/// <summary>
/// Decrypts Login Key from Client
/// </summary>
/// <param name="LoginKey">
/// Login Key from Client
/// </param>
/// <param name="UserName">
/// Username stored in Login Key
/// </param>
/// <param name="ServerSalt">
/// Server Salt stored in Login Key
/// </param>
/// <param name="Password">
/// Password stored in Login Key
/// </param>
public void DecryptLoginKey(string LoginKey, out string UserName, out string ServerSalt, out string Password)
{
string[] LoginKeySplit = LoginKey.Split('-');
BigInteger ClientPublicKey = new BigInteger(LoginKeySplit[0], 16);
string EncryptedBlock = LoginKeySplit[1];
// These should really be in a config file, but for now hardcoded
BigInteger ServerPrivateKey =
new BigInteger("7ad852c6494f664e8df21446285ecd6f400cf20e1d872ee96136d7744887424b", 16);
BigInteger Prime =
new BigInteger(
"eca2e8c85d863dcdc26a429a71a9815ad052f6139669dd659f98ae159d313d13c6bf2838e10a69b6478b64a24bd054ba8248e8fa778703b418408249440b2c1edd28853e240d8a7e49540b76d120d3b1ad2878b1b99490eb4a2a5e84caa8a91cecbdb1aa7c816e8be343246f80c637abc653b893fd91686cf8d32d6cfe5f2a6f",
16);
string TeaKey = ClientPublicKey.modPow(ServerPrivateKey, Prime).ToString(16).ToLower();
if (TeaKey.Length < 32)
{
// If TeaKey is not at least 128bits, pad to the left with 0x00
TeaKey.PadLeft(32, '0');
}
else
{
// If TeaKey is more than 128bits, truncate
TeaKey = TeaKey.Substring(0, 32);
}
string DecryptedBlock = this.DecryptTea(EncryptedBlock, TeaKey);
DecryptedBlock = DecryptedBlock.Substring(8); // Strip first 8 bytes of padding
int DataLength = this.ConvertStringToIntSwapEndian(DecryptedBlock.Substring(0, 4));
DecryptedBlock = DecryptedBlock.Substring(4);
string[] BlockParts = DecryptedBlock.Split(new[] { '|' }, 2);
UserName = BlockParts[0];
ServerSalt = string.Empty;
for (int i = 0; i < 32; i += 4)
{
ServerSalt += string.Format("{0:x8}", this.ConvertStringToIntSwapEndian(BlockParts[1].Substring(i, 4)));
}
Password = BlockParts[1].Substring(33, DataLength - 34 - UserName.Length);
}
/// <summary>
/// </summary>
/// <param name="bi1">
/// </param>
/// <param name="shiftVal">
/// </param>
/// <returns>
/// </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>
/// </summary>
/// <param name="bits">
/// </param>
/// <param name="confidence">
/// </param>
/// <param name="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;
}
// ***********************************************************************
// 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>
/// </summary>
/// <param name="rounds">
/// </param>
public static void RSATest2(int rounds)
{
Random rand = new Random();
byte[] val = new byte[64];
byte[] pseudoPrime1 =
{
0x85, 0x84, 0x64, 0xFD, 0x70, 0x6A, 0x9F, 0xF0, 0x94, 0x0C, 0x3E, 0x2C, 0x74, 0x34,
0x05, 0xC9, 0x55, 0xB3, 0x85, 0x32, 0x98, 0x71, 0xF9, 0x41, 0x21, 0x5F, 0x02, 0x9E,
0xEA, 0x56, 0x8D, 0x8C, 0x44, 0xCC, 0xEE, 0xEE, 0x3D, 0x2C, 0x9D, 0x2C, 0x12, 0x41,
0x1E, 0xF1, 0xC5, 0x32, 0xC3, 0xAA, 0x31, 0x4A, 0x52, 0xD8, 0xE8, 0xAF, 0x42, 0xF4,
0x72, 0xA1, 0x2A, 0x0D, 0x97, 0xB1, 0x31, 0xB3,
};
