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/*
* 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(-100)}. 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(-t)}.
* @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() {
}
}
