| /* |
| * 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 org.apache.commons.numbers.primes; |
| |
| import java.math.BigInteger; |
| import java.util.AbstractMap.SimpleImmutableEntry; |
| import java.util.ArrayList; |
| import java.util.Arrays; |
| import java.util.HashSet; |
| import java.util.List; |
| import java.util.Map.Entry; |
| import java.util.Set; |
| |
| /** |
| * Utility methods to work on primes within the <code>int</code> range. |
| */ |
| final class SmallPrimes { |
| /** |
| * The first 512 prime numbers. |
| * <p> |
| * It contains all primes smaller or equal to the cubic square of Integer.MAX_VALUE. |
| * As a result, <code>int</code> numbers which are not reduced by those primes are guaranteed |
| * to be either prime or semi prime. |
| */ |
| 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, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, |
| 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, |
| 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, |
| 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, |
| 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, |
| 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, |
| 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, |
| 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, |
| 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, |
| 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, |
| 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, |
| 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, |
| 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, |
| 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, |
| 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, |
| 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, |
| 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, |
| 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, |
| 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, |
| 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671}; |
| |
| /** The last number in {@link #PRIMES}. */ |
| static final int PRIMES_LAST = PRIMES[PRIMES.length - 1]; |
| |
| /** |
| * A set of prime numbers mapped to an array of all integers between |
| * 0 (inclusive) and the least common multiple, i.e. the product, of those |
| * prime numbers (exclusive) that are not divisible by any of these prime |
| * numbers. The prime numbers in the set are among the first 512 prime |
| * numbers, and the {@code int} array containing the numbers undivisible by |
| * these prime numbers is sorted in ascending order. |
| * |
| * <p>The purpose of this field is to speed up trial division by skipping |
| * multiples of individual prime numbers, specifically those contained |
| * in the key of this {@code Entry}, by only trying integers that are equivalent |
| * to one of the integers contained in the value of this {@code Entry} modulo |
| * the least common multiple of the prime numbers in the set.</p> |
| * |
| * <p>Note that, if {@code product} is the product of the prime numbers, |
| * the last number in the array of coprime integers is necessarily |
| * {@code product - 1}, because if {@code product ≡ 0 mod p}, then |
| * {@code product - 1 ≡ -1 mod p}, and {@code 0 ≢ -1 mod p} for any prime number p.</p> |
| */ |
| static final Entry<Set<Integer>, int[]> PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES; |
| |
| static { |
| /* |
| According to the Chinese Remainder Theorem, for every combination of |
| congruence classes modulo distinct, pairwise coprime moduli, there |
| exists exactly one congruence class modulo the product of these |
| moduli that is contained in every one of the former congruence |
| classes. Since the number of congruence classes coprime to a prime |
| number p is p-1, the number of congruence classes coprime to all |
| prime numbers p_1, p_2, p_3 … is (p_1 - 1) * (p_2 - 1) * (p_3 - 1) … |
| |
| Therefore, when using the first five prime numbers as those whose multiples |
| are to be skipped in trial division, the array containing the coprime |
| equivalence classes will have to hold (2-1)*(3-1)*(5-1)*(7-1)*(11-1) = 480 |
| values. As a consequence, the amount of integers to be tried in |
| trial division is reduced to 480/(2*3*5*7*11), which is about 20.78%, |
| of all integers. |
| */ |
| final Set<Integer> primeNumbers = new HashSet<>(); |
| primeNumbers.add(Integer.valueOf(2)); |
| primeNumbers.add(Integer.valueOf(3)); |
| primeNumbers.add(Integer.valueOf(5)); |
| primeNumbers.add(Integer.valueOf(7)); |
| primeNumbers.add(Integer.valueOf(11)); |
| |
| final int product = primeNumbers.stream().reduce(1, (a, b) -> a * b); |
| final int[] equivalenceClasses = new int[primeNumbers.stream().mapToInt(a -> a - 1).reduce(1, (a, b) -> a * b)]; |
| |
| int equivalenceClassIndex = 0; |
| for (int i = 0; i < product; i++) { |
| boolean foundPrimeFactor = false; |
| for (final Integer prime : primeNumbers) { |
| if (i % prime == 0) { |
| foundPrimeFactor = true; |