byte[] pseudoPrime2 =
{
0x99, 0x98, 0xCA, 0xB8, 0x5E, 0xD7, 0xE5, 0xDC, 0x28, 0x5C, 0x6F, 0x0E, 0x15, 0x09,
0x59, 0x6E, 0x84, 0xF3, 0x81, 0xCD, 0xDE, 0x42, 0xDC, 0x93, 0xC2, 0x7A, 0x62, 0xAC,
0x6C, 0xAF, 0xDE, 0x74, 0xE3, 0xCB, 0x60, 0x20, 0x38, 0x9C, 0x21, 0xC3, 0xDC, 0xC8,
0xA2, 0x4D, 0xC6, 0x2A, 0x35, 0x7F, 0xF3, 0xA9, 0xE8, 0x1D, 0x7B, 0x2C, 0x78, 0xFA,
0xB8, 0x02, 0x55, 0x80, 0x9B, 0xC2, 0xA5, 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>
/// </summary>
/// <param name="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>
/// </summary>
/// <param name="a">
/// </param>
/// <param name="b">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArgumentException">
/// </exception>
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>
/// </summary>
/// <param name="x">
/// </param>
/// <param name="n">
/// </param>
/// <param name="constant">
/// </param>
/// <returns>
/// </returns>
private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
{
int k = n.dataLength, kPlusOne = k + 1, kMinusOne = k - 1;
BigInteger q1 = new BigInteger();
// q1 = x / b^(k-1)
for (int i = kMinusOne, j = 0; i < x.dataLength; i++, j++)
{
q1.data[j] = x.data[i];
}
q1.dataLength = x.dataLength - kMinusOne;
if (q1.dataLength <= 0)
{
q1.dataLength = 1;
}
BigInteger q2 = q1 * constant;
BigInteger q3 = new BigInteger();
// q3 = q2 / b^(k+1)
for (int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++)
{
q3.data[j] = q2.data[i];
}
q3.dataLength = q2.dataLength - kPlusOne;
if (q3.dataLength <= 0)
{
q3.dataLength = 1;
}
// r1 = x mod b^(k+1)
// i.e. keep the lowest (k+1) words
BigInteger r1 = new BigInteger();
int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength;
for (int i = 0; i < lengthToCopy; i++)
{
r1.data[i] = x.data[i];
}
r1.dataLength = lengthToCopy;
// r2 = (q3 * n) mod b^(k+1)
// partial multiplication of q3 and n
BigInteger r2 = new BigInteger();
for (int i = 0; i < q3.dataLength; i++)
{
if (q3.data[i] == 0)
{
continue;
}
ulong mcarry = 0;
int t = i;
for (int j = 0; j < n.dataLength && t < kPlusOne; j++, t++)
{
// t = i + j
ulong val = (q3.data[i] * (ulong)n.data[j]) + r2.data[t] + mcarry;
r2.data[t] = (uint)(val & 0xFFFFFFFF);
mcarry = val >> 32;
}
if (t < kPlusOne)
{
r2.data[t] = (uint)mcarry;
}
}
r2.dataLength = kPlusOne;
while (r2.dataLength > 1 && r2.data[r2.dataLength - 1] == 0)
{
r2.dataLength--;
}
r1 -= r2;
if ((r1.data[maxLength - 1] & 0x80000000) != 0)
{
// negative
BigInteger val = new BigInteger();
val.data[kPlusOne] = 0x00000001;
val.dataLength = kPlusOne + 1;
r1 += val;
}
while (r1 >= n)
{
r1 -= n;
}
return r1;
}
/// <summary>
/// </summary>
/// <param name="bi1">
/// </param>
/// <param name="bi2">
/// </param>
/// <param name="outQuotient">
/// </param>
/// <param name="outRemainder">
/// </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) + 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;
}
}
// ***********************************************************************
// Returns a string representing the BigInteger in sign-and-magnitude
// format in the specified radix.
