课程作业四 生成随机数并求和,大数运算
1.设计思路:生成随机数,分别赋值给数组。再将其求和输出
程序流程图:
源程序代码:
package test; import javax.swing.JOptionPane; public class Arrays { public static void main(String args[]) { int intVal[] = new int[ 10 ];//定义新数组 String output="数组元素分别为:"; int sum=0; for(int i=0;i<10;i++){ intVal[i] = (int)(Math.random()*100+1);//生成随机数 output+=intVal[i] + " "; //将其加到output后边 } output+="\n"; for(int j=0;j<10;j++){//求和 sum+=intVal[j]; } output+="所有数字的和为:" + sum; JOptionPane.showMessageDialog(null,output,"Results", JOptionPane.PLAIN_MESSAGE);//输出 System.exit(0);//结束程序 } }
结果截图:
编程总结:多吸取以前编程的经验
2.大数
源代码
/* * Copyright (c) 2006, Oracle and/or its affiliates. All rights reserved. * ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms. */ /* * %W% %E% */ package java.math; import java.util.Random; import java.io.*; /** * Immutable arbitrary-precision integers. All operations behave as if * BigIntegers were represented in two's-complement notation (like Java's * primitive integer types). BigInteger provides analogues to all of Java's * primitive integer operators, and all relevant methods from java.lang.Math. * Additionally, BigInteger provides operations for modular arithmetic, GCD * calculation, primality testing, prime generation, bit manipulation, * and a few other miscellaneous operations. * <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. * All of the details in the Spec concerning overflow are ignored, as * BigIntegers are made as large as necessary to accommodate the results of an * operation. * <p> * Semantics of shift operations extend those of Java's shift operators * to allow for negative shift distances. A right-shift with a negative * shift distance results in a left shift, and vice-versa. The unsigned * right shift operator ({@code >>>}) is omitted, as this operation makes * little sense in combination with the "infinite word size" abstraction * provided by this class. * <p> * Semantics of bitwise logical operations exactly mimic those of Java's * bitwise integer operators. The binary operators ({@code and}, * {@code or}, {@code xor}) implicitly perform sign extension on the shorter * of the two operands prior to performing the operation. * <p> * Comparison operations perform signed integer comparisons, analogous to * those performed by Java's relational and equality operators. * <p> * Modular arithmetic operations are provided to compute residues, perform * exponentiation, and compute multiplicative inverses. These methods always * return a non-negative result, between {@code 0} and {@code (modulus - 1)}, * inclusive. * <p> * Bit operations operate on a single bit of the two's-complement * representation of their operand. If necessary, the operand is sign- * extended so that it contains the designated bit. None of the single-bit * operations can produce a BigInteger with a different sign from the * BigInteger being operated on, as they affect only a single bit, and the * "infinite word size" abstraction provided by this class ensures that there * are infinitely many "virtual sign bits" preceding each BigInteger. * <p> * For the sake of brevity and clarity, pseudo-code is used throughout the * descriptions of BigInteger methods. The pseudo-code expression * {@code (i + j)} is shorthand for "a BigInteger whose value is * that of the BigInteger {@code i} plus that of the BigInteger {@code j}." * The pseudo-code expression {@code (i == j)} is shorthand for * "{@code true} if and only if the BigInteger {@code i} represents the same * value as the BigInteger {@code j}." Other pseudo-code expressions are * interpreted similarly. * <p> * All methods and constructors in this class throw * {@code NullPointerException} when passed * a null object reference for any input parameter. * * @see BigDecimal * @author Josh Bloch * @author Michael McCloskey * @since JDK1.1 */ public class BigInteger extends Number implements Comparable<BigInteger> { /** * The signum of this BigInteger: -1 for negative, 0 for zero, or * 1 for positive. Note that the BigInteger zero <i>must</i> have * a signum of 0. This is necessary to ensures that there is exactly one * representation for each BigInteger value. * * @serial */ final int signum; /** * The magnitude of this BigInteger, in <i>big-endian</i> order: the * zeroth element of this array is the most-significant int of the * magnitude. The magnitude must be "minimal" in that the most-significant * int ({@code mag[0]}) must be non-zero. This is necessary to * ensure that there is exactly one representation for each BigInteger * value. Note that this implies that the BigInteger zero has a * zero-length mag array. */ final int[] mag; // These "redundant fields" are initialized with recognizable nonsense // values, and cached the first time they are needed (or never, if they // aren't needed). /** * One plus the bitCount of this BigInteger. Zeros means unitialized. * * @serial * @see #bitCount * @deprecated Deprecated since logical value is offset from stored * value and correction factor is applied in accessor method. */ @Deprecated private int bitCount; /** * One plus the bitLength of this BigInteger. Zeros means unitialized. * (either value is acceptable). * * @serial * @see #bitLength() * @deprecated Deprecated since logical value is offset from stored * value and correction factor is applied in accessor method. */ @Deprecated private int bitLength; /** * Two plus the lowest set bit of this BigInteger, as returned by * getLowestSetBit(). * * @serial * @see #getLowestSetBit * @deprecated Deprecated since logical value is offset from stored * value and correction factor is applied in accessor method. */ @Deprecated private int lowestSetBit; /** * Two plus the index of the lowest-order int in the magnitude of this * BigInteger that contains a nonzero int, or -2 (either value is acceptable). * The least significant int has int-number 0, the next int in order of * increasing significance has int-number 1, and so forth. * @deprecated Deprecated since logical value is offset from stored * value and correction factor is applied in accessor method. */ @Deprecated private int firstNonzeroIntNum; /** * This mask is used to obtain the value of an int as if it were unsigned. */ final static long LONG_MASK = 0xffffffffL; //Constructors /** * Translates a byte array containing the two's-complement binary * representation of a BigInteger into a BigInteger. The input array is * assumed to be in <i>big-endian</i> byte-order: the most significant * byte is in the zeroth element. * * @param val big-endian two's-complement binary representation of * BigInteger. * @throws NumberFormatException {@code val} is zero bytes long. */ public BigInteger(byte[] val) { if (val.length == 0) throw new NumberFormatException("Zero length BigInteger"); if (val[0] < 0) { mag = makePositive(val); signum = -1; } else { mag = stripLeadingZeroBytes(val); signum = (mag.length == 0 ? 0 : 1); } } /** * This private constructor translates an int array containing the * two's-complement binary representation of a BigInteger into a * BigInteger. The input array is assumed to be in <i>big-endian</i> * int-order: the most significant int is in the zeroth element. */ private BigInteger(int[] val) { if (val.length == 0) throw new NumberFormatException("Zero length BigInteger"); if (val[0] < 0) { mag = makePositive(val); signum = -1; } else { mag = trustedStripLeadingZeroInts(val); signum = (mag.length == 0 ? 0 : 1); } } /** * Translates the sign-magnitude representation of a BigInteger into a * BigInteger. The sign is represented as an integer signum value: -1 for * negative, 0 for zero, or 1 for positive. The magnitude is a byte array * in <i>big-endian</i> byte-order: the most significant byte is in the * zeroth element. A zero-length magnitude array is permissible, and will * result in a BigInteger value of 0, whether signum is -1, 0 or 1. * * @param signum signum of the number (-1 for negative, 0 for zero, 1 * for positive). * @param magnitude big-endian binary representation of the magnitude of * the number. * @throws NumberFormatException {@code signum} is not one of the three * legal values (-1, 0, and 1), or {@code signum} is 0 and * {@code magnitude} contains one or more non-zero bytes. */ public BigInteger(int signum, byte[] magnitude) { this.mag = stripLeadingZeroBytes(magnitude); if (signum < -1 || signum > 1) throw(new NumberFormatException("Invalid signum value")); if (this.mag.length==0) { this.signum = 0; } else { if (signum == 0) throw(new NumberFormatException("signum-magnitude mismatch")); this.signum = signum; } } /** * A constructor for internal use that translates the sign-magnitude * representation of a BigInteger into a BigInteger. It checks the * arguments and copies the magnitude so this