Java tutorial
/* * Copyright (c) 1996, 2018, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this * particular file as subject to the "Classpath" exception as provided * by Oracle in the LICENSE file that accompanied this code. * * This code is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License * version 2 for more details (a copy is included in the LICENSE file that * accompanied this code). * * You should have received a copy of the GNU General Public License version * 2 along with this work; if not, write to the Free Software Foundation, * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. * * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ /* * Portions Copyright (c) 1995 Colin Plumb. All rights reserved. */ package java.math; import java.io.IOException; import java.io.ObjectInputStream; import java.io.ObjectOutputStream; import java.io.ObjectStreamField; import java.util.Arrays; import java.util.Objects; import java.util.Random; import java.util.concurrent.ThreadLocalRandom; import jdk.internal.math.DoubleConsts; import jdk.internal.math.FloatConsts; import jdk.internal.HotSpotIntrinsicCandidate; import jdk.internal.vm.annotation.Stable; /** * Immutable arbitrary-precision integers. All operations behave as if * BigIntegers were represented in two's-complement notation (like Java's * primitive integer types). BigInteger provides analogues to all of Java's * primitive integer operators, and all relevant methods from java.lang.Math. * Additionally, BigInteger provides operations for modular arithmetic, GCD * calculation, primality testing, prime generation, bit manipulation, * and a few other miscellaneous operations. * * <p>Semantics of arithmetic operations exactly mimic those of Java's integer * arithmetic operators, as defined in <i>The Java™ Language Specification</i>. * For example, division by zero throws an {@code ArithmeticException}, and * division of a negative by a positive yields a negative (or zero) remainder. * * <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 since this operation * only makes sense for a fixed sized word and not for a * representation conceptually having an infinite number of leading * virtual sign bits. * * <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 * arbitrarily large abstraction provided by this class ensures that conceptually * there are infinitely many "virtual sign bits" preceding each BigInteger. * * <p>For the sake of brevity and clarity, pseudo-code is used throughout the * descriptions of BigInteger methods. The pseudo-code expression * {@code (i + j)} is shorthand for "a BigInteger whose value is * that of the BigInteger {@code i} plus that of the BigInteger {@code j}." * The pseudo-code expression {@code (i == j)} is shorthand for * "{@code true} if and only if the BigInteger {@code i} represents the same * value as the BigInteger {@code j}." Other pseudo-code expressions are * interpreted similarly. * * <p>All methods and constructors in this class throw * {@code NullPointerException} when passed * a null object reference for any input parameter. * * BigInteger must support values in the range * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) * and may support values outside of that range. * * An {@code ArithmeticException} is thrown when a BigInteger * constructor or method would generate a value outside of the * supported range. * * The range of probable prime values is limited and may be less than * the full supported positive range of {@code BigInteger}. * The range must be at least 1 to 2<sup>500000000</sup>. * * @implNote * In the reference implementation, BigInteger constructors and * operations throw {@code ArithmeticException} when the result is out * of the supported range of * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive). * * @see BigDecimal * @jls 4.2.2 Integer Operations * @author Josh Bloch * @author Michael McCloskey * @author Alan Eliasen * @author Timothy Buktu * @since 1.1 */ 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 <em>must</em> have * a signum of 0. This is necessary to ensures that there is exactly one * representation for each BigInteger value. */ 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; // The following fields are stable variables. A stable variable's value // changes at most once from the default zero value to a non-zero stable // value. A stable value is calculated lazily on demand. /** * One plus the bitCount of this BigInteger. This is a stable variable. * * @see #bitCount */ private int bitCountPlusOne; /** * One plus the bitLength of this BigInteger. This is a stable variable. * (either value is acceptable). * * @see #bitLength() */ private int bitLengthPlusOne; /** * Two plus the lowest set bit of this BigInteger. This is a stable variable. * * @see #getLowestSetBit */ private int lowestSetBitPlusTwo; /** * Two plus the index of the lowest-order int in the magnitude of this * BigInteger that contains a nonzero int. This is a stable variable. The * least significant int has int-number 0, the next int in order of * increasing significance has int-number 1, and so forth. * * <p>Note: never used for a BigInteger with a magnitude of zero. * * @see #firstNonzeroIntNum() */ private int firstNonzeroIntNumPlusTwo; /** * This mask is used to obtain the value of an int as if it were unsigned. */ static final long LONG_MASK = 0xffffffffL; /** * This constant limits {@code mag.length} of BigIntegers to the supported * range. */ private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26) /** * Bit lengths larger than this constant can cause overflow in searchLen * calculation and in BitSieve.singleSearch method. */ private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000; /** * The threshold value for using Karatsuba multiplication. If the number * of ints in both mag arrays are greater than this number, then * Karatsuba multiplication will be used. This value is found * experimentally to work well. */ private static final int KARATSUBA_THRESHOLD = 80; /** * The threshold value for using 3-way Toom-Cook multiplication. * If the number of ints in each mag array is greater than the * Karatsuba threshold, and the number of ints in at least one of * the mag arrays is greater than this threshold, then Toom-Cook * multiplication will be used. */ private static final int TOOM_COOK_THRESHOLD = 240; /** * The threshold value for using Karatsuba squaring. If the number * of ints in the number are larger than this value, * Karatsuba squaring will be used. This value is found * experimentally to work well. */ private static final int KARATSUBA_SQUARE_THRESHOLD = 128; /** * The threshold value for using Toom-Cook squaring. If the number * of ints in the number are larger than this value, * Toom-Cook squaring will be used. This value is found * experimentally to work well. */ private static final int TOOM_COOK_SQUARE_THRESHOLD = 216; /** * The threshold value for using Burnikel-Ziegler division. If the number * of ints in the divisor are larger than this value, Burnikel-Ziegler * division may be used. This value is found experimentally to work well. */ static final int BURNIKEL_ZIEGLER_THRESHOLD = 80; /** * The offset value for using Burnikel-Ziegler division. If the number * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the * number of ints in the dividend is greater than the number of ints in the * divisor plus this value, Burnikel-Ziegler division will be used. This * value is found experimentally to work well. */ static final int BURNIKEL_ZIEGLER_OFFSET = 40; /** * The threshold value for using Schoenhage recursive base conversion. If * the number of ints in the number are larger than this value, * the Schoenhage algorithm will be used. In practice, it appears that the * Schoenhage routine is faster for any threshold down to 2, and is * relatively flat for thresholds between 2-25, so this choice may be * varied within this range for very small effect. */ private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20; /** * The threshold value for using squaring code to perform multiplication * of a {@code BigInteger} instance by itself. If the number of ints in * the number are larger than this value, {@code multiply(this)} will * return {@code square()}. */ private static final int MULTIPLY_SQUARE_THRESHOLD = 20; /** * The threshold for using an intrinsic version of * implMontgomeryXXX to perform Montgomery multiplication. If the * number of ints in the number is more than this value we do not * use the intrinsic. */ private static final int MONTGOMERY_INTRINSIC_THRESHOLD = 512; // Constructors /** * Translates a byte sub-array containing the two's-complement binary * representation of a BigInteger into a BigInteger. The sub-array is * specified via an offset into the array and a length. The sub-array is * assumed to be in <i>big-endian</i> byte-order: the most significant * byte is the element at index {@code off}. The {@code val} array is * assumed to be unchanged for the duration of the constructor call. * * An {@code IndexOutOfBoundsException} is thrown if the length of the array * {@code val} is non-zero and either {@code off} is negative, {@code len} * is negative, or {@code off+len} is greater than the length of * {@code val}. * * @param val byte array containing a sub-array which is the big-endian * two's-complement binary representation of a BigInteger. * @param off the start offset of the binary representation. * @param len the number of bytes to use. * @throws NumberFormatException {@code val} is zero bytes long. * @throws IndexOutOfBoundsException if the provided array offset and * length would cause an index into the byte array to be * negative or greater than or equal to the array length. * @since 9 */ public BigInteger(byte[] val, int off, int len) { if (val.length == 0) { throw new NumberFormatException("Zero length BigInteger"); } Objects.checkFromIndexSize(off, len, val.length); if (val[off] < 0) { mag = makePositive(val, off, len); signum = -1; } else { mag = stripLeadingZeroBytes(val, off, len); signum = (mag.length == 0 ? 0 : 1); } if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * 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. The {@code val} array is assumed to be * unchanged for the duration of the constructor call. * * @param val big-endian two's-complement binary representation of a * BigInteger. * @throws NumberFormatException {@code val} is zero bytes long. */ public BigInteger(byte[] val) { this(val, 0, val.length); } /** * 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. The * {@code val} array is assumed to be unchanged for the duration of * the constructor call. */ private BigInteger(int[] val) { if (val.length == 0) throw new NumberFormatException("Zero length BigInteger"); if (val[0] < 0) { mag = makePositive(val); signum = -1; } else { mag = trustedStripLeadingZeroInts(val); signum = (mag.length == 0 ? 0 : 1); } if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * Translates the sign-magnitude representation of a BigInteger into a * BigInteger. The sign is represented as an integer signum value: -1 for * negative, 0 for zero, or 1 for positive. The magnitude is a sub-array of * a byte array in <i>big-endian</i> byte-order: the most significant byte * is the element at index {@code off}. A zero value of the length * {@code len} is permissible, and will result in a BigInteger value of 0, * whether signum is -1, 0 or 1. The {@code magnitude} array is assumed to * be unchanged for the duration of the constructor call. * * An {@code IndexOutOfBoundsException} is thrown if the length of the array * {@code magnitude} is non-zero and either {@code off} is negative, * {@code len} is negative, or {@code off+len} is greater than the length of * {@code magnitude}. * * @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. * @param off the start offset of the binary representation. * @param len the number of bytes to use. * @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. * @throws IndexOutOfBoundsException if the provided array offset and * length would cause an index into the byte array to be * negative or greater than or equal to the array length. * @since 9 */ public BigInteger(int signum, byte[] magnitude, int off, int len) { if (signum < -1 || signum > 1) { throw (new NumberFormatException("Invalid signum value")); } Objects.checkFromIndexSize(off, len, magnitude.length); // stripLeadingZeroBytes() returns a zero length array if len == 0 this.mag = stripLeadingZeroBytes(magnitude, off, len); if (this.mag.length == 0) { this.signum = 0; } else { if (signum == 0) throw (new NumberFormatException("signum-magnitude mismatch")); this.signum = signum; } if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * Translates the 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 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. The * {@code magnitude} array is assumed to be unchanged for the duration of * the constructor call. * * @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(signum, magnitude, 0, magnitude.length); } /** * 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. The {@code magnitude} array is assumed to be * unchanged for the duration of the constructor call. */ private BigInteger(int signum, int[] magnitude) { this.mag = stripLeadingZeroInts(magnitude); if (signum < -1 || signum > 1) throw (new NumberFormatException("Invalid signum value")); if (this.mag.length == 0) { this.signum = 0; } else { if (signum == 0) throw (new NumberFormatException("signum-magnitude mismatch")); this.signum = signum; } if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * Translates the String representation of a BigInteger in the * specified radix into a BigInteger. The String representation * consists of an optional minus or plus sign followed by a * sequence of one or more digits in the specified radix. The * character-to-digit mapping is provided by {@code * Character.digit}. The String may not contain any extraneous * characters (whitespace, for example). * * @param val String representation of BigInteger. * @param radix radix to be used in interpreting {@code val}. * @throws NumberFormatException {@code val} is not a valid representation * of a BigInteger in the specified radix, or {@code radix} is * outside the range from {@link Character#MIN_RADIX} to * {@link Character#MAX_RADIX}, inclusive. * @see Character#digit */ public BigInteger(String val, int radix) { int cursor = 0, numDigits; final int len = val.length(); if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) throw new NumberFormatException("Radix out of range"); if (len == 0) throw new NumberFormatException("Zero length BigInteger"); // Check for at most one leading sign int sign = 1; int index1 = val.lastIndexOf('-'); int index2 = val.lastIndexOf('+'); if (index1 >= 0) { if (index1 != 0 || index2 >= 0) { throw new NumberFormatException("Illegal embedded sign character"); } sign = -1; cursor = 1; } else if (index2 >= 0) { if (index2 != 0) { throw new NumberFormatException("Illegal embedded sign character"); } cursor = 1; } if (cursor == len) throw new NumberFormatException("Zero length BigInteger"); // Skip leading zeros and compute number of digits in magnitude while (cursor < len && Character.digit(val.charAt(cursor), radix) == 0) { cursor++; } if (cursor == len) { signum = 0; mag = ZERO.mag; return; } numDigits = len - cursor; signum = sign; // Pre-allocate array of expected size. May be too large but can // never be too small. Typically exact. long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1; if (numBits + 31 >= (1L << 32)) { reportOverflow(); } int numWords = (int) (numBits + 31) >>> 5; int[] magnitude = new int[numWords]; // Process first (potentially short) digit group int firstGroupLen = numDigits % digitsPerInt[radix]; if (firstGroupLen == 0) firstGroupLen = digitsPerInt[radix]; String group = val.substring(cursor, cursor += firstGroupLen); magnitude[numWords - 1] = Integer.parseInt(group, radix); if (magnitude[numWords - 1] < 0) throw new NumberFormatException("Illegal digit"); // Process remaining digit groups int superRadix = intRadix[radix]; int groupVal = 0; while (cursor < len) { group = val.substring(cursor, cursor += digitsPerInt[radix]); groupVal = Integer.parseInt(group, radix); if (groupVal < 0) throw new NumberFormatException("Illegal digit"); destructiveMulAdd(magnitude, superRadix, groupVal); } // Required for cases where the array was overallocated. mag = trustedStripLeadingZeroInts(magnitude); if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /* * Constructs a new BigInteger using a char array with radix=10. * Sign is precalculated outside and not allowed in the val. The {@code val} * array is assumed to be unchanged for the duration of the constructor * call. */ BigInteger(char[] val, int sign, int len) { int cursor = 0, numDigits; // Skip leading zeros and compute number of digits in magnitude while (cursor < len && Character.digit(val[cursor], 10) == 0) { cursor++; } if (cursor == len) { signum = 0; mag = ZERO.mag; return; } numDigits = len - cursor; signum = sign; // Pre-allocate array of expected size int numWords; if (len < 10) { numWords = 1; } else { long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1; if (numBits + 31 >= (1L << 32)) { reportOverflow(); } numWords = (int) (numBits + 31) >>> 5; } int[] magnitude = new int[numWords]; // Process first (potentially short) digit group int firstGroupLen = numDigits % digitsPerInt[10]; if (firstGroupLen == 0) firstGroupLen = digitsPerInt[10]; magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen); // Process remaining digit groups while (cursor < len) { int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]); destructiveMulAdd(magnitude, intRadix[10], groupVal); } mag = trustedStripLeadingZeroInts(magnitude); if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } // Create an integer with the digits between the two indexes // Assumes start < end. The result may be negative, but it // is to be treated as an unsigned value. private int parseInt(char[] source, int start, int end) { int result = Character.digit(source[start++], 10); if (result == -1) throw new NumberFormatException(new String(source)); for (int index = start; index < end; index++) { int nextVal = Character.digit(source[index], 10); if (nextVal == -1) throw new NumberFormatException(new String(source)); result = 10 * result + nextVal; } return result; } // bitsPerDigit in the given radix times 1024 // Rounded up to avoid underallocation. private static long bitsPerDigit[] = { 0, 0, 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672, 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633, 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210, 5253, 5295 }; // Multiply x array times word y in place, and add word z private static void destructiveMulAdd(int[] x, int y, int z) { // Perform the multiplication word by word long ylong = y & LONG_MASK; long zlong = z & LONG_MASK; int len = x.length; long product = 0; long carry = 0; for (int i = len - 1; i >= 0; i--) { product = ylong * (x[i] & LONG_MASK) + carry; x[i] = (int) product; carry = product >>> 32; } // Perform the addition long sum = (x[len - 1] & LONG_MASK) + zlong; x[len - 1] = (int) sum; carry = sum >>> 32; for (int i = len - 2; i >= 0; i--) { sum = (x[i] & LONG_MASK) + carry; x[i] = (int) sum; carry = sum >>> 32; } } /** * Translates the decimal String representation of a BigInteger into a * BigInteger. The String representation consists of an optional minus * sign followed by a sequence of one or more decimal digits. The * character-to-digit mapping is provided by {@code Character.digit}. * The String may not contain any extraneous characters (whitespace, for * example). * * @param val decimal String representation of BigInteger. * @throws NumberFormatException {@code val} is not a valid representation * of a BigInteger. * @see Character#digit */ public BigInteger(String val) { this(val, 10); } /** * Constructs a randomly generated BigInteger, uniformly distributed over * the range 0 to (2<sup>{@code numBits}</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. * * @apiNote 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} or {@code bitLength} is too large. * @see #bitLength() */ public BigInteger(int bitLength, int certainty, Random rnd) { BigInteger prime; if (bitLength < 2) throw new ArithmeticException("bitLength < 2"); prime = (bitLength < SMALL_PRIME_THRESHOLD ? smallPrime(bitLength, certainty, rnd) : largePrime(bitLength, certainty, rnd)); signum = 1; mag = prime.mag; } // Minimum size in bits that the requested prime number has // before we use the large prime number generating algorithms. // The cutoff of 95 was chosen empirically for best performance. private static final int SMALL_PRIME_THRESHOLD = 95; // Certainty required to meet the spec of probablePrime private static final int DEFAULT_PRIME_CERTAINTY = 100; /** * Returns a positive BigInteger that is probably prime, with the * specified bitLength. The probability that a BigInteger returned * by this method is composite does not exceed 2<sup>-100</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} or {@code bitLength} is too large. * @see #bitLength() * @since 1.4 */ public static BigInteger probablePrime(int bitLength, Random rnd) { if (bitLength < 2) throw new ArithmeticException("bitLength < 2"); return (bitLength < SMALL_PRIME_THRESHOLD ? smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) : largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd)); } /** * Find a random number of the specified bitLength that is probably prime. * This method is used for smaller primes, its performance degrades on * larger bitlengths. * * This method assumes bitLength > 1. */ private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) { int magLen = (bitLength + 31) >>> 5; int temp[] = new int[magLen]; int highBit = 1 << ((bitLength + 31) & 0x1f); // High bit of high int int highMask = (highBit << 1) - 1; // Bits to keep in high int while (true) { // Construct a candidate for (int i = 0; i < magLen; i++) temp[i] = rnd.nextInt(); temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length if (bitLength > 2) temp[magLen - 1] |= 1; // Make odd if bitlen > 2 BigInteger p = new BigInteger(temp, 1); // Do cheap "pre-test" if applicable if (bitLength > 6) { long r = p.remainder(SMALL_PRIME_PRODUCT).longValue(); if ((r % 3 == 0) || (r % 5 == 0) || (r % 7 == 0) || (r % 11 == 0) || (r % 13 == 0) || (r % 17 == 0) || (r % 19 == 0) || (r % 23 == 0) || (r % 29 == 0) || (r % 31 == 0) || (r % 37 == 0) || (r % 41 == 0)) continue; // Candidate is composite; try another } // All candidates of bitLength 2 and 3 are prime by this point if (bitLength < 4) return p; // Do expensive test if we survive pre-test (or it's inapplicable) if (p.primeToCertainty(certainty, rnd)) return p; } } private static final BigInteger SMALL_PRIME_PRODUCT = valueOf( 3L * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41); /** * Find a random number of the specified bitLength that is probably prime. * This method is more appropriate for larger bitlengths since it uses * a sieve to eliminate most composites before using a more expensive * test. */ private static BigInteger largePrime(int bitLength, int certainty, Random rnd) { BigInteger p; p = new BigInteger(bitLength, rnd).setBit(bitLength - 1); p.mag[p.mag.length - 1] &= 0xfffffffe; // Use a sieve length likely to contain the next prime number int searchLen = getPrimeSearchLen(bitLength); BitSieve searchSieve = new BitSieve(p, searchLen); BigInteger candidate = searchSieve.retrieve(p, certainty, rnd); while ((candidate == null) || (candidate.bitLength() != bitLength)) { p = p.add(BigInteger.valueOf(2 * searchLen)); if (p.bitLength() != bitLength) p = new BigInteger(bitLength, rnd).setBit(bitLength - 1); p.mag[p.mag.length - 1] &= 0xfffffffe; searchSieve = new BitSieve(p, searchLen); candidate = searchSieve.retrieve(p, certainty, rnd); } return candidate; } /** * Returns the first integer greater than this {@code BigInteger} that * is probably prime. The probability that the number returned by this * method is composite does not exceed 2<sup>-100</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} or {@code this} is too large. * @since 1.5 */ public BigInteger nextProbablePrime() { if (this.signum < 0) throw new ArithmeticException("start < 0: " + this); // Handle trivial cases if ((this.signum == 0) || this.equals(ONE)) return TWO; BigInteger result = this.add(ONE); // Fastpath for small numbers if (result.bitLength() < SMALL_PRIME_THRESHOLD) { // Ensure an odd number if (!result.testBit(0)) result = result.add(ONE); while (true) { // Do cheap "pre-test" if applicable if (result.bitLength() > 6) { long r = result.remainder(SMALL_PRIME_PRODUCT).longValue(); if ((r % 3 == 0) || (r % 5 == 0) || (r % 7 == 0) || (r % 11 == 0) || (r % 13 == 0) || (r % 17 == 0) || (r % 19 == 0) || (r % 23 == 0) || (r % 29 == 0) || (r % 31 == 0) || (r % 37 == 0) || (r % 41 == 0)) { result = result.add(TWO); continue; // Candidate is composite; try another } } // All candidates of bitLength 2 and 3 are prime by this point if (result.bitLength() < 4) return result; // The expensive test if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null)) return result; result = result.add(TWO); } } // Start at previous even number if (result.testBit(0)) result = result.subtract(ONE); // Looking for the next large prime int searchLen = getPrimeSearchLen(result.bitLength()); while (true) { BitSieve searchSieve = new BitSieve(result, searchLen); BigInteger candidate = searchSieve.retrieve(result, DEFAULT_PRIME_CERTAINTY, null); if (candidate != null) return candidate; result = result.add(BigInteger.valueOf(2 * searchLen)); } } private static int getPrimeSearchLen(int bitLength) { if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) { throw new ArithmeticException("Prime search implementation restriction on bitLength"); } return bitLength / 20 * 64; } /** * Returns {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. * * This method assumes bitLength > 2. * * @param certainty a measure of the uncertainty that the caller is * willing to tolerate: if the call returns {@code true} * the probability that this BigInteger is prime exceeds * {@code (1 - 1/2<sup>certainty</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; } /** * Returns true iff this BigInteger passes the specified number of * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS * 186-2). * * The following assumptions are made: * This BigInteger is a positive, odd number greater than 2. * iterations<=50. */ private boolean passesMillerRabin(int iterations, Random rnd) { // Find a and m such that m is odd and this == 1 + 2**a * m BigInteger thisMinusOne = this.subtract(ONE); BigInteger m = thisMinusOne; int a = m.getLowestSetBit(); m = m.shiftRight(a); // Do the tests if (rnd == null) { rnd = ThreadLocalRandom.current(); } for (int i = 0; i < iterations; i++) { // Generate a uniform random on (1, this) BigInteger b; do { b = new BigInteger(this.bitLength(), rnd); } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0); int j = 0; BigInteger z = b.modPow(m, this); while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) { if (j > 0 && z.equals(ONE) || ++j == a) return false; z = z.modPow(TWO, this); } } return true; } /** * This internal constructor differs from its public cousin * with the arguments reversed in two ways: it assumes that its * arguments are correct, and it doesn't copy the magnitude array. */ BigInteger(int[] magnitude, int signum) { this.signum = (magnitude.length == 0 ? 0 : signum); this.mag = magnitude; if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * This private constructor is for internal use and assumes that its * arguments are correct. The {@code magnitude} array is assumed to be * unchanged for the duration of the constructor call. */ private BigInteger(byte[] magnitude, int signum) { this.signum = (magnitude.length == 0 ? 0 : signum); this.mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length); if (mag.length >= MAX_MAG_LENGTH) { checkRange(); } } /** * Throws an {@code ArithmeticException} if the {@code BigInteger} would be * out of the supported range. * * @throws ArithmeticException if {@code this} exceeds the supported range. */ private void checkRange() { if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) { reportOverflow(); } } private static void reportOverflow() { throw new ArithmeticException("BigInteger would overflow supported range"); } //Static Factory Methods /** * Returns a BigInteger whose value is equal to that of the * specified {@code long}. * * @apiNote 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 static final int MAX_CONSTANT = 16; @Stable private static final BigInteger[] posConst = new BigInteger[MAX_CONSTANT + 1]; @Stable private static final BigInteger[] negConst = new BigInteger[MAX_CONSTANT + 1]; /** * The cache of powers of each radix. This allows us to not have to * recalculate powers of radix^(2^n) more than once. This speeds * Schoenhage recursive base conversion significantly. */ private static volatile BigInteger[][] powerCache; /** The cache of logarithms of radices for base conversion. */ private static final double[] logCache; /** The natural log of 2. This is used in computing cache indices. */ private static final double LOG_TWO = Math.log(2.0); static { assert 0 < KARATSUBA_THRESHOLD && KARATSUBA_THRESHOLD < TOOM_COOK_THRESHOLD && TOOM_COOK_THRESHOLD < Integer.MAX_VALUE && 0 < KARATSUBA_SQUARE_THRESHOLD && KARATSUBA_SQUARE_THRESHOLD < TOOM_COOK_SQUARE_THRESHOLD && TOOM_COOK_SQUARE_THRESHOLD < Integer.MAX_VALUE : "Algorithm thresholds are inconsistent"; for (int i = 1; i <= MAX_CONSTANT; i++) { int[] magnitude = new int[1]; magnitude[0] = i; posConst[i] = new BigInteger(magnitude, 1); negConst[i] = new BigInteger(magnitude, -1); } /* * Initialize the cache of radix^(2^x) values used for base conversion * with just the very first value. Additional values will be created * on demand. */ powerCache = new BigInteger[Character.MAX_RADIX + 1][]; logCache = new double[Character.MAX_RADIX + 1]; for (int i = Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) { powerCache[i] = new BigInteger[] { BigInteger.valueOf(i) }; logCache[i] = Math.log(i); } } /** * The BigInteger constant zero. * * @since 1.2 */ public static final BigInteger ZERO = new BigInteger(new int[0], 0); /** * The BigInteger constant one. * * @since 1.2 */ public static final BigInteger ONE = valueOf(1); /** * The BigInteger constant two. * * @since 9 */ public static final BigInteger TWO = valueOf(2); /** * The BigInteger constant -1. (Not exported.) */ private static final BigInteger NEGATIVE_ONE = valueOf(-1); /** * The BigInteger constant ten. * * @since 1.5 */ public static final BigInteger TEN = valueOf(10); // Arithmetic Operations /** * Returns a BigInteger whose value is {@code (this + val)}. * * @param val value to be added to this BigInteger. * @return {@code this + val} */ public BigInteger add(BigInteger val) { if (val.signum == 0) return this; if (signum == 0) return val; if (val.signum == signum) return new BigInteger(add(mag, val.mag), signum); int cmp = compareMagnitude(val); if (cmp == 0) return ZERO; int[] resultMag = (cmp > 0 ? subtract(mag, val.mag) : subtract(val.mag, mag)); resultMag = trustedStripLeadingZeroInts(resultMag); return new BigInteger(resultMag, cmp == signum ? 