/// <summary>
/// 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).
/// </summary>
/// <remarks>
/// this method is fast but fails for Carmichael numbers except
/// when the randomly chosen base is a factor of the number.
/// </remarks>
/// <param name="confidence">The confidence.</param>
/// <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.</returns>
public bool FermatLittleTest(int confidence)
{
BigInteger thisVal;
if ((this.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;
else 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();
BigInteger a = new BigInteger();
BigInteger p_sub1 = thisVal - (new BigInteger(1));
Random rand = new Random();
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.NextDouble() * 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(p_sub1, 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;
}
/// <summary>
/// Overloading of unary >> operators
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="shiftVal">The shift val.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator >>(BigInteger bi1, int shiftVal)
{
BigInteger result = new BigInteger(bi1);
result.dataLength = shiftRight(result.data, shiftVal);
if ((bi1.data[maxLength - 1] & 0x80000000) != 0) // negative
{
for (int i = maxLength - 1; i >= result.dataLength; i--)
result.data[i] = 0xFFFFFFFF;
uint mask = 0x80000000;
for (int i = 0; i < 32; i++)
{
if ((result.data[result.dataLength - 1] & mask) != 0)
break;
result.data[result.dataLength - 1] |= mask;
mask >>= 1;
}
result.dataLength = maxLength;
}
return result;
}
/// <summary>
/// Implements the operator -.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator -(BigInteger bi1)
{
// handle neg of zero separately since it'll cause an overflow
// if we proceed.
if (bi1.dataLength == 1 && bi1.data[0] == 0)
return (new BigInteger());
BigInteger result = new BigInteger(bi1);
// 1's complement
for (int i = 0; i < maxLength; i++)
result.data[i] = (uint)(~(bi1.data[i]));
// add one to result of 1's complement
long val, carry = 1;
int index = 0;
while (carry != 0 && index < maxLength)
{
val = (long)(result.data[index]);
val++;
result.data[index] = (uint)(val & 0xFFFFFFFF);
carry = val >> 32;
index++;
}
if ((bi1.data[maxLength - 1] & 0x80000000) == (result.data[maxLength - 1] & 0x80000000))
throw (new ArithmeticException("Overflow in negation.\n"));
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
return result;
}
/// <summary>
/// Implements the operator -.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="bi2">The bi2.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
{
BigInteger result = new BigInteger();
result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
long carryIn = 0;
for (int i = 0; i < result.dataLength; i++)
{
long diff;
diff = (long)bi1.data[i] - (long)bi2.data[i] - carryIn;
result.data[i] = (uint)(diff & 0xFFFFFFFF);
if (diff < 0)
carryIn = 1;
else
carryIn = 0;
}
// roll over to negative
if (carryIn != 0)
{
for (int i = result.dataLength; i < maxLength; i++)
result.data[i] = 0xFFFFFFFF;
result.dataLength = maxLength;
}
// fixed in v1.03 to give correct datalength for a - (-b)
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException());
}
return result;
}
/// <summary>
/// Implements the operator *.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="bi2">The bi2.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
{
int lastPos = maxLength - 1;
bool bi1Neg = false, bi2Neg = false;
// take the absolute value of the inputs
try
{
if ((bi1.data[lastPos] & 0x80000000) != 0) // bi1 negative
{
bi1Neg = true; bi1 = -bi1;
}
if ((bi2.data[lastPos] & 0x80000000) != 0) // bi2 negative
{
bi2Neg = true; bi2 = -bi2;
}
}
catch (Exception) { }
BigInteger result = new BigInteger();
// multiply the absolute values
try
{
for (int i = 0; i < bi1.dataLength; i++)
{
if (bi1.data[i] == 0) continue;
ulong mcarry = 0;
for (int j = 0, k = i; j < bi2.dataLength; j++, k++)
{
// k = i + j
ulong val = ((ulong)bi1.data[i] * (ulong)bi2.data[j]) +
(ulong)result.data[k] + mcarry;
result.data[k] = (uint)(val & 0xFFFFFFFF);
mcarry = (val >> 32);
}
if (mcarry != 0)
result.data[i + bi2.dataLength] = (uint)mcarry;
}
}
catch (Exception)
{
throw (new ArithmeticException("Multiplication overflow."));
}
result.dataLength = bi1.dataLength + bi2.dataLength;
if (result.dataLength > maxLength)
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check (result is -ve)
if ((result.data[lastPos] & 0x80000000) != 0)
{
if (bi1Neg != bi2Neg && result.data[lastPos] == 0x80000000) // different sign
{
// handle the special case where multiplication produces
// a max negative number in 2's complement.
