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package org.bouncycastle.math;
import java.math.BigInteger;
import java.security.SecureRandom;
import org.bouncycastle.crypto.Digest;
import org.bouncycastle.util.Arrays;
import org.bouncycastle.util.BigIntegers;
/**
* Utility methods for generating primes and testing for primality.
*/
public abstract class Primes
{
public static final int SMALL_FACTOR_LIMIT = 211;
private static final BigInteger ONE = BigInteger.valueOf(1);
private static final BigInteger TWO = BigInteger.valueOf(2);
private static final BigInteger THREE = BigInteger.valueOf(3);
/**
* Used to return the output from the
* {@linkplain Primes#enhancedMRProbablePrimeTest(BigInteger, SecureRandom, int) Enhanced
* Miller-Rabin Probabilistic Primality Test}
*/
public static class MROutput
{
private static MROutput probablyPrime()
{
return new MROutput(false, null);
}
private static MROutput provablyCompositeWithFactor(BigInteger factor)
{
return new MROutput(true, factor);
}
private static MROutput provablyCompositeNotPrimePower()
{
return new MROutput(true, null);
}
private boolean provablyComposite;
private BigInteger factor;
private MROutput(boolean provablyComposite, BigInteger factor)
{
this.provablyComposite = provablyComposite;
this.factor = factor;
}
public BigInteger getFactor()
{
return factor;
}
public boolean isProvablyComposite()
{
return provablyComposite;
}
public boolean isNotPrimePower()
{
return provablyComposite && factor == null;
}
}
/**
* Used to return the output from the
* {@linkplain Primes#generateSTRandomPrime(Digest, int, byte[]) Shawe-Taylor Random_Prime
* Routine}
*/
public static class STOutput
{
private BigInteger prime;
private byte[] primeSeed;
private int primeGenCounter;
private STOutput(BigInteger prime, byte[] primeSeed, int primeGenCounter)
{
this.prime = prime;
this.primeSeed = primeSeed;
this.primeGenCounter = primeGenCounter;
}
public BigInteger getPrime()
{
return prime;
}
public byte[] getPrimeSeed()
{
return primeSeed;
}
public int getPrimeGenCounter()
{
return primeGenCounter;
}
}
/**
* FIPS 186-4 C.6 Shawe-Taylor Random_Prime Routine
*
* Construct a provable prime number using a hash function.
*
* @param hash
* the {@link Digest} instance to use (as "Hash()"). Cannot be null.
* @param length
* the length (in bits) of the prime to be generated. Must be at least 2.
* @param inputSeed
* the seed to be used for the generation of the requested prime. Cannot be null or
* empty.
* @return an {@link STOutput} instance containing the requested prime.
*/
public static STOutput generateSTRandomPrime(Digest hash, int length, byte[] inputSeed)
{
if (hash == null)
{
throw new IllegalArgumentException("'hash' cannot be null");
}
if (length < 2)
{
throw new IllegalArgumentException("'length' must be >= 2");
}
if (inputSeed == null || inputSeed.length == 0)
{
throw new IllegalArgumentException("'inputSeed' cannot be null or empty");
}
return implSTRandomPrime(hash, length, Arrays.clone(inputSeed));
}
/**
* FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test
*
* Run several iterations of the Miller-Rabin algorithm with randomly-chosen bases. This is an
* alternative to {@link #isMRProbablePrime(BigInteger, SecureRandom, int)} that provides more
* information about a composite candidate, which may be useful when generating or validating
* RSA moduli.
*
* @param candidate
* the {@link BigInteger} instance to test for primality.
* @param random
* the source of randomness to use to choose bases.
* @param iterations
* the number of randomly-chosen bases to perform the test for.
* @return an {@link MROutput} instance that can be further queried for details.
