mailiverse/gwt/jre/java/math/Primality.java

/*
 * Copyright 2009 Google Inc.
 * 
 * Licensed under the Apache License, Version 2.0 (the "License"); you may not
 * use this file except in compliance with the License. You may obtain a copy of
 * the License at
 * 
 * http://www.apache.org/licenses/LICENSE-2.0
 * 
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
 * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
 * License for the specific language governing permissions and limitations under
 * the License.
 */

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements. See the NOTICE file distributed with this
 * work for additional information regarding copyright ownership. The ASF
 * licenses this file to You under the Apache License, Version 2.0 (the
 * "License"); you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 * 
 * http://www.apache.org/licenses/LICENSE-2.0
 * 
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
 * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
 * License for the specific language governing permissions and limitations under
 * the License.
 * 
 * INCLUDES MODIFICATIONS BY RICHARD ZSCHECH AS WELL AS GOOGLE.
 */
package java.math;

import java.util.Arrays;
import java.util.Random;

/**
 * Provides primality probabilistic methods.
 */
class Primality {

  /**
   * It encodes how many iterations of Miller-Rabin test are need to get an
   * error bound not greater than {@code 2<sup>(-100)</sup>}. For example: for a
   * {@code 1000}-bit number we need {@code 4} iterations, since {@code BITS[3]
   * < 1000 <= BITS[4]}.
   */
  private static final int[] BITS = {
      0, 0, 1854, 1233, 927, 747, 627, 543, 480, 431, 393, 361, 335, 314, 295,
      279, 265, 253, 242, 232, 223, 216, 181, 169, 158, 150, 145, 140, 136,
      132, 127, 123, 119, 114, 110, 105, 101, 96, 92, 87, 83, 78, 73, 69, 64,
      59, 54, 49, 44, 38, 32, 26, 1};

  /**
   * It encodes how many i-bit primes there are in the table for {@code
   * i=2,...,10}. For example {@code offsetPrimes[6]} says that from index
   * {@code 11} exists {@code 7} consecutive {@code 6}-bit prime numbers in the
   * array.
   */
  private static final int[][] offsetPrimes = {
      null, null, {0, 2}, {2, 2}, {4, 2}, {6, 5}, {11, 7}, {18, 13}, {31, 23},
      {54, 43}, {97, 75}};

  /**
   * All prime numbers with bit length lesser than 10 bits.
   */
  private static final int primes[] = {
      2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
      71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149,
      151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
      229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307,
      311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389,
      397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467,
      479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571,
      577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653,
      659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751,
      757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853,
      857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
      953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021};

  /**
   * All {@code BigInteger} prime numbers with bit length lesser than 8 bits.
   */
  private static final BigInteger BIprimes[] = new BigInteger[primes.length];

  static {
    // To initialize the dual table of BigInteger primes
    for (int i = 0; i < primes.length; i++) {
      BIprimes[i] = BigInteger.valueOf(primes[i]);
    }
  }

  /**
   * A random number is generated until a probable prime number is found.
   * 
   * @see BigInteger#BigInteger(int,int,Random)
   * @see BigInteger#probablePrime(int,Random)
   * @see #isProbablePrime(BigInteger, int)
   */
  static BigInteger consBigInteger(int bitLength, int certainty, Random rnd) {
    // PRE: bitLength >= 2;
    // For small numbers get a random prime from the prime table
    if (bitLength <= 10) {
      int rp[] = offsetPrimes[bitLength];
      return BIprimes[rp[0] + rnd.nextInt(rp[1])];
    }
    int shiftCount = (-bitLength) & 31;
    int last = (bitLength + 31) >> 5;
    BigInteger n = new BigInteger(1, last, new int[last]);

    last--;
    do {
      // To fill the array with random integers
      for (int i = 0; i < n.numberLength; i++) {
        n.digits[i] = rnd.nextInt();
      }
      // To fix to the correct bitLength
      n.digits[last] |= 0x80000000;
      n.digits[last] >>>= shiftCount;
      // To create an odd number
      n.digits[0] |= 1;
    } while (!isProbablePrime(n, certainty));
    return n;
  }

  /**
   * @see BigInteger#isProbablePrime(int)
   * @see #millerRabin(BigInteger, int)
   * @ar.org.fitc.ref Optimizations: "A. Menezes - Handbook of applied
   *                  Cryptography, Chapter 4".
   */
  static boolean isProbablePrime(BigInteger n, int certainty) {
    // PRE: n >= 0;
    if ((certainty <= 0) || ((n.numberLength == 1) && (n.digits[0] == 2))) {
      return true;
    }
    // To discard all even numbers
    if (!n.testBit(0)) {
      return false;
    }
    // To check if 'n' exists in the table (it fit in 10 bits)
    if ((n.numberLength == 1) && ((n.digits[0] & 0XFFFFFC00) == 0)) {
      return (Arrays.binarySearch(primes, n.digits[0]) >= 0);
    }
    // To check if 'n' is divisible by some prime of the table
    for (int i = 1; i < primes.length; i++) {
      if (Division.remainderArrayByInt(n.digits, n.numberLength, primes[i]) == 0) {
        return false;
      }
    }
    // To set the number of iterations necessary for Miller-Rabin test
    int i;
    int bitLength = n.bitLength();