// Example
// -------
// If the value of BigInteger is -255 in base 10, then
// ToString(16) returns "-FF"
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="radix">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArgumentException">
/// </exception>
public string ToString(int radix)
{
if (radix < 2 || radix > 36)
{
throw new ArgumentException("Radix must be >= 2 and <= 36");
}
string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
string result = string.Empty;
BigInteger a = this;
bool negative = false;
if ((a.data[maxLength - 1] & 0x80000000) != 0)
{
negative = true;
try
{
a = -a;
}
catch (Exception)
{
}
}
BigInteger quotient = new BigInteger();
BigInteger remainder = new BigInteger();
BigInteger biRadix = new BigInteger(radix);
if (a.dataLength == 1 && a.data[0] == 0)
{
result = "0";
}
else
{
while (a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0))
{
singleByteDivide(a, biRadix, quotient, remainder);
if (remainder.data[0] < 10)
{
result = remainder.data[0] + result;
}
else
{
result = charSet[(int)remainder.data[0] - 10] + result;
}
a = quotient;
}
if (negative)
{
result = "-" + result;
}
}
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.
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="confidence">
/// </param>
/// <returns>
/// </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;
}
// ***********************************************************************
// Constructor (Default value provided by BigInteger)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="bi">
/// </param>
public BigInteger(BigInteger bi)
{
this.data = new uint[maxLength];
this.dataLength = bi.dataLength;
for (int i = 0; i < this.dataLength; i++)
{
this.data[i] = bi.data[i];
}
}
/// <summary>
/// </summary>
/// <param name="confidence">
/// </param>
/// <returns>
/// </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>
/// </summary>
/// <param name="confidence">
/// </param>
/// <returns>
/// </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;
}
// ***********************************************************************
// Overloading of the NEGATE operator (2's complement)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="bi1">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArithmeticException">
/// </exception>
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] = ~bi1.data[i];
}
// add one to result of 1's complement
long val, carry = 1;
int index = 0;
while (carry != 0 && index < maxLength)
{
val = 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>
/// </summary>
/// <param name="bi1">
/// </param>
/// <param name="bi2">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArithmeticException">
/// </exception>
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 = 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;
}
// ***********************************************************************
// Sets the value of the specified bit to 0
// The Least Significant Bit position is 0.
// ***********************************************************************
// ***********************************************************************
// Returns a value that is equivalent to the integer square root
// of the BigInteger.
// The integer square root of "this" is defined as the largest integer n
// such that (n * n) <= this
// ***********************************************************************
/// <summary>
/// </summary>
/// <returns>
/// </returns>
public BigInteger sqrt()
{
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 * result) > this)
{
result.data[i] ^= mask;
}
mask >>= 1;
}
mask = 0x80000000;
}
return result;
}
// ***********************************************************************
// 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
// ***********************************************************************
// ***********************************************************************
// 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
// ***********************************************************************
#region Methods
/// <summary>
/// </summary>
/// <param name="P">
/// </param>
/// <param name="Q">
/// </param>
/// <param name="k">
/// </param>
/// <param name="n">
/// </param>
/// <param name="constant">
/// </param>
/// <param name="s">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArgumentException">
/// </exception>
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;
}
// ***********************************************************************
// Constructor (Default value provided by a string of digits of the
// specified base)
// Example (base 10)
// -----------------
// To initialize "a" with the default value of 1234 in base 10
// BigInteger a = new BigInteger("1234", 10)
// To initialize "a" with the default value of -1234
// BigInteger a = new BigInteger("-1234", 10)
// Example (base 16)
// -----------------
// To initialize "a" with the default value of 0x1D4F in base 16
// BigInteger a = new BigInteger("1D4F", 16)
// To initialize "a" with the default value of -0x1D4F
// BigInteger a = new BigInteger("-1D4F", 16)
// Note that string values are specified in the <sign><magnitude>
// format.