*/
public static MROutput enhancedMRProbablePrimeTest(BigInteger candidate, SecureRandom random, int iterations)
{
checkCandidate(candidate, "candidate");
if (random == null)
{
throw new IllegalArgumentException("'random' cannot be null");
}
if (iterations < 1)
{
throw new IllegalArgumentException("'iterations' must be > 0");
}
if (candidate.bitLength() == 2)
{
return MROutput.probablyPrime();
}
if (!candidate.testBit(0))
{
return MROutput.provablyCompositeWithFactor(TWO);
}
BigInteger w = candidate;
BigInteger wSubOne = candidate.subtract(ONE);
BigInteger wSubTwo = candidate.subtract(TWO);
int a = wSubOne.getLowestSetBit();
BigInteger m = wSubOne.shiftRight(a);
for (int i = 0; i < iterations; ++i)
{
BigInteger b = BigIntegers.createRandomInRange(TWO, wSubTwo, random);
BigInteger g = b.gcd(w);
if (g.compareTo(ONE) > 0)
{
return MROutput.provablyCompositeWithFactor(g);
}
BigInteger z = b.modPow(m, w);
if (z.equals(ONE) || z.equals(wSubOne))
{
continue;
}
boolean primeToBase = false;
BigInteger x = z;
for (int j = 1; j < a; ++j)
{
z = z.modPow(TWO, w);
if (z.equals(wSubOne))
{
primeToBase = true;
break;
}
if (z.equals(ONE))
{
break;
}
x = z;
}
if (!primeToBase)
{
if (!z.equals(ONE))
{
x = z;
z = z.modPow(TWO, w);
if (!z.equals(ONE))
{
x = z;
}
}
g = x.subtract(ONE).gcd(w);
if (g.compareTo(ONE) > 0)
{
return MROutput.provablyCompositeWithFactor(g);
}
return MROutput.provablyCompositeNotPrimePower();
}
}
return MROutput.probablyPrime();
}
/**
* A fast check for small divisors, up to some implementation-specific limit.
*
* @param candidate
* the {@link BigInteger} instance to test for division by small factors.
*
* @return true if the candidate is found to have any small factors,
* false otherwise.
*/
public static boolean hasAnySmallFactors(BigInteger candidate)
{
checkCandidate(candidate, "candidate");
return implHasAnySmallFactors(candidate);
}
/**
* FIPS 186-4 C.3.1 Miller-Rabin Probabilistic Primality Test
*
* Run several iterations of the Miller-Rabin algorithm with randomly-chosen bases.
*
* @param candidate
* the {@link BigInteger} instance to test for primality.
* @param random
* the source of randomness to use to choose bases.
* @param iterations
* the number of randomly-chosen bases to perform the test for.
* @return false if any witness to compositeness is found amongst the chosen bases
* (so candidate is definitely NOT prime), or else true
* (indicating primality with some probability dependent on the number of iterations
* that were performed).
*/
public static boolean isMRProbablePrime(BigInteger candidate, SecureRandom random, int iterations)
{
checkCandidate(candidate, "candidate");
if (random == null)
{
throw new IllegalArgumentException("'random' cannot be null");
}
if (iterations < 1)
{
throw new IllegalArgumentException("'iterations' must be > 0");
}
if (candidate.bitLength() == 2)
{
return true;
}
if (!candidate.testBit(0))
{
return false;
}
BigInteger w = candidate;
BigInteger wSubOne = candidate.subtract(ONE);
BigInteger wSubTwo = candidate.subtract(TWO);
int a = wSubOne.getLowestSetBit();
BigInteger m = wSubOne.shiftRight(a);
for (int i = 0; i < iterations; ++i)
{
BigInteger b = BigIntegers.createRandomInRange(TWO, wSubTwo, random);
if (!implMRProbablePrimeToBase(w, wSubOne, m, a, b))
{
return false;
}
}
return true;
}
/**
* FIPS 186-4 C.3.1 Miller-Rabin Probabilistic Primality Test (to a fixed base).
*
* Run a single iteration of the Miller-Rabin algorithm against the specified base.
*
* @param candidate
* the {@link BigInteger} instance to test for primality.
* @param base
* the base value to use for this iteration.
* @return false if the specified base is a witness to compositeness (so
* candidate is definitely NOT prime), or else true.