if (result.dataLength == 1)
return result;
else
{
bool isMaxNeg = true;
for (int i = 0; i < result.dataLength - 1 && isMaxNeg; i++)
{
if (result.data[i] != 0)
isMaxNeg = false;
}
if (isMaxNeg)
return result;
}
}
throw (new ArithmeticException("Multiplication overflow."));
}
// if input has different signs, then result is -ve
if (bi1Neg != bi2Neg)
return -result;
return result;
}
/// <summary>
/// Tests the correct implementation of the modulo exponential and
/// inverse modulo functions using RSA encryption and decryption. The two
/// pseudoprimes p and q are fixed, but the two RSA keys are generated
/// for each round of testing.
/// </summary>
/// <param name="rounds">The rounds.</param>
public static void RSATest2(int rounds)
{
Random rand = new Random();
byte[] val = new byte[64];
byte[] pseudoPrime1 = {
(byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
(byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
(byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
(byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
(byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
(byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
(byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
(byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
(byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
(byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
(byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
};
byte[] pseudoPrime2 = {
(byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
(byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
(byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
(byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
(byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
(byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
(byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
(byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
(byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
(byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
(byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
};
BigInteger bi_p = new BigInteger(pseudoPrime1);
BigInteger bi_q = new BigInteger(pseudoPrime2);
BigInteger bi_pq = (bi_p - 1) * (bi_q - 1);
BigInteger bi_n = bi_p * bi_q;
for (int count = 0; count < rounds; count++)
{
// generate private and public key
BigInteger bi_e = bi_pq.genCoPrime(512, rand);
BigInteger bi_d = bi_e.modInverse(bi_pq);
Console.WriteLine("\ne =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
Console.Write("Round = " + count);
// encrypt and decrypt data
BigInteger bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);
// compare
if (bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/// <summary>
/// Implements the operator +.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="bi2">The bi2.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
{
BigInteger result = new BigInteger();
result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
long carry = 0;
for (int i = 0; i < result.dataLength; i++)
{
long sum = (long)bi1.data[i] + (long)bi2.data[i] + carry;
carry = sum >> 32;
result.data[i] = (uint)(sum & 0xFFFFFFFF);
}
if (carry != 0 && result.dataLength < maxLength)
{
result.data[result.dataLength] = (uint)(carry);
result.dataLength++;
}
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
if ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException());
}
return result;
}
/// <summary>
/// Generates a random positive BigInteger with the specified number of bits such that gcd(number, this) = 1
/// </summary>
/// <param name="bits">The bits.</param>
/// <param name="rand">The rand.</param>
/// <returns></returns>
public BigInteger genCoPrime(int bits, Random rand)
{
bool done = false;
BigInteger 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;
}
/// <summary>
/// Returns the modulo inverse of this.
/// The modulus inverse is defined as the unique number x such that (this * x) mod modulus = 1
/// </summary>
/// <exception cref="ArithmeticException">
/// Throws ArithmeticException ifthe inverse does not exist. (i.e. gcd(this, modulus) != 1).