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="value">
/// </param>
/// <param name="radix">
/// </param>
/// <exception cref="ArithmeticException">
/// </exception>
public BigInteger(string value, int radix)
{
BigInteger multiplier = new BigInteger(1);
BigInteger result = new BigInteger();
value = value.ToUpper().Trim();
int limit = 0;
if (value[0] == '-')
{
limit = 1;
}
for (int i = value.Length - 1; i >= limit; i--)
{
int posVal = value[i];
if (posVal >= '0' && posVal <= '9')
{
posVal -= '0';
}
else if (posVal >= 'A' && posVal <= 'Z')
{
posVal = (posVal - 'A') + 10;
}
else
{
posVal = 9999999; // arbitrary large
}
if (posVal >= radix)
{
throw new ArithmeticException("Invalid string in constructor.");
}
else
{
if (value[0] == '-')
{
posVal = -posVal;
}
result = result + (multiplier * posVal);
if ((i - 1) >= limit)
{
multiplier = multiplier * radix;
}
}
}
if (value[0] == '-')
{
// negative values
if ((result.data[maxLength - 1] & 0x80000000) == 0)
{
throw new ArithmeticException("Negative underflow in constructor.");
}
}
else
{
// positive values
if ((result.data[maxLength - 1] & 0x80000000) != 0)
{
throw new ArithmeticException("Positive overflow in constructor.");
}
}
this.data = new uint[maxLength];
for (int i = 0; i < result.dataLength; i++)
{
this.data[i] = result.data[i];
}
this.dataLength = result.dataLength;
}
/// <summary>
/// </summary>
/// <param name="bi1">
/// </param>
/// <param name="bi2">
/// </param>
/// <param name="outQuotient">
/// </param>
/// <param name="outRemainder">
/// </param>
private static void singleByteDivide(
BigInteger bi1,
BigInteger bi2,
BigInteger outQuotient,
BigInteger outRemainder)
{
uint[] result = new uint[maxLength];
int resultPos = 0;
// copy dividend to reminder
for (int i = 0; i < maxLength; i++)
{
outRemainder.data[i] = bi1.data[i];
}
outRemainder.dataLength = bi1.dataLength;
while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0)
{
outRemainder.dataLength--;
}
ulong divisor = bi2.data[0];
int pos = outRemainder.dataLength - 1;
ulong dividend = outRemainder.data[pos];
// Console.WriteLine("divisor = " + divisor + " dividend = " + dividend);
// Console.WriteLine("divisor = " + bi2 + "\ndividend = " + bi1);
if (dividend >= divisor)
{
ulong quotient = dividend / divisor;
result[resultPos++] = (uint)quotient;
outRemainder.data[pos] = (uint)(dividend % divisor);
}
pos--;
while (pos >= 0)
{
// Console.WriteLine(pos);
dividend = ((ulong)outRemainder.data[pos + 1] << 32) + outRemainder.data[pos];
ulong quotient = dividend / divisor;
result[resultPos++] = (uint)quotient;
outRemainder.data[pos + 1] = 0;
outRemainder.data[pos--] = (uint)(dividend % divisor);
// Console.WriteLine(">>>> " + bi1);
}
outQuotient.dataLength = resultPos;
int j = 0;
for (int i = outQuotient.dataLength - 1; i >= 0; i--, j++)
{
outQuotient.data[j] = result[i];
}
for (; j < maxLength; j++)
{
outQuotient.data[j] = 0;
}
while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0)
{
outQuotient.dataLength--;
}
if (outQuotient.dataLength == 0)
{
outQuotient.dataLength = 1;
}
while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0)
{
outRemainder.dataLength--;
}
}
// ***********************************************************************
// Returns a hex string showing the contains of the BigInteger
// Examples
// -------
// 1) If the value of BigInteger is 255 in base 10, then
// ToHexString() returns "FF"
// 2) If the value of BigInteger is -255 in base 10, then
// ToHexString() returns ".....FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF01",
// which is the 2's complement representation of -255.