1 : -1); } /** * Package private methods used by BigDecimal code to add a BigInteger * with a long. Assumes val is not equal to INFLATED. */ BigInteger add(long val) { if (val == 0) return this; if (signum == 0) return valueOf(val); if (Long.signum(val) == signum) return new BigInteger(add(mag, Math.abs(val)), signum); int cmp = compareMagnitude(val); if (cmp == 0) return ZERO; int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag)); resultMag = trustedStripLeadingZeroInts(resultMag); return new BigInteger(resultMag, cmp == signum ? 1 : -1); } /** * Adds the contents of the int array x and long value val. This * method allocates a new int array to hold the answer and returns * a reference to that array. Assumes x.length > 0 and val is * non-negative */ private static int[] add(int[] x, long val) { int[] y; long sum = 0; int xIndex = x.length; int[] result; int highWord = (int) (val >>> 32); if (highWord == 0) { result = new int[xIndex]; sum = (x[--xIndex] & LONG_MASK) + val; result[xIndex] = (int) sum; } else { if (xIndex == 1) { result = new int[2]; sum = val + (x[0] & LONG_MASK); result[1] = (int) sum; result[0] = (int) (sum >>> 32); return result; } else { result = new int[xIndex]; sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK); result[xIndex] = (int) sum; sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32); result[xIndex] = (int) sum; } } // Copy remainder of longer number while carry propagation is required boolean carry = (sum >>> 32 != 0); while (xIndex > 0 && carry) carry = ((result[--xIndex] = x[xIndex] + 1) == 0); // Copy remainder of longer number while (xIndex > 0) result[--xIndex] = x[xIndex]; // Grow result if necessary if (carry) { int bigger[] = new int[result.length + 1]; System.arraycopy(result, 0, bigger, 1, result.length); bigger[0] = 0x01; return bigger; } return result; } /** * Adds the contents of the int arrays x and y. This method allocates * a new int array to hold the answer and returns a reference to that * array. */ private static int[] add(int[] x, int[] y) { // If x is shorter, swap the two arrays if (x.length < y.length) { int[] tmp = x; x = y; y = tmp; } int xIndex = x.length; int yIndex = y.length; int result[] = new int[xIndex]; long sum = 0; if (yIndex == 1) { sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK); result[xIndex] = (int) sum; } else { // Add common parts of both numbers while (yIndex > 0) { sum = (x[--xIndex] & LONG_MASK) + (y[--yIndex] & LONG_MASK) + (sum >>> 32); result[xIndex] = (int) sum; } } // Copy remainder of longer number while carry propagation is required boolean carry = (sum >>> 32 != 0); while (xIndex > 0 && carry) carry = ((result[--xIndex] = x[xIndex] + 1) == 0); // Copy remainder of longer number while (xIndex > 0) result[--xIndex] = x[xIndex]; // Grow result if necessary if (carry) { int bigger[] = new int[result.length + 1]; System.arraycopy(result, 0, bigger, 1, result.length); bigger[0] = 0x01; return bigger; } return result; } private static int[] subtract(long val, int[] little) { int highWord = (int) (val >>> 32); if (highWord == 0) { int result[] = new int[1]; result[0] = (int) (val - (little[0] & LONG_MASK)); return result; } else { int result[] = new int[2]; if (little.length == 1) { long difference = ((int) val & LONG_MASK) - (little[0] & LONG_MASK); result[1] = (int) difference; // Subtract remainder of longer number while borrow propagates boolean borrow = (difference >> 32 != 0); if (borrow) { result[0] = highWord - 1; } else { // Copy remainder of longer number result[0] = highWord; } return result; } else { // little.length == 2 long difference = ((int) val & LONG_MASK) - (little[1] & LONG_MASK); result[1] = (int) difference; difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32); result[0] = (int) difference; return result; } } } /** * Subtracts the contents of the second argument (val) from the * first (big). The first int array (big) must represent a larger number * than the second. This method allocates the space necessary to hold the * answer. * assumes val >= 0 */ private static int[] subtract(int[] big, long val) { int highWord = (int) (val >>> 32); int bigIndex = big.length; int result[] = new int[bigIndex]; long difference = 0; if (highWord == 0) { difference = (big[--bigIndex] & LONG_MASK) - val; result[bigIndex] = (int) difference; } else { difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK); result[bigIndex] = (int) difference; difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32); result[bigIndex] = (int) difference; } // Subtract remainder of longer number while borrow propagates boolean borrow = (difference >> 32 != 0); while (bigIndex > 0 && borrow) borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1); // Copy remainder of longer number while (bigIndex > 0) result[--bigIndex] = big[bigIndex]; return result; } /** * Returns a BigInteger whose value is {@code (this - val)}. * * @param val value to be subtracted from this BigInteger. * @return {@code this - val} */ public BigInteger subtract(BigInteger val) { if (val.signum == 0) return this; if (signum == 0) return val.negate(); if (val.signum != signum) return new BigInteger(add(mag, val.mag), signum); int cmp = compareMagnitude(val); if (cmp == 0) return ZERO; int[] resultMag = (cmp > 0 ? subtract(mag, val.mag) : subtract(val.mag, mag)); resultMag = trustedStripLeadingZeroInts(resultMag); return new BigInteger(resultMag, cmp == signum ? 1 : -1); } /** * Subtracts the contents of the second int arrays (little) from the * first (big). The first int array (big) must represent a larger number * than the second. This method allocates the space necessary to hold the * answer. */ private static int[] subtract(int[] big, int[] little) { int bigIndex = big.length; int result[] = new int[bigIndex]; int littleIndex = little.length; long difference = 0; // Subtract common parts of both numbers while (littleIndex > 0) { difference = (big[--bigIndex] & LONG_MASK) - (little[--littleIndex] & LONG_MASK) + (difference >> 32); result[bigIndex] = (int) difference; } // Subtract remainder of longer number while borrow propagates boolean borrow = (difference >> 32 != 0); while (bigIndex > 0 && borrow) borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1); // Copy remainder of longer number while (bigIndex > 0) result[--bigIndex] = big[bigIndex]; return result; } /** * Returns a BigInteger whose value is {@code (this * val)}. * * @implNote An implementation may offer better algorithmic * performance when {@code val == this}. * * @param val value to be multiplied by this BigInteger. * @return {@code this * val} */ public BigInteger multiply(BigInteger val) { return multiply(val, false); } /** * Returns a BigInteger whose value is {@code (this * val)}. If * the invocation is recursive certain overflow checks are skipped. * * @param val value to be multiplied by this BigInteger. * @param isRecursion whether this is a recursive invocation * @return {@code this * val} */ private BigInteger multiply(BigInteger val, boolean isRecursion) { if (val.signum == 0 || signum == 0) return ZERO; int xlen = mag.length; if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) { return square(); } int ylen = val.mag.length; if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) { int resultSign = signum == val.signum ? 1 : -1; if (val.mag.length == 1) { return multiplyByInt(mag, val.mag[0], resultSign); } if (mag.length == 1) { return multiplyByInt(val.mag, mag[0], resultSign); } int[] result = multiplyToLen(mag, xlen, val.mag, ylen, null); result = trustedStripLeadingZeroInts(result); return new BigInteger(result, resultSign); } else { if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) { return multiplyKaratsuba(this, val); } else { // // In "Hacker's Delight" section 2-13, p.33, it is explained // that if x and y are unsigned 32-bit quantities and m and n // are their respective numbers of leading zeros within 32 bits, // then the number of leading zeros within their product as a // 64-bit unsigned quantity is either m + n or m + n + 1. If // their product is not to overflow, it cannot exceed 32 bits, // and so the number of leading zeros of the product within 64 // bits must be at least 32, i.e., the leftmost set bit is at // zero-relative position 31 or less. // // From the above there are three cases: // // m + n leftmost set bit condition // ----- ---------------- --------- // >= 32 x <= 64 - 32 = 32 no overflow // == 31 x >= 64 - 32 = 32 possible overflow // <= 30 x >= 64 - 31 = 33 definite overflow // // The "possible overflow" condition cannot be detected by // examning data lengths alone and requires further calculation. // // By analogy, if 'this' and 'val' have m and n as their // respective numbers of leading zeros within 32*MAX_MAG_LENGTH // bits, then: // // m + n >= 32*MAX_MAG_LENGTH no overflow // m + n == 32*MAX_MAG_LENGTH - 1 possible overflow // m + n <= 32*MAX_MAG_LENGTH - 2 definite overflow // // Note however that if the number of ints in the result // were to be MAX_MAG_LENGTH and mag[0] < 0, then there would // be overflow. As a result the leftmost bit (of mag[0]) cannot // be used and the constraints must be adjusted by one bit to: // // m + n > 32*MAX_MAG_LENGTH no overflow // m + n == 32*MAX_MAG_LENGTH possible overflow // m + n < 32*MAX_MAG_LENGTH definite overflow // // The foregoing leading zero-based discussion is for clarity // only. The actual calculations use the estimated bit length // of the product as this is more natural to the internal // array representation of the magnitude which has no leading // zero elements. // if (!isRecursion) { // The bitLength() instance method is not used here as we // are only considering the magnitudes as non-negative. The // Toom-Cook multiplication algorithm determines the sign // at its end from the two signum values. if (bitLength(mag, mag.length) + bitLength(val.mag, val.mag.length) > 32L * MAX_MAG_LENGTH) { reportOverflow(); } } return multiplyToomCook3(this, val); } } } private static BigInteger multiplyByInt(int[] x, int y, int sign) { if (Integer.bitCount(y) == 1) { return new BigInteger(shiftLeft(x, Integer.numberOfTrailingZeros(y)), sign); } int xlen = x.length; int[] rmag = new int[xlen + 1]; long carry = 0; long yl = y & LONG_MASK; int rstart = rmag.length - 1; for (int i = xlen - 1; i >= 0; i--) { long product = (x[i] & LONG_MASK) * yl + carry; rmag[rstart--] = (int) product; carry = product >>> 32; } if (carry == 0L) { rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length); } else { rmag[rstart] = (int) carry; } return new BigInteger(rmag, sign); } /** * Package private methods used by BigDecimal code to multiply a BigInteger * with a long. Assumes v is not equal to INFLATED. */ BigInteger multiply(long v) { if (v == 0 || signum == 0) return ZERO; if (v == BigDecimal.INFLATED) return multiply(BigInteger.valueOf(v)); int rsign = (v > 0 ? signum : -signum); if (v < 0) v = -v; long dh = v >>> 32; // higher order bits long dl = v & LONG_MASK; // lower order bits int xlen = mag.length; int[] value = mag; int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]); long carry = 0; int rstart = rmag.length - 1; for (int i = xlen - 1; i >= 0; i--) { long product = (value[i] & LONG_MASK) * dl + carry; rmag[rstart--] = (int) product; carry = product >>> 32; } rmag[rstart] = (int) carry; if (dh != 0L) { carry = 0; rstart = rmag.length - 2; for (int i = xlen - 1; i >= 0; i--) { long product = (value[i] & LONG_MASK) * dh + (rmag[rstart] & LONG_MASK) + carry; rmag[rstart--] = (int) product; carry = product >>> 32; } rmag[0] = (int) carry; } if (carry == 0L) rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length); return new BigInteger(rmag, rsign); } /** * Multiplies int arrays x and y to the specified lengths and places * the result into z. There will be no leading zeros in the resultant array. */ private static int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) { multiplyToLenCheck(x, xlen); multiplyToLenCheck(y, ylen); return implMultiplyToLen(x, xlen, y, ylen, z); } @HotSpotIntrinsicCandidate private static int[] implMultiplyToLen(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; } private static void multiplyToLenCheck(int[] array, int length) { if (length <= 0) { return; // not an error because multiplyToLen won't execute if len <= 0 } Objects.requireNonNull(array); if (length > array.length) { throw new ArrayIndexOutOfBoundsException(length - 1); } } /** * Multiplies two BigIntegers using the Karatsuba multiplication * algorithm. This is a recursive divide-and-conquer algorithm which is * more efficient for large numbers than what is commonly called the * "grade-school" algorithm used in multiplyToLen. If the numbers to be * multiplied have length n, the "grade-school" algorithm has an * asymptotic complexity of O(n^2). In contrast, the Karatsuba algorithm * has complexity of O(n^(log2(3))), or O(n^1.585). It achieves this * increased performance by doing 3 multiplies instead of 4 when * evaluating the product. As it has some overhead, should be used when * both numbers are larger than a certain threshold (found * experimentally). * * See: http://en.wikipedia.org/wiki/Karatsuba_algorithm */ private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) { int xlen = x.mag.length; int ylen = y.mag.length; // The number of ints in each half of the number. int half = (Math.max(xlen, ylen) + 1) / 2; // xl and yl are the lower halves of x and y respectively, // xh and yh are the upper halves. BigInteger xl = x.getLower(half); BigInteger xh = x.getUpper(half); BigInteger yl = y.getLower(half); BigInteger yh = y.getUpper(half); BigInteger p1 = xh.multiply(yh); // p1 = xh*yh BigInteger p2 = xl.multiply(yl); // p2 = xl*yl // p3=(xh+xl)*(yh+yl) BigInteger p3 = xh.add(xl).multiply(yh.add(yl)); // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2 BigInteger result = p1.shiftLeft(32 * half).add(p3.subtract(p1).subtract(p2)).shiftLeft(32 * half).add(p2); if (x.signum != y.signum) { return result.negate(); } else { return result; } } /** * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication * algorithm. This is a recursive divide-and-conquer algorithm which is * more efficient for large numbers than what is commonly called the * "grade-school" algorithm used in multiplyToLen. If the numbers to be * multiplied have length n, the "grade-school" algorithm has an * asymptotic complexity of O(n^2). In contrast, 3-way Toom-Cook has a * complexity of about O(n^1.465). It achieves this increased asymptotic * performance by breaking each number into three parts and by doing 5 * multiplies instead of 9 when evaluating the product. Due to overhead * (additions, shifts, and one division) in the Toom-Cook algorithm, it * should only be used when both numbers are larger than a certain * threshold (found experimentally). This threshold is generally larger * than that for Karatsuba multiplication, so this algorithm is generally * only used when numbers become significantly larger. * * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined * by Marco Bodrato. * * See: http://bodrato.it/toom-cook/ * http://bodrato.it/papers/#WAIFI2007 * * "Towards Optimal Toom-Cook Multiplication for Univariate and * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO; * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133, * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007. * */ private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) { int