/// </exception>
/// <param name="modulus">The modulus.</param>
/// <returns></returns>
public BigInteger modInverse(BigInteger modulus)
{
BigInteger[] p = { 0, 1 };
BigInteger[] q = new BigInteger[2]; // quotients
BigInteger[] r = { 0, 0 }; // remainders
int step = 0;
BigInteger a = modulus;
BigInteger b = this;
while (b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0))
{
BigInteger quotient = new BigInteger();
BigInteger remainder = new BigInteger();
if (step > 1)
{
BigInteger pval = (p[0] - (p[1] * q[0])) % modulus;
p[0] = p[1];
p[1] = pval;
}
if (b.dataLength == 1)
singleByteDivide(a, b, quotient, remainder);
else
multiByteDivide(a, b, quotient, remainder);
/*
Console.WriteLine(quotient.dataLength);
Console.WriteLine("{0} = {1}({2}) + {3} p = {4}", a.ToString(10),
b.ToString(10), quotient.ToString(10), remainder.ToString(10),
p[1].ToString(10));
*/
q[0] = q[1];
r[0] = r[1];
q[1] = quotient; r[1] = remainder;
a = b;
b = remainder;
step++;
}
if (r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1))
throw (new ArithmeticException("No inverse!"));
BigInteger result = ((p[0] - (p[1] * q[0])) % modulus);
if ((result.data[maxLength - 1] & 0x80000000) != 0)
result += modulus; // get the least positive modulus
return result;
}
/// <summary>
/// Computes the Jacobi Symbol for a and b.
/// </summary>
/// <exception cref="ArgumentException">Throws ArgumentException if b is not odd.</exception>
/// <param name="a">A.</param>
/// <param name="b">The b.</param>
/// <returns></returns>
public static int Jacobi(BigInteger a, BigInteger b)
{
// Jacobi defined only for odd integers
if ((b.data[0] & 0x1) == 0)
throw (new ArgumentException("Jacobi defined only for odd integers."));
if (a >= b) a %= b;
if (a.dataLength == 1 && a.data[0] == 0) return 0; // a == 0
if (a.dataLength == 1 && a.data[0] == 1) return 1; // a == 1
if (a < 0)
{
if ((((b - 1).data[0]) & 0x2) == 0) //if( (((b-1) >> 1).data[0] & 0x1) == 0)
return Jacobi(-a, b);
else
return -Jacobi(-a, b);
}
int e = 0;
for (int index = 0; index < a.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((a.data[index] & mask) != 0)
{
index = a.dataLength; // to break the outer loop
break;
}
mask <<= 1;
e++;
}
}
BigInteger a1 = a >> e;
int s = 1;
if ((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5))
s = -1;
if ((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3)
s = -s;
if (a1.dataLength == 1 && a1.data[0] == 1)
return s;
else
return (s * Jacobi(b % a1, a1));
}
/// <summary>
/// Generates a random positive BigInteger with the specified number of bits and is possibly prime.
/// </summary>
/// <param name="bits">The bits.</param>
/// <param name="confidence">The confidence.</param>
/// <param name="rand">The rand.</param>
/// <returns></returns>
public static BigInteger genPseudoPrime(int bits, int confidence, Random rand)
{
BigInteger 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;
}
/// <summary>
/// Private method called by the public LucasStrongTest method to perform a Lucas strong pseudoprime test on thisVal.
