public static BigInteger FindSmallPrimes(int bits)
{
if (bits > 20)
{
throw new ArgumentException("bits > 20");
}
Log.MethodEnter("TrialDivision.CheckForSmallComposites", nameof(bits), bits);
BigInteger result = 0;
using (executionTimer.StartTimer())
{
long n = 0;
bool composite = true;
while (composite)
{
n = 1 << (bits - 1);
for (int i = 0; i < bits - 1; i++)
{
n |= CryptoRandomSingleton.Next(2) << i;
}
composite = !Eratosthenes.IsPrime(n);
}
result = n;
}
Log.MethodLeave();
return(result);
}
public static bool IsProbablyPrime(BigInteger primeHopeful, int testCount)
{
bool result = false;
using (outsideExecutionTime.StartTimer())
{
Log.MethodEnter("MillerRabinPrimalityTest", nameof(primeHopeful), $"{primeHopeful}, {nameof(testCount)}: {testCount}");
if (primeHopeful == 2 || primeHopeful == 3 || primeHopeful == 5 || primeHopeful == 7)
{
FoundFactor(primeHopeful, BigInteger.GreatestCommonDivisor(primeHopeful, 210));
result = true;
goto exit;
}
BigInteger remainder = primeHopeful & 1; // % Two;
if (remainder == 0)
{
FoundFactor(primeHopeful, 2);
result = false;
goto exit;
}
BigInteger hopefulLess1 = primeHopeful - 1;
BigInteger quotientD = hopefulLess1;
remainder = quotientD & 1; // % Two;
long divisionCountR = 0;
while (remainder == 0)
{
quotientD = quotientD >> 1; // / Two;
remainder = quotientD & 1; // % Two;
divisionCountR++;
}
BigInteger hopefulLess2 = primeHopeful - Two;
BigInteger residueX;
BigInteger priorResidueX;
BigInteger randomA = 0;
// Tracks previous random values, prevents testing with the same value twice,
// which would dramatically reduce our confidence in the primality of a probable prime.
// Seems unlikely to happen, but its not a hit to performance to check.
List <BigInteger> previousRandomValues = new List <BigInteger>();
int counter = 1;
int modCount = 1;
for (counter = 1; counter <= testCount; counter++)
{
do
{
randomA = CryptoRandomSingleton.RandomRange(Two, hopefulLess2);
}while (previousRandomValues.Contains(randomA));
previousRandomValues.Add(randomA);
Log.Message($"MillerRabin check prime hopeful (P) against random n: {randomA}");
using (insideExecutionTime.StartTimer())
{
residueX = BigInteger.ModPow(randomA, quotientD, primeHopeful);
Log.Message($"MillerRabin check a^d ≡ r (mod primeHopeful): {randomA}^{quotientD} ≡ {residueX} (mod {primeHopeful})");
}
if (residueX == 1 || residueX == hopefulLess1)
{
BigInteger gcd = BigInteger.GreatestCommonDivisor(residueX - 1, primeHopeful); // Chance to factor: GCD((a^d mod n) − 1, n)
Log.Message($"A) Chance to factor: GCD((a^d mod n) − 1, primeHopeful)");
if (gcd != 1 && gcd != primeHopeful)
{
FoundFactor(primeHopeful, gcd);
result = false;
goto exit;
}
continue;
}
modCount = 1;
while (modCount <= divisionCountR)
{
priorResidueX = residueX;
residueX = BigInteger.ModPow(residueX, 2, primeHopeful);
Log.Message($"MillerRabin check a^2 ≡ r (mod primeHopeful): {priorResidueX}^2 ≡ {residueX} (mod {primeHopeful})");
if (residueX == 1)
{
// Chance to factor: a^(d*2^r)-1
BigInteger twoR = BigInteger.Pow(2, (int)divisionCountR);
BigInteger dTwoR = BigInteger.Multiply(quotientD, twoR);
if (dTwoR < int.MaxValue)
{
int pow = (int)dTwoR;
BigInteger potentialFactor = BigInteger.Pow(randomA, pow) - 1;
BigInteger gcd = BigInteger.GreatestCommonDivisor(potentialFactor, primeHopeful);
Log.Message($"B) Chance to factor: GCD(a^(d*(2^r))-1, primeHopeful): GCD({randomA}^({quotientD}*2^{divisionCountR})-1 == {potentialFactor - 1}, primeHopeful) == {gcd}");
if (gcd != 1 && gcd != primeHopeful)
{
FoundFactor(primeHopeful, gcd);
result = false;
goto exit;
}
}
Log.MethodLeave();
return(false);
}
if (residueX == hopefulLess1)
{
// Chance to factor: GCD((a^(d*2^r) mod n) − 1, n)
BigInteger twoR = BigInteger.Pow(2, (int)divisionCountR);
BigInteger dTwoR = BigInteger.Multiply(quotientD, twoR);
BigInteger potentialFactor = BigInteger.ModPow(randomA, dTwoR, primeHopeful);
BigInteger gcd = BigInteger.GreatestCommonDivisor(potentialFactor - 1, primeHopeful);
Log.Message($"C) Chance to factor: GCD( a^(d*(2^r)) ≡ m−1 (mod primeHopeful) , primeHopeful): GCD( {randomA}^({quotientD}*(2^{divisionCountR})) ≡ {potentialFactor}-1 (mod primeHopeful) == {potentialFactor - 1}, primeHopeful) == {gcd}");
if (gcd != 1 && gcd != primeHopeful)
{
FoundFactor(primeHopeful, gcd);
result = false;
goto exit;
}
break;
}
modCount++;
}
if (residueX != hopefulLess1)
{
// ??? Chance to factor ???
BigInteger twoR = BigInteger.Pow(2, (int)divisionCountR);
BigInteger dTwoR = BigInteger.Multiply(quotientD, twoR);
BigInteger potentialFactor1 = BigInteger.ModPow(randomA, dTwoR, primeHopeful);
BigInteger gcd1 = BigInteger.GreatestCommonDivisor(potentialFactor1 - 1, primeHopeful);
Log.Message($"D) Chance to factor: GCD( a^(d*(2^r)) ≡ m−1 (mod primeHopeful) , primeHopeful): GCD( {randomA}^({quotientD}*(2^{divisionCountR})) ≡ {potentialFactor1}-1 (mod primeHopeful) == {potentialFactor1 - 1}, primeHopeful) == {gcd1}");
if (gcd1 != 1 && gcd1 != primeHopeful)
{
FoundFactor(primeHopeful, gcd1);
result = false;
goto exit;
}
result = false;
goto exit;
}
}
result = true;
goto exit;
}
exit:
Log.MethodLeave();
return(result);
}