*/
public static boolean isMRProbablePrimeToBase(BigInteger candidate, BigInteger base)
{
checkCandidate(candidate, "candidate");
checkCandidate(base, "base");
if (base.compareTo(candidate.subtract(ONE)) >= 0)
{
throw new IllegalArgumentException("'base' must be < ('candidate' - 1)");
}
if (candidate.bitLength() == 2)
{
return true;
}
BigInteger w = candidate;
BigInteger wSubOne = candidate.subtract(ONE);
int a = wSubOne.getLowestSetBit();
BigInteger m = wSubOne.shiftRight(a);
return implMRProbablePrimeToBase(w, wSubOne, m, a, base);
}
private static void checkCandidate(BigInteger n, String name)
{
if (n == null || n.signum() < 1 || n.bitLength() < 2)
{
throw new IllegalArgumentException("'" + name + "' must be non-null and >= 2");
}
}
private static boolean implHasAnySmallFactors(BigInteger x)
{
/*
* Bundle trial divisors into ~32-bit moduli then use fast tests on the ~32-bit remainders.
*/
int m = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23;
int r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 2) == 0 || (r % 3) == 0 || (r % 5) == 0 || (r % 7) == 0 || (r % 11) == 0 || (r % 13) == 0
|| (r % 17) == 0 || (r % 19) == 0 || (r % 23) == 0)
{
return true;
}
m = 29 * 31 * 37 * 41 * 43;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 29) == 0 || (r % 31) == 0 || (r % 37) == 0 || (r % 41) == 0 || (r % 43) == 0)
{
return true;
}
m = 47 * 53 * 59 * 61 * 67;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 47) == 0 || (r % 53) == 0 || (r % 59) == 0 || (r % 61) == 0 || (r % 67) == 0)
{
return true;
}
m = 71 * 73 * 79 * 83;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 71) == 0 || (r % 73) == 0 || (r % 79) == 0 || (r % 83) == 0)
{
return true;
}
m = 89 * 97 * 101 * 103;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 89) == 0 || (r % 97) == 0 || (r % 101) == 0 || (r % 103) == 0)
{
return true;
}
m = 107 * 109 * 113 * 127;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 107) == 0 || (r % 109) == 0 || (r % 113) == 0 || (r % 127) == 0)
{
return true;
}
m = 131 * 137 * 139 * 149;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 131) == 0 || (r % 137) == 0 || (r % 139) == 0 || (r % 149) == 0)
{
return true;
}
m = 151 * 157 * 163 * 167;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 151) == 0 || (r % 157) == 0 || (r % 163) == 0 || (r % 167) == 0)
{
return true;
}
m = 173 * 179 * 181 * 191;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 173) == 0 || (r % 179) == 0 || (r % 181) == 0 || (r % 191) == 0)
{
return true;
}
m = 193 * 197 * 199 * 211;
r = x.mod(BigInteger.valueOf(m)).intValue();
if ((r % 193) == 0 || (r % 197) == 0 || (r % 199) == 0 || (r % 211) == 0)
{
return true;
}
/*
* NOTE: Unit tests depend on SMALL_FACTOR_LIMIT matching the
* highest small factor tested here.
*/
return false;
}
private static boolean implMRProbablePrimeToBase(BigInteger w, BigInteger wSubOne, BigInteger m, int a, BigInteger b)
{
BigInteger z = b.modPow(m, w);
if (z.equals(ONE) || z.equals(wSubOne))
{
return true;
}
boolean result = false;
for (int j = 1; j < a; ++j)
{
z = z.modPow(TWO, w);
if (z.equals(wSubOne))
{
result = true;
break;
}
if (z.equals(ONE))
{
return false;
}
}
return result;
}
private static STOutput implSTRandomPrime(Digest d, int length, byte[] primeSeed)
{
int dLen = d.getDigestSize();
if (length < 33)
{
int primeGenCounter = 0;
byte[] c0 = new byte[dLen];
byte[] c1 = new byte[dLen];
for (;;)
{
hash(d, primeSeed, c0, 0);
inc(primeSeed, 1);
hash(d, primeSeed, c1, 0);
inc(primeSeed, 1);
int c = extract32(c0) ^ extract32(c1);
c &= (-1 >>> (32 - length));
c |= (1 << (length - 1)) | 1;
++primeGenCounter;
long c64 = c & 0xFFFFFFFFL;
if (isPrime32(c64))
{
return new STOutput(BigInteger.valueOf(c64), primeSeed, primeGenCounter);
}
if (primeGenCounter > (4 * length))
{
throw new IllegalStateException("Too many iterations in Shawe-Taylor Random_Prime Routine");
}
}
}
STOutput rec = implSTRandomPrime(d, (length + 3) / 2, primeSeed);
BigInteger c0 = rec.getPrime();
primeSeed = rec.getPrimeSeed();
int primeGenCounter = rec.getPrimeGenCounter();
int outlen = 8 * dLen;
int iterations = (length - 1) / outlen;
int oldCounter = primeGenCounter;
BigInteger x = hashGen(d, primeSeed, iterations + 1);
x = x.mod(ONE.shiftLeft(length - 1)).setBit(length - 1);
BigInteger c0x2 = c0.shiftLeft(1);
BigInteger tx2 = x.subtract(ONE).divide(c0x2).add(ONE).shiftLeft(1);
int dt = 0;
BigInteger c = tx2.multiply(c0).add(ONE);
/*
* TODO Since the candidate primes are generated by constant steps ('c0x2'), sieving could
* be used here in place of the 'hasAnySmallFactors' approach.
