SmallPrimes.java
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* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
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* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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* See the License for the specific language governing permissions and
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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} 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} 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(2);
primeNumbers.add(3);
primeNumbers.add(5);
primeNumbers.add(7);
primeNumbers.add(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 (n % p == 0) {
n /= p;
factors.add(p);
}
}
return n;
}
/**
* Extract factors between {@link #PRIMES_LAST} {@code + 2} and {@code maxFactors}.
*
* @param n Number to factorize, must be larger than {@link #PRIMES_LAST} {@code + 2}
* and must not contain any factor below {@link #PRIMES_LAST} {@code + 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.
*/
static void 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[] a = PRIME_NUMBERS_AND_COPRIME_EQUIVALENCE_CLASSES.getValue();
final int m = a[a.length - 1] + 1;
int km = m * (minFactor / m);
int currentEquivalenceClassIndex = Arrays.binarySearch(a, 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 + a[currentEquivalenceClassIndex];
if (f > maxFactor) {
done = true;
} else if (n % f == 0) {
n /= f;
factors.add(f);
done = true;
} else {
if (currentEquivalenceClassIndex == a.length - 1) {
km += m;
currentEquivalenceClassIndex = 0;
} else {
currentEquivalenceClassIndex++;
}
}
}
// Note: When this method is used after the small trial division n != 1
// for all n in [2, MAX_VALUE] (tested using an exhaustive enumeration).
factors.add(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 (n == 1) {
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 {@code 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
final int t;
if (n >= 25326001) {
// works up to 3.2 billion, int range stops at 2.1 so we are safe :-)
t = 4;
} else if (n >= 1373653) {
t = 3;
} else if (n >= 2047) {
t = 2;
} else {
t = 1;
}
final BigInteger br = BigInteger.valueOf(r);
final BigInteger bn = BigInteger.valueOf(n);
for (int i = 0; i < t; i++) {
final BigInteger a = BigInteger.valueOf(PRIMES[i]);
final BigInteger bPow = a.modPow(br, bn);
int y = bPow.intValue();
if (y != 1 && y != nMinus1) {
int j = 1;
while (j <= s - 1 && y != nMinus1) {
final long square = ((long) y) * y;
y = (int) (square % n);
if (y == 1) {
return false;
} // definitely composite
j++;
}
if (y != nMinus1) {
return false;
} // definitely composite
}
}
return true; // definitely prime
}
}