Java tutorial
/* * 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. */ package java.math; import java.util.Arrays; import com.google.gwt.user.client.Random; /** * Provides primality probabilistic methods. */ class Primality { /** Just to denote that this class can't be instantiated. */ private Primality() { } /** 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]; /** * 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 } }; static {// To initialize the dual table of BigInteger primes for (int i = 0; i < primes.length; i++) { BIprimes[i] = BigInteger.valueOf(primes[i]); } } /** * 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++) { ; } 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++) { ; } // 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); } } /** * 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++) { ; } certainty = Math.min(i, 1 + ((certainty - 1) >> 1)); return millerRabin(n, certainty); } /** * 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 n_minus_1 = n.subtract(BigInteger.ONE); // n-1 int bitLength = n_minus_1.bitLength(); // ~ log2(n-1) // (q,k) such that: n-1 = q * 2^k and q is odd int k = n_minus_1.getLowestSetBit(); BigInteger q = n_minus_1.shiftRight(k); 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, (Random) null); } while ((x.compareTo(n) >= BigInteger.EQUALS) || (x.sign == 0) || x.isOne()); } y = x.modPow(q, n); if (y.isOne() || y.equals(n_minus_1)) { continue; } for (int j = 1; j < k; j++) { if (y.equals(n_minus_1)) { continue; } y = y.multiply(y).mod(n); if (y.isOne()) { return false; } } if (!y.equals(n_minus_1)) { return false; } } return true; } }