constructor would be * safe for external use. */ private BigInteger(int signum, int[] magnitude) { this.mag = stripLeadingZeroInts(magnitude); if (signum < -1 || signum > 1) throw(new NumberFormatException("Invalid signum value")); if (this.mag.length==0) { this.signum = 0; } else { if (signum == 0) throw(new NumberFormatException("signum-magnitude mismatch")); this.signum = signum; } } /** * Translates the String representation of a BigInteger in the specified * radix into a BigInteger. The String representation consists of an * optional minus sign followed by a sequence of one or more digits in the * specified radix. The character-to-digit mapping is provided by * {@code Character.digit}. The String may not contain any extraneous * characters (whitespace, for example). * * @param val String representation of BigInteger. * @param radix radix to be used in interpreting {@code val}. * @throws NumberFormatException {@code val} is not a valid representation * of a BigInteger in the specified radix, or {@code radix} is * outside the range from {@link Character#MIN_RADIX} to * {@link Character#MAX_RADIX}, inclusive. * @see Character#digit */ public BigInteger(String val, int radix) { int cursor = 0, numDigits; int len = val.length(); if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) throw new NumberFormatException("Radix out of range"); if (val.length() == 0) throw new NumberFormatException("Zero length BigInteger"); // Check for minus sign int sign = 1; int index = val.lastIndexOf("-"); if (index != -1) { if (index == 0) { if (val.length() == 1) throw new NumberFormatException("Zero length BigInteger"); sign = -1; cursor = 1; } else { throw new NumberFormatException("Illegal embedded minus sign"); } } // Skip leading zeros and compute number of digits in magnitude while (cursor < len && Character.digit(val.charAt(cursor), radix) == 0) cursor++; if (cursor == len) { mag = ZERO.mag; signum = 0; return; } numDigits = len - cursor; signum = sign; // Pre-allocate array of expected size. May be too large but can // never be too small. Typically exact. int numBits = (int)(((numDigits * bitsPerDigit[radix]) >>> 10) + 1); int numWords = (numBits + 31) >>> 5; int[] magnitude = new int[numWords]; // Process first (potentially short) digit group int firstGroupLen = numDigits % digitsPerInt[radix]; if (firstGroupLen == 0) firstGroupLen = digitsPerInt[radix]; String group = val.substring(cursor, cursor += firstGroupLen); magnitude[magnitude.length - 1] = Integer.parseInt(group, radix); if (magnitude[magnitude.length - 1] < 0) throw new NumberFormatException("Illegal digit"); // Process remaining digit groups int superRadix = intRadix[radix]; int groupVal = 0; while (cursor < val.length()) { group = val.substring(cursor, cursor += digitsPerInt[radix]); groupVal = Integer.parseInt(group, radix); if (groupVal < 0) throw new NumberFormatException("Illegal digit"); destructiveMulAdd(magnitude, superRadix, groupVal); } // Required for cases where the array was overallocated. mag = trustedStripLeadingZeroInts(magnitude); } // Constructs a new BigInteger using a char array with radix=10 BigInteger(char[] val) { int cursor = 0, numDigits; int len = val.length; // Check for leading minus sign int sign = 1; if (val[0] == '-') { if (len == 1) throw new NumberFormatException("Zero length BigInteger"); sign = -1; cursor = 1; } // Skip leading zeros and compute number of digits in magnitude while (cursor < len && Character.digit(val[cursor], 10) == 0) cursor++; if (cursor == len) { signum = 0; mag = ZERO.mag; return; } numDigits = len - cursor; signum = sign; // Pre-allocate array of expected size int numWords; if (len < 10) { numWords = 1; } else { int numBits = (int)(((numDigits * bitsPerDigit[10]) >>> 10) + 1); numWords = (numBits + 31) >>> 5; } int magnitude[] = new int[numWords]; // Process first (potentially short) digit group int firstGroupLen = numDigits % digitsPerInt[10]; if (firstGroupLen == 0) firstGroupLen = digitsPerInt[10]; magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen); // Process remaining digit groups while (cursor < len) { int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]); destructiveMulAdd(magnitude, intRadix[10], groupVal); } mag = trustedStripLeadingZeroInts(magnitude); } // Create an integer with the digits between the two indexes // Assumes start < end. The result may be negative, but it // is to be treated as an unsigned value. private int parseInt(char[] source, int start, int end) { int result = Character.digit(source[start++], 10); if (result == -1) throw new NumberFormatException(new String(source)); for (int index = start; index<end; index++) { int nextVal = Character.digit(source[index], 10); if (nextVal == -1) throw new NumberFormatException(new String(source)); result = 10*result + nextVal; } return result; } // bitsPerDigit in the given radix times 1024 // Rounded up to avoid underallocation. private static long bitsPerDigit[] = { 0, 0, 1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672, 3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633, 4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210, 5253, 5295}; // Multiply x array times word y in place, and add word z private static void destructiveMulAdd(int[] x, int y, int z) { // Perform the multiplication word by word long ylong = y & LONG_MASK; long zlong = z & LONG_MASK; int len = x.length; long product = 0; long carry = 0; for (int i = len-1; i >= 0; i--) { product = ylong * (x[i] & LONG_MASK) + carry; x[i] = (int)product; carry = product >>> 32; } // Perform the addition long sum = (x[len-1] & LONG_MASK) + zlong; x[len-1] = (int)sum; carry = sum >>> 32; for (int i = len-2; i >= 0; i--) { sum = (x[i] & LONG_MASK) + carry; x[i] = (int)sum; carry = sum >>> 32; } } /** * Translates the decimal String representation of a BigInteger into a * BigInteger. The String representation consists of an optional minus * sign followed by a sequence of one or more decimal digits. The * character-to-digit mapping is provided by {@code Character.digit}. * The String may not contain any extraneous characters (whitespace, for * example). * * @param val decimal String representation of BigInteger. * @throws NumberFormatException {@code val} is not a valid representation * of a BigInteger. * @see Character#digit */ public BigInteger(String val) { this(val, 10); } /** * Constructs a randomly generated BigInteger, uniformly distributed over * the range {@code 0} to (2<sup>{@code numBits}</sup> - 1), inclusive. * The uniformity of the distribution assumes that a fair source of random * bits is provided in {@code rnd}. Note that this constructor always * constructs a non-negative BigInteger. * * @param numBits maximum bitLength of the new BigInteger. * @param rnd source of randomness to be used in computing the new * BigInteger. * @throws IllegalArgumentException {@code numBits} is negative. * @see #bitLength() */ public BigInteger(int numBits, Random rnd) { this(1, randomBits(numBits, rnd)); } private static byte[] randomBits(int numBits, Random rnd) { if (numBits < 0) throw new IllegalArgumentException("numBits must be non-negative"); int numBytes = (int)(((long)numBits+7)/8); // avoid overflow byte[] randomBits = new byte[numBytes]; // Generate random bytes and mask out any excess bits if (numBytes > 0) { rnd.nextBytes(randomBits); int excessBits = 8*numBytes - numBits; randomBits[0] &= (1 << (8-excessBits)) - 1; } return randomBits; } /** * Constructs a randomly generated positive BigInteger that is probably * prime, with the specified bitLength.