/// </summary>
/// <param name="thisVal">The this val.</param>
/// <returns></returns>
private bool LucasStrongTestHelper(BigInteger thisVal)
{
// Do the test (selects D based on Selfridge)
// Let D be the first element of the sequence
// 5, -7, 9, -11, 13, ... for which J(D,n) = -1
// Let P = 1, Q = (1-D) / 4
long D = 5, sign = -1, dCount = 0;
bool done = false;
while (!done)
{
int Jresult = BigInteger.Jacobi(D, thisVal);
if (Jresult == -1)
done = true; // J(D, this) = 1
else
{
if (Jresult == 0 && Math.Abs(D) < thisVal) // divisor found
return false;
if (dCount == 20)
{
// check for square
BigInteger root = thisVal.sqrt();
if (root * root == thisVal)
return false;
}
//Console.WriteLine(D);
D = (Math.Abs(D) + 2) * sign;
sign = -sign;
}
dCount++;
}
long Q = (1 - D) >> 2;
/*
Console.WriteLine("D = " + D);
Console.WriteLine("Q = " + Q);
Console.WriteLine("(n,D) = " + thisVal.gcd(D));
Console.WriteLine("(n,Q) = " + thisVal.gcd(Q));
Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal));
*/
BigInteger p_add1 = thisVal + 1;
int s = 0;
for (int index = 0; index < p_add1.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((p_add1.data[index] & mask) != 0)
{
index = p_add1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_add1 >> s;
// calculate constant = b^(2k) / m
// for Barrett Reduction
BigInteger constant = new BigInteger();
int nLen = thisVal.dataLength << 1;
constant.data[nLen] = 0x00000001;
constant.dataLength = nLen + 1;
constant = constant / thisVal;
BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
bool isPrime = false;
if ((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
(lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
{
// u(t) = 0 or V(t) = 0
isPrime = true;
}
for (int i = 1; i < s; i++)
{
if (!isPrime)
{
// doubling of index
lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant);
lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal;
//lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal;
if ((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
isPrime = true;
}
lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant); //Q^k
}
if (isPrime) // additional checks for composite numbers
{
// If n is prime and gcd(n, Q) == 1, then
// Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n
BigInteger g = thisVal.gcd(Q);
if (g.dataLength == 1 && g.data[0] == 1) // gcd(this, Q) == 1
{
if ((lucas[2].data[maxLength - 1] & 0x80000000) != 0)
lucas[2] += thisVal;
BigInteger temp = (Q * BigInteger.Jacobi(Q, thisVal)) % thisVal;
if ((temp.data[maxLength - 1] & 0x80000000) != 0)
temp += thisVal;
if (lucas[2] != temp)
isPrime = false;
}
}
return isPrime;
}
/// <summary>
/// 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.
/// </summary>
/// <param name="confidence">The confidence.</param>
/// <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.</returns>
public bool SolovayStrassenTest(int confidence)
{
BigInteger thisVal;
if ((this.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;
else 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();
BigInteger a = new BigInteger();
BigInteger p_sub1 = thisVal - 1;
BigInteger p_sub1_shift = p_sub1 >> 1;
Random rand = new Random();
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.NextDouble() * 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(p_sub1_shift, thisVal);
if (expResult == p_sub1)
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;
}
/// <summary>
/// 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.
/// </summary>
/// <param name="confidence">The confidence.</param>
/// <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.</returns>
public bool RabinMillerTest(int confidence)
{
BigInteger thisVal;
if ((this.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;
else 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 p_sub1 = thisVal - (new BigInteger(1));
int s = 0;
for (int index = 0; index < p_sub1.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((p_sub1.data[index] & mask) != 0)
{
index = p_sub1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_sub1 >> s;
int bits = thisVal.bitCount();
BigInteger a = new BigInteger();
Random rand = new Random();
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.NextDouble() * 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);
/*
Console.WriteLine("a = " + a.ToString(10));
Console.WriteLine("b = " + b.ToString(10));
Console.WriteLine("t = " + t.ToString(10));
Console.WriteLine("s = " + s);
*/
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 == p_sub1) // 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;
}
/// <summary>
/// Tests the correct implementation of the /, %, * and + operators
/// </summary>
/// <param name="rounds">The rounds.</param>
public static void MulDivTest(int rounds)
{
Random rand = new Random();
byte[] val = new byte[64];
byte[] val2 = new byte[64];
for (int count = 0; count < rounds; count++)
{
// generate 2 numbers of random length
int t1 = 0;
while (t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
int t2 = 0;
while (t2 == 0)
t2 = (int)(rand.NextDouble() * 65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t2)
val2[i] = (byte)(rand.NextDouble() * 256);
else
val2[i] = 0;
if (val2[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
while (val2[0] == 0)
val2[0] = (byte)(rand.NextDouble() * 256);
Console.WriteLine(count);
BigInteger bn1 = new BigInteger(val, t1);
BigInteger bn2 = new BigInteger(val2, t2);
// Determine the quotient and remainder by dividing
// the first number by the second.
