/// <summary>
/// Determines whether <paramref name="n"/> is probably a prime number. Uses the
/// [Miller–Rabin primality test](https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test).
/// </summary>
/// <remarks>This is a probabilistic test. The probability to return <b>true</b> for a composite number
/// is 4^(−<paramref name="k"/>). </remarks>
/// <param name="n">Number to test (must be odd).</param>
/// <param name="k">The number of random tests to do.</param>
/// <param name="rand">The random number generator to use.</param>
/// <returns>
/// <b>true</b> if the number <paramref name="n"/> is probable a prime number; <b>false</b>, if the number is composite.
/// </returns>
public static bool IsProbablyPrime_MillerRabinTest(BigInteger n, BigInteger k, RandomXorShift rand)
{
var n1 = n - 1;
var d = FactorPow2(n1, out long r);
for (int i = 0; i < k; i++)
{
var a = rand.NextValue(n - 3) + 2;
var x = IntegerMath.PowMod(a, d, n);
if (x == 1 || x == n1)
{
continue;
}
bool continueI = false;
for (int j = 0; j < r - 1; j++)
{
x = IntegerMath.PowMod(x, 2, n);
if (x == 1)
{
return(false);
}
if (x == n1)
{
continueI = true; break;
}
}
if (continueI)
{
continue;
}
return(false);
}
return(true);
}
/// <summary>
/// Tests a number for primality. The result is always exact.
/// </summary>
/// <param name="num">The number to test.</param>
/// <returns><b>true</b> if the number is a prime number; <b>false</b> if it is composite.</returns>
public static bool IsPrime(this long num)
{
if (num <= 1)
{
return(false);
}
if (num <= 3)
{
return(true);
}
if ((num & 1) == 0)
{
return(false);
}
var end = IntegerMath.IntSqrt(num);
for (int i = 2; i < SmallPrimes.Length; i++)
{
var testNum = SmallPrimes[i];
if (testNum > end)
{
return(true);
}
if (num % testNum == 0)
{
return(false);
}
}
return(SPRPLong(num));
}
/// <summary>
/// Guesses a prime factor of the number <paramref name="num"/>.
/// </summary>
/// <param name="num">The number.</param>
/// <returns>A possible prime factor.</returns>
public static BigInteger GuessPrimeFactor(BigInteger num)
{
var d = FactorPow2(num - 1, out long r);
var pm = IntegerMath.PowMod(2, d, num);
var gcd = IntegerMath.Gcd(pm - 1, num);
return(gcd);
}
/// <summary>
/// Deterministic Miller Rabin test for primality.
/// </summary>
/// <param name="n">Number to test (must be odd).</param>
/// <param name="b">Base to test.</param>
/// <returns>
/// <b>true</b> if the number <paramref name="n"/> is probable a prime number; <b>false</b>, if the number is composite.
/// </returns>
public static bool SPRP(BigInteger n, BigInteger b)
{
var nMinus1 = n - 1;
var t = FactorPow2(nMinus1, out long r);
var test = IntegerMath.PowMod(b, t, n);
if (test == 1 || test == nMinus1)
{
return(true);
}
while (--r > 0)
{
test = IntegerMath.MulMod(test, test, n);
if (test == nMinus1)
{
return(true);
}
}
return(false);
}