alen = a.mag.length; int blen = b.mag.length; int largest = Math.max(alen, blen); // k is the size (in ints) of the lower-order slices. int k = (largest + 2) / 3; // Equal to ceil(largest/3) // r is the size (in ints) of the highest-order slice. int r = largest - 2 * k; // Obtain slices of the numbers. a2 and b2 are the most significant // bits of the numbers a and b, and a0 and b0 the least significant. BigInteger a0, a1, a2, b0, b1, b2; a2 = a.getToomSlice(k, r, 0, largest); a1 = a.getToomSlice(k, r, 1, largest); a0 = a.getToomSlice(k, r, 2, largest); b2 = b.getToomSlice(k, r, 0, largest); b1 = b.getToomSlice(k, r, 1, largest); b0 = b.getToomSlice(k, r, 2, largest); BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1; v0 = a0.multiply(b0, true); da1 = a2.add(a0); db1 = b2.add(b0); vm1 = da1.subtract(a1).multiply(db1.subtract(b1), true); da1 = da1.add(a1); db1 = db1.add(b1); v1 = da1.multiply(db1, true); v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(db1.add(b2).shiftLeft(1).subtract(b0), true); vinf = a2.multiply(b2, true); // The algorithm requires two divisions by 2 and one by 3. // All divisions are known to be exact, that is, they do not produce // remainders, and all results are positive. The divisions by 2 are // implemented as right shifts which are relatively efficient, leaving // only an exact division by 3, which is done by a specialized // linear-time algorithm. t2 = v2.subtract(vm1).exactDivideBy3(); tm1 = v1.subtract(vm1).shiftRight(1); t1 = v1.subtract(v0); t2 = t2.subtract(t1).shiftRight(1); t1 = t1.subtract(tm1).subtract(vinf); t2 = t2.subtract(vinf.shiftLeft(1)); tm1 = tm1.subtract(t2); // Number of bits to shift left. int ss = k * 32; BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss) .add(v0); if (a.signum != b.signum) { return result.negate(); } else { return result; } } /** * Returns a slice of a BigInteger for use in Toom-Cook multiplication. * * @param lowerSize The size of the lower-order bit slices. * @param upperSize The size of the higher-order bit slices. * @param slice The index of which slice is requested, which must be a * number from 0 to size-1. Slice 0 is the highest-order bits, and slice * size-1 are the lowest-order bits. Slice 0 may be of different size than * the other slices. * @param fullsize The size of the larger integer array, used to align * slices to the appropriate position when multiplying different-sized * numbers. */ private BigInteger getToomSlice(int lowerSize, int upperSize, int slice, int fullsize) { int start, end, sliceSize, len, offset; len = mag.length; offset = fullsize - len; if (slice == 0) { start = 0 - offset; end = upperSize - 1 - offset; } else { start = upperSize + (slice - 1) * lowerSize - offset; end = start + lowerSize - 1; } if (start < 0) { start = 0; } if (end < 0) { return ZERO; } sliceSize = (end - start) + 1; if (sliceSize <= 0) { return ZERO; } // While performing Toom-Cook, all slices are positive and // the sign is adjusted when the final number is composed. if (start == 0 && sliceSize >= len) { return this.abs(); } int intSlice[] = new int[sliceSize]; System.arraycopy(mag, start, intSlice, 0, sliceSize); return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1); } /** * Does an exact division (that is, the remainder is known to be zero) * of the specified number by 3. This is used in Toom-Cook * multiplication. This is an efficient algorithm that runs in linear * time. If the argument is not exactly divisible by 3, results are * undefined. Note that this is expected to be called with positive * arguments only. */ private BigInteger exactDivideBy3() { int len = mag.length; int[] result = new int[len]; long x, w, q, borrow; borrow = 0L; for (int i = len - 1; i >= 0; i--) { x = (mag[i] & LONG_MASK); w = x - borrow; if (borrow > x) { // Did we make the number go negative? borrow = 1L; } else { borrow = 0L; } // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32). Thus, // the effect of this is to divide by 3 (mod 2^32). // This is much faster than division on most architectures. q = (w * 0xAAAAAAABL) & LONG_MASK; result[i] = (int) q; // Now check the borrow. The second check can of course be // eliminated if the first fails. if (q >= 0x55555556L) { borrow++; if (q >= 0xAAAAAAABL) borrow++; } } result = trustedStripLeadingZeroInts(result); return new BigInteger(result, signum); } /** * Returns a new BigInteger representing n lower ints of the number. * This is used by Karatsuba multiplication and Karatsuba squaring. */ private BigInteger getLower(int n) { int len = mag.length; if (len <= n) { return abs(); } int lowerInts[] = new int[n]; System.arraycopy(mag, len - n, lowerInts, 0, n); return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1); } /** * Returns a new BigInteger representing mag.length-n upper * ints of the number. This is used by Karatsuba multiplication and * Karatsuba squaring. */ private BigInteger getUpper(int n) { int len = mag.length; if (len <= n) { return ZERO; } int upperLen = len - n; int upperInts[] = new int[upperLen]; System.arraycopy(mag, 0, upperInts, 0, upperLen); return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1); } // Squaring /** * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. * * @return {@code this<sup>2</sup>} */ private BigInteger square() { return square(false); } /** * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. If * the invocation is recursive certain overflow checks are skipped. * * @param isRecursion whether this is a recursive invocation * @return {@code this<sup>2</sup>} */ private BigInteger square(boolean isRecursion) { if (signum == 0) { return ZERO; } int len = mag.length; if (len < KARATSUBA_SQUARE_THRESHOLD) { int[] z = squareToLen(mag, len, null); return new BigInteger(trustedStripLeadingZeroInts(z), 1); } else { if (len < TOOM_COOK_SQUARE_THRESHOLD) { return squareKaratsuba(); } else { // // For a discussion of overflow detection see multiply() // if (!isRecursion) { if (bitLength(mag, mag.length) > 16L * MAX_MAG_LENGTH) { reportOverflow(); } } return squareToomCook3(); } } } /** * Squares the contents of the int array x. The result is placed into the * int array z. The contents of x are not changed. */ private static final int[] squareToLen(int[] x, int len, int[] z) { int zlen = len << 1; if (z == null || z.length < zlen) z = new int[zlen]; // Execute checks before calling intrinsified method. implSquareToLenChecks(x, len, z, zlen); return implSquareToLen(x, len, z, zlen); } /** * Parameters validation. */ private static void implSquareToLenChecks(int[] x, int len, int[] z, int zlen) throws RuntimeException { if (len < 1) { throw new IllegalArgumentException("invalid input length: " + len); } if (len > x.length) { throw new IllegalArgumentException("input length out of bound: " + len + " > " + x.length); } if (len * 2 > z.length) { throw new IllegalArgumentException("input length out of bound: " + (len * 2) + " > " + z.length); } if (zlen < 1) { throw new IllegalArgumentException("invalid input length: " + zlen); } if (zlen > z.length) { throw new IllegalArgumentException("input length out of bound: " + len + " > " + z.length); } } /** * Java Runtime may use intrinsic for this method. */ @HotSpotIntrinsicCandidate private static final int[] implSquareToLen(int[] x, int len, int[] z, int zlen) { /* * 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. */ // Store the squares, right shifted one bit (i.e., divided by 2) int lastProductLowWord = 0; for (int j = 0, i = 0; j < len; j++) { long piece = (x[j] & LONG_MASK); long product = piece * piece; z[i++] = (lastProductLowWord << 31) | (int) (product >>> 33); z[i++] = (int) (product >>> 1); lastProductLowWord = (int) product; } // Add in off-diagonal sums for (int i = len, offset = 1; i > 0; i--, offset += 2) { int t = x[i - 1]; t = mulAdd(z, x, offset, i - 1, t); addOne(z, offset - 1, i, t); } // Shift back up and set low bit primitiveLeftShift(z, zlen, 1); z[zlen - 1] |= x[len - 1] & 1; return z; } /** * Squares a BigInteger using the Karatsuba squaring algorithm. It should * be used when both numbers are larger than a certain threshold (found * experimentally). It is a recursive divide-and-conquer algorithm that * has better asymptotic performance than the algorithm used in * squareToLen. */ private BigInteger squareKaratsuba() { int half = (mag.length + 1) / 2; BigInteger xl = getLower(half); BigInteger xh = getUpper(half); BigInteger xhs = xh.square(); // xhs = xh^2 BigInteger xls = xl.square(); // xls = xl^2 // xh^2 << 64 + (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2 return xhs.shiftLeft(half * 32).add(xl.add(xh).square().subtract(xhs.add(xls))).shiftLeft(half * 32) .add(xls); } /** * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm. It * should be used when both numbers are larger than a certain threshold * (found experimentally). It is a recursive divide-and-conquer algorithm * that has better asymptotic performance than the algorithm used in * squareToLen or squareKaratsuba. */ private BigInteger squareToomCook3() { int len = mag.length; // k is the size (in ints) of the lower-order slices. int k = (len + 2) / 3; // Equal to ceil(largest/3) // r is the size (in ints) of the highest-order slice. int r = len - 2 * k; // Obtain slices of the numbers. a2 is the most significant // bits of the number, and a0 the least significant. BigInteger a0, a1, a2; a2 = getToomSlice(k, r, 0, len); a1 = getToomSlice(k, r, 1, len); a0 = getToomSlice(k, r, 2, len); BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1; v0 = a0.square(true); da1 = a2.add(a0); vm1 = da1.subtract(a1).square(true); da1 = da1.add(a1); v1 = da1.square(true); vinf = a2.square(true); v2 = da1.add(a2).shiftLeft(1).subtract(a0).square(true); // The algorithm requires two divisions by 2 and one by 3. // All divisions are known to be exact, that is, they do not produce // remainders, and all results are positive. The divisions by 2 are // implemented as right shifts which are relatively efficient, leaving // only a division by 3. // The division by 3 is done by an optimized algorithm for this case. t2 = v2.subtract(vm1).exactDivideBy3(); tm1 = v1.subtract(vm1).shiftRight(1); t1 = v1.subtract(v0); t2 = t2.subtract(t1).shiftRight(1); t1 = t1.subtract(tm1).subtract(vinf); t2 = t2.subtract(vinf.shiftLeft(1)); tm1 = tm1.subtract(t2); // Number of bits to shift left. int ss = k * 32; return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss).add(v0); } // Division /** * Returns a BigInteger whose value is {@code (this / val)}. * * @param val value by which this BigInteger is to be divided. * @return {@code this / val} * @throws ArithmeticException if {@code val} is zero. */ public BigInteger divide(BigInteger val) { if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { return divideKnuth(val); } else { return divideBurnikelZiegler(val); } } /** * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth. * * @param val value by which this BigInteger is to be divided. * @return {@code this / val} * @throws ArithmeticException if {@code val} is zero. * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean) */ private BigInteger divideKnuth(BigInteger val) { MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); a.divideKnuth(b, q, false); return q.toBigInteger(this.signum * val.signum); } /** * Returns an array of two BigIntegers containing {@code (this / val)} * followed by {@code (this % val)}. * * @param val value by which this BigInteger is to be divided, and the * remainder computed. * @return an array of two BigIntegers: the quotient {@code (this / val)} * is the initial element, and the remainder {@code (this % val)} * is the final element. * @throws ArithmeticException if {@code val} is zero. */ public BigInteger[] divideAndRemainder(BigInteger val) { if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { return divideAndRemainderKnuth(val); } else { return divideAndRemainderBurnikelZiegler(val); } } /** Long division */ private BigInteger[] divideAndRemainderKnuth(BigInteger val) { BigInteger[] result = new BigInteger[2]; MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); MutableBigInteger r = a.divideKnuth(b, q); result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1); result[1] = r.toBigInteger(this.signum); return result; } /** * Returns a BigInteger whose value is {@code (this % val)}. * * @param val value by which this BigInteger is to be divided, and the * remainder computed. * @return {@code this % val} * @throws ArithmeticException if {@code val} is zero. */ public BigInteger remainder(BigInteger val) { if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD || mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) { return remainderKnuth(val); } else { return remainderBurnikelZiegler(val); } } /** Long division */ private BigInteger remainderKnuth(BigInteger val) { MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); return a.divideKnuth(b, q).toBigInteger(this.signum); } /** * Calculates {@code this / val} using the Burnikel-Ziegler algorithm. * @param val the divisor * @return {@code this / val} */ private BigInteger divideBurnikelZiegler(BigInteger val) { return divideAndRemainderBurnikelZiegler(val)[0]; } /** * Calculates {@code this % val} using the Burnikel-Ziegler algorithm. * @param val the divisor * @return {@code this % val} */ private BigInteger remainderBurnikelZiegler(BigInteger val) { return divideAndRemainderBurnikelZiegler(val)[1]; } /** * Computes {@code this / val} and {@code this % val} using the * Burnikel-Ziegler algorithm. * @param val the divisor * @return an array containing the quotient and remainder */ private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) { MutableBigInteger q = new MutableBigInteger(); MutableBigInteger r = new MutableBigInteger(this) .divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q); BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum * val.signum); BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum); return new BigInteger[] { qBigInt, rBigInt }; } /** * Returns a BigInteger whose value is <code>(this<sup>exponent</sup>)</code>. * Note that {@code exponent} is an integer rather than a BigInteger. * * @param exponent exponent to which this BigInteger is to be raised. * @return <code>this<sup>exponent</sup></code> * @throws ArithmeticException {@code exponent} is negative. (This would * cause the operation to yield a non-integer value.) */ public BigInteger pow(int exponent) { if (exponent < 0) { throw new ArithmeticException("Negative exponent"); } if (signum == 0) { return (exponent == 0 ? ONE : this); } BigInteger partToSquare = this.abs(); // Factor out powers of two from the base, as the exponentiation of // these can be done by left shifts only. // The remaining part can then be exponentiated faster. The // powers of two will be multiplied back at the end. int powersOfTwo = partToSquare.getLowestSetBit(); long bitsToShiftLong = (long) powersOfTwo * exponent; if (bitsToShiftLong > Integer.MAX_VALUE) { reportOverflow(); } int bitsToShift = (int) bitsToShiftLong; int remainingBits; // Factor the powers of two out quickly by shifting right, if needed. if (powersOfTwo > 0) { partToSquare = partToSquare.shiftRight(powersOfTwo); remainingBits = partToSquare.bitLength(); if (remainingBits == 1) { // Nothing left but +/- 1? if (signum < 0 && (exponent & 1) == 1) { return NEGATIVE_ONE.shiftLeft(bitsToShift); } else { return ONE.shiftLeft(bitsToShift); } } } else { remainingBits = partToSquare.bitLength(); if (remainingBits == 1) { // Nothing left but +/- 1? if (signum < 0 && (exponent & 1) == 1) { return NEGATIVE_ONE; } else { return