*/
for (;;)
{
if (c.bitLength() > length)
{
tx2 = ONE.shiftLeft(length - 1).subtract(ONE).divide(c0x2).add(ONE).shiftLeft(1);
c = tx2.multiply(c0).add(ONE);
}
++primeGenCounter;
/*
* This is an optimization of the original algorithm, using trial division to screen out
* many non-primes quickly.
*
* NOTE: 'primeSeed' is still incremented as if we performed the full check!
*/
if (!implHasAnySmallFactors(c))
{
BigInteger a = hashGen(d, primeSeed, iterations + 1);
a = a.mod(c.subtract(THREE)).add(TWO);
tx2 = tx2.add(BigInteger.valueOf(dt));
dt = 0;
BigInteger z = a.modPow(tx2, c);
if (c.gcd(z.subtract(ONE)).equals(ONE) && z.modPow(c0, c).equals(ONE))
{
return new STOutput(c, primeSeed, primeGenCounter);
}
}
else
{
inc(primeSeed, iterations + 1);
}
if (primeGenCounter >= ((4 * length) + oldCounter))
{
throw new IllegalStateException("Too many iterations in Shawe-Taylor Random_Prime Routine");
}
dt += 2;
c = c.add(c0x2);
}
}
private static int extract32(byte[] bs)
{
int result = 0;
int count = Math.min(4, bs.length);
for (int i = 0; i < count; ++i)
{
int b = bs[bs.length - (i + 1)] & 0xFF;
result |= (b << (8 * i));
}
return result;
}
private static void hash(Digest d, byte[] input, byte[] output, int outPos)
{
d.update(input, 0, input.length);
d.doFinal(output, outPos);
}
private static BigInteger hashGen(Digest d, byte[] seed, int count)
{
int dLen = d.getDigestSize();
int pos = count * dLen;
byte[] buf = new byte[pos];
for (int i = 0; i < count; ++i)
{
pos -= dLen;
hash(d, seed, buf, pos);
inc(seed, 1);
}
return new BigInteger(1, buf);
}
private static void inc(byte[] seed, int c)
{
int pos = seed.length;
while (c > 0 && --pos >= 0)
{
c += (seed[pos] & 0xFF);
seed[pos] = (byte)c;
c >>>= 8;
}
}
private static boolean isPrime32(long x)
{
if (x >>> 32 != 0L)
{
throw new IllegalArgumentException("Size limit exceeded");
}
/*
* Use wheel factorization with 2, 3, 5 to select trial divisors.
*/
if (x <= 5L)
{
return x == 2L || x == 3L || x == 5L;
}
if ((x & 1L) == 0L || (x % 3L) == 0L || (x % 5L) == 0L)
{
return false;
}
long[] ds = new long[]{ 1L, 7L, 11L, 13L, 17L, 19L, 23L, 29L };
long base = 0L;
for (int pos = 1;; pos = 0)
{
/*
* Trial division by wheel-selected divisors
*/
while (pos < ds.length)
{
long d = base + ds[pos];
if (x % d == 0L)
{
return x < 30L;
}
++pos;
}
base += 30L;
if (base * base >= x)
{
return true;
}
}
}
}