// ***********************************************************************
// ***********************************************************************
// Returns gcd(this, bi)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="bi">
/// </param>
/// <returns>
/// </returns>
public BigInteger gcd(BigInteger bi)
{
BigInteger x;
BigInteger y;
if ((this.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;
}
/// <summary>
/// </summary>
/// <param name="thisVal">
/// </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 = 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 * Jacobi(Q, thisVal)) % thisVal;
if ((temp.data[maxLength - 1] & 0x80000000) != 0)
{
temp += thisVal;
}
if (lucas[2] != temp)
{
isPrime = false;
}
}
}
return isPrime;
}
/// <summary>
/// </summary>
/// <param name="bits">
/// </param>
/// <param name="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>
/// </summary>
/// <param name="P">
/// </param>
/// <param name="Q">
/// </param>
/// <param name="k">
/// </param>
/// <param name="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>
/// </summary>
/// <param name="bi">
/// </param>
/// <returns>
/// </returns>
public BigInteger max(BigInteger bi)
{
if (this > bi)
{
return new BigInteger(this);
}
else
{
return new BigInteger(bi);
}
}
// ***********************************************************************
// Tests the correct implementation of the modulo exponential function
// using RSA encryption and decryption (using pre-computed encryption and
// decryption keys).
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="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>.");
}
}
// ***********************************************************************
// Returns min(this, bi)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="bi">
/// </param>
/// <returns>
/// </returns>
public BigInteger min(BigInteger bi)
{
if (this < bi)
{
return new BigInteger(this);
}
else
{
return new BigInteger(bi);
}
}
// ***********************************************************************
// Tests the correct implementation of sqrt() method.
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="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>.");
}
}
// ***********************************************************************
// Returns the lowest 4 bytes of the BigInteger as an int.
// ***********************************************************************
// ***********************************************************************
// Returns the modulo inverse of this. Throws ArithmeticException if
// the inverse does not exist. (i.e. gcd(this, modulus) != 1)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="modulus">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArithmeticException">
/// </exception>
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>
/// </summary>
/// <param name="exp">
/// </param>
/// <param name="n">
/// </param>
/// <returns>
/// </returns>
/// <exception cref="ArithmeticException">
/// </exception>
public BigInteger modPow(BigInteger exp, BigInteger n)
{
if ((exp.data[maxLength - 1] & 0x80000000) != 0)
{
throw new ArithmeticException("Positive exponents only.");
}
BigInteger resultNum = 1;
BigInteger tempNum;
bool thisNegative = false;
if ((this.data[maxLength - 1] & 0x80000000) != 0)
{
// negative this
tempNum = -this % n;
thisNegative = true;
}
else
{
tempNum = this % n; // ensures (tempNum * tempNum) < b^(2k)
}
if ((n.data[maxLength - 1] & 0x80000000) != 0)
{
// negative n
n = -n;
}
// calculate constant = b^(2k) / m
BigInteger constant = new BigInteger();
int i = n.dataLength << 1;
constant.data[i] = 0x00000001;
constant.dataLength = i + 1;
constant = constant / n;
int totalBits = exp.bitCount();
int count = 0;
// perform squaring and multiply exponentiation
for (int pos = 0; pos < exp.dataLength; pos++)
{
uint mask = 0x01;
// Console.WriteLine("pos = " + pos);
for (int index = 0; index < 32; index++)
{
if ((exp.data[pos] & mask) != 0)
{
resultNum = this.BarrettReduction(resultNum * tempNum, n, constant);
}
mask <<= 1;
tempNum = this.BarrettReduction(tempNum * tempNum, n, constant);
if (tempNum.dataLength == 1 && tempNum.data[0] == 1)
{
if (thisNegative && (exp.data[0] & 0x1) != 0)
{
// odd exp
return -resultNum;
}
return resultNum;
}
count++;
if (count == totalBits)
{
break;
}
}
}
if (thisNegative && (exp.data[0] & 0x1) != 0)
{
// odd exp
return -resultNum;
}
return resultNum;
}
// least significant bits at lower part of buffer
// ***********************************************************************
// Overloading of the NOT operator (1's complement)
// ***********************************************************************
/// <summary>
/// </summary>
/// <param name="bi1">
/// </param>
/// <returns>
/// </returns>
public static BigInteger operator ~(BigInteger bi1)
{
BigInteger result = new BigInteger(bi1);
for (int i = 0; i < maxLength; i++)
{
result.data[i] = ~bi1.data[i];
}
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
{
result.dataLength--;
}
return result;
}