ONE; } } } // This is a quick way to approximate the size of the result, // similar to doing log2[n] * exponent. This will give an upper bound // of how big the result can be, and which algorithm to use. long scaleFactor = (long) remainingBits * exponent; // Use slightly different algorithms for small and large operands. // See if the result will safely fit into a long. (Largest 2^63-1) if (partToSquare.mag.length == 1 && scaleFactor <= 62) { // Small number algorithm. Everything fits into a long. int newSign = (signum < 0 && (exponent & 1) == 1 ? -1 : 1); long result = 1; long baseToPow2 = partToSquare.mag[0] & LONG_MASK; int workingExponent = exponent; // Perform exponentiation using repeated squaring trick while (workingExponent != 0) { if ((workingExponent & 1) == 1) { result = result * baseToPow2; } if ((workingExponent >>>= 1) != 0) { baseToPow2 = baseToPow2 * baseToPow2; } } // Multiply back the powers of two (quickly, by shifting left) if (powersOfTwo > 0) { if (bitsToShift + scaleFactor <= 62) { // Fits in long? return valueOf((result << bitsToShift) * newSign); } else { return valueOf(result * newSign).shiftLeft(bitsToShift); } } else { return valueOf(result * newSign); } } else { if ((long) bitLength() * exponent / Integer.SIZE > MAX_MAG_LENGTH) { reportOverflow(); } // Large number algorithm. This is basically identical to // the algorithm above, but calls multiply() and square() // which may use more efficient algorithms for large numbers. BigInteger answer = ONE; int workingExponent = exponent; // Perform exponentiation using repeated squaring trick while (workingExponent != 0) { if ((workingExponent & 1) == 1) { answer = answer.multiply(partToSquare); } if ((workingExponent >>>= 1) != 0) { partToSquare = partToSquare.square(); } } // Multiply back the (exponentiated) powers of two (quickly, // by shifting left) if (powersOfTwo > 0) { answer = answer.shiftLeft(bitsToShift); } if (signum < 0 && (exponent & 1) == 1) { return answer.negate(); } else { return answer; } } } /** * Returns the integer square root of this BigInteger. The integer square * root of the corresponding mathematical integer {@code n} is the largest * mathematical integer {@code s} such that {@code s*s <= n}. It is equal * to the value of {@code floor(sqrt(n))}, where {@code sqrt(n)} denotes the * real square root of {@code n} treated as a real. Note that the integer * square root will be less than the real square root if the latter is not * representable as an integral value. * * @return the integer square root of {@code this} * @throws ArithmeticException if {@code this} is negative. (The square * root of a negative integer {@code val} is * {@code (i * sqrt(-val))} where <i>i</i> is the * <i>imaginary unit</i> and is equal to * {@code sqrt(-1)}.) * @since 9 */ public BigInteger sqrt() { if (this.signum < 0) { throw new ArithmeticException("Negative BigInteger"); } return new MutableBigInteger(this.mag).sqrt().toBigInteger(); } /** * Returns an array of two BigIntegers containing the integer square root * {@code s} of {@code this} and its remainder {@code this - s*s}, * respectively. * * @return an array of two BigIntegers with the integer square root at * offset 0 and the remainder at offset 1 * @throws ArithmeticException if {@code this} is negative. (The square * root of a negative integer {@code val} is * {@code (i * sqrt(-val))} where <i>i</i> is the * <i>imaginary unit</i> and is equal to * {@code sqrt(-1)}.) * @see #sqrt() * @since 9 */ public BigInteger[] sqrtAndRemainder() { BigInteger s = sqrt(); BigInteger r = this.subtract(s.square()); assert r.compareTo(BigInteger.ZERO) >= 0; return new BigInteger[] { s, r }; } /** * Returns a BigInteger whose value is the greatest common divisor of * {@code abs(this)} and {@code abs(val)}. Returns 0 if * {@code this == 0 && val == 0}. * * @param val value with which the GCD is to be computed. * @return {@code GCD(abs(this), abs(val))} */ public BigInteger gcd(BigInteger val) { if (val.signum == 0) return this.abs(); else if (this.signum == 0) return val.abs(); MutableBigInteger a = new MutableBigInteger(this); MutableBigInteger b = new MutableBigInteger(val); MutableBigInteger result = a.hybridGCD(b); return result.toBigInteger(1); } /** * Package private method to return bit length for an integer. */ static int bitLengthForInt(int n) { return 32 - Integer.numberOfLeadingZeros(n); } /** * Left shift int array a up to len by n bits. Returns the array that * results from the shift since space may have to be reallocated. */ private static int[] leftShift(int[] a, int len, int n) { int nInts = n >>> 5; int nBits = n & 0x1F; int bitsInHighWord = bitLengthForInt(a[0]); // If shift can be done without recopy, do so if (n <= (32 - bitsInHighWord)) { primitiveLeftShift(a, len, nBits); return a; } else { // Array must be resized if (nBits <= (32 - bitsInHighWord)) { int result[] = new int[nInts + len]; System.arraycopy(a, 0, result, 0, len); primitiveLeftShift(result, result.length, nBits); return result; } else { int result[] = new int[nInts + len + 1]; System.arraycopy(a, 0, result, 0, len); primitiveRightShift(result, result.length, 32 - nBits); return result; } } } // shifts a up to len right n bits assumes no leading zeros, 0<n<32 static void primitiveRightShift(int[] a, int len, int n) { int n2 = 32 - n; for (int i = len - 1, c = a[i]; i > 0; i--) { int b = c; c = a[i - 1]; a[i] = (c << n2) | (b >>> n); } a[0] >>>= n; } // shifts a up to len left n bits assumes no leading zeros, 0<=n<32 static void primitiveLeftShift(int[] a, int len, int n) { if (len == 0 || n == 0) return; int n2 = 32 - n; for (int i = 0, c = a[i], m = i + len - 1; i < m; i++) { int b = c; c = a[i + 1]; a[i] = (b << n) | (c >>> n2); } a[len - 1] <<= n; } /** * Calculate bitlength of contents of the first len elements an int array, * assuming there are no leading zero ints. */ private static int bitLength(int[] val, int len) { if (len == 0) return 0; return ((len - 1) << 5) + bitLengthForInt(val[0]); } /** * Returns a BigInteger whose value is the absolute value of this * BigInteger. * * @return {@code abs(this)} */ public BigInteger abs() { return (signum >= 0 ? this : this.negate()); } /** * Returns a BigInteger whose value is {@code (-this)}. * * @return {@code -this} */ public BigInteger negate() { return new BigInteger(this.mag, -this.signum); } /** * Returns the signum function of this BigInteger. * * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or * positive. */ public int signum() { return this.signum; } // Modular Arithmetic Operations /** * Returns a BigInteger whose value is {@code (this mod m}). This method * differs from {@code remainder} in that it always returns a * <i>non-negative</i> BigInteger. * * @param m the modulus. * @return {@code this mod m} * @throws ArithmeticException {@code m} ≤ 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 * <code>(this<sup>exponent</sup> mod m)</code>. (Unlike {@code pow}, this * method permits negative exponents.) * * @param exponent the exponent. * @param m the modulus. * @return <code>this<sup>exponent</sup> mod m</code> * @throws ArithmeticException {@code m} ≤ 0 or the exponent is * negative and this BigInteger is not <i>relatively * prime</i> to {@code m}. * @see #modInverse */ public BigInteger modPow(BigInteger exponent, BigInteger m) { if (m.signum <= 0) throw new ArithmeticException("BigInteger: modulus not positive"); // Trivial cases if (exponent.signum == 0) return (m.equals(ONE) ? ZERO : ONE); if (this.equals(ONE)) return (m.equals(ONE) ? ZERO : ONE); if (this.equals(ZERO) && exponent.signum >= 0) return ZERO; if (this.equals(negConst[1]) && (!exponent.testBit(0))) return (m.equals(ONE) ? ZERO : ONE); boolean invertResult; if ((invertResult = (exponent.signum < 0))) exponent = exponent.negate(); BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0 ? this.mod(m) : this); BigInteger result; if (m.testBit(0)) { // odd modulus result = base.oddModPow(exponent, m); } else { /* * Even modulus. Tear it into an "odd part" (m1) and power of two * (m2), exponentiate mod m1, manually exponentiate mod m2, and * use Chinese Remainder Theorem to combine results. */ // Tear m apart into odd part (m1) and power of 2 (m2) int p = m.getLowestSetBit(); // Max pow of 2 that divides m BigInteger m1 = m.shiftRight(p); // m/2**p BigInteger m2 = ONE.shiftLeft(p); // 2**p // Calculate new base from m1 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0 ? this.mod(m1) : this); // Caculate (base ** exponent) mod m1. BigInteger a1 = (m1.equals(ONE) ? ZERO : base2.oddModPow(exponent, m1)); // Calculate (this ** exponent) mod m2 BigInteger a2 = base.modPow2(exponent, p); // Combine results using Chinese Remainder Theorem BigInteger y1 = m2.modInverse(m1); BigInteger y2 = m1.modInverse(m2); if (m.mag.length < MAX_MAG_LENGTH / 2) { result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m); } else { MutableBigInteger t1 = new MutableBigInteger(); new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1); MutableBigInteger t2 = new MutableBigInteger(); new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2); t1.add(t2); MutableBigInteger q = new MutableBigInteger(); result = t1.divide(new MutableBigInteger(m), q).toBigInteger(); } } return (invertResult ? result.modInverse(m) : result); } // Montgomery multiplication. These are wrappers for // implMontgomeryXX routines which are expected to be replaced by // virtual machine intrinsics. We don't use the intrinsics for // very large operands: MONTGOMERY_INTRINSIC_THRESHOLD should be // larger than any reasonable crypto key. private static int[] montgomeryMultiply(int[] a, int[] b, int[] n, int len, long inv, int[] product) { implMontgomeryMultiplyChecks(a, b, n, len, product); if (len > MONTGOMERY_INTRINSIC_THRESHOLD) { // Very long argument: do not use an intrinsic product = multiplyToLen(a, len, b, len, product); return montReduce(product, n, len, (int) inv); } else { return implMontgomeryMultiply(a, b, n, len, inv, materialize(product, len)); } } private static int[] montgomerySquare(int[] a, int[] n, int len, long inv, int[] product) { implMontgomeryMultiplyChecks(a, a, n, len, product); if (len > MONTGOMERY_INTRINSIC_THRESHOLD) { // Very long argument: do not use an intrinsic product = squareToLen(a, len, product); return montReduce(product, n, len, (int) inv); } else { return implMontgomerySquare(a, n, len, inv, materialize(product, len)); } } // Range-check everything. private static void implMontgomeryMultiplyChecks(int[] a, int[] b, int[] n, int len, int[] product) throws RuntimeException { if (len % 2 != 0) { throw new IllegalArgumentException("input array length must be even: " + len); } if (len < 1) { throw new IllegalArgumentException("invalid input length: " + len); } if (len > a.length || len > b.length || len > n.length || (product != null && len > product.length)) { throw new IllegalArgumentException("input array length out of bound: " + len); } } // Make sure that the int array z (which is expected to contain // the result of a Montgomery multiplication) is present and // sufficiently large. private static int[] materialize(int[] z, int len) { if (z == null || z.length < len) z = new int[len]; return z; } // These methods are intended to be replaced by virtual machine // intrinsics. @HotSpotIntrinsicCandidate private static int[] implMontgomeryMultiply(int[] a, int[] b, int[] n, int len, long inv, int[] product) { product = multiplyToLen(a, len, b, len, product); return montReduce(product, n, len, (int) inv); } @HotSpotIntrinsicCandidate private static int[] implMontgomerySquare(int[] a, int[] n, int len, long inv, int[] product) { product = squareToLen(a, len, product); return montReduce(product, n, len, (int) inv); } 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; // Make modLen even. It is conventional to use a cryptographic // modulus that is 512, 768, 1024, or 2048 bits, so this code // will not normally be executed. However, it is necessary for // the correct functioning of the HotSpot intrinsics. if ((modLen & 1) != 0) { int[] x = new int[modLen + 1]; System.arraycopy(mod, 0, x, 1, modLen); mod = x; modLen++; } // Select an appropriate window size int wbits = 0; int ebits = bitLength(exp, exp.length); // if exponent is 65537 (0x10001), use minimum window size if ((ebits != 17) || (exp[0] != 65537)) { while (ebits > bnExpModThreshTable[wbits]) { wbits++; } } // Calculate appropriate table size int tblmask = 1 << wbits; // Allocate table for precomputed odd powers of base in Montgomery form int[][] table = new int[tblmask][]; for (int i = 0; i < tblmask; i++) table[i] = new int[modLen]; // Compute the modular inverse of the least significant 64-bit // digit of the modulus long n0 = (mod[modLen - 1] & LONG_MASK) + ((mod[modLen - 2] & LONG_MASK) << 32); long inv = -MutableBigInteger.inverseMod64(n0); // Convert base to Montgomery form int[] a = leftShift(base, base.length, modLen << 5); MutableBigInteger q = new MutableBigInteger(), a2 = new MutableBigInteger(a), b2 = new MutableBigInteger(mod); b2.normalize(); // MutableBigInteger.divide() assumes that its // divisor is in normal form. MutableBigInteger r = a2.divide(b2, q); table[0] = r.toIntArray(); // Pad table[0] with leading zeros so its length is at least modLen if (table[0].length < modLen) { int offset = modLen - table[0].length; int[] t2 = new int[modLen]; System.arraycopy(table[0], 0, t2, offset, table[0].length); table[0] = t2; } // Set b to the square of the base int[] b = montgomerySquare(table[0], mod, modLen, inv, null); // Set t to high half of b int[] t = Arrays.copyOf(b, modLen); // Fill in the table with odd powers of the base for (int i = 1; i < tblmask; i++) { table[i] = montgomeryMultiply(t, table[i - 1], mod, modLen, inv, null); } // Pre load the window that slides over the exponent int bitpos = 1 << ((ebits - 1) & (32 - 1)); int buf = 0; int elen = exp.length; int eIndex = 0; for (int i = 0; i <= wbits; i++) { buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0) ? 