BigInteger bn3 = bn1 / bn2;
BigInteger bn4 = bn1 % bn2;
// Recalculate the number
BigInteger bn5 = (bn3 * bn2) + bn4;
// Make sure they're the same
if (bn5 != bn1)
{
Console.WriteLine("Error at " + count);
Console.WriteLine(bn1 + "\n");
Console.WriteLine(bn2 + "\n");
Console.WriteLine(bn3 + "\n");
Console.WriteLine(bn4 + "\n");
Console.WriteLine(bn5 + "\n");
return;
}
}
}
/// <summary>
///Returns the k_th number in the Lucas Sequence reduced modulo n.
///
/// Uses index doubling to speed up the process. For example, to calculate V(k),
/// we maintain two numbers in the sequence V(n) and V(n+1).
///
/// To obtain V(2n), we use the identity
/// V(2n) = (V(n) * V(n)) - (2 * Q^n)
/// To obtain V(2n+1), we first write it as
/// V(2n+1) = V((n+1) + n)
/// and use the identity
/// V(m+n) = V(m) * V(n) - Q * V(m-n)
/// Hence,
/// V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n)
/// = V(n+1) * V(n) - Q^n * V(1)
/// = V(n+1) * V(n) - Q^n * P
///
/// We use k in its binary expansion and perform index doubling for each
/// bit position. For each bit position that is set, we perform an
/// index doubling followed by an index addition. This means that for V(n),
/// we need to update it to V(2n+1). For V(n+1), we need to update it to
/// V((2n+1)+1) = V(2*(n+1))
///
/// This function returns
/// [0] = U(k)
/// [1] = V(k)
/// [2] = Q^n
///
/// Where U(0) = 0 % n, U(1) = 1 % n
/// V(0) = 2 % n, V(1) = P % n
/// </summary>
/// <param name="P">The P.</param>
/// <param name="Q">The Q.</param>
/// <param name="k">The k.</param>
/// <param name="n">The n.</param>
/// <returns></returns>
public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
BigInteger k, BigInteger n)
{
if (k.dataLength == 1 && k.data[0] == 0)
{
BigInteger[] result = new BigInteger[3];
result[0] = 0; result[1] = 2 % n; result[2] = 1 % n;
return result;
}
// calculate constant = b^(2k) / m
// for Barrett Reduction
BigInteger constant = new BigInteger();
int nLen = n.dataLength << 1;
constant.data[nLen] = 0x00000001;
constant.dataLength = nLen + 1;
constant = constant / n;
// calculate values of s and t
int s = 0;
for (int index = 0; index < k.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((k.data[index] & mask) != 0)
{
index = k.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = k >> s;
//Console.WriteLine("s = " + s + " t = " + t);
return LucasSequenceHelper(P, Q, t, n, constant, s);
}
/// <summary>
/// Tests the correct implementation of the modulo exponential function
/// using RSA encryption and decryption (using pre-computed encryption and
/// decryption keys).
/// </summary>
/// <param name="rounds">The rounds.</param>
public static void RSATest(int rounds)
{
Random rand = new Random(1);
byte[] val = new byte[64];
// private and public key
BigInteger bi_e = new BigInteger("a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7", 16);
BigInteger bi_d = new BigInteger("4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7", 16);
BigInteger bi_n = new BigInteger("e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f", 16);
Console.WriteLine("e =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
for (int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
Console.Write("Round = " + count);
// encrypt and decrypt data
BigInteger bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);
// compare
if (bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/// <summary>
/// Performs the calculation of the kth term in the Lucas Sequence.