<p> * * It is recommended that the {@link #probablePrime probablePrime} * method be used in preference to this constructor unless there * is a compelling need to specify a certainty. * * @param bitLength bitLength of the returned BigInteger. * @param certainty a measure of the uncertainty that the caller is * willing to tolerate. The probability that the new BigInteger * represents a prime number will exceed * (1 - 1/2<sup>{@code certainty}</sup>). The execution time of * this constructor is proportional to the value of this parameter. * @param rnd source of random bits used to select candidates to be * tested for primality. * @throws ArithmeticException {@code bitLength < 2}. * @see #bitLength() */ public BigInteger(int bitLength, int certainty, Random rnd) { BigInteger prime; if (bitLength < 2) throw new ArithmeticException("bitLength < 2"); // The cutoff of 95 was chosen empirically for best performance prime = (bitLength < 95 ? smallPrime(bitLength, certainty, rnd) : largePrime(bitLength, certainty, rnd)); signum = 1; mag = prime.mag; } // Minimum size in bits that the requested prime number has // before we use the large prime number generating algorithms private static final int SMALL_PRIME_THRESHOLD = 95; // Certainty required to meet the spec of probablePrime private static final int DEFAULT_PRIME_CERTAINTY = 100; /** * Returns a positive BigInteger that is probably prime, with the * specified bitLength. The probability that a BigInteger returned * by this method is composite does not exceed 2<sup>-100</sup>. * * @param bitLength bitLength of the returned BigInteger. * @param rnd source of random bits used to select candidates to be * tested for primality. * @return a BigInteger of {@code bitLength} bits that is probably prime * @throws ArithmeticException {@code bitLength < 2}. * @see #bitLength() * @since 1.4 */ public static BigInteger probablePrime(int bitLength, Random rnd) { if (bitLength < 2) throw new ArithmeticException("bitLength < 2"); // The cutoff of 95 was chosen empirically for best performance return (bitLength < SMALL_PRIME_THRESHOLD ? smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) : largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd)); } /** * Find a random number of the specified bitLength that is probably prime. * This method is used for smaller primes, its performance degrades on * larger bitlengths. * * This method assumes bitLength > 1. */ private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) { int magLen = (bitLength + 31) >>> 5; int temp[] = new int[magLen]; int highBit = 1 << ((bitLength+31) & 0x1f); // High bit of high int int highMask = (highBit << 1) - 1; // Bits to keep in high int while(true) { // Construct a candidate for (int i=0; i<magLen; i++) temp[i] = rnd.nextInt(); temp[0] = (temp[0] & highMask) | highBit; // Ensure exact length if (bitLength > 2) temp[magLen-1] |= 1; // Make odd if bitlen > 2 BigInteger p = new BigInteger(temp, 1); // Do cheap "pre-test" if applicable if (bitLength > 6) { long r = p.remainder(SMALL_PRIME_PRODUCT).longValue(); if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) || (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) || (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) continue; // Candidate is composite; try another } // All candidates of bitLength 2 and 3 are prime by this point if (bitLength < 4) return p; // Do expensive test if we survive pre-test (or it's inapplicable) if (p.primeToCertainty(certainty, rnd)) return p; } } private static final BigInteger SMALL_PRIME_PRODUCT = valueOf(3L*5*7*11*13*17*19*23*29*31*37*41); /** * Find a random number of the specified bitLength that is probably prime. * This method is more appropriate for larger bitlengths since it uses * a sieve to eliminate most composites before using a more expensive * test. */ private static BigInteger largePrime(int bitLength, int certainty, Random rnd) { BigInteger p; p = new BigInteger(bitLength, rnd).setBit(bitLength-1); p.mag[p.mag.length-1] &= 0xfffffffe; // Use a sieve length likely to contain the next prime number int searchLen = (bitLength / 20) * 64; BitSieve searchSieve = new BitSieve(p, searchLen); BigInteger candidate = searchSieve.retrieve(p, certainty, rnd); while ((candidate == null) || (candidate.bitLength() != bitLength)) { p = p.add(BigInteger.valueOf(2*searchLen)); if (p.bitLength() != bitLength) p = new BigInteger(bitLength, rnd).setBit(bitLength-1); p.mag[p.mag.length-1] &= 0xfffffffe; searchSieve = new BitSieve(p, searchLen); candidate = searchSieve.retrieve(p, certainty, rnd); } return candidate; } /** * Returns the first integer greater than this {@code BigInteger} that * is probably prime. The probability that the number returned by this * method is composite does not exceed 2<sup>-100</sup>. This method will * never skip over a prime when searching: if it returns {@code p}, there * is no prime {@code q} such that {@code this < q < p}. * * @return the first integer greater than this {@code BigInteger} that * is probably prime. * @throws ArithmeticException {@code this < 0}. * @since 1.5 */ public BigInteger nextProbablePrime() { if (this.signum < 0) throw new ArithmeticException("start < 0: " + this); // Handle trivial cases if ((this.signum == 0) || this.equals(ONE)) return TWO; BigInteger result = this.add(ONE); // Fastpath for small numbers if (result.bitLength() < SMALL_PRIME_THRESHOLD) { // Ensure an odd number if (!result.testBit(0)) result = result.add(ONE); while(true) { // Do cheap "pre-test" if applicable if (result.bitLength() > 6) { long r = result.remainder(SMALL_PRIME_PRODUCT).longValue(); if ((r%3==0) || (r%5==0) || (r%7==0) || (r%11==0) || (r%13==0) || (r%17==0) || (r%19==0) || (r%23==0) || (r%29==0) || (r%31==0) || (r%37==0) || (r%41==0)) { result = result.add(TWO); continue; // Candidate is composite; try another } } // All candidates of bitLength 2 and 3 are prime by this point if (result.bitLength() < 4) return result; // The expensive test if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null)) return result; result = result.add(TWO); } } // Start at previous even number if (result.testBit(0)) result = result.subtract(ONE); // Looking for the next large prime int searchLen = (result.bitLength() / 20) * 64; while(true) { BitSieve searchSieve = new BitSieve(result, searchLen); BigInteger candidate = searchSieve.retrieve(result, DEFAULT_PRIME_CERTAINTY, null); if (candidate != null) return candidate; result = result.add(BigInteger.valueOf(2 * searchLen)); } } /** * Returns {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. * * This method assumes bitLength > 2. * * @param certainty a measure of the uncertainty that the caller is * willing to tolerate: if the call returns {@code true} * the probability that this BigInteger is prime exceeds * {@code (1 - 1/2<sup>certainty</sup>)}. The execution time of * this method is proportional to the value of this parameter. * @return {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. */ boolean primeToCertainty(int certainty, Random random) { int rounds = 0; int n = (Math.min(certainty, Integer.MAX_VALUE-1)+1)/2; // The relationship between the certainty and the number of rounds // we perform is given in the draft standard ANSI X9.80, "PRIME // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES". int sizeInBits = this.bitLength(); if (sizeInBits < 100) { rounds = 50; rounds = n < rounds ? n : rounds; return passesMillerRabin(rounds, random); } if (sizeInBits < 256) { rounds = 27; } else if (sizeInBits < 512) { rounds = 15; } else if (sizeInBits < 768) { rounds = 8; } else if (sizeInBits < 1024) { rounds = 4; } else { rounds = 2; } rounds = n < rounds ? n : rounds; return passesMillerRabin(rounds, random) && passesLucasLehmer(); } /** * Returns true iff this BigInteger is a Lucas-Lehmer probable prime. * * The following assumptions are made: * This BigInteger is a positive, odd number. */ private boolean passesLucasLehmer() { BigInteger thisPlusOne = this.add(ONE); // Step 1 int d = 5; while (jacobiSymbol(d, this) != -1) { // 5, -7, 9, -11, ... d = (d<0) ? Math.abs(d)+2 : -(d+2); } // Step 2 BigInteger u = lucasLehmerSequence(d, thisPlusOne, this); // Step 3 return u.mod(this).equals(ZERO); } /** * Computes Jacobi(p,n). * Assumes n positive, odd, n>=3. */ private static int jacobiSymbol(int p, BigInteger n) { if (p == 0) return 0; // Algorithm and comments adapted from Colin Plumb's C library. int j = 1; int u = n.mag[n.mag.length-1]; // Make p positive if (p < 0) { p = -p; int n8 = u & 7; if ((n8 == 3) || (n8 == 7)) j = -j; // 3 (011) or 7 (111) mod 8 } // Get rid of factors of 2 in p while ((p & 3) == 0) p >>= 2; if ((p & 1) == 0) { p >>= 1; if (((u ^ (u>>1)) & 2) != 0) j = -j; // 3 (011) or 5 (101) mod 8 } if (p == 1) return j; // Then, apply quadratic reciprocity if ((p & u & 2) != 0) // p = u = 3 (mod 4)? j = -j; // And reduce u mod p u = n.mod(BigInteger.valueOf(p)).intValue(); // Now compute Jacobi(u,p), u < p while (u != 0) { while ((u & 3) == 0) u >>= 2; if ((u & 1) == 0) { u >>= 1; if (((p ^ (p>>1)) & 2) != 0) j = -j; // 3 (011) or 5 (101) mod 8 } if (u == 1) return j; // Now both u and p are odd, so use quadratic reciprocity assert (u < p); int t = u; u = p; p = t; if ((u & p & 2) != 0) // u = p = 3 (mod 4)? j = -j; // Now u >= p, so it can be reduced u %= p; } return 0; } private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) { BigInteger d = BigInteger.valueOf(z); BigInteger u = ONE; BigInteger u2; BigInteger v = ONE; BigInteger v2; for (int i=k.bitLength()-2; i>=0; i--) { u2 = u.multiply(v).mod(n); v2 = v.square().add(d.multiply(u.square())).mod(n); if (v2.testBit(0)) v2 = v2.subtract(n); v2 = v2.shiftRight(1); u = u2; v = v2; if (k.testBit(i)) { u2 = u.add(v).mod(n); if (u2.testBit(0)) u2 = u2.subtract(n); u2 = u2.shiftRight(1); v2 = v.add(d.multiply(u)).mod(n); if (v2.testBit(0)) v2 = v2.subtract(n); v2 = v2.shiftRight(1); u = u2; v = v2; } } return u; } private static volatile Random staticRandom; private static Random getSecureRandom() { if (staticRandom == null) { staticRandom = new java.security.SecureRandom(); } return staticRandom; } /** * Returns true iff this BigInteger passes the specified number of * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS * 186-2). * * The following assumptions are made: * This BigInteger is a positive, odd number greater than 2. * iterations<=50. */ private boolean passesMillerRabin(int iterations, Random rnd) { // Find a and m such that m is odd and this == 1 + 2**a * m BigInteger thisMinusOne = this.subtract(ONE); BigInteger m = thisMinusOne; int a = m.getLowestSetBit(); m = m.shiftRight(a); // Do the tests if (rnd == null) { rnd = getSecureRandom(); } for (int i=0; i<iterations; i++) { // Generate a uniform random on (1, this) BigInteger b; do { b = new BigInteger(this.bitLength(), rnd); } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0); int j = 0; BigInteger z = b.modPow(m, this); while(!