the maximum is to be computed. + * @return the BigInteger whose value is the greater of this and + * {@code val}. If they are equal, either may be returned. + */ + public BigInteger max(BigInteger val) { + return (compareTo(val)>0 ? this : val); + } + + + // Hash Function + + /** + * Returns the hash code for this BigInteger. + * + * @return hash code for this BigInteger. + */ + public int hashCode() { + int hashCode = 0; + + for (int i=0; i Character.MAX_RADIX) + radix = 10; + + // Compute upper bound on number of digit groups and allocate space + int maxNumDigitGroups = (4*mag.length + 6)/7; + String digitGroup[] = new String[maxNumDigitGroups]; + + // Translate number to string, a digit group at a time + BigInteger tmp = this.abs(); + int numGroups = 0; + while (tmp.signum != 0) { + BigInteger d = longRadix[radix]; + + MutableBigInteger q = new MutableBigInteger(), + a = new MutableBigInteger(tmp.mag), + b = new MutableBigInteger(d.mag); + MutableBigInteger r = a.divide(b, q); + BigInteger q2 = q.toBigInteger(tmp.signum * d.signum); + BigInteger r2 = r.toBigInteger(tmp.signum * d.signum); + + digitGroup[numGroups++] = Long.toString(r2.longValue(), radix); + tmp = q2; + } + + // Put sign (if any) and first digit group into result buffer + StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1); + if (signum<0) + buf.append('-'); + buf.append(digitGroup[numGroups-1]); + + // Append remaining digit groups padded with leading zeros + for (int i=numGroups-2; i>=0; i--) { + // Prepend (any) leading zeros for this digit group + int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length(); + if (numLeadingZeros != 0) + buf.append(zeros[numLeadingZeros]); + buf.append(digitGroup[i]); + } + return buf.toString(); + } + + /* zero[i] is a string of i consecutive zeros. */ + private static String zeros[] = new String[64]; + static { + zeros[63] = + "000000000000000000000000000000000000000000000000000000000000000"; + for (int i=0; i<63; i++) + zeros[i] = zeros[63].substring(0, i); + } + + /** + * Returns the decimal String representation of this BigInteger. + * The digit-to-character mapping provided by + * {@code Character.forDigit} is used, and a minus sign is + * prepended if appropriate. (This representation is compatible + * with the {@link #BigInteger(String) (String)} constructor, and + * allows for String concatenation with Java's + operator.) + * + * @return decimal String representation of this BigInteger. + * @see Character#forDigit + * @see #BigInteger(java.lang.String) + */ + public String toString() { + return toString(10); + } + + /** + * Returns a byte array containing the two's-complement + * representation of this BigInteger. The byte array will be in + * big-endian byte-order: the most significant byte is in + * the zeroth element. The array will contain the minimum number + * of bytes required to represent this BigInteger, including at + * least one sign bit, which is {@code (ceil((this.bitLength() + + * 1)/8))}. (This representation is compatible with the + * {@link #BigInteger(byte[]) (byte[])} constructor.) + * + * @return a byte array containing the two's-complement representation of + * this BigInteger. + * @see #BigInteger(byte[]) + */ + public byte[] toByteArray() { + int byteLen = bitLength()/8 + 1; + byte[] byteArray = new byte[byteLen]; + + for (int i=byteLen-1, bytesCopied=4, nextInt=0, intIndex=0; i>=0; i--) { + if (bytesCopied == 4) { + nextInt = getInt(intIndex++); + bytesCopied = 1; + } else { + nextInt >>>= 8; + bytesCopied++; + } + byteArray[i] = (byte)nextInt; + } + return byteArray; + } + + /** + * Converts this BigInteger to an {@code int}. This + * conversion is analogous to a + * narrowing primitive conversion from {@code long} to + * {@code int} as defined in section 5.1.3 of + * The Java™ Language Specification: + * if this BigInteger is too big to fit in an + * {@code int}, only the low-order 32 bits are returned. + * Note that this conversion can lose information about the + * overall magnitude of the BigInteger value as well as return a + * result with the opposite sign. + * + * @return this BigInteger converted to an {@code int}. + */ + public int intValue() { + int result = 0; + result = getInt(0); + return result; + } + + /** + * Converts this BigInteger to a {@code long}. This + * conversion is analogous to a + * narrowing primitive conversion from {@code long} to + * {@code int} as defined in section 5.1.3 of + * The Java™ Language Specification: + * if this BigInteger is too big to fit in a + * {@code long}, only the low-order 64 bits are returned. + * Note that this conversion can lose information about the + * overall magnitude of the BigInteger value as well as return a + * result with the opposite sign. + * + * @return this BigInteger converted to a {@code long}. + */ + public long longValue() { + long result = 0; + + for (int i=1; i>=0; i--) + result = (result << 32) + (getInt(i) & LONG_MASK); + return result; + } + + /** + * Converts this BigInteger to a {@code float}. This + * conversion is similar to the + * narrowing primitive conversion from {@code double} to + * {@code float} as