/// For details of the algorithm, see reference [9].
/// k must be odd. i.e LSB == 1
/// </summary>
/// <param name="P">The P.</param>
/// <param name="Q">The Q.</param>
/// <param name="k">The k.</param>
/// <param name="n">The n.</param>
/// <param name="constant">The constant.</param>
/// <param name="s">The s.</param>
/// <returns></returns>
private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q,
BigInteger k, BigInteger n,
BigInteger constant, int s)
{
BigInteger[] result = new BigInteger[3];
if ((k.data[0] & 0x00000001) == 0)
throw (new ArgumentException("Argument k must be odd."));
int numbits = k.bitCount();
uint mask = (uint)0x1 << ((numbits & 0x1F) - 1);
// v = v0, v1 = v1, u1 = u1, Q_k = Q^0
BigInteger v = 2 % n, Q_k = 1 % n,
v1 = P % n, u1 = Q_k;
bool flag = true;
for (int i = k.dataLength - 1; i >= 0; i--) // iterate on the binary expansion of k
{
//Console.WriteLine("round");
while (mask != 0)
{
if (i == 0 && mask == 0x00000001) // last bit
break;
if ((k.data[i] & mask) != 0) // bit is set
{
// index doubling with addition
u1 = (u1 * v1) % n;
v = ((v * v1) - (P * Q_k)) % n;
v1 = n.BarrettReduction(v1 * v1, n, constant);
v1 = (v1 - ((Q_k * Q) << 1)) % n;
if (flag)
flag = false;
else
Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
Q_k = (Q_k * Q) % n;
}
else
{
// index doubling
u1 = ((u1 * v) - Q_k) % n;
v1 = ((v * v1) - (P * Q_k)) % n;
v = n.BarrettReduction(v * v, n, constant);
v = (v - (Q_k << 1)) % n;
if (flag)
{
Q_k = Q % n;
flag = false;
}
else
Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
}
mask >>= 1;
}
mask = 0x80000000;
}
// at this point u1 = u(n+1) and v = v(n)
// since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1)
u1 = ((u1 * v) - Q_k) % n;
v = ((v * v1) - (P * Q_k)) % n;
if (flag)
flag = false;
else
Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
Q_k = (Q_k * Q) % n;
for (int i = 0; i < s; i++)
{
// index doubling
u1 = (u1 * v) % n;
v = ((v * v) - (Q_k << 1)) % n;
if (flag)
{
Q_k = Q % n;
flag = false;
}
else
Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
}
result[0] = u1;
result[1] = v;
result[2] = Q_k;
return result;
}
/// <summary>
/// Tests the correct implementation of sqrt() method.
/// </summary>
/// <param name="rounds">The rounds.</param>
public static void SqrtTest(int rounds)
{
Random rand = new Random();
for (int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int)(rand.NextDouble() * 1024);
Console.Write("Round = " + count);
BigInteger a = new BigInteger();
a.genRandomBits(t1, rand);
BigInteger b = a.sqrt();
BigInteger c = (b + 1) * (b + 1);
// check that b is the largest integer such that b*b <= a
if (c <= a)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(a + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/// <summary>
/// Raises the current number to the power specified.
