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* Unless required by applicable law or agreed to in writing, software
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package org.apache.commons.math3.primes;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;
import org.apache.commons.math3.util.FastMath;
Utility methods to work on primes within the int range.
Since: 3.2
/**
* Utility methods to work on primes within the <code>int</code> range.
* @since 3.2
*/
class SmallPrimes {
The first 512 prime numbers.
It contains all primes smaller or equal to the cubic square of Integer.MAX_VALUE.
As a result, int numbers which are not reduced by those primes are guaranteed
to be either prime or semi prime.
/**
* 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.
*/
public 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 PRIMES. /** The last number in PRIMES. */
public static final int PRIMES_LAST = PRIMES[PRIMES.length - 1];
Hide utility class.
/**
* Hide utility class.
*/
private SmallPrimes() {
}
Extract small factors.
Params: - n – the number to factor, must be > 0.
- factors – the list where to add the factors.
Returns: the part of n which remains to be factored, it is either a prime or a semi-prime
/**
* Extract small factors.
* @param n the number to factor, must be > 0.
* @param factors the list where to add the factors.
* @return the part of n which remains to be factored, it is either a prime or a semi-prime
*/
public static int smallTrialDivision(int n, final List<Integer> factors) {
for (int p : PRIMES) {
while (0 == n % p) {
n /= p;
factors.add(p);
}
}
return n;
}
Extract factors in the range PRIME_LAST+2 to maxFactors.
Params: - n – the number to factorize, must be >= PRIME_LAST+2 and must not contain any factor below PRIME_LAST+2
- maxFactor – the upper bound of trial division: if it is reached, the method gives up and returns n.
- factors – the list where to add the factors.
Returns: n or 1 if factorization is completed.
/**
* Extract factors in the range <code>PRIME_LAST+2</code> to <code>maxFactors</code>.
* @param n the number to factorize, must be >= PRIME_LAST+2 and must not contain any factor below PRIME_LAST+2
* @param maxFactor the upper bound of trial division: if it is reached, the method gives up and returns n.
* @param factors the list where to add the factors.
* @return n or 1 if factorization is completed.
*/
public static int boundedTrialDivision(int n, int maxFactor, List<Integer> factors) {
int f = PRIMES_LAST + 2;
// no check is done about n >= f
while (f <= maxFactor) {
if (0 == n % f) {
n /= f;
factors.add(f);
break;
}
f += 4;
if (0 == n % f) {
n /= f;
factors.add(f);
break;
}
f += 2;
}
if (n != 1) {
factors.add(n);
}
return n;
}
Factorization by trial division.
Params: - n – the number to factor
Returns: the list of prime factors of n
/**
* Factorization by trial division.
* @param n the number to factor
* @return the list of prime factors of n
*/
public static List<Integer> trialDivision(int n){
final List<Integer> factors = new ArrayList<Integer>(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) FastMath.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.
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)
Params: - n – number to test: an odd integer ≥ 3
Returns: true if n is prime. false if n is definitely composite.
/**
* 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 n is prime. false if n is definitely composite.
*/
public 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 :-)
BigInteger br = BigInteger.valueOf(r);
BigInteger bn = BigInteger.valueOf(n);
for (int i = 0; i < t; i++) {
BigInteger a = BigInteger.valueOf(SmallPrimes.PRIMES[i]);
BigInteger bPow = a.modPow(br, bn);
int y = bPow.intValue();
if ((1 != y) && (y != nMinus1)) {
int j = 1;
while ((j <= s - 1) && (nMinus1 != y)) {
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
}
}