1 : 0); bitpos >>>= 1; if (bitpos == 0) { eIndex++; bitpos = 1 << (32 - 1); elen--; } } int multpos = ebits; // The first iteration, which is hoisted out of the main loop ebits--; boolean isone = true; multpos = ebits - wbits; while ((buf & 1) == 0) { buf >>>= 1; multpos++; } int[] mult = table[buf >>> 1]; buf = 0; if (multpos == ebits) isone = false; // The main loop while (true) { ebits--; // Advance the window buf <<= 1; if (elen != 0) { buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0; bitpos >>>= 1; if (bitpos == 0) { eIndex++; bitpos = 1 << (32 - 1); elen--; } } // Examine the window for pending multiplies if ((buf & tblmask) != 0) { multpos = ebits - wbits; while ((buf & 1) == 0) { buf >>>= 1; multpos++; } mult = table[buf >>> 1]; buf = 0; } // Perform multiply if (ebits == multpos) { if (isone) { b = mult.clone(); isone = false; } else { t = b; a = montgomeryMultiply(t, mult, mod, modLen, inv, a); t = a; a = b; b = t; } } // Check if done if (ebits == 0) break; // Square the input if (!isone) { t = b; a = montgomerySquare(t, mod, modLen, inv, a); t = a; a = b; b = t; } } // Convert result out of Montgomery form and return int[] t2 = new int[2 * modLen]; System.arraycopy(b, 0, t2, modLen, modLen); b = montReduce(t2, mod, modLen, (int) inv); t2 = Arrays.copyOf(b, modLen); return new BigInteger(1, t2); } /** * Montgomery reduce n, modulo mod. This reduces modulo mod and divides * by 2^(32*mlen). Adapted from Colin Plumb's C library. */ private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) { int c = 0; int len = mlen; int offset = 0; do { int nEnd = n[n.length - 1 - offset]; int carry = mulAdd(n, mod, offset, mlen, inv * nEnd); c += addOne(n, offset, mlen, carry); offset++; } while (--len > 0); while (c > 0) c += subN(n, mod, mlen); while (intArrayCmpToLen(n, mod, mlen) >= 0) subN(n, mod, mlen); return n; } /* * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than, * equal to, or greater than arg2 up to length len. */ private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) { for (int i = 0; i < len; i++) { long b1 = arg1[i] & LONG_MASK; long b2 = arg2[i] & LONG_MASK; if (b1 < b2) return -1; if (b1 > b2) return 1; } return 0; } /** * Subtracts two numbers of same length, returning borrow. */ private static int subN(int[] a, int[] b, int len) { long sum = 0; while (--len >= 0) { sum = (a[len] & LONG_MASK) - (b[len] & LONG_MASK) + (sum >> 32); a[len] = (int) sum; } return (int) (sum >> 32); } /** * Multiply an array by one word k and add to result, return the carry */ static int mulAdd(int[] out, int[] in, int offset, int len, int k) { implMulAddCheck(out, in, offset, len, k); return implMulAdd(out, in, offset, len, k); } /** * Parameters validation. */ private static void implMulAddCheck(int[] out, int[] in, int offset, int len, int k) { if (len > in.length) { throw new IllegalArgumentException("input length is out of bound: " + len + " > " + in.length); } if (offset < 0) { throw new IllegalArgumentException("input offset is invalid: " + offset); } if (offset > (out.length - 1)) { throw new IllegalArgumentException( "input offset is out of bound: " + offset + " > " + (out.length - 1)); } if (len > (out.length - offset)) { throw new IllegalArgumentException("input len is out of bound: " + len + " > " + (out.length - offset)); } } /** * Java Runtime may use intrinsic for this method. */ @HotSpotIntrinsicCandidate private static int implMulAdd(int[] out, int[] in, int offset, int len, int k) { long kLong = k & LONG_MASK; long carry = 0; offset = out.length - offset - 1; for (int j = len - 1; j >= 0; j--) { long product = (in[j] & LONG_MASK) * kLong + (out[offset] & LONG_MASK) + carry; out[offset--] = (int) product; carry = product >>> 32; } return (int) carry; } /** * Add one word to the number a mlen words into a. Return the resulting * carry. */ static int addOne(int[] a, int offset, int mlen, int carry) { offset = a.length - 1 - mlen - offset; long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK); a[offset] = (int) t; if ((t >>> 32) == 0) return 0; while (--mlen >= 0) { if (--offset < 0) { // Carry out of number return 1; } else { a[offset]++; if (a[offset] != 0) return 0; } } return 1; } /** * Returns a BigInteger whose value is (this ** exponent) mod (2**p) */ private BigInteger modPow2(BigInteger exponent, int p) { /* * Perform exponentiation using repeated squaring trick, chopping off * high order bits as indicated by modulus. */ BigInteger result = ONE; BigInteger baseToPow2 = this.mod2(p); int expOffset = 0; int limit = exponent.bitLength(); if (this.testBit(0)) limit = (p - 1) < limit ? (p - 1) : limit; while (expOffset < limit) { if (exponent.testBit(expOffset)) result = result.multiply(baseToPow2).mod2(p); expOffset++; if (expOffset < limit) baseToPow2 = baseToPow2.square().mod2(p); } return result; } /** * Returns a BigInteger whose value is this mod(2**p). * Assumes that this {@code BigInteger >= 0} and {@code p > 0}. */ private BigInteger mod2(int p) { if (bitLength() <= p) return this; // Copy remaining ints of mag int numInts = (p + 31) >>> 5; int[] mag = new int[numInts]; System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts); // Mask out any excess bits int excessBits = (numInts << 5) - p; mag[0] &= (1L << (32 - excessBits)) - 1; return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1)); } /** * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}. * * @param m the modulus. * @return {@code this}<sup>-1</sup> {@code mod m}. * @throws ArithmeticException {@code m} ≤ 0, or this BigInteger * has no multiplicative inverse mod m (that is, this BigInteger * is not <i>relatively prime</i> to m). */ public BigInteger modInverse(BigInteger m) { if (m.signum != 1) throw new ArithmeticException("BigInteger: modulus not positive"); if (m.equals(ONE)) return ZERO; // Calculate (this mod m) BigInteger modVal = this; if (signum < 0 || (this.compareMagnitude(m) >= 0)) modVal = this.mod(m); if (modVal.equals(ONE)) return ONE; MutableBigInteger a = new MutableBigInteger(modVal); MutableBigInteger b = new MutableBigInteger(m); MutableBigInteger result = a.mutableModInverse(b); return result.toBigInteger(1); } // Shift Operations /** * Returns a BigInteger whose value is {@code (this << n)}. * The shift distance, {@code n}, may be negative, in which case * this method performs a right shift. * (Computes <code>floor(this * 2<sup>n</sup>)</code>.) * * @param n shift distance, in bits. * @return {@code this << n} * @see #shiftRight */ public BigInteger shiftLeft(int n) { if (signum == 0) return ZERO; if (n > 0) { return new BigInteger(shiftLeft(mag, n), signum); } else if (n == 0) { return this; } else { // Possible int overflow in (-n) is not a trouble, // because shiftRightImpl considers its argument unsigned return shiftRightImpl(-n); } } /** * Returns a magnitude array whose value is {@code (mag << n)}. * The shift distance, {@code n}, is considered unnsigned. * (Computes <code>this * 2<sup>n</sup></code>.) * * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero. * @param n unsigned shift distance, in bits. * @return {@code mag << n} */ private static int[] shiftLeft(int[] mag, int n) { int nInts = n >>> 5; int nBits = n & 0x1f; int magLen = mag.length; int newMag[] = null; if (nBits == 0) { newMag = new int[magLen + nInts]; System.arraycopy(mag, 0, newMag, 0, magLen); } else { int i = 0; int nBits2 = 32 - nBits; int highBits = mag[0] >>> nBits2; if (highBits != 0) { newMag = new int[magLen + nInts + 1]; newMag[i++] = highBits; } else { newMag = new int[magLen + nInts]; } int j = 0; while (j < magLen - 1) newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2; newMag[i] = mag[j] << nBits; } return newMag; } /** * Returns a BigInteger whose value is {@code (this >> n)}. Sign * extension is performed. The shift distance, {@code n}, may be * negative, in which case this method performs a left shift. * (Computes <code>floor(this / 2<sup>n</sup>)</code>.) * * @param n shift distance, in bits. * @return {@code this >> n} * @see #shiftLeft */ public BigInteger shiftRight(int n) { if (signum == 0) return ZERO; if (n > 0) { return shiftRightImpl(n); } else if (n == 0) { return this; } else { // Possible int overflow in {@code -n} is not a trouble, // because shiftLeft considers its argument unsigned return new BigInteger(shiftLeft(mag, -n), signum); } } /** * Returns a BigInteger whose value is {@code (this >> n)}. The shift * distance, {@code n}, is considered unsigned. * (Computes <code>floor(this * 2<sup>-n</sup>)</code>.) * * @param n unsigned shift distance, in bits. * @return {@code this >> n} */ private BigInteger shiftRightImpl(int n) { int nInts = n >>> 5; int nBits = n & 0x1f; int magLen = mag.length; int newMag[] = null; // Special case: entire contents shifted off the end if (nInts >= magLen) return (signum >= 0 ? ZERO : negConst[1]); if (nBits == 0) { int newMagLen = magLen - nInts; newMag = Arrays.copyOf(mag, newMagLen); } else { int i = 0; int highBits = mag[0] >>> nBits; if (highBits != 0) { newMag = new int[magLen - nInts]; newMag[i++] = highBits; } else { newMag = new int[magLen - nInts - 1]; } int nBits2 = 32 - nBits; int j = 0; while (j < magLen - nInts - 1) newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits); } if (signum < 0) { // Find out whether any one-bits were shifted off the end. boolean onesLost = false; for (int i = magLen - 1, j = magLen - nInts; i >= j && !onesLost; i--) onesLost = (mag[i] != 0); if (!onesLost && nBits != 0) onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0); if (onesLost) newMag = javaIncrement(newMag); } return new BigInteger(newMag, signum); } int[] javaIncrement(int[] val) { int lastSum = 0; for (int i = val.length - 1; i >= 0 && lastSum == 0; i--) lastSum = (val[i] += 1); if (lastSum == 0) { val = new int[val.length + 1]; val[0] = 1; } return val; } // Bitwise Operations /** * Returns a BigInteger whose value is {@code (this & val)}. (This * method returns a negative BigInteger if and only if this and val are * both negative.) * * @param val value to be AND'ed with this BigInteger. * @return {@code this & val} */ public BigInteger and(BigInteger val) { int[] result = new int[Math.max(intLength(), val.intLength())]; for (int i = 0; i < result.length; i++) result[i] = (getInt(result.length - i - 1) & val.getInt(result.length - i - 1)); return valueOf(result); } /** * Returns a BigInteger whose value is {@code (this | val)}. (This method * returns a negative BigInteger if and only if either this or val is * negative.) * * @param val value to be OR'ed with this BigInteger. * @return {@code this | val} */ public BigInteger or(BigInteger val) { int[] result = new int[Math.max(intLength(), val.intLength())]; for (int i = 0; i < result.length; i++) result[i] = (getInt(result.length - i - 1) | val.getInt(result.length - i - 1)); return valueOf(result); } /** * Returns a BigInteger whose value is {@code (this ^ val)}. (This method * returns a negative BigInteger if and only if exactly one of this and * val are negative.) * * @param val value to be XOR'ed with this BigInteger. * @return {@code this ^ val} */ public BigInteger xor(BigInteger val) { int[] result = new int[Math.max(intLength(), val.intLength())]; for (int i = 0; i < result.length; i++) result[i] = (getInt(result.length - i - 1) ^ val.getInt(result.length - i - 1)); return valueOf(result); } /** * Returns a BigInteger whose value is {@code (~this)}. (This method * returns a negative value if and only if this BigInteger is * non-negative.) * * @return {@code ~this} */ public BigInteger not() { int[] result = new int[intLength()]; for (int i = 0; i < result.length; i++) result[i] = ~getInt(result.length - i - 1); return valueOf(result); } /** * Returns a BigInteger whose value is {@code (this & ~val)}. This * method, which is equivalent to {@code and(val.not())}, is provided as * a convenience for masking operations. (This method returns a negative * BigInteger if and only if {@code this} is negative and {@code val} is * positive.) * * @param val value to be complemented and AND'ed with this BigInteger. * @return {@code this & ~val} */ public BigInteger andNot(BigInteger val) { int[] result = new int[Math.max(intLength(), val.intLength())]; for (int i = 0; i < result.length; i++) result[i] = (getInt(result.length - i - 1) & ~val.getInt(result.length - i - 1)); return valueOf(result); } // Single Bit Operations /** * Returns {@code true} if and only if the designated bit is set. * (Computes {@code ((this & (1<<n)) != 0)}.) * * @param n index of bit to test. * @return {@code true} if and only if the designated bit is set. * @throws ArithmeticException {@code n} is negative. */ public boolean testBit(int n) { if (n < 0) throw new ArithmeticException("Negative bit address"); return (getInt(n >>> 5) & (1 << (n & 31))) != 0; } /** * Returns a BigInteger whose value is equivalent to this BigInteger * with the designated bit set. (Computes {@code (this | (1<<n))}.) * * @param n index of bit to set. * @return {@code this | (1<<n)} * @throws ArithmeticException {@code n} is negative. */ public BigInteger setBit(int n) { if (n < 0) throw new ArithmeticException("Negative bit address"); int intNum = n >>> 5; int[] result = new int[Math.max(intLength(), intNum + 2)]; for (int i = 0; i < result.length; i++) result[result.length - i - 1] = getInt(i); result[result.length - intNum - 1] |= (1 << (n & 31)); return valueOf(result); } /** * Returns a BigInteger whose value is equivalent to this BigInteger * with the designated bit cleared. * (Computes {@code (this & ~(1<<n))}.) * * @param n index of bit to clear. * @return {@code this & ~(1<<n)} * @throws ArithmeticException {@code n} is negative. */ public BigInteger clearBit(int n) { if (n < 0) throw new ArithmeticException("Negative bit address"); int intNum = n >>> 5; int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)]; for (int i = 0; i < result.length; i++) result[result.length - i - 1] = getInt(i); result[result.length - intNum - 1] &= ~(1 << (n & 31)); return valueOf(result); } /** * Returns a BigInteger whose value is equivalent to this BigInteger * with the designated bit flipped. * (Computes {@code (this ^ (1<<n))}.) * * @param n index of bit to flip. * @return {@code this ^ (1<<n)} * @throws ArithmeticException {@code n} is negative. */ public BigInteger flipBit(int n) { if (n < 0) throw new ArithmeticException("Negative bit address"); int intNum = n >>> 5; int[] result = new int[Math.max(intLength(), intNum + 2)]; for (int i = 0; i < result.length; i++) result[result.length - i - 1] = getInt(i); result[result.length - intNum - 1] ^= (1 << (n & 31)); return valueOf(result); } /** * Returns the index of the rightmost (lowest-order) one bit in this * BigInteger (the number of zero bits to the right of the rightmost * one bit). Returns -1 if this BigInteger contains no one bits. * (Computes {@code (this == 0? -1 : log2(this & -this))}.) * * @return index of the rightmost one bit in this BigInteger. */ public int getLowestSetBit() { int lsb = lowestSetBitPlusTwo - 2; if (lsb == -2) { // lowestSetBit not initialized yet lsb = 0; if (signum == 0) { lsb -= 1; } else { // Search for lowest order nonzero int int i, b; for (i = 0; (b = getInt(i)) == 0; i++) ; lsb += (i << 5) + Integer.numberOfTrailingZeros(b); } lowestSetBitPlusTwo = lsb + 2; } return lsb; } // Miscellaneous Bit Operations /** * Returns the number of bits in the minimal two's-complement * representation of this BigInteger, <em>excluding</em> a sign bit. * For positive BigIntegers, this is equivalent to the number of bits in * the ordinary binary representation. For zero this method returns * {@code 0}. (Computes {@code (ceil(log2(this < 0 ? -this : this+1)))}.) * * @return number of bits in the minimal two's-complement * representation of this BigInteger, <em>excluding</em> a sign bit. */ public int bitLength() { int n = bitLengthPlusOne - 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; } } bitLengthPlusOne = 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() { int bc = bitCountPlusOne - 1; if (bc == -1) { // bitCount not initialized yet bc = 0; // offset by one to initialize // Count the bits in the magnitude for (int i = 0; i < mag.length; i++) bc += Integer.bitCount(mag[i]); if (signum < 0) { // Count the trailing zeros in the magnitude int magTrailingZeroCount = 0, j; for (j = mag.length - 1; mag[j] == 0; j--) magTrailingZeroCount += 32; magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]); bc += magTrailingZeroCount - 1; } bitCountPlusOne = bc + 1; } return bc; } // Primality Testing /** * Returns {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. If * {@code certainty} is ≤ 0, {@code true} is * returned. * * @param certainty a measure of the uncertainty that the caller is * willing to tolerate: if the call returns {@code true} * the probability that this BigInteger is prime exceeds * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of * this method is proportional to the value of this parameter. * @return {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. */ public boolean isProbablePrime(int certainty) { if (certainty <= 0) return true; BigInteger w = this.abs(); if (w.equals(TWO)) return true; if (!w.testBit(0) || w.equals(ONE)) return false; return w.primeToCertainty(certainty, null); } // Comparison Operations /** * Compares this BigInteger with the specified BigInteger. This * method is provided in preference to individual methods for each * of the six boolean comparison operators ({@literal <}, ==, * {@literal >}, {@literal >=}, !