((j==0 && z.equals(ONE)) || z.equals(thisMinusOne))) { if (j>0 && z.equals(ONE) || ++j==a) return false; z = z.modPow(TWO, this); } } return true; } /** * This internal constructor differs from its public cousin * with the arguments reversed in two ways: it assumes that its * arguments are correct, and it doesn't copy the magnitude array. */ BigInteger(int[] magnitude, int signum) { this.signum = (magnitude.length==0 ? 0 : signum); this.mag = magnitude; } /** * This private constructor is for internal use and assumes that its * arguments are correct. */ private BigInteger(byte[] magnitude, int signum) { this.signum = (magnitude.length==0 ? 0 : signum); this.mag = stripLeadingZeroBytes(magnitude); } //Static Factory Methods /** * Returns a BigInteger whose value is equal to that of the * specified {@code long}. This "static factory method" is * provided in preference to a ({@code long}) constructor * because it allows for reuse of frequently used BigIntegers. * * @param val value of the BigInteger to return. * @return a BigInteger with the specified value. */ public static BigInteger valueOf(long val) { // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant if (val == 0) return ZERO; if (val > 0 && val <= MAX_CONSTANT) return posConst[(int) val]; else if (val < 0 && val >= -MAX_CONSTANT) return negConst[(int) -val]; return new BigInteger(val); } /** * Constructs a BigInteger with the specified value, which may not be zero. */ private BigInteger(long val) { if (val < 0) { val = -val; signum = -1; } else { signum = 1; } int highWord = (int)(val >>> 32); if (highWord==0) { mag = new int[1]; mag[0] = (int)val; } else { mag = new int[2]; mag[0] = highWord; mag[1] = (int)val; } } /** * Returns a BigInteger with the given two's complement representation. * Assumes that the input array will not be modified (the returned * BigInteger will reference the input array if feasible). */ private static BigInteger valueOf(int val[]) { return (val[0]>0 ? new BigInteger(val, 1) : new BigInteger(val)); } // Constants /** * Initialize static constant array when class is loaded. */ private final static int MAX_CONSTANT = 16; private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT+1]; private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT+1]; static { for (int i = 1; i <= MAX_CONSTANT; i++) { int[] magnitude = new int[1]; magnitude[0] = i; posConst[i] = new BigInteger(magnitude, 1); negConst[i] = new BigInteger(magnitude, -1); } } /** * The BigInteger constant zero. * * @since 1.2 */ public static final BigInteger ZERO = new BigInteger(new int[0], 0); /** * The BigInteger constant one. * * @since 1.2 */ public static final BigInteger ONE = valueOf(1); /** * The BigInteger constant two. (Not exported.) */ private static final BigInteger TWO = valueOf(2); /** * The BigInteger constant ten. * * @since 1.5 */ public static final BigInteger TEN = valueOf(10); // Arithmetic Operations /** * Returns a BigInteger whose value is {@code (this + val)}. * * @param val value to be added to this BigInteger. * @return {@code this + val} */ public BigInteger add(BigInteger val) { if (val.signum == 0) return this; if (signum == 0) return val; if (val.signum == signum) return new BigInteger(add(mag, val.mag), signum); int cmp = compareMagnitude(val); if (cmp==0) return ZERO; int[] resultMag = (cmp>0 ? subtract(mag, val.mag) : subtract(val.mag, mag)); resultMag = trustedStripLeadingZeroInts(resultMag); return new BigInteger(resultMag, cmp == signum ? 1 : -1); } /** * Adds the contents of the int arrays x and y. This method allocates * a new int array to hold the answer and returns a reference to that * array. */ private static int[] add(int[] x, int[] y) { // If x is shorter, swap the two arrays if (x.length < y.length) { int[] tmp = x; x = y; y = tmp; } int xIndex = x.length; int yIndex = y.length; int result[] = new int[xIndex]; long sum = 0; // Add common parts of both numbers while(yIndex > 0) { sum = (x[--xIndex] & LONG_MASK) + (y[--yIndex] & LONG_MASK) + (sum >>> 32); result[xIndex] = (int)sum; } // Copy remainder of longer number while carry propagation is required boolean carry = (sum >>> 32 != 0); while (xIndex > 0 && carry) carry = ((result[--xIndex] = x[xIndex] + 1) == 0); // Copy remainder of longer number while (xIndex > 0) result[--xIndex] = x[xIndex]; // Grow result if necessary if (carry) { int bigger[] = new int[result.length + 1]; System.arraycopy(result, 0, bigger, 1, result.length); bigger[0] = 0x01; return bigger; } return result; } /** * Returns a BigInteger whose value is {@code (this - val)}. * * @param val value to be subtracted from this BigInteger. * @return {@code this - val} */ public BigInteger subtract(BigInteger val) { if (val.signum == 0) return this; if (signum == 0) return val.negate(); if (val.signum != signum) return new BigInteger(add(mag, val.mag), signum); int cmp = compareMagnitude(val); if (cmp==0) return ZERO; int[] resultMag = (cmp>0 ? subtract(mag, val.mag) : subtract(val.mag, mag)); resultMag = trustedStripLeadingZeroInts(resultMag); return new BigInteger(resultMag, (cmp == signum) ? 1 : -1); } /** * Subtracts the contents of the second int arrays (little) from the * first (big). The first int array (big) must represent a larger number * than the second. This method allocates the space necessary to hold the * answer. */ private static int[] subtract(int[] big, int[] little) { int bigIndex = big.length; int result[] = new int[bigIndex]; int littleIndex = little.length; long difference = 0; // Subtract common parts of both numbers while(littleIndex > 0) { difference = (big[--bigIndex] & LONG_MASK) - (little[--littleIndex] & LONG_MASK) + (difference >> 32); result[bigIndex] = (int)difference; } // Subtract remainder of longer number while borrow propagates boolean borrow = (difference >> 32 != 0); while (bigIndex > 0 && borrow) borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1); // Copy remainder of longer number while (bigIndex > 0) result[--bigIndex] = big[bigIndex]; return result; } /** * Returns a BigInteger whose value is {@code (this * val)}. * * @param val value to be multiplied by this BigInteger. * @return {@code this * val} */ public BigInteger multiply(BigInteger val) { if (val.signum == 0 || signum == 0) return ZERO; int[] res = multiplyToLen(mag, mag.length, val.mag, val.mag.length, null); res = trustedStripLeadingZeroInts(res); return new BigInteger(res, signum == val.signum ? 