/// </summary>
/// <param name="number">The number to be raised.</param>
/// <param name="raisedTo">The power to be raised to.</param>
/// <returns>
/// A BigInteger representing this raised to a power
/// </returns>
public static BigInteger Pow(BigInteger number, BigInteger raisedTo)
{
return number.Pow(raisedTo);
}
/// <summary>
/// Implements the operator ++.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator ++(BigInteger bi1)
{
BigInteger result = new BigInteger(bi1);
long val, carry = 1;
int index = 0;
while (carry != 0 && index < maxLength)
{
val = (long)(result.data[index]);
val++;
result.data[index] = (uint)(val & 0xFFFFFFFF);
carry = val >> 32;
index++;
}
if (index > result.dataLength)
result.dataLength = index;
else
{
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
}
// overflow check
int lastPos = maxLength - 1;
// overflow if initial value was +ve but ++ caused a sign
// change to negative.
if ((bi1.data[lastPos] & 0x80000000) == 0 &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException("Overflow in ++."));
}
return result;
}
public BigInteger Pow(BigInteger exp)
{
return power(this, exp);
}
/// <summary>
/// Implements the operator --.
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator --(BigInteger bi1)
{
BigInteger result = new BigInteger(bi1);
long val;
bool carryIn = true;
int index = 0;
while (carryIn && index < maxLength)
{
val = (long)(result.data[index]);
val--;
result.data[index] = (uint)(val & 0xFFFFFFFF);
if (val >= 0)
carryIn = false;
index++;
}
if (index > result.dataLength)
result.dataLength = index;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
// overflow if initial value was -ve but -- caused a sign
// change to positive.
if ((bi1.data[lastPos] & 0x80000000) != 0 &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException("Underflow in --."));
}
return result;
}
private static BigInteger power(BigInteger number, BigInteger exponent)
{
if (exponent == 0)
return 1;
if (exponent == 1)
return number;
if (exponent % 2 == 0)
return square(power(number, exponent / 2));
else
return number * square(power(number, (exponent - 1) / 2));
}
/// <summary>
/// Overloading of unary << operators
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="shiftVal">The shift val.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator <<(BigInteger bi1, int shiftVal)
{
BigInteger result = new BigInteger(bi1);
result.dataLength = shiftLeft(result.data, shiftVal);
return result;
}
private static BigInteger square(BigInteger num)
{
return num * num;
}
/// <summary>
/// Overloading of the NOT operator (1's complement)
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <returns>The result of the operator.</returns>
public static BigInteger operator ~(BigInteger bi1)
{
BigInteger result = new BigInteger(bi1);
for (int i = 0; i < maxLength; i++)
result.data[i] = (uint)(~(bi1.data[i]));
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
return result;
}
//root function
public BigInteger root(int order)
{
uint numBits = (uint)this.bitCount();
if ((numBits & 0x1) != 0) // odd number of bits
numBits = (numBits >> 1) + 1;
else
numBits = (numBits >> 1);
uint bytePos = numBits >> 5;
byte bitPos = (byte)(numBits & 0x1F);
uint mask;
BigInteger result = new BigInteger();
if (bitPos == 0)
mask = 0x80000000;
else
{
mask = (uint)1 << bitPos;
bytePos++;
}
result.dataLength = (int)bytePos;
for (int i = (int)bytePos - 1; i >= 0; i--)
{
while (mask != 0)
{
// guess
result.data[i] ^= mask;
// undo the guess if its square is larger than this
if ((result.Pow(order)) > this)
result.data[i] ^= mask;
mask >>= 1;
}
mask = 0x80000000;
}
return result;
}
/// <summary>
/// Private function that supports the division of two numbers with
/// a divisor that has more than 1 digit.