=, {@literal <=}). The suggested * idiom for performing these comparisons is: {@code * (x.compareTo(y)} <<i>op</i>> {@code 0)}, where * <<i>op</i>> is one of the six comparison operators. * * @param val BigInteger to which this BigInteger is to be compared. * @return -1, 0 or 1 as this BigInteger is numerically less than, equal * to, or greater than {@code val}. */ public int compareTo(BigInteger val) { if (signum == val.signum) { switch (signum) { case 1: return compareMagnitude(val); case -1: return val.compareMagnitude(this); default: return 0; } } return signum > val.signum ? 1 : -1; } /** * Compares the magnitude array of this BigInteger with the specified * BigInteger's. This is the version of compareTo ignoring sign. * * @param val BigInteger whose magnitude array to be compared. * @return -1, 0 or 1 as this magnitude array is less than, equal to or * greater than the magnitude aray for the specified BigInteger's. */ final int compareMagnitude(BigInteger val) { int[] m1 = mag; int len1 = m1.length; int[] m2 = val.mag; int len2 = m2.length; if (len1 < len2) return -1; if (len1 > len2) return 1; for (int i = 0; i < len1; i++) { int a = m1[i]; int b = m2[i]; if (a != b) return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; } return 0; } /** * Version of compareMagnitude that compares magnitude with long value. * val can't be Long.MIN_VALUE. */ final int compareMagnitude(long val) { assert val != Long.MIN_VALUE; int[] m1 = mag; int len = m1.length; if (len > 2) { return 1; } if (val < 0) { val = -val; } int highWord = (int) (val >>> 32); if (highWord == 0) { if (len < 1) return -1; if (len > 1) return 1; int a = m1[0]; int b = (int) val; if (a != b) { return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; } return 0; } else { if (len < 2) return -1; int a = m1[0]; int b = highWord; if (a != b) { return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; } a = m1[1]; b = (int) val; if (a != b) { return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1; } return 0; } } /** * Compares this BigInteger with the specified Object for equality. * * @param x Object to which this BigInteger is to be compared. * @return {@code true} if and only if the specified Object is a * BigInteger whose value is numerically equal to this BigInteger. */ public boolean equals(Object x) { // This test is just an optimization, which may or may not help if (x == this) return true; if (!(x instanceof BigInteger)) return false; BigInteger xInt = (BigInteger) x; if (xInt.signum != signum) return false; int[] m = mag; int len = m.length; int[] xm = xInt.mag; if (len != xm.length) return false; for (int i = 0; i < len; i++) if (xm[i] != m[i]) return false; return true; } /** * Returns the minimum of this BigInteger and {@code val}. * * @param val value with which the minimum is to be computed. * @return the BigInteger whose value is the lesser of this BigInteger and * {@code val}. If they are equal, either may be returned. */ public BigInteger min(BigInteger val) { return (compareTo(val) < 0 ? this : val); } /** * Returns the maximum of this BigInteger and {@code val}. * * @param val value with which the maximum is to be computed. * @return the BigInteger whose value is the greater of this and * {@code val}. If they are equal, either may be returned. */ public BigInteger max(BigInteger val) { return (compareTo(val) > 0 ? this : val); } // Hash Function /** * Returns the hash code for this BigInteger. * * @return hash code for this BigInteger. */ public int hashCode() { int hashCode = 0; for (int i = 0; i < mag.length; i++) hashCode = (int) (31 * hashCode + (mag[i] & LONG_MASK)); return hashCode * signum; } /** * Returns the String representation of this BigInteger in the * given radix. If the radix is outside the range from {@link * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive, * it will default to 10 (as is the case for * {@code Integer.toString}). The digit-to-character mapping * provided by {@code Character.forDigit} is used, and a minus * sign is prepended if appropriate. (This representation is * compatible with the {@link #BigInteger(String, int) (String, * int)} constructor.) * * @param radix radix of the String representation. * @return String representation of this BigInteger in the given radix. * @see Integer#toString * @see Character#forDigit * @see #BigInteger(java.lang.String, int) */ public String toString(int radix) { if (signum == 0) return "0"; if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) radix = 10; // If it's small enough, use smallToString. if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) return smallToString(radix); // Otherwise use recursive toString, which requires positive arguments. // The results will be concatenated into this StringBuilder StringBuilder sb = new StringBuilder(); if (signum < 0) { toString(this.negate(), sb, radix, 0); sb.insert(0, '-'); } else toString(this, sb, radix, 0); return sb.toString(); } /** This method is used to perform toString when arguments are small. */ private String smallToString(int radix) { if (signum == 0) { return "0"; } // Compute upper bound on number of digit groups and allocate space int maxNumDigitGroups = (4 * mag.length + 6) / 7; String digitGroup[] = new String[maxNumDigitGroups]; // Translate number to string, a digit group at a time BigInteger tmp = this.abs(); int numGroups = 0; while (tmp.signum != 0) { BigInteger d = longRadix[radix]; MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(tmp.mag), b = new MutableBigInteger(d.mag); MutableBigInteger r = a.divide(b, q); BigInteger q2 = q.toBigInteger(tmp.signum * d.signum); BigInteger r2 = r.toBigInteger(tmp.signum * d.signum); digitGroup[numGroups++] = Long.toString(r2.longValue(), radix); tmp = q2; } // Put sign (if any) and first digit group into result buffer StringBuilder buf = new StringBuilder(numGroups * digitsPerLong[radix] + 1); if (signum < 0) { buf.append('-'); } buf.append(digitGroup[numGroups - 1]); // Append remaining digit groups padded with leading zeros for (int i = numGroups - 2; i >= 0; i--) { // Prepend (any) leading zeros for this digit group int numLeadingZeros = digitsPerLong[radix] - digitGroup[i].length(); if (numLeadingZeros != 0) { buf.append(zeros[numLeadingZeros]); } buf.append(digitGroup[i]); } return buf.toString(); } /** * Converts the specified BigInteger to a string and appends to * {@code sb}. This implements the recursive Schoenhage algorithm * for base conversions. * <p> * See Knuth, Donald, _The Art of Computer Programming_, Vol. 2, * Answers to Exercises (4.4) Question 14. * * @param u The number to convert to a string. * @param sb The StringBuilder that will be appended to in place. * @param radix The base to convert to. * @param digits The minimum number of digits to pad to. */ private static void toString(BigInteger u, StringBuilder sb, int radix, int digits) { // If we're smaller than a certain threshold, use the smallToString // method, padding with leading zeroes when necessary. if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) { String s = u.smallToString(radix); // Pad with internal zeros if necessary. // Don't pad if we're at the beginning of the string. if ((s.length() < digits) && (sb.length() > 0)) { for (int i = s.length(); i < digits; i++) { sb.append('0'); } } sb.append(s); return; } int b, n; b = u.bitLength(); // Calculate a value for n in the equation radix^(2^n) = u // and subtract 1 from that value. This is used to find the // cache index that contains the best value to divide u. n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0); BigInteger v = getRadixConversionCache(radix, n); BigInteger[] results; results = u.divideAndRemainder(v); int expectedDigits = 1 << n; // Now recursively build the two halves of each number. toString(results[0], sb, radix, digits - expectedDigits); toString(results[1], sb, radix, expectedDigits); } /** * Returns the value radix^(2^exponent) from the cache. * If this value doesn't already exist in the cache, it is added. * <p> * This could be changed to a more complicated caching method using * {@code Future}. */ private static BigInteger getRadixConversionCache(int radix, int exponent) { BigInteger[] cacheLine = powerCache[radix]; // volatile read if (exponent < cacheLine.length) { return cacheLine[exponent]; } int oldLength = cacheLine.length; cacheLine = Arrays.copyOf(cacheLine, exponent + 1); for (int i = oldLength; i <= exponent; i++) { cacheLine[i] = cacheLine[i - 1].pow(2); } BigInteger[][] pc = powerCache; // volatile read again if (exponent >= pc[radix].length) { pc = pc.clone(); pc[radix] = cacheLine; powerCache = pc; // volatile write, publish } return cacheLine[exponent]; } /* zero[i] is a string of i consecutive zeros. */ private static String zeros[] = new String[64]; static { zeros[63] = "000000000000000000000000000000000000000000000000000000000000000"; for (int i = 0; i < 63; i++) zeros[i] = zeros[63].substring(0, i); } /** * Returns the decimal String representation of this BigInteger. * The digit-to-character mapping provided by * {@code Character.forDigit} is used, and a minus sign is * prepended if appropriate. (This representation is compatible * with the {@link #BigInteger(String) (String)} constructor, and * allows for String concatenation with Java's + operator.) * * @return decimal String representation of this BigInteger. * @see Character#forDigit * @see #BigInteger(java.lang.String) */ public String toString() { return toString(10); } /** * Returns a byte array containing the two's-complement * representation of this BigInteger. The byte array will be in * <i>big-endian</i> byte-order: the most significant byte is in * the zeroth element. The array will contain the minimum number * of bytes required to represent this BigInteger, including at * least one sign bit, which is {@code (ceil((this.bitLength() + * 1)/8))}. (This representation is compatible with the * {@link #BigInteger(byte[]) (byte[])} constructor.) * * @return a byte array containing the two's-complement representation of * this BigInteger. * @see #BigInteger(byte[]) */ public byte[] toByteArray() { int byteLen = bitLength() / 8 + 1; byte[] byteArray = new byte[byteLen]; for (int i = byteLen - 1, bytesCopied = 4, nextInt = 0, intIndex = 0; i >= 0; i--) { if (bytesCopied == 4) { nextInt = getInt(intIndex++); bytesCopied = 1; } else { nextInt >>>= 8; bytesCopied++; } byteArray[i] = (byte) nextInt; } return byteArray; } /** * Converts this BigInteger to an {@code int}. This * conversion is analogous to a * <i>narrowing primitive conversion</i> from {@code long} to * {@code int} as defined in * <cite>The Java™ Language Specification</cite>: * if this BigInteger is too big to fit in an * {@code int}, only the low-order 32 bits are returned. * Note that this conversion can lose information about the * overall magnitude of the BigInteger value as well as return a * result with the opposite sign. * * @return this BigInteger converted to an {@code int}. * @see #intValueExact() * @jls 5.1.3 Narrowing Primitive Conversion */ public int intValue() { int result = 0; result = getInt(0); return result; } /** * Converts this BigInteger to a {@code long}. This * conversion is analogous to a * <i>narrowing primitive conversion</i> from {@code long} to * {@code int} as defined in * <cite>The Java™ Language Specification</cite>: * if this BigInteger is too big to fit in a * {@code long}, only the low-order 64 bits are returned. * Note that this conversion can lose information about the * overall magnitude of the BigInteger value as well as return a * result with the opposite sign. * * @return this BigInteger converted to a {@code long}. * @see #longValueExact() * @jls 5.1.3 Narrowing Primitive Conversion */ public long longValue() { long result = 0; for (int i = 1; i >= 0; i--) result = (result << 32) + (getInt(i) & LONG_MASK); return result; } /** * Converts this BigInteger to a {@code float}. This * conversion is similar to the * <i>narrowing primitive conversion</i> from {@code double} to * {@code float} as defined in * <cite>The Java™ Language Specification</cite>: * if this BigInteger has too great a magnitude * to represent as a {@code float}, it will be converted to * {@link Float#NEGATIVE_INFINITY} or {@link * Float#POSITIVE_INFINITY} as appropriate. Note that even when * the return value is finite, this conversion can lose * information about the precision of the BigInteger value. * * @return this BigInteger converted to a {@code float}. * @jls 5.1.3 Narrowing Primitive Conversion */ public float floatValue() { if (signum == 0) { return 0.0f; } int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1; // exponent == floor(log2(abs(this))) if (exponent < Long.SIZE - 1) { return longValue(); } else if (exponent > Float.MAX_EXPONENT) { return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY; } /* * We need the top SIGNIFICAND_WIDTH bits, including the "implicit" * one bit. To make rounding easier, we pick out the top * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1 * bits, and signifFloor the top SIGNIFICAND_WIDTH. * * It helps to consider the real number signif = abs(this) * * 2^(SIGNIFICAND_WIDTH - 1 - exponent). */ int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH; int twiceSignifFloor; // twiceSignifFloor will be == abs().shiftRight(shift).intValue() // We do the shift into an int directly to improve performance. int nBits = shift & 0x1f; int nBits2 = 32 - nBits; if (nBits == 0) { twiceSignifFloor = mag[0]; } else { twiceSignifFloor = mag[0] >>> nBits; if (twiceSignifFloor == 0) { twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits); } } int signifFloor = twiceSignifFloor >> 1; signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit /* * We round up if either the fractional part of signif is strictly * greater than 0.5 (which is true if the 0.5 bit is set and any lower * bit is set), or if the fractional part of signif is >= 0.5 and * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit * are set). This is equivalent to the desired HALF_EVEN rounding. */ boolean increment = (twiceSignifFloor & 1) != 0 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift); int signifRounded = increment ? signifFloor + 1 : signifFloor; int bits = ((exponent + FloatConsts.EXP_BIAS)) << (FloatConsts.SIGNIFICAND_WIDTH - 1); bits += signifRounded; /* * If signifRounded == 2^24, we'd need to set all of the significand * bits to zero and add 1 to the exponent. This is exactly the behavior * we get from just adding signifRounded to bits directly. If the * exponent is Float.MAX_EXPONENT, we round up (correctly) to * Float.POSITIVE_INFINITY. */ bits |= signum & FloatConsts.SIGN_BIT_MASK; return Float.intBitsToFloat(bits); } /** * Converts this BigInteger to a {@code double}. This * conversion is similar to the * <i>narrowing primitive conversion</i> from {@code double} to * {@code float} as defined in * <cite>The Java™ Language Specification</cite>: * if this BigInteger has too great a magnitude * to represent as a {@code double}, it will be converted to * {@link Double#NEGATIVE_INFINITY} or {@link * Double#POSITIVE_INFINITY} as appropriate. Note that even when * the return value is finite, this conversion can lose * information about the precision of the BigInteger value. * * @return this BigInteger converted to a {@code double}. * @jls 5.1.3 Narrowing Primitive