1 : -1); } /** * Package private methods used by BigDecimal code to multiply a BigInteger * with a long. Assumes v is not equal to INFLATED. */ BigInteger multiply(long v) { if (v == 0 || signum == 0) return ZERO; assert v != BigDecimal.INFLATED; int rsign = (v > 0 ? signum : -signum); if (v < 0) v = -v; long dh = v >>> 32; // higher order bits long dl = v & LONG_MASK; // lower order bits int xlen = mag.length; int[] value = mag; int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]); long carry = 0; int rstart = rmag.length - 1; for (int i = xlen - 1; i >= 0; i--) { long product = (value[i] & LONG_MASK) * dl + carry; rmag[rstart--] = (int)product; carry = product >>> 32; } rmag[rstart] = (int)carry; if (dh != 0L) { carry = 0; rstart = rmag.length - 2; for (int i = xlen - 1; i >= 0; i--) { long product = (value[i] & LONG_MASK) * dh + (rmag[rstart] & LONG_MASK) + carry; rmag[rstart--] = (int)product; carry = product >>> 32; } rmag[0] = (int)carry; } if (carry == 0L) rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length); return new BigInteger(rmag, rsign); } /** * Multiplies int arrays x and y to the specified lengths and places * the result into z. There will be no leading zeros in the resultant array. */ private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) { int xstart = xlen - 1; int ystart = ylen - 1; if (z == null || z.length < (xlen+ ylen)) z = new int[xlen+ylen]; long carry = 0; for (int j=ystart, k=ystart+1+xstart; j>=0; j--, k--) { long product = (y[j] & LONG_MASK) * (x[xstart] & LONG_MASK) + carry; z[k] = (int)product; carry = product >>> 32; } z[xstart] = (int)carry; for (int i = xstart-1; i >= 0; i--) { carry = 0; for (int j=ystart, k=ystart+1+i; j>=0; j--, k--) { long product = (y[j] & LONG_MASK) * (x[i] & LONG_MASK) + (z[k] & LONG_MASK) + carry; z[k] = (int)product; carry = product >>> 32; } z[i] = (int)carry; } return z; } /** * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}. * * @return {@code this<sup>2</sup>} */ private BigInteger square() { if (signum == 0) return ZERO; int[] z = squareToLen(mag, mag.length, null); return new BigInteger(trustedStripLeadingZeroInts(z), 1); } /** * Squares the contents of the int array x. The result is placed into the * int array z. The contents of x are not changed. */ private static final int[] squareToLen(int[] x, int len, int[] z) { /* * The algorithm used here is adapted from Colin Plumb's C library. * Technique: Consider the partial products in the multiplication * of "abcde" by itself: * * a b c d e * * a b c d e * ================== * ae be ce de ee * ad bd cd dd de * ac bc cc cd ce * ab bb bc bd be * aa ab ac ad ae * * Note that everything above the main diagonal: * ae be ce de = (abcd) * e * ad bd cd = (abc) * d * ac bc = (ab) * c * ab = (a) * b * * is a copy of everything below the main diagonal: * de * cd ce * bc bd be * ab ac ad ae * * Thus, the sum is 2 * (off the diagonal) + diagonal. * * This is accumulated beginning with the diagonal (which * consist of the squares of the digits of the input), which is then * divided by two, the off-diagonal added, and multiplied by two * again. The low bit is simply a copy of the low bit of the * input, so it doesn't need special care. */ int zlen = len << 1; if (z == null || z.length < zlen) z = new int[zlen]; // Store the squares, right shifted one bit (i.e., divided by 2) int lastProductLowWord = 0; for (int j=0, i=0; j<len; j++) { long piece = (x[j] & LONG_MASK); long product = piece * piece; z[i++] = (lastProductLowWord << 31) | (int)(product >>> 33); z[i++] = (int)(product >>> 1); lastProductLowWord = (int)product; } // Add in off-diagonal sums for (int i=len, offset=1; i>0; i--, offset+=2) { int t = x[i-1]; t = mulAdd(z, x, offset, i-1, t); addOne(z, offset-1, i, t); } // Shift back up and set low bit primitiveLeftShift(z, zlen, 1); z[zlen-1] |= x[len-1] & 1; return z; } /** * Returns a BigInteger whose value is {@code (this / val)}. * * @param val value by which this BigInteger is to be divided. * @return {@code this / val} * @throws ArithmeticException {@code val==0} */ public BigInteger divide(BigInteger val) { MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); a.divide(b, q); return q.toBigInteger(this.signum * val.signum); } /** * Returns an array of two BigIntegers containing {@code (this / val)} * followed by {@code (this % val)}. * * @param val value by which this BigInteger is to be divided, and the * remainder computed. * @return an array of two BigIntegers: the quotient {@code (this / val)} * is the initial element, and the remainder {@code (this % val)} * is the final element. * @throws ArithmeticException {@code val==0} */ public BigInteger[] divideAndRemainder(BigInteger val) { BigInteger[] result = new BigInteger[2]; MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); MutableBigInteger r = a.divide(b, q); result[0] = q.toBigInteger(this.signum * val.signum); result[1] = r.toBigInteger(this.signum); return result; } /** * Returns a BigInteger whose value is {@code (this % val)}. * * @param val value by which this BigInteger is to be divided, and the * remainder computed. * @return {@code this % val} * @throws ArithmeticException {@code val==0} */ public BigInteger remainder(BigInteger val) { MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(this.mag), b = new MutableBigInteger(val.mag); return a.divide(b, q).toBigInteger(this.signum); } /** * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>. * Note that {@code exponent} is an integer rather than a BigInteger. * * @param exponent exponent to which this BigInteger is to be raised. * @return <tt>this<sup>exponent</sup></tt> * @throws ArithmeticException {@code exponent} is negative. (This would * cause the operation to yield a non-integer value.) */ public BigInteger pow(int exponent) { if (exponent < 0) throw new ArithmeticException("Negative exponent"); if (signum==0) return (exponent==0 ? ONE : this); // Perform exponentiation using repeated squaring trick int newSign = (signum<0 && (exponent&1)==1 ? -1 : 1); int[] baseToPow2 = this.mag; int[] result = {1}; while (exponent != 0) { if ((exponent & 1)==1) { result = multiplyToLen(result, result.length, baseToPow2, baseToPow2.length, null); result = trustedStripLeadingZeroInts(result); } if ((exponent >>>= 1) != 0) { baseToPow2 = squareToLen(baseToPow2, baseToPow2.length, null); baseToPow2 = trustedStripLeadingZeroInts(baseToPow2); } } return new BigInteger(result, newSign); } /** * Returns a BigInteger whose value is the greatest common divisor of * {@code abs(this)} and {@code abs(val)}. Returns 0 if * {@code this==0 && val==0}. * * @param val value with which the GCD is to be computed. * @return {@code GCD(abs(this), abs(val))} */ public BigInteger gcd(BigInteger val) { if (val.signum == 0) return this.abs(); else if (this.signum == 0) return val.abs(); MutableBigInteger a = new MutableBigInteger(this); MutableBigInteger b = new MutableBigInteger(val); MutableBigInteger result = a.hybridGCD(b); return result.toBigInteger(1); } /** * Package private method to return bit length for an integer. * */ static int bitLengthForInt(int n) { return 32 - Integer.numberOfLeadingZeros(n); } /** * Left shift int array a up to len by n bits. Returns the array that * results from the shift since space may have to be reallocated. */ private static int[] leftShift(int[] a, int len, int n) { int nInts = n >>> 5; int nBits = n&0x1F; int bitsInHighWord = bitLengthForInt(a[0]); // If shift can be done without recopy, do so if (n <= (32-bitsInHighWord)) { primitiveLeftShift(a, len, nBits); return a; } else { // Array must be resized if (nBits <= (32-bitsInHighWord)) { int result[] = new int[nInts+len]; for (int i=0; i<len; i++) result[i] = a[i]; primitiveLeftShift(result, result.length, nBits); return result; } else { int result[] = new int[nInts+len+1]; for (int i=0; i<len; i++) result[i] = a[i]; primitiveRightShift(result, result.length, 32 - nBits); return result; } } } // shifts a up to len right n bits assumes no leading zeros, 0<n<32 static void primitiveRightShift(int[] a, int len, int n) { int n2 = 32 - n; for (int i=len-1, c=a[i]; i>0; i--) { int b = c; c = a[i-1]; a[i] = (c << n2) | (b >>> n); } a[0] >>>= n; } // shifts a up to len left n bits assumes no leading zeros, 0<=n<32 static void primitiveLeftShift(int[] a, int len, int n) { if (len == 0 || n == 0) return; int n2 = 32 - n; for (int i=0, c=a[i], m=i+len-1; i<m; i++) { int b = c; c = a[i+1]; a[i] = (b << n) | (c >>> n2); } a[len-1] <<= n; } /** * Calculate bitlength 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 * <tt>(this<sup>exponent</sup> mod m)</tt>. (Unlike {@code pow}, this * method permits negative exponents.) * * @param exponent the exponent. * @param m the modulus. * @return <tt>this<sup>exponent</sup> mod m</tt> * @throws ArithmeticException {@code m <= 0} * @see #modInverse */ public BigInteger modPow(BigInteger exponent, BigInteger m) { if (m.signum <= 0) throw new ArithmeticException("BigInteger: modulus not positive"); // Trivial cases if (exponent.signum == 0) return (m.equals(ONE) ? ZERO : ONE); if (this.equals(ONE)) return (m.equals(ONE) ? ZERO : ONE); if (this.equals(ZERO) && exponent.signum >= 0) return ZERO; if (this.equals(negConst[1]) && (!exponent.testBit(0))) return (m.equals(ONE) ? ZERO : ONE); boolean invertResult; if ((invertResult = (exponent.signum < 0))) exponent = exponent.negate(); BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0 ? this.mod(m) : this); BigInteger result; if (m.testBit(0)) { // odd modulus result = base.oddModPow(exponent, m); } else { /* * Even modulus. Tear it into an "odd part" (m1) and power of two * (m2), exponentiate mod m1, manually exponentiate mod m2, and * use Chinese Remainder Theorem to combine results. */ // Tear m apart into odd part (m1) and power of 2 (m2) int p = m.getLowestSetBit(); // Max pow of 2 that divides m BigInteger m1 = m.shiftRight(p); // m/2**p BigInteger m2 = ONE.shiftLeft(p); // 2**p // Calculate new base from m1 BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0 ? this.mod(m1) : this); // Caculate (base ** exponent) mod m1. BigInteger a1 = (m1.equals(ONE) ? ZERO : base2.oddModPow(exponent, m1)); // Calculate (this ** exponent) mod m2 BigInteger a2 = base.modPow2(exponent, p); // Combine results using Chinese Remainder Theorem BigInteger y1 = m2.modInverse(m1); BigInteger y2 = m1.modInverse(m2); result = a1.multiply(m2).multiply(y1).add (a2.multiply(m1).multiply(y2)).mod(m); } return (invertResult ? result.modInverse(m) : result); } static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793, Integer.MAX_VALUE}; // Sentinel /** * Returns a BigInteger whose value is x to the power of y mod z. * Assumes: z is odd && x < z. */ private BigInteger oddModPow(BigInteger y, BigInteger z) { /* * The algorithm is adapted from Colin Plumb's C library. * * The window algorithm: * The idea is to keep a running product of b1 = n^(high-order bits of exp) * and then keep appending exponent bits to it. The following patterns * apply to a 3-bit window (k = 3): * To append 0: square * To append 1: square, multiply by n^1 * To append 10: square, multiply by n^1, square * To append 11: square, square, multiply by n^3 * To append 100: square, multiply by n^1, square, square * To append 101: square, square, square, multiply by n^5 * To append 110: square, square, multiply by n^3, square * To append 111: square, square, square, multiply by n^7 * * Since each pattern involves only one multiply, the longer the pattern * the better, except that a 0 (no multiplies) can be appended directly. * We precompute a table of odd powers of n, up to 2^k, and can then * multiply k bits of exponent at a time. Actually, assuming random * exponents, there is on average one zero bit between needs to * multiply (1/2 of the time there's none, 1/4 of the time there's 1, * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so * you have to do one multiply per k+1 bits of exponent. * * The loop walks down the exponent, squaring the result buffer as * it goes. There is a wbits+1 bit lookahead buffer, buf, that is * filled with the upcoming exponent bits. (What is read after the * end of the exponent is unimportant, but it is filled with zero here.) * When the most-significant bit of this buffer becomes set, i.e. * (buf & tblmask) != 0, we have to decide what pattern to multiply * by, and when to do it. We decide, remember to do it in future * after a suitable number of squarings have passed (e.g. a pattern * of "100" in the buffer requires that we multiply by n^1 immediately; * a pattern of "110" calls for multiplying by n^3 after one more * squaring), clear the buffer, and continue. * * When we start, there is one more optimization: the result buffer * is implcitly one, so squaring it or multiplying by it can be * optimized away. Further, if we start with a pattern like "100" * in the lookahead window, rather than placing n into the buffer * and then starting to square it, we have already computed n^2 * to compute the odd-powers table, so we can place that into * the buffer and save a squaring. * * This means that if you have a k-bit window, to compute n^z, * where z is the high k bits of the exponent, 1/2 of the time * it requires no squarings. 1/4 of the time, it requires 1 * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings. * And the remaining 1/2^(k-1) of the time, the top k bits are a * 1 followed by k-1 0 bits, so it again only requires k-2 * squarings, not k-1. The average of these is 1. Add that * to the one squaring we have to do to compute the table, * and you'll see that a k-bit window saves k-2 squarings * as well as reducing the multiplies. (It actually doesn't * hurt in the case k = 1, either.) */ // Special case for exponent of one if (y.equals(ONE)) return this; // Special case for base of zero if (signum==0) return ZERO; int[] base = mag.clone(); int[] exp = y.mag; int[] mod = z.mag; int modLen = mod.length; // Select an appropriate window size int wbits = 0; int ebits = bitLength(exp, exp.length); // if exponent is 65537 (0x10001), use minimum window size if ((ebits != 17) || (exp[0] != 65537)) { while (ebits > bnExpModThreshTable[wbits]) { wbits++; } } // Calculate appropriate table size int tblmask = 1 << wbits; // Allocate table for precomputed odd powers of base in Montgomery form int[][] table = new int[tblmask][]; for (int i=0; i<tblmask; i++) table[i] = new int[modLen]; // Compute the modular inverse int inv = -MutableBigInteger.inverseMod32(mod[modLen-1]); // Convert base to Montgomery form int[] a = leftShift(base, base.length, modLen << 5); MutableBigInteger q = new MutableBigInteger(), a2 = new MutableBigInteger(a), b2 = new MutableBigInteger(mod); MutableBigInteger r = a2.divide(b2, q); table[0] = r.toIntArray(); // Pad table[0] with leading zeros so its length is at least modLen if (table[0].length < modLen) { int offset = modLen - table[0].length; int[] t2 = new int[modLen]; for (int i=0; i<table[0].length; i++) t2[i+offset] = table[0][i]; table[0] = t2; } // Set b to the square of the base int[] b = squareToLen(table[0], modLen, null); b = montReduce(b, mod, modLen, inv); // Set t to high half of b int[] t = new int[modLen]; for(int i=0; i<modLen; i++) t[i] = b[i]; // Fill in the table with odd powers of the base for (int i=1; i<tblmask; i++) { int[] prod = multiplyToLen(t, modLen, table[i-1], modLen, null); table[i] = montReduce(prod, mod, modLen, inv); } // Pre load the window that slides over the exponent int bitpos = 1 << ((ebits-1) & (32-1)); int buf = 0; int elen = exp.length; int eIndex = 0; for (int i = 0; i <= wbits; i++) { buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0)?1:0); bitpos >>>= 1; if (bitpos == 0) { eIndex++; bitpos = 1 << (32-1); elen--; } } int multpos = ebits; // The first iteration, which is hoisted out of the main loop ebits--; boolean isone = true; multpos = ebits - wbits; while ((buf & 1) == 0) { buf >>>= 1; multpos++; } int[] mult = table[buf >>> 1]; buf = 0; if (multpos == ebits) isone = false; // The main loop while(true) { ebits--; // Advance the window buf <<= 1; if (elen != 0) { buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0; bitpos >>>= 1; if (bitpos == 0) { eIndex++; bitpos = 1 << (32-1); elen--; } } // Examine the window for pending multiplies if ((buf & tblmask) != 0) { multpos = ebits - wbits; while ((buf & 1) == 0) { buf >>>= 1; multpos++; } mult = table[buf >>> 1]; buf = 0; } // Perform multiply if (ebits == multpos) { if (isone) { b = mult.clone(); isone = false; } else { t = b; a = multiplyToLen(t, modLen, mult, modLen, a); a = montReduce(a, mod, modLen, inv); t = a; a = b; b = t; } } // Check if done if (ebits == 0) break; // Square the input if (!isone) { t = b; a = squareToLen(t, modLen, a); a = montReduce(a, mod, modLen, inv); t = a; a = b; b = t; } } // Convert result out of Montgomery form and return int[] t2 = new int[2*modLen]; for(int i=0; i<modLen; i++) t2[i+modLen] = b[i]; b = montReduce(t2, mod, modLen, inv); t2 = new int[modLen]; for(int i=0; i<modLen; i++) t2[i] = b[i]; return new BigInteger(1, t2); } /** * Montgomery reduce n, modulo mod. This reduces modulo mod and divides * by 2^(32*mlen). Adapted from Colin Plumb's C library. */ private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) { int c=0; int len = mlen; int offset=0; do { int nEnd = n[n.length-1-offset]; int carry = mulAdd(n, mod, offset, mlen, inv * nEnd); c += addOne(n, offset, mlen, carry); offset++; } while(--len > 0); while(c>0) c += subN(n, mod, mlen); while (intArrayCmpToLen(n, mod, mlen) >= 0) subN(n, mod, mlen); return n; } /* * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than, * equal to, or greater than arg2 up to length len. */ private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) { for (int i=0; i<len; i++) { long b1 = arg1[i] & LONG_MASK; long b2 = arg2[i] & LONG_MASK; if (b1 < b2) return -1; if (b1 > b2) return 1; } return 0; } /** * Subtracts two numbers of same length, returning borrow. */ private static int subN(int[] a, int[] b, int len) { long sum = 0; while(--len >= 0) { sum = (a[len] & LONG_MASK) - (b[len] & LONG_MASK) + (sum >> 32); a[len] = (int)sum; } return (int)(sum >> 32); } /** * Multiply an array by one word k and add to result, return the carry */ static int mulAdd(int[] out, int[] in, int offset, int len, int k) { long kLong = k & LONG_MASK; long carry = 0; offset = out.length-offset - 1; for (int j=len-1; j >= 0; j--) { long product = (in[j] & LONG_MASK) * kLong + (out[offset] & LONG_MASK) + carry; out[offset--] = (int)product; carry = product >>> 32; } return (int)carry; } /** * Add one word to the number a mlen words into a. Return the resulting * carry. */ static int addOne(int[] a, int offset, int mlen, int carry) { offset = a.length-1-mlen-offset; long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK); a[offset] = (int)t; if ((t >>> 32) == 0) return 0; while (--mlen >= 0) { if (--offset < 0) { // Carry out of number return 1; } else { a[offset]++; if (a[offset] != 0) return 0; } } return 1; } /** * Returns a BigInteger whose value is (this ** exponent) mod (2**p) */ private BigInteger modPow2(BigInteger exponent, int p) { /* * Perform exponentiation using repeated squaring trick, chopping off * high order bits as indicated by modulus. */ BigInteger result = valueOf(1); BigInteger baseToPow2 = this.mod2(p); int expOffset = 0; int limit = exponent.bitLength(); if (this.testBit(0)) limit = (p-1) < limit ? (p-1) : limit; while (expOffset < limit) { if (exponent.testBit(expOffset)) result = result.multiply(baseToPow2).mod2(p); expOffset++; if (expOffset < limit) baseToPow2 = baseToPow2.square().mod2(p); } return result; } /** * Returns a BigInteger whose value is this mod(2**p). * Assumes that this {@code BigInteger >= 0} and {@code p > 0}. */ private BigInteger mod2(int p) { if (bitLength() <= p) return this; // Copy remaining ints of mag int numInts = (p + 31) >>> 5; int[] mag = new int[numInts]; for (int i=0; i<numInts; i++) mag[i] = this.mag[i + (this.mag.length - 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 <tt>floor(this * 2<sup>n</sup>)</tt>.) * * @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 this; if (n<0) return shiftRight(-n); int nInts = n >>> 5; int nBits = n & 0x1f; int magLen = mag.length; int newMag[] = null; if (nBits == 0) { newMag = new int[magLen + nInts]; for (int i=0; i<magLen; i++) newMag[i] = mag[i]; } 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 new BigInteger(newMag, signum); } /** * Returns a BigInteger whose value is {@code (this >> n)}. Sign * extension is performed. The shift distance, {@code n}, may be * negative, in which case this method performs a left shift. * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.) * * @param n shift distance, in bits. * @return {@code this >> n} * @see #shiftLeft */ public BigInteger shiftRight(int n) { if (n==0) return this; if (n<0) return shiftLeft(-n); int nInts = n >>> 5; int nBits = n & 0x1f; int magLen = mag.length; int newMag[] = null; // Special case: entire contents shifted off the end if (nInts >= magLen) return (signum >= 0 ? ZERO : negConst[1]); if (nBits == 0) { int newMagLen = magLen - nInts; newMag = new int[newMagLen]; for (int i=0; i<newMagLen; i++) newMag[i] = mag[i]; } 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() { @SuppressWarnings("deprecation") int lsb = lowestSetBit - 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); } lowestSetBit = lsb + 2; } return lsb; } // Miscellaneous Bit Operations /** * Returns the number of bits in the minimal two's-complement * representation of this BigInteger, <i>excluding</i> a sign bit. * For positive BigIntegers, this is equivalent to the number of bits in * the ordinary binary representation. (Computes * {@code (ceil(log2(this < 0 ? -this : this+1)))}.) * * @return number of bits in the minimal two's-complement * representation of this BigInteger, <i>excluding</i> a sign bit. */ public int bitLength() { @SuppressWarnings("deprecation") int n = bitLength - 1; if (n == -1) { // bitLength not initialized yet int[] m = mag; int len = m.length; if (len == 0) { n = 0; // offset by one to initialize } else { // Calculate the bit length of the magnitude int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]); if (signum < 0) { // Check if magnitude is a power of two boolean pow2 = (bitCnt(mag[0]) == 1); for(int i=1; i< len && pow2; i++) pow2 = (mag[i] == 0); n = (pow2 ? magBitLength -1 : magBitLength); } else { n = magBitLength; } } bitLength = n + 1; } return n; } /** * Returns the number of bits in the two's complement representation * of this BigInteger that differ from its sign bit. This method is * useful when implementing bit-vector style sets atop BigIntegers. * * @return number of bits in the two's complement representation * of this BigInteger that differ from its sign bit. */ public int bitCount() { @SuppressWarnings("deprecation") int bc = bitCount - 1; if (bc == -1) { // bitCount not initialized yet bc = 0; // offset by one to initialize // Count the bits in the magnitude for (int i=0; i<mag.length; i++) bc += bitCnt(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; } bitCount = bc + 1; } return bc; } static int bitCnt(int val) { val -= (0xaaaaaaaa & val) >>> 1; val = (val & 0x33333333) + ((val >>> 2) & 0x33333333); val = val + (val >>> 4) & 0x0f0f0f0f; val += val >>> 8; val += val >>> 16; return val & 0xff; } // Primality Testing /** * Returns {@code true} if this BigInteger is probably prime, * {@code false} if it's definitely composite. If * {@code certainty} is {@code <= 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; } /** * Version of compareTo that ignores sign. */ final int compareMagnitude(BigInteger val) { int[] m1 = mag; int len = m1.length; int[] m2 = val.mag; int m2len = m2.length; if (len < m2len) return -1; if (len > m2len) return 1; for (int i = 0; i < len; ++i) { int a = m1[i]; int b = m2[i]; if (a != b) { return ((a & LONG_MASK) < (b & LONG_MASK))? -1 : 1; } } return 0; } /** * Compares this BigInteger with the specified Object for equality. * * @param x Object to which this BigInteger is to be compared. * @return {@code true} if and only if the specified Object is a * BigInteger whose value is numerically equal to this BigInteger. */ public boolean equals(Object x) { // This test is just an optimization, which may or may not help if (x == this) return true; if (!(x instanceof BigInteger)) return false; BigInteger xInt = (BigInteger) x; if (xInt.signum != signum) return false; int[] m = mag; int len = m.length; int[] xm = xInt.mag; if (xm.length != len) 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; // Compute upper bound on number of digit groups and allocate space int maxNumDigitGroups = (4*mag.length + 6)/7; String digitGroup[] = new String[maxNumDigitGroups]; // Translate number to string, a digit group at a time BigInteger tmp = this.abs(); int numGroups = 0; while (tmp.signum != 0) { BigInteger d = longRadix[radix]; MutableBigInteger q = new MutableBigInteger(), a = new MutableBigInteger(tmp.mag), b = new MutableBigInteger(d.mag); MutableBigInteger r = a.divide(b, q); BigInteger q2 = q.toBigInteger(tmp.signum * d.signum); BigInteger r2 = r.toBigInteger(tmp.signum * d.signum); digitGroup[numGroups++] = Long.toString(r2.longValue(), radix); tmp = q2; } // Put sign (if any) and first digit group into result buffer StringBuilder buf = new StringBuilder(numGroups*digitsPerLong[radix]+1); if (signum<0) buf.append('-'); buf.append(digitGroup[numGroups-1]); // Append remaining digit groups padded with leading zeros for (int i=numGroups-2; i>=0; i--) { // Prepend (any) leading zeros for this digit group int numLeadingZeros = digitsPerLong[radix]-digitGroup[i].length(); if (numLeadingZeros != 0) buf.append(zeros[numLeadingZeros]); buf.append(digitGroup[i]); } return buf.toString(); } /* zero[i] is a string of i consecutive zeros. */ private static String zeros[] = new String[64]; static { zeros[63] = "000000000000000000000000000000000000000000000000000000000000000"; for (int i=0; i<63; i++) zeros[i] = zeros[63].substring(0, i); } /** * Returns the decimal String representation of this BigInteger. * The digit-to-character mapping provided by * {@code Character.forDigit} is used, and a minus sign is * prepended if appropriate. (This representation is compatible * with the {@link #BigInteger(String) (String)} constructor, and * allows for String concatenation with Java's + operator.) * * @return decimal String representation of this BigInteger. * @see Character#forDigit * @see #BigInteger(java.lang.String) */ public String toString() { return toString(10); } /** * Returns a byte array containing the two's-complement * representation of this BigInteger. The byte array will be in * <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 <a * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing * primitive conversion</i></a> from {@code long} to * {@code int} as defined in the <a * href="http://java.sun.com/docs/books/jls/html/">Java Language * Specification</a>: if this BigInteger is too big to fit in an * {@code int}, only the low-order 32 bits are returned. * Note that this conversion can lose information about the * overall magnitude of the BigInteger value as well as return a * result with the opposite sign. * * @return this BigInteger converted to an {@code int}. */ public int intValue() { int result = 0; result = getInt(0); return result; } /** * Converts this BigInteger to a {@code