defined in section 5.1.3 of + * The Java™ Language Specification: + * if this BigInteger has too great a magnitude + * to represent as a {@code float}, it will be converted to + * {@link Float#NEGATIVE_INFINITY} or {@link + * Float#POSITIVE_INFINITY} as appropriate. Note that even when + * the return value is finite, this conversion can lose + * information about the precision of the BigInteger value. + * + * @return this BigInteger converted to a {@code float}. + */ + public float floatValue() { + // Somewhat inefficient, but guaranteed to work. + return Float.parseFloat(this.toString()); + } + + /** + * Converts this BigInteger to a {@code double}. This + * conversion is similar to the + * narrowing primitive conversion from {@code double} to + * {@code float} as defined in section 5.1.3 of + * The Java™ Language Specification: + * if this BigInteger has too great a magnitude + * to represent as a {@code double}, it will be converted to + * {@link Double#NEGATIVE_INFINITY} or {@link + * Double#POSITIVE_INFINITY} as appropriate. Note that even when + * the return value is finite, this conversion can lose + * information about the precision of the BigInteger value. + * + * @return this BigInteger converted to a {@code double}. + */ + public double doubleValue() { + // Somewhat inefficient, but guaranteed to work. + return Double.parseDouble(this.toString()); + } + + /** + * Returns a copy of the input array stripped of any leading zero bytes. + */ + private static int[] stripLeadingZeroInts(int val[]) { + int vlen = val.length; + int keep; + + // Find first nonzero byte + for (keep = 0; keep < vlen && val[keep] == 0; keep++) + ; + return java.util.Arrays.copyOfRange(val, keep, vlen); + } + + /** + * Returns the input array stripped of any leading zero bytes. + * Since the source is trusted the copying may be skipped. + */ + private static int[] trustedStripLeadingZeroInts(int val[]) { + int vlen = val.length; + int keep; + + // Find first nonzero byte + for (keep = 0; keep < vlen && val[keep] == 0; keep++) + ; + return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen); + } + + /** + * Returns a copy of the input array stripped of any leading zero bytes. + */ + private static int[] stripLeadingZeroBytes(byte a[]) { + int byteLength = a.length; + int keep; + + // Find first nonzero byte + for (keep = 0; keep < byteLength && a[keep]==0; keep++) + ; + + // Allocate new array and copy relevant part of input array + int intLength = ((byteLength - keep) + 3) >>> 2; + int[] result = new int[intLength]; + int b = byteLength - 1; + for (int i = intLength-1; i >= 0; i--) { + result[i] = a[b--] & 0xff; + int bytesRemaining = b - keep + 1; + int bytesToTransfer = Math.min(3, bytesRemaining); + for (int j=8; j <= (bytesToTransfer << 3); j += 8) + result[i] |= ((a[b--] & 0xff) << j); + } + return result; + } + + /** + * Takes an array a representing a negative 2's-complement number and + * returns the minimal (no leading zero bytes) unsigned whose value is -a. + */ + private static int[] makePositive(byte a[]) { + int keep, k; + int byteLength = a.length; + + // Find first non-sign (0xff) byte of input + for (keep=0; keep= 0; i--) { + result[i] = a[b--] & 0xff; + int numBytesToTransfer = Math.min(3, b-keep+1); + if (numBytesToTransfer < 0) + numBytesToTransfer = 0; + for (int j=8; j <= 8*numBytesToTransfer; j += 8) + result[i] |= ((a[b--] & 0xff) << j); + + // Mask indicates which bits must be complemented + int mask = -1 >>> (8*(3-numBytesToTransfer)); + result[i] = ~result[i] & mask; + } + + // Add one to one's complement to generate two's complement + for (int i=result.length-1; i>=0; i--) { + result[i] = (int)((result[i] & LONG_MASK) + 1); + if (result[i] != 0) + break; + } + + return result; + } + + /** + * Takes an array a representing a negative 2's-complement number and + * returns the minimal (no leading zero ints) unsigned whose value is -a. + */ + private static int[] makePositive(int a[]) { + int keep, j; + + // Find first non-sign (0xffffffff) int of input + for (keep=0; keep>> 5) + 1; + } + + /* Returns sign bit */ + private int signBit() { + return signum < 0 ? 1 : 0; + } + + /* Returns an int of sign bits */ + private int signInt() { + return signum < 0 ? -1 : 0; + } + + /** + * Returns the specified int of the little-endian two's complement + * representation (int 0 is the least significant). The int number can + * be arbitrarily high (values are logically preceded by infinitely many + * sign ints). + */ + private int getInt(int n) { + if (n < 0) + return 0; + if (n >= mag.length) + return signInt(); + + int magInt = mag[mag.length-n-1]; + + return (signum >= 0 ? magInt : + (n <= firstNonzeroIntNum() ? -magInt : ~magInt)); + } + + /** + * Returns the index of the int that contains the first nonzero int in the + * little-endian binary representation of the magnitude (int 0 is the + * least significant). If the magnitude is zero, return value is undefined. + */ + private int firstNonzeroIntNum() { + int fn = firstNonzeroIntNum - 2; + if (fn == -2) { // firstNonzeroIntNum not initialized yet + fn = 0; + + // Search for the first nonzero