Prime (GNU cryptographic primitives and tools)
Keywords: gnu.crypto.util.Prime,Prime class

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## gnu.crypto.util
Class Prime

    java.lang.Object
      |
      +--gnu.crypto.util.Prime

public class Prime
extends java.lang.Object

A collection of prime number related utilities used in this library.

Method Summary
`static boolean` | `hasSmallPrimeDivisor(java.math.BigInteger w)` Trial division for the first 1000 small primes.
`static boolean` | `isProbablePrime(java.math.BigInteger w)`
`static boolean` | `isProbablePrime(java.math.BigInteger w, int certainty)` This implementation does not rely solely on the Miller-Rabin strong probabilistic primality test to claim the primality of the designated number.
`static boolean` | `passEulerCriterion(java.math.BigInteger w)` Java port of Colin Plumb primality test (Euler Criterion) implementation for a base of 2 --from bnlib-1.1 release, function primeTest() in prime.c.
`static boolean` | `passFermatLittleTheorem(java.math.BigInteger w, int t)` Checks Fermat's Little Theorem for base b; i.e.
`static boolean` | `passMillerRabin(java.math.BigInteger n, int t)` Applies the Miller-Rabin strong probabilistic primality test.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Method Detail

### hasSmallPrimeDivisor

    public static boolean hasSmallPrimeDivisor(java.math.BigInteger w)

  Trial division for the first 1000 small primes.

  Returns `true` if at least one small prime, among the first 1000 ones, was found to divide the designated number. Retuens `false` otherwise.

  Parameters:
    `w` - the number to test.
  Returns:
    `true` if at least one small prime was found to divide the designated number.

### passEulerCriterion

    public static boolean passEulerCriterion(java.math.BigInteger w)

  Java port of Colin Plumb primality test (Euler Criterion) implementation for a base of 2 --from bnlib-1.1 release, function primeTest() in prime.c. this is his comments; (bn is our w).

  "Now, check that bn is prime. If it passes to the base 2, it's prime beyond all reasonable doubt, and everything else is just gravy, but it gives people warm fuzzies to do it.

  This starts with verifying Euler's criterion for a base of 2. This is the fastest pseudoprimality test that I know of, saving a modular squaring over a Fermat test, as well as being stronger. 7/8 of the time, it's as strong as a strong pseudoprimality test, too. (The exception being when `bn == 1 mod 8` and `2` is a quartic residue, i.e. `bn` is of the form `a^2 + (8*b)^2`.) The precise series of tricks used here is not documented anywhere, so here's an explanation. Euler's criterion states that if `p` is prime then `a^((p-1)/2)` is congruent to `Jacobi(a,p)`, modulo `p`. `Jacobi(a, p)` is a function which is `+1` if a is a square modulo `p`, and `-1` if it is not. For `a = 2`, this is particularly simple. It's `+1` if `p == +/-1 (mod 8)`, and `-1` if `m == +/-3 (mod 8)`. If `p == 3 (mod 4)`, then all a strong test does is compute `2^((p-1)/2)`. and see if it's `+1` or `-1`. (Euler's criterion says which it should be.) If `p == 5 (mod 8)`, then `2^((p-1)/2)` is `-1`, so the initial step in a strong test, looking at `2^((p-1)/4)`, is wasted --you're not going to find a `+/-1` before then if it is prime, and it shouldn't have either of those values if it isn't. So don't bother.

  The remaining case is `p == 1 (mod 8)`. In this case, we expect `2^((p-1)/2) == 1 (mod p)`, so we expect that the square root of this, `2^((p-1)/4)`, will be `+/-1 (mod p)` . Evaluating this saves us a modular squaring 1/4 of the time. If it's `-1`, a strong pseudoprimality test would call `p` prime as well. Only if the result is `+1`, indicating that `2` is not only a quadratic residue, but a quartic one as well, does a strong pseudoprimality test verify more things than this test does. Good enough.

  We could back that down another step, looking at `2^((p-1)/8)` if there was a cheap way to determine if `2` were expected to be a quartic residue or not. Dirichlet proved that `2` is a quadratic residue iff `p` is of the form `a^2 + (8*b^2)`. All primes `== 1 (mod 4)` can be expressed as `a^2 + (2*b)^2`, but I see no cheap way to evaluate this condition."

  Parameters:
    `w` - the number to test.
  Returns:
    `true` iff the designated number passes Euler criterion as implemented by Colin Plumb in his bnlib version 1.1.

### passFermatLittleTheorem

    public static boolean passFermatLittleTheorem(java.math.BigInteger w,
                                                  int t)

  Checks Fermat's Little Theorem for base b; i.e. `b**(w-1) == 1 (mod w)`.

  Parameters:
    `w` - the number to test.
    `t` - the number of random bases to test.
  Returns:
    `true` iff `b**(w-1) == 1 (mod w)`, for some b.

### passMillerRabin

    public static boolean passMillerRabin(java.math.BigInteger n,
                                          int t)

  Applies the Miller-Rabin strong probabilistic primality test.

  The HAC (Handbook of Applied Cryptography), Alfred Menezes & al. Note 4.57 states that for `q`, `n=18` is enough while for `p`, `n=6` (512 bits) or `n=3 (1024 bits) are enough to yield robust primality tests. The values used are from table 4.4 given in Note 4.49.`

  Returns:
    `true` iff the designated number passes the Miller- Rabin probabilistic primality test for a computed number of rounds.

### isProbablePrime

    public static boolean isProbablePrime(java.math.BigInteger w)

### isProbablePrime

    public static boolean isProbablePrime(java.math.BigInteger w,
                                          int certainty)

  This implementation does not rely solely on the Miller-Rabin strong probabilistic primality test to claim the primality of the designated number. It instead, tries dividing the designated number by the first 1000 small primes, and if no divisor was found, invokes a port of Colin Plumb's implementation of the Euler Criterion, then tries the Miller-Rabin test.

  Parameters:
    `w` - the integer to test.
    `certainty` - the certainty with which to compute the test.
  Returns:
    `true` iff the designated number has no small prime divisor passes the Euler criterion, and optionally a Miller-Rabin test.

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