| break; |
| } |
| } |
| if (!foundPrimeFactor) { |
| equivalenceClasses[equivalenceClassIndex] = i; |
| equivalenceClassIndex++; |
| } |
| } |
| |
| PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES = new SimpleImmutableEntry<>(primeNumbers, equivalenceClasses); |
| } |
| |
| /** |
| * Utility class. |
| */ |
| private SmallPrimes() {} |
| |
| /** |
| * Extract small factors. |
| * |
| * @param n Number to factor, must be > 0. |
| * @param factors List where to add the factors. |
| * @return the part of {@code n} which remains to be factored, it is either |
| * a prime or a semi-prime. |
| */ |
| static int smallTrialDivision(int n, |
| final List<Integer> factors) { |
| for (final int p : PRIMES) { |
| while (0 == n % p) { |
| n /= p; |
| factors.add(p); |
| } |
| } |
| return n; |
| } |
| |
| /** |
| * Extract factors between {@code PRIME_LAST + 2} and {@code maxFactors}. |
| * |
| * @param n Number to factorize, must be larger than {@code PRIME_LAST + 2} |
| * and must not contain any factor below {@code PRIME_LAST + 2}. |
| * @param maxFactor Upper bound of trial division: if it is reached, the |
| * method gives up and returns {@code n}. |
| * @param factors the list where to add the factors. |
| * @return {@code n} (or 1 if factorization is completed). |
| */ |
| static int boundedTrialDivision(int n, |
| int maxFactor, |
| List<Integer> factors) { |
| final int minFactor = PRIMES_LAST + 2; |
| |
| /* |
| only trying integers of the form k*m + c, where k >= 0, m is the |
| product of some prime numbers which n is required not to contain |
| as prime factors, and c is an integer undivisible by all of those |
| prime numbers; in other words, skipping multiples of these primes |
| */ |
| final int m = PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue() |
| [PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue().length - 1] + 1; |
| int km = m * (minFactor / m); |
| int currentEquivalenceClassIndex = Arrays.binarySearch( |
| PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue(), |
| minFactor % m); |
| |
| /* |
| Since minFactor is the next smallest prime number after the |
| first 512 primes, it cannot be a multiple of one of them, therefore, |
| the index returned by the above binary search must be non-negative. |
| */ |
| |
| boolean done = false; |
| while (!done) { |
| // no check is done about n >= f |
| final int f = km + PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue()[currentEquivalenceClassIndex]; |
| if (f > maxFactor) { |
| done = true; |
| } else if (0 == n % f) { |
| n /= f; |
| factors.add(f); |
| done = true; |
| } else { |
| if (currentEquivalenceClassIndex == |
| PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue().length - 1) { |
| km += m; |
| currentEquivalenceClassIndex = 0; |
| } else { |
| currentEquivalenceClassIndex++; |
| } |
| } |
| } |
| if (n != 1) { |
| factors.add(n); |
| } |
| return n; |
| } |
| |
| /** |
| * Factorization by trial division. |
| * |
| * @param n Number to factor. |
| * @return the list of prime factors of {@code n}. |
| */ |
| static List<Integer> trialDivision(int n) { |
| final List<Integer> factors = new ArrayList<>(32); |
| n = smallTrialDivision(n, factors); |
| if (1 == n) { |
| return factors; |
| } |
| // here we are sure that n is either a prime or a semi prime |
| final int bound = (int) Math.sqrt(n); |
| boundedTrialDivision(n, bound, factors); |
| return factors; |
| } |
| |
| /** |
| * Miller-Rabin probabilistic primality test for int type, used in such |
| * a way that a result is always guaranteed. |
| * <p> |
| * It uses the prime numbers as successive base therefore it is guaranteed |
| * to be always correct (see Handbook of applied cryptography by Menezes, |
| * table 4.1). |
| * |
| * @param n Number to test: an odd integer ≥ 3. |
| * @return true if {@code n} is prime, false if it is definitely composite. |
| */ |
| static boolean millerRabinPrimeTest(final int n) { |
| final int nMinus1 = n - 1; |
| final int s = Integer.numberOfTrailingZeros(nMinus1); |
| final int r = nMinus1 >> s; |
| // r must be odd, it is not checked here |
| int t = 1; |
| if (n >= 2047) { |
| t = 2; |
| } |
| if (n >= 1373653) { |
| t = 3; |
| } |
| if (n >= 25326001) { |
| t = 4; |
| } // works up to 3.2 billion, int range stops at 2.7 so we are safe :-) |
| final BigInteger br = BigInteger.valueOf(r); |
| final BigInteger bn = BigInteger.valueOf(n); |
| |
| for (int i = 0; i < t; i++) { |
| final BigInteger a = BigInteger.valueOf(SmallPrimes.PRIMES[i]); |
| final BigInteger bPow = a.modPow(br, bn); |
| int y = bPow.intValue(); |
| if (1 != y && y != nMinus1) { |
| int j = 1; |
| while (j <= s - 1 && nMinus1 != y) { |
| final long square = ((long) y) * y; |
| y = (int) (square % n); |
| if (1 == y) { |
| return false; |
| } // definitely composite |
| j++; |
| } |
| if (nMinus1 != y) { |
| return false; |
| } // definitely composite |
| } |
| } |
| return true; // definitely prime |
| } |
| } |