int + int i; + int mlen = mag.length; + for (i = mlen - 1; i >= 0 && mag[i] == 0; i--) + ; + fn = mlen - i - 1; + firstNonzeroIntNum = fn + 2; // offset by two to initialize + } + return fn; + } + + /** use serialVersionUID from JDK 1.1. for interoperability */ + private static final long serialVersionUID = -8287574255936472291L; + + /** + * Serializable fields for BigInteger. + * + * @serialField signum int + * signum of this BigInteger. + * @serialField magnitude int[] + * magnitude array of this BigInteger. + * @serialField bitCount int + * number of bits in this BigInteger + * @serialField bitLength int + * the number of bits in the minimal two's-complement + * representation of this BigInteger + * @serialField lowestSetBit int + * lowest set bit in the twos complement representation + */ + private static final ObjectStreamField[] serialPersistentFields = { + new ObjectStreamField("signum", Integer.TYPE), + new ObjectStreamField("magnitude", byte[].class), + new ObjectStreamField("bitCount", Integer.TYPE), + new ObjectStreamField("bitLength", Integer.TYPE), + new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE), + new ObjectStreamField("lowestSetBit", Integer.TYPE) + }; + + /** + * Reconstitute the {@code BigInteger} instance from a stream (that is, + * deserialize it). The magnitude is read in as an array of bytes + * for historical reasons, but it is converted to an array of ints + * and the byte array is discarded. + * Note: + * The current convention is to initialize the cache fields, bitCount, + * bitLength and lowestSetBit, to 0 rather than some other marker value. + * Therefore, no explicit action to set these fields needs to be taken in + * readObject because those fields already have a 0 value be default since + * defaultReadObject is not being used. + */ + private void readObject(java.io.ObjectInputStream s) + throws java.io.IOException, ClassNotFoundException { + /* + * In order to maintain compatibility with previous serialized forms, + * the magnitude of a BigInteger is serialized as an array of bytes. + * The magnitude field is used as a temporary store for the byte array + * that is deserialized. The cached computation fields should be + * transient but are serialized for compatibility reasons. + */ + + // prepare to read the alternate persistent fields + ObjectInputStream.GetField fields = s.readFields(); + + // Read the alternate persistent fields that we care about + int sign = fields.get("signum", -2); + byte[] magnitude = (byte[])fields.get("magnitude", null); + + // Validate signum + if (sign < -1 || sign > 1) { + String message = "BigInteger: Invalid signum value"; + if (fields.defaulted("signum")) + message = "BigInteger: Signum not present in stream"; + throw new java.io.StreamCorruptedException(message); + } + if ((magnitude.length == 0) != (sign == 0)) { + String message = "BigInteger: signum-magnitude mismatch"; + if (fields.defaulted("magnitude")) + message = "BigInteger: Magnitude not present in stream"; + throw new java.io.StreamCorruptedException(message); + } + + // Commit final fields via Unsafe + unsafe.putIntVolatile(this, signumOffset, sign); + + // Calculate mag field from magnitude and discard magnitude + unsafe.putObjectVolatile(this, magOffset, + stripLeadingZeroBytes(magnitude)); + } + + // Support for resetting final fields while deserializing + private static final sun.misc.Unsafe unsafe = sun.misc.Unsafe.getUnsafe(); + private static final long signumOffset; + private static final long magOffset; + static { + try { + signumOffset = unsafe.objectFieldOffset + (BigInteger.class.getDeclaredField("signum")); + magOffset = unsafe.objectFieldOffset + (BigInteger.class.getDeclaredField("mag")); + } catch (Exception ex) { + throw new Error(ex); + } + } + + /** + * Save the {@code BigInteger} instance to a stream. + * The magnitude of a BigInteger is serialized as a byte array for + * historical reasons. + * + * @serialData two necessary fields are written as well as obsolete + * fields for compatibility with older versions. + */ + private void writeObject(ObjectOutputStream s) throws IOException { + // set the values of the Serializable fields + ObjectOutputStream.PutField fields = s.putFields(); + fields.put("signum", signum); + fields.put("magnitude", magSerializedForm()); + // The values written for cached fields are compatible with older + // versions, but are ignored in readObject so don't otherwise matter. + fields.put("bitCount", -1); + fields.put("bitLength", -1); + fields.put("lowestSetBit", -2); + fields.put("firstNonzeroByteNum", -2); + + // save them + s.writeFields(); +} + + /** + * Returns the mag array as an array of bytes. + */ + private byte[] magSerializedForm() { + int len = mag.length; + + int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0])); + int byteLen = (bitLen + 7) >>> 3; + byte[] result = new byte[byteLen]; + + for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0; + i>=0; i--) { + if (bytesCopied == 4) { + nextInt = mag[intIndex--]; + bytesCopied = 1; + } else { + nextInt >>>= 8; + bytesCopied++; + } + result[i] = (byte)nextInt; + } + return result; + } +}