///
/// Algorithm taken from [1]
/// </summary>
/// <param name="bi1">The bi1.</param>
/// <param name="bi2">The bi2.</param>
/// <param name="outQuotient">The out quotient.</param>
/// <param name="outRemainder">The out remainder.</param>
private static void multiByteDivide(BigInteger bi1, BigInteger bi2,
BigInteger outQuotient, BigInteger outRemainder)
{
uint[] result = new uint[maxLength];
int remainderLen = bi1.dataLength + 1;
uint[] remainder = new uint[remainderLen];
uint mask = 0x80000000;
uint val = bi2.data[bi2.dataLength - 1];
int shift = 0, resultPos = 0;
while (mask != 0 && (val & mask) == 0)
{
shift++; mask >>= 1;
}
//Console.WriteLine("shift = {0}", shift);
//Console.WriteLine("Before bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);
for (int i = 0; i < bi1.dataLength; i++)
remainder[i] = bi1.data[i];
shiftLeft(remainder, shift);
bi2 = bi2 << shift;
/*
Console.WriteLine("bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);
Console.WriteLine("dividend = " + bi1 + "\ndivisor = " + bi2);
for(int q = remainderLen - 1; q >= 0; q--)
Console.Write("{0:x2}", remainder[q]);
Console.WriteLine();
*/
int j = remainderLen - bi2.dataLength;
int pos = remainderLen - 1;
ulong firstDivisorByte = bi2.data[bi2.dataLength - 1];
ulong secondDivisorByte = bi2.data[bi2.dataLength - 2];
int divisorLen = bi2.dataLength + 1;
uint[] dividendPart = new uint[divisorLen];
while (j > 0)
{
ulong dividend = ((ulong)remainder[pos] << 32) + (ulong)remainder[pos - 1];
//Console.WriteLine("dividend = {0}", dividend);
ulong q_hat = dividend / firstDivisorByte;
ulong r_hat = dividend % firstDivisorByte;
//Console.WriteLine("q_hat = {0:X}, r_hat = {1:X}", q_hat, r_hat);
bool done = false;
while (!done)
{
done = true;
if (q_hat == 0x100000000 ||
(q_hat * secondDivisorByte) > ((r_hat << 32) + remainder[pos - 2]))
{
q_hat--;
r_hat += firstDivisorByte;
if (r_hat < 0x100000000)
done = false;
}
}
for (int h = 0; h < divisorLen; h++)
dividendPart[h] = remainder[pos - h];
BigInteger kk = new BigInteger(dividendPart);
BigInteger ss = bi2 * (long)q_hat;
//Console.WriteLine("ss before = " + ss);
while (ss > kk)
{
q_hat--;
ss -= bi2;
//Console.WriteLine(ss);
}
BigInteger yy = kk - ss;
//Console.WriteLine("ss = " + ss);
//Console.WriteLine("kk = " + kk);
//Console.WriteLine("yy = " + yy);
for (int h = 0; h < divisorLen; h++)
remainder[pos - h] = yy.data[bi2.dataLength - h];
/*
Console.WriteLine("dividend = ");
for(int q = remainderLen - 1; q >= 0; q--)
Console.Write("{0:x2}", remainder[q]);
Console.WriteLine("\n************ q_hat = {0:X}\n", q_hat);
*/
result[resultPos++] = (uint)q_hat;
pos--;
j--;
}
outQuotient.dataLength = resultPos;
int y = 0;
for (int x = outQuotient.dataLength - 1; x >= 0; x--, y++)
outQuotient.data[y] = result[x];
for (; y < maxLength; y++)
outQuotient.data[y] = 0;
while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0)
outQuotient.dataLength--;
if (outQuotient.dataLength == 0)
outQuotient.dataLength = 1;
outRemainder.dataLength = shiftRight(remainder, shift);
for (y = 0; y < outRemainder.dataLength; y++)
outRemainder.data[y] = remainder[y];
for (; y < maxLength; y++)
outRemainder.data[y] = 0;
}
/// <summary>
/// Returns the Greatest Common Divisor of this and bi.
/// </summary>
/// <param name="bi">The bi.</param>
/// <returns>the GCD</returns>
public BigInteger gcd(BigInteger bi)
{
BigInteger x;
BigInteger y;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative
x = -this;
else
x = this;
if ((bi.data[maxLength - 1] & 0x80000000) != 0) // negative
y = -bi;
else
y = bi;
BigInteger g = y;
while (x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0))
{
g = x;
x = y % x;
y = g;
}
return g;
}