long}. This * conversion is analogous to a <a * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing * primitive conversion</i></a> from {@code long} to * {@code int} as defined in the <a * href="http://java.sun.com/docs/books/jls/html/">Java Language * Specification</a>: if this BigInteger is too big to fit in a * {@code long}, only the low-order 64 bits are returned. * Note that this conversion can lose information about the * overall magnitude of the BigInteger value as well as return a * result with the opposite sign. * * @return this BigInteger converted to a {@code long}. */ public long longValue() { long result = 0; for (int i=1; i>=0; i--) result = (result << 32) + (getInt(i) & LONG_MASK); return result; } /** * Converts this BigInteger to a {@code float}. This * conversion is similar to the <a * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing * primitive conversion</i></a> from {@code double} to * {@code float} defined in the <a * href="http://java.sun.com/docs/books/jls/html/">Java Language * Specification</a>: if this BigInteger has too great a magnitude * to represent as a {@code float}, it will be converted to * {@link Float#NEGATIVE_INFINITY} or {@link * Float#POSITIVE_INFINITY} as appropriate. Note that even when * the return value is finite, this conversion can lose * information about the precision of the BigInteger value. * * @return this BigInteger converted to a {@code float}. */ public float floatValue() { // Somewhat inefficient, but guaranteed to work. return Float.parseFloat(this.toString()); } /** * Converts this BigInteger to a {@code double}. This * conversion is similar to the <a * href="http://java.sun.com/docs/books/jls/second_edition/html/conversions.doc.html#25363"><i>narrowing * primitive conversion</i></a> from {@code double} to * {@code float} defined in the <a * href="http://java.sun.com/docs/books/jls/html/">Java Language * Specification</a>: if this BigInteger has too great a magnitude * to represent as a {@code double}, it will be converted to * {@link Double#NEGATIVE_INFINITY} or {@link * Double#POSITIVE_INFINITY} as appropriate. Note that even when * the return value is finite, this conversion can lose * information about the precision of the BigInteger value. * * @return this BigInteger converted to a {@code double}. */ public double doubleValue() { // Somewhat inefficient, but guaranteed to work. return Double.parseDouble(this.toString()); } /** * Returns a copy of the input array stripped of any leading zero bytes. */ private static int[] stripLeadingZeroInts(int val[]) { int len = val.length; int keep; // Find first nonzero byte for (keep=0; keep<len && val[keep]==0; keep++) ; return java.util.Arrays.copyOfRange(val, keep, len); } /** * 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 len = val.length; int keep; // Find first nonzero byte for (keep=0; keep<len && val[keep]==0; keep++) ; return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, len); } /** * Returns a copy of the input array stripped of any leading zero bytes. */ private static int[] stripLeadingZeroBytes(byte a[]) { int byteLength = a.length; int keep; // Find first nonzero byte for (keep=0; keep<byteLength && a[keep]==0; keep++) ; // Allocate new array and copy relevant part of input array int intLength = ((byteLength - keep) + 3) >>> 2; int[] result = new int[intLength]; int b = byteLength - 1; for (int i = intLength-1; i >= 0; i--) { result[i] = a[b--] & 0xff; int bytesRemaining = b - keep + 1; int bytesToTransfer = Math.min(3, bytesRemaining); for (int j=8; j <= (bytesToTransfer << 3); j += 8) result[i] |= ((a[b--] & 0xff) << j); } return result; } /** * Takes an array a representing a negative 2's-complement number and * returns the minimal (no leading zero bytes) unsigned whose value is -a. */ private static int[] makePositive(byte a[]) { int keep, k; int byteLength = a.length; // Find first non-sign (0xff) byte of input for (keep=0; keep<byteLength && 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<byteLength && a[k]==0; k++) ; int extraByte = (k==byteLength) ? 1 : 0; int intLength = ((byteLength - keep + extraByte) + 3)/4; int result[] = new int[intLength]; /* Copy one's complement of input into output, leaving extra * byte (if it exists) == 0x00 */ int b = byteLength - 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. */ private int firstNonzeroIntNum() { int fn = firstNonzeroIntNum - 2; if (fn == -2) { // firstNonzeroIntNum not initialized yet fn = 0; // Search for the first nonzero int int i; for (i=mag.length-1; i>=0 && mag[i]==0; i--) ; fn = mag.length - i - 1; firstNonzeroIntNum = fn + 2; // offset by two to initialize } return fn; } /** use serialVersionUID from JDK 1.1. for interoperability */ private static final long serialVersionUID = -8287574255936472291L; /** * Serializable fields for BigInteger. * * @serialField signum int * signum of this BigInteger. * @serialField magnitude int[] * magnitude array of this BigInteger. * @serialField bitCount int * number of bits in this BigInteger * @serialField bitLength int * the number of bits in the minimal two's-complement * representation of this BigInteger * @serialField lowestSetBit int * lowest set bit in the twos complement representation */ private static final ObjectStreamField[] serialPersistentFields = { new ObjectStreamField("signum", Integer.TYPE), new ObjectStreamField("magnitude", byte[].class), new ObjectStreamField("bitCount", Integer.TYPE), new ObjectStreamField("bitLength", Integer.TYPE), new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE), new ObjectStreamField("lowestSetBit", Integer.TYPE) }; /** * Reconstitute the {@code BigInteger} instance from a stream (that is, * deserialize it). The magnitude is read in as an array of bytes * for historical reasons, but it is converted to an array of ints * and the byte array is discarded. * Note: * The current convention is to initialize the cache fields, bitCount, * bitLength and lowestSetBit, to 0 rather than some other marker value. * Therefore, no explicit action to set these fields needs to be taken in * readObject because those fields already have a 0 value be default * since defaultReadObject is not being used. */ private void readObject(java.io.ObjectInputStream s) throws java.io.IOException, ClassNotFoundException { /* * In order to maintain compatibility with previous serialized forms, * the magnitude of a BigInteger is serialized as an array of bytes. * The magnitude field is used as a temporary store for the byte array * that is deserialized. The cached computation fields should be * transient but are serialized for compatibility reasons. */ // prepare to read the alternate persistent fields ObjectInputStream.GetField fields = s.readFields(); // Read the alternate persistent fields that we care about int sign = fields.get("signum", -2); byte[] magnitude = (byte[])fields.get("magnitude", null); // Validate signum if (sign < -1 || sign > 1) { String message = "BigInteger: Invalid signum value"; if (fields.defaulted("signum")) message = "BigInteger: Signum not present in stream"; throw new java.io.StreamCorruptedException(message); } if ((magnitude.length==0) != (sign==0)) { String message = "BigInteger: signum-magnitude mismatch"; if (fields.defaulted("magnitude")) message = "BigInteger: Magnitude not present in stream"; throw new java.io.StreamCorruptedException(message); } // Commit final fields via Unsafe unsafe.putIntVolatile(this, signumOffset, sign); // Calculate mag field from magnitude and discard magnitude unsafe.putObjectVolatile(this, magOffset, stripLeadingZeroBytes(magnitude)); } // Support for resetting final fields while deserializing private static final sun.misc.Unsafe unsafe = sun.misc.Unsafe.getUnsafe(); private static final long signumOffset; private static final long magOffset; static { try { signumOffset = unsafe.objectFieldOffset (BigInteger.class.getDeclaredField("signum")); magOffset = unsafe.objectFieldOffset (BigInteger.class.getDeclaredField("mag")); } catch (Exception ex) { throw new Error(ex); } } /** * Save the {@code BigInteger} instance to a stream. * The magnitude of a BigInteger is serialized as a byte array for * historical reasons. * * @serialData two necessary fields are written as well as obsolete * fields for compatibility with older versions. */ private void writeObject(ObjectOutputStream s) throws IOException { // set the values of the Serializable fields ObjectOutputStream.PutField fields = s.putFields(); fields.put("signum", signum); fields.put("magnitude", magSerializedForm()); // The values written for cached fields are compatible with old // 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)/8; 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; } }