    for (i = 2; bitLength < BITS[i]; i++) {
      // empty
    }
    certainty = Math.min(i, 1 + ((certainty - 1) >> 1));

    return millerRabin(n, certainty);
  }

  /**
   * It uses the sieve of Eratosthenes to discard several composite numbers in
   * some appropriate range (at the moment {@code [this, this + 1024]}). After
   * this process it applies the Miller-Rabin test to the numbers that were not
   * discarded in the sieve.
   * 
   * @see BigInteger#nextProbablePrime()
   * @see #millerRabin(BigInteger, int)
   */
  static BigInteger nextProbablePrime(BigInteger n) {
    // PRE: n >= 0
    int i, j;
    int certainty;
    int gapSize = 1024; // for searching of the next probable prime number
    int modules[] = new int[primes.length];
    boolean isDivisible[] = new boolean[gapSize];
    BigInteger startPoint;
    BigInteger probPrime;
    // If n < "last prime of table" searches next prime in the table
    if ((n.numberLength == 1) && (n.digits[0] >= 0)
        && (n.digits[0] < primes[primes.length - 1])) {
      for (i = 0; n.digits[0] >= primes[i]; i++) {
        // empty
      }
      return BIprimes[i];
    }
    /*
     * Creates a "N" enough big to hold the next probable prime Note that: N <
     * "next prime" < 2*N
     */
    startPoint = new BigInteger(1, n.numberLength, new int[n.numberLength + 1]);
    System.arraycopy(n.digits, 0, startPoint.digits, 0, n.numberLength);
    // To fix N to the "next odd number"
    if (n.testBit(0)) {
      Elementary.inplaceAdd(startPoint, 2);
    } else {
      startPoint.digits[0] |= 1;
    }
    // To set the improved certainly of Miller-Rabin
    j = startPoint.bitLength();
    for (certainty = 2; j < BITS[certainty]; certainty++) {
      // empty
    }
    // To calculate modules: N mod p1, N mod p2, ... for first primes.
    for (i = 0; i < primes.length; i++) {
      modules[i] = Division.remainder(startPoint, primes[i]) - gapSize;
    }
    while (true) {
      // At this point, all numbers in the gap are initialized as
      // probably primes
      Arrays.fill(isDivisible, false);
      // To discard multiples of first primes
      for (i = 0; i < primes.length; i++) {
        modules[i] = (modules[i] + gapSize) % primes[i];
        j = (modules[i] == 0) ? 0 : (primes[i] - modules[i]);
        for (; j < gapSize; j += primes[i]) {
          isDivisible[j] = true;
        }
      }
      // To execute Miller-Rabin for non-divisible numbers by all first
      // primes
      for (j = 0; j < gapSize; j++) {
        if (!isDivisible[j]) {
          probPrime = startPoint.copy();
          Elementary.inplaceAdd(probPrime, j);

          if (millerRabin(probPrime, certainty)) {
            return probPrime;
          }
        }
      }
      Elementary.inplaceAdd(startPoint, gapSize);
    }
  }

  /**
   * The Miller-Rabin primality test.
   * 
   * @param n the input number to be tested.
   * @param t the number of trials.
   * @return {@code false} if the number is definitely compose, otherwise
   *         {@code true} with probability {@code 1 - 4<sup>(-t)</sup>}.
   * @ar.org.fitc.ref "D. Knuth, The Art of Computer Programming Vo.2, Section
   *                  4.5.4., Algorithm P"
   */
  private static boolean millerRabin(BigInteger n, int t) {
    // PRE: n >= 0, t >= 0
    BigInteger x; // x := UNIFORM{2...n-1}
    BigInteger y; // y := x^(q * 2^j) mod n
    BigInteger nMinus1 = n.subtract(BigInteger.ONE); // n-1
    int bitLength = nMinus1.bitLength(); // ~ log2(n-1)
    // (q,k) such that: n-1 = q * 2^k and q is odd
    int k = nMinus1.getLowestSetBit();
    BigInteger q = nMinus1.shiftRight(k);
    Random rnd = new Random();

    for (int i = 0; i < t; i++) {
      // To generate a witness 'x', first it use the primes of table
      if (i < primes.length) {
        x = BIprimes[i];
      } else {
        /*
         * It generates random witness only if it's necesssary. Note that all
         * methods would call Miller-Rabin with t <= 50 so this part is only to
         * do more robust the algorithm
         */
        do {
          x = new BigInteger(bitLength, rnd);
        } while ((x.compareTo(n) >= BigInteger.EQUALS) || (x.sign == 0)
            || x.isOne());
      }
      y = x.modPow(q, n);
      if (y.isOne() || y.equals(nMinus1)) {
        continue;
      }
      for (int j = 1; j < k; j++) {
        if (y.equals(nMinus1)) {
          continue;
        }
        y = y.multiply(y).mod(n);
        if (y.isOne()) {
          return false;
        }
      }
      if (!y.equals(nMinus1)) {
        return false;
      }
    }
    return true;
  }

  /**
   * Just to denote that this class can't be instantiated.
   */
  private Primality() {
  }

}