Conversion */ public double doubleValue() { if (signum == 0) { return 0.0; } int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1; // exponent == floor(log2(abs(this))Double) if (exponent < Long.SIZE - 1) { return longValue(); } else if (exponent > Double.MAX_EXPONENT) { return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY; } /* * We need the top SIGNIFICAND_WIDTH bits, including the "implicit" * one bit. To make rounding easier, we pick out the top * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1 * bits, and signifFloor the top SIGNIFICAND_WIDTH. * * It helps to consider the real number signif = abs(this) * * 2^(SIGNIFICAND_WIDTH - 1 - exponent). */ int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH; long twiceSignifFloor; // twiceSignifFloor will be == abs().shiftRight(shift).longValue() // We do the shift into a long directly to improve performance. int nBits = shift & 0x1f; int nBits2 = 32 - nBits; int highBits; int lowBits; if (nBits == 0) { highBits = mag[0]; lowBits = mag[1]; } else { highBits = mag[0] >>> nBits; lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits); if (highBits == 0) { highBits = lowBits; lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits); } } twiceSignifFloor = ((highBits & LONG_MASK) << 32) | (lowBits & LONG_MASK); long signifFloor = twiceSignifFloor >> 1; signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit /* * We round up if either the fractional part of signif is strictly * greater than 0.5 (which is true if the 0.5 bit is set and any lower * bit is set), or if the fractional part of signif is >= 0.5 and * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit * are set). This is equivalent to the desired HALF_EVEN rounding. */ boolean increment = (twiceSignifFloor & 1) != 0 && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift); long signifRounded = increment ? signifFloor + 1 : signifFloor; long bits = (long) ((exponent + DoubleConsts.EXP_BIAS)) << (DoubleConsts.SIGNIFICAND_WIDTH - 1); bits += signifRounded; /* * If signifRounded == 2^53, we'd need to set all of the significand * bits to zero and add 1 to the exponent. This is exactly the behavior * we get from just adding signifRounded to bits directly. If the * exponent is Double.MAX_EXPONENT, we round up (correctly) to * Double.POSITIVE_INFINITY. */ bits |= signum & DoubleConsts.SIGN_BIT_MASK; return Double.longBitsToDouble(bits); } /** * Returns a copy of the input array stripped of any leading zero bytes. */ private static int[] stripLeadingZeroInts(int val[]) { int vlen = val.length; int keep; // Find first nonzero byte for (keep = 0; keep < vlen && val[keep] == 0; keep++) ; return java.util.Arrays.copyOfRange(val, keep, vlen); } /** * Returns the input array stripped of any leading zero bytes. * Since the source is trusted the copying may be skipped. */ private static int[] trustedStripLeadingZeroInts(int val[]) { int vlen = val.length; int keep; // Find first nonzero byte for (keep = 0; keep < vlen && val[keep] == 0; keep++) ; return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen); } /** * Returns a copy of the input array stripped of any leading zero bytes. */ private static int[] stripLeadingZeroBytes(byte a[], int off, int len) { int indexBound = off + len; int keep; // Find first nonzero byte for (keep = off; keep < indexBound && a[keep] == 0; keep++) ; // Allocate new array and copy relevant part of input array int intLength = ((indexBound - keep) + 3) >>> 2; int[] result = new int[intLength]; int b = indexBound - 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 off, int len) { int keep, k; int indexBound = off + len; // Find first non-sign (0xff) byte of input for (keep = off; keep < indexBound && a[keep] == -1; keep++) ; /* Allocate output array. If all non-sign bytes are 0x00, we must * allocate space for one extra output byte. */ for (k = keep; k < indexBound && a[k] == 0; k++) ; int extraByte = (k == indexBound) ? 1 : 0; int intLength = ((indexBound - keep + extraByte) + 3) >>> 2; int result[] = new int[intLength]; /* Copy one's complement of input into output, leaving extra * byte (if it exists) == 0x00 */ int b = indexBound - 1; for (int i = intLength - 1; i >= 0; i--) { result[i] = a[b--] & 0xff; int numBytesToTransfer = Math.min(3, b - keep + 1); if (numBytesToTransfer < 0) numBytesToTransfer = 0; for (int j = 8; j <= 8 * numBytesToTransfer; j += 8) result[i] |= ((a[b--] & 0xff) << j); // Mask indicates which bits must be complemented int mask = -1 >>> (8 * (3 - numBytesToTransfer)); result[i] = ~result[i] & mask; } // Add one to one's complement to generate two's complement for (int i = result.length - 1; i >= 0; i--) { result[i] = (int) ((result[i] & LONG_MASK) + 1); if (result[i] != 0) break; } return result; } /** * Takes an array a representing a negative 2's-complement number and * returns the minimal (no leading zero ints) unsigned whose value is -a. */ private static int[] makePositive(int a[]) { int keep, j; // Find first non-sign (0xffffffff) int of input for (keep = 0; keep < a.length && a[keep] == -1; keep++) ; /* Allocate output array. If all non-sign ints are 0x00, we must * allocate space for one extra output int. */ for (j = keep; j < a.length && a[j] == 0; j++) ; int extraInt = (j == a.length ? 1 : 0); int result[] = new int[a.length - keep + extraInt]; /* Copy one's complement of input into output, leaving extra * int (if it exists) == 0x00 */ for (int i = keep; i < a.length; i++) result[i - keep + extraInt] = ~a[i]; // Add one to one's complement to generate two's complement for (int i = result.length - 1; ++result[i] == 0; i--) ; return result; } /* * The following two arrays are used for fast String conversions. Both * are indexed by radix. The first is the number of digits of the given * radix that can fit in a Java long without "going negative", i.e., the * highest integer n such that radix**n < 2**63. The second is the * "long radix" that tears each number into "long digits", each of which * consists of the number of digits in the corresponding element in * digitsPerLong (longRadix[i] = i**digitPerLong[i]). Both arrays have * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not * used. */ private static int digitsPerLong[] = { 0, 0, 62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14, 14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12 }; private static BigInteger longRadix[] = { null, null, valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL), valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL), valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L), valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L), valueOf(0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L), valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL), valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L), valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L), valueOf(0x5da0e1e53c5c8000L), valueOf(0xb16a458ef403f19L), valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L), valueOf(0x5658597bcaa24000L), valueOf(0x6feb266931a75b7L), valueOf(0xc29e98000000000L), valueOf(0x14adf4b7320334b9L), valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL), valueOf(0x5a3c23e39c000000L), valueOf(0x4e900abb53e6b71L), valueOf(0x7600ec618141000L), valueOf(0xaee5720ee830681L), valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L), valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L), valueOf(0x41c21cb8e1000000L) }; /* * These two arrays are the integer analogue of above. */ private static int digitsPerInt[] = { 0, 0, 30, 19, 15, 13, 11, 11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5 }; private static int intRadix[] = { 0, 0, 0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800, 0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61, 0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000, 0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d, 0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40, 0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41, 0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400 }; /** * These routines provide access to the two's complement representation * of BigIntegers. */ /** * Returns the length of the two's complement representation in ints, * including space for at least one sign bit. */ private int intLength() { return (bitLength() >>> 5) + 1; } /* Returns sign bit */ private int signBit() { return signum < 0 ? 1 : 0; } /* Returns an int of sign bits */ private int signInt() { return signum < 0 ? -1 : 0; } /** * Returns the specified int of the little-endian two's complement * representation (int 0 is the least significant). The int number can * be arbitrarily high (values are logically preceded by infinitely many * sign ints). */ private int getInt(int n) { if (n < 0) return 0; if (n >= mag.length) return signInt(); int magInt = mag[mag.length - n - 1]; return (signum >= 0 ? magInt : (n <= firstNonzeroIntNum() ? -magInt : ~magInt)); } /** * Returns the index of the int that contains the first nonzero int in the * little-endian binary representation of the magnitude (int 0 is the * least significant). If the magnitude is zero, return value is undefined. * * <p>Note: never used for a BigInteger with a magnitude of zero. * @see #getInt. */ private int firstNonzeroIntNum() { int fn = firstNonzeroIntNumPlusTwo - 2; if (fn == -2) { // firstNonzeroIntNum not initialized yet // 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; firstNonzeroIntNumPlusTwo = 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 byte[] * magnitude array of this BigInteger * @serialField bitCount int * appears in the serialized form for backward compatibility * @serialField bitLength int * appears in the serialized form for backward compatibility * @serialField firstNonzeroByteNum int * appears in the serialized form for backward compatibility * @serialField lowestSetBit int * appears in the serialized form for backward compatibility */ 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, bitCountPlusOne, * bitLengthPlusOne and lowestSetBitPlusTwo, 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 by * default since defaultReadObject is not being used. */ private void readObject(java.io.ObjectInputStream s) throws java.io.IOException, ClassNotFoundException { // 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); } int[] mag = stripLeadingZeroBytes(magnitude, 0, magnitude.length); if ((mag.length == 0) != (sign == 0)) { String message = "BigInteger: signum-magnitude mismatch"; if (fields.defaulted("magnitude")) message = "BigInteger: Magnitude not present in stream"; throw new java.io.StreamCorruptedException(message); } // Commit final fields via Unsafe UnsafeHolder.putSign(this, sign); // Calculate mag field from magnitude and discard magnitude UnsafeHolder.putMag(this, mag); if (mag.length >= MAX_MAG_LENGTH) { try { checkRange(); } catch (ArithmeticException e) { throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range"); } } } // Support for resetting final fields while deserializing private static class UnsafeHolder { private static final jdk.internal.misc.Unsafe unsafe = jdk.internal.misc.Unsafe.getUnsafe(); private static final long signumOffset = unsafe.objectFieldOffset(BigInteger.class, "signum"); private static final long magOffset = unsafe.objectFieldOffset(BigInteger.class, "mag"); static void putSign(BigInteger bi, int sign) { unsafe.putInt(bi, signumOffset, sign); } static void putMag(BigInteger bi, int[] magnitude) { unsafe.putReference(bi, magOffset, magnitude); } } /** * Save the {@code BigInteger} instance to a stream. The magnitude of a * {@code BigInteger} is serialized as a byte array for historical reasons. * To maintain compatibility with older implementations, the integers * -1, -1, -2, and -2 are written as the values of the obsolete fields * {@code bitCount}, {@code bitLength}, {@code lowestSetBit}, and * {@code firstNonzeroByteNum}, respectively. These values are compatible * with older implementations, but will be ignored by current * implementations. */ private void writeObject(ObjectOutputStream s) throws IOException { // set the values of the Serializable fields ObjectOutputStream.PutField fields = s.putFields(); fields.put("signum", signum); fields.put("magnitude", magSerializedForm()); // The values written for cached fields are compatible with older // versions, but are ignored in readObject so don't otherwise matter. fields.put("bitCount", -1); fields.put("bitLength", -1); fields.put("lowestSetBit", -2); fields.put("firstNonzeroByteNum", -2); // save them s.writeFields(); } /** * Returns the mag array as an array of bytes. */ private byte[] magSerializedForm() { int len = mag.length; int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0])); int byteLen = (bitLen + 7) >>> 3; byte[] result = new byte[byteLen]; for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0; i >= 0; i--) { if (bytesCopied == 4) { nextInt = mag[intIndex--]; bytesCopied = 1; } else { nextInt >>>= 8; bytesCopied++; } result[i] = (byte) nextInt; } return result; } /** * Converts this {@code BigInteger} to a {@code long}, checking * for lost information. If the value of this {@code BigInteger} * is out of the range of the {@code long} type, then an * {@code ArithmeticException} is thrown. * * @return this {@code BigInteger} converted to a {@code long}. * @throws ArithmeticException if the value of {@code this} will * not exactly fit in a {@code long}. * @see BigInteger#longValue * @since 1.8 */ public long longValueExact() { if (mag.length <= 2 && bitLength() <= 63) return longValue(); else throw new ArithmeticException("BigInteger out of long range"); } /** * Converts this {@code BigInteger} to an {@code int}, checking * for lost information. If the value of this {@code BigInteger} * is out of the range of the {@code int} type, then an * {@code ArithmeticException} is thrown. * * @return this {@code BigInteger} converted to an {@code int}. * @throws ArithmeticException if the value of {@code this} will * not exactly fit in an {@code int}. * @see BigInteger#intValue * @since 1.8 */ public int intValueExact() { if (mag.length <= 1 && bitLength() <= 31) return intValue(); else throw new ArithmeticException("BigInteger out of int range"); } /** * Converts this {@code BigInteger} to a {@code short}, checking * for lost information. If the value of this {@code BigInteger} * is out of the range of the {@code short} type, then an * {@code ArithmeticException} is thrown. * * @return this {@code BigInteger} converted to a {@code short}. * @throws ArithmeticException if the value of {@code this} will * not exactly fit in a {@code short}. * @see BigInteger#shortValue * @since 1.8 */ public short shortValueExact() { if (mag.length <= 1 && bitLength() <= 31) { int value = intValue(); if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE) return shortValue(); } throw new ArithmeticException("BigInteger out of short range"); } /** * Converts this {@code BigInteger} to a {@code byte}, checking * for lost information. If the value of this {@code BigInteger} * is out of the range of the {@code byte} type, then an * {@code ArithmeticException} is thrown. * * @return this {@code BigInteger} converted to a {@code byte}. * @throws ArithmeticException if the value of {@code this} will * not exactly fit in a {@code byte}. * @see BigInteger#byteValue * @since 1.8 */ public byte byteValueExact() { if (mag.length <= 1 && bitLength() <= 31) { int value = intValue(); if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE) return byteValue(); } throw new ArithmeticException("BigInteger out of byte range"); } }