//***********************************************************************
// Generates a random number with the specified number of bits such
// that gcd(number, this) = 1
//***********************************************************************
public BigInteger GenCoPrime(int bits, StrongNumberProvider rand)
{
bool done = false;
var result = new BigInteger();
while (!done)
{
result.GenRandomBits(bits, rand);
//Console.WriteLine(result.ToString(16));
// gcd test
BigInteger g = result.Gcd(this);
if (g.DataLength == 1 && g._data[0] == 1)
done = true;
}
return result;
}
//***********************************************************************
// Generates a positive BigInteger that is probably prime.
//***********************************************************************
public static BigInteger GenPseudoPrime(int bits, int confidence, StrongNumberProvider rand)
{
var result = new BigInteger();
bool done = false;
while (!done)
{
result.GenRandomBits(bits, rand);
result._data[0] |= 0x01; // make it odd
// prime test
done = result.IsProbablePrime(confidence);
}
return result;
}
//***********************************************************************
// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
//
// p is probably prime if for any a < p (a is not multiple of p),
// a^((p-1)/2) mod p = J(a, p)
//
// where J is the Jacobi symbol.
//
// Otherwise, p is composite.
//
// Returns
// -------
// True if "this" is a Euler pseudoprime to randomly chosen
// bases. The number of chosen bases is given by the "confidence"
// parameter.
//
// False if "this" is definitely NOT prime.
//
//***********************************************************************
public bool SolovayStrassenTest(int confidence)
{
BigInteger thisVal;
if ((_data[MaxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.DataLength == 1)
{
// test small numbers
if (thisVal._data[0] == 0 || thisVal._data[0] == 1)
return false;
if (thisVal._data[0] == 2 || thisVal._data[0] == 3)
return true;
}
if ((thisVal._data[0] & 0x1) == 0) // even numbers
return false;
int bits = thisVal.BitCount();
var a = new BigInteger();
BigInteger pSub1 = thisVal - 1;
BigInteger pSub1Shift = pSub1 >> 1;
var rand = new StrongNumberProvider();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = (int) (rand.GetNextSingle()*bits);
a.GenRandomBits(testBits, rand);
int byteLen = a.DataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a._data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.Gcd(thisVal);
if (gcdTest.DataLength == 1 && gcdTest._data[0] != 1)
return false;
// calculate a^((p-1)/2) mod p
BigInteger expResult = a.ModPow(pSub1Shift, thisVal);
if (expResult == pSub1)
expResult = -1;
// calculate Jacobi symbol
BigInteger jacob = Jacobi(a, thisVal);
//Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
//Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));
// if they are different then it is not prime
if (expResult != jacob)
return false;
}
return true;
}
//***********************************************************************
// Probabilistic prime test based on Rabin-Miller's
//
// for any p > 0 with p - 1 = 2^s * t
//
// p is probably prime (strong pseudoprime) if for any a < p,
// 1) a^t mod p = 1 or
// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
//
// Otherwise, p is composite.
//
// Returns
// -------
// True if "this" is a strong pseudoprime to randomly chosen
// bases. The number of chosen bases is given by the "confidence"
// parameter.
//
// False if "this" is definitely NOT prime.
//
//***********************************************************************
public bool RabinMillerTest(int confidence)
{
BigInteger thisVal;
if ((_data[MaxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.DataLength == 1)
{
// test small numbers
if (thisVal._data[0] == 0 || thisVal._data[0] == 1)
return false;
if (thisVal._data[0] == 2 || thisVal._data[0] == 3)
return true;
}
if ((thisVal._data[0] & 0x1) == 0) // even numbers
return false;
// calculate values of s and t
BigInteger pSub1 = thisVal - (new BigInteger(1));
int s = 0;
for (int index = 0; index < pSub1.DataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((pSub1._data[index] & mask) != 0)
{
index = pSub1.DataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = pSub1 >> s;
int bits = thisVal.BitCount();
var a = new BigInteger();
var rand = new StrongNumberProvider();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = (int) (rand.GetNextSingle()*bits);
a.GenRandomBits(testBits, rand);
int byteLen = a.DataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a._data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.Gcd(thisVal);
if (gcdTest.DataLength == 1 && gcdTest._data[0] != 1)
return false;
BigInteger b = a.ModPow(t, thisVal);
bool result = false;
if (b.DataLength == 1 && b._data[0] == 1) // a^t mod p = 1
result = true;
for (int j = 0; result == false && j < s; j++)
{
if (b == pSub1) // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
{
result = true;
break;
}
b = (b*b)%thisVal;
}
if (result == false)
return false;
}
return true;
}
//***********************************************************************
// Probabilistic prime test based on Fermat's little theorem
//
// for any a < p (p does not divide a) if
// a^(p-1) mod p != 1 then p is not prime.
//
// Otherwise, p is probably prime (pseudoprime to the chosen base).
//
// Returns
// -------
// True if "this" is a pseudoprime to randomly chosen
// bases. The number of chosen bases is given by the "confidence"
// parameter.
//
// False if "this" is definitely NOT prime.
//
// Note - this method is fast but fails for Carmichael numbers except
// when the randomly chosen base is a factor of the number.
//
//***********************************************************************
public bool FermatLittleTest(int confidence)
{
BigInteger thisVal;
if ((_data[MaxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.DataLength == 1)
{
// test small numbers
if (thisVal._data[0] == 0 || thisVal._data[0] == 1)
return false;
if (thisVal._data[0] == 2 || thisVal._data[0] == 3)
return true;
}
if ((thisVal._data[0] & 0x1) == 0) // even numbers
return false;
int bits = thisVal.BitCount();
var a = new BigInteger();
BigInteger pSub1 = thisVal - (new BigInteger(1));
var rand = new StrongNumberProvider();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = rand.GetNextInt()*bits;
a.GenRandomBits(testBits, rand);
int byteLen = a.DataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a._data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.Gcd(thisVal);
if (gcdTest.DataLength == 1 && gcdTest._data[0] != 1)
return false;
// calculate a^(p-1) mod p
BigInteger expResult = a.ModPow(pSub1, thisVal);
int resultLen = expResult.DataLength;
// is NOT prime is a^(p-1) mod p != 1
if (resultLen > 1 || (resultLen == 1 && expResult._data[0] != 1))
{
//Console.WriteLine("a = " + a.ToString());
return false;
}
}
return true;
}
//***********************************************************************
// Populates "this" with the specified amount of random bits
//***********************************************************************
public void GenRandomBits(int bits, StrongNumberProvider rand)
{
int dwords = bits >> 5;
int remBits = bits & 0x1F;
if (remBits != 0)
dwords++;
if (dwords > MaxLength)
throw (new ArithmeticException("Number of required bits > maxLength."));
for (int i = 0; i < dwords; i++)
_data[i] = rand.GetNextUInt32();
for (int i = dwords; i < MaxLength; i++)
_data[i] = 0;
if (remBits != 0)
{
var mask = (uint) (0x01 << (remBits - 1));
_data[dwords - 1] |= mask;
mask = (0xFFFFFFFF >> (32 - remBits));
_data[dwords - 1] &= mask;
}
else
_data[dwords - 1] |= 0x80000000;
DataLength = dwords;
if (DataLength == 0)
DataLength = 1;
}