//***********************************************************************
// Overloading of bitwise XOR operator
//***********************************************************************
public static BigInteger operator ^(BigInteger bi1, BigInteger bi2)
{
var result = new BigInteger();
int len = (bi1.DataLength > bi2.DataLength) ? bi1.DataLength : bi2.DataLength;
for (int i = 0; i < len; i++)
{
uint sum = (bi1._data[i] ^ bi2._data[i]);
result._data[i] = sum;
}
result.DataLength = MaxLength;
while (result.DataLength > 1 && result._data[result.DataLength - 1] == 0)
result.DataLength--;
return result;
}
//***********************************************************************
// Overloading of unary << operators
//***********************************************************************
public static BigInteger operator <<(BigInteger bi1, int shiftVal)
{
var result = new BigInteger(bi1);
result.DataLength = ShiftLeft(result._data, shiftVal);
return result;
}
//***********************************************************************
// Overloading of the NOT operator (1's complement)
//***********************************************************************
public static BigInteger operator ~(BigInteger bi1)
{
var 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;
}
//***********************************************************************
// Overloading of the unary ++ operator
//***********************************************************************
public static BigInteger operator ++(BigInteger bi1)
{
var result = new BigInteger(bi1);
long carry = 1;
int index = 0;
while (carry != 0 && index < MaxLength)
{
long val = (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
const 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;
}
//***********************************************************************
// Overloading of the unary -- operator
//***********************************************************************
public static BigInteger operator --(BigInteger bi1)
{
var result = new BigInteger(bi1);
bool carryIn = true;
int index = 0;
while (carryIn && index < MaxLength)
{
long val = (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
const 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;
}
//***********************************************************************
// 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
//***********************************************************************
public static BigInteger[] LucasSequence(BigInteger p, BigInteger q,
BigInteger k, BigInteger n)
{
if (k.DataLength == 1 && k._data[0] == 0)
{
var 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
var 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);
}
//***********************************************************************
// 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)
//
// string values are specified in the <sign><magnitude>
// format.
//
//***********************************************************************
public BigInteger(string value, int radix)
{
var multiplier = new BigInteger(1);
var 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."));
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."));
}
_data = new uint[MaxLength];
for (int i = 0; i < result.DataLength; i++)
_data[i] = result._data[i];
DataLength = result.DataLength;
}
//***********************************************************************
// Returns gcd(this, bi)
//***********************************************************************
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;
}
//***********************************************************************
// 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;
}
//***********************************************************************
// Modulo Exponentiation
//***********************************************************************
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 ((_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
var 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 = BarrettReduction(resultNum*tempNum, n, constant);
mask <<= 1;
tempNum = 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;
}
//***********************************************************************
// Fast calculation of modular reduction using Barrett's reduction.
// Requires x < b^(2k), where b is the base. In this case, base is
// 2^32 (uint).
//
// Reference [4]
//***********************************************************************
public static BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
{
int k = n.DataLength,
kPlusOne = k + 1,
kMinusOne = k - 1;
var 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;
var 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
var 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
var 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
{
var val = new BigInteger();
val._data[kPlusOne] = 0x00000001;
val.DataLength = kPlusOne + 1;
r1 += val;
}
while (r1 >= n)
r1 -= n;
return r1;
}
//***********************************************************************
// 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"
//
//***********************************************************************
public string ToString(int radix)
{
if (radix < 2 || radix > 36)
throw (new ArgumentException("Radix must be >= 2 and <= 36"));
const string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
string result = "";
BigInteger a = this;
bool negative = false;
if ((a._data[MaxLength - 1] & 0x80000000) != 0)
{
negative = true;
try
{
a = -a;
}
catch
{
}
}
var quotient = new BigInteger();
var remainder = new BigInteger();
var 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;
}
//***********************************************************************
// Returns min(this, bi)
//***********************************************************************
public BigInteger Min(BigInteger bi)
{
return this < bi ? (new BigInteger(this)) : (new BigInteger(bi));
}
//***********************************************************************
// Returns max(this, bi)
//***********************************************************************
public BigInteger Max(BigInteger bi)
{
return this > bi ? (new BigInteger(this)) : (new BigInteger(bi));
}
//***********************************************************************
// Constructor (Default value provided by BigInteger)
//***********************************************************************
public BigInteger(BigInteger bi)
{
_data = new uint[MaxLength];
DataLength = bi.DataLength;
for (int i = 0; i < DataLength; i++)
_data[i] = bi._data[i];
}
//***********************************************************************
// 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;
}
//***********************************************************************
// 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
//
//***********************************************************************
public BigInteger Sqrt()
{
var numBits = (uint) BitCount();
if ((numBits & 0x1) != 0) // odd number of bits
numBits = (numBits >> 1) + 1;
else
numBits = (numBits >> 1);
uint bytePos = numBits >> 5;
var bitPos = (byte) (numBits & 0x1F);
uint mask;
var 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;
}
//***********************************************************************
// 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;
}
//***********************************************************************
// 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
//***********************************************************************
private static BigInteger[] LucasSequenceHelper(BigInteger p, BigInteger q,
BigInteger k, BigInteger n,
BigInteger constant, int s)
{
var 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,
qK = 1%n,
v1 = p%n,
u1 = qK;
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*qK))%n;
v1 = BarrettReduction(v1*v1, n, constant);
v1 = (v1 - ((qK*q) << 1))%n;
if (flag)
flag = false;
else
qK = BarrettReduction(qK*qK, n, constant);
qK = (qK*q)%n;
}
else
{
// index doubling
u1 = ((u1*v) - qK)%n;
v1 = ((v*v1) - (p*qK))%n;
v = BarrettReduction(v*v, n, constant);
v = (v - (qK << 1))%n;
if (flag)
{
qK = q%n;
flag = false;
}
else
qK = BarrettReduction(qK*qK, 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) - qK)%n;
v = ((v*v1) - (p*qK))%n;
//if (flag)
// flag = false;
//else
qK = BarrettReduction(qK*qK, n, constant);
qK = (qK*q)%n;
for (int i = 0; i < s; i++)
{
// index doubling
u1 = (u1*v)%n;
v = ((v*v) - (qK << 1))%n;
qK = BarrettReduction(qK*qK, n, constant);
}
result[0] = u1;
result[1] = v;
result[2] = qK;
return result;
}
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;
BigInteger pAdd1 = thisVal + 1;
int s = 0;
for (int index = 0; index < pAdd1.DataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((pAdd1._data[index] & mask) != 0)
{
index = pAdd1.DataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = pAdd1 >> s;
// calculate constant = b^(2k) / m
// for Barrett Reduction
var 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] = 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] = 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;
}
//***********************************************************************
// Overloading of addition operator
//***********************************************************************
public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
{
var result = new BigInteger
{DataLength = (bi1.DataLength > bi2.DataLength) ? bi1.DataLength : bi2.DataLength};
long carry = 0;
for (int i = 0; i < result.DataLength; i++)
{
long sum = 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
const 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;
}
//***********************************************************************
// Computes the Jacobi Symbol for a and b.
// Algorithm adapted from [3] and [4] with some optimizations
//***********************************************************************
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);
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;
return a1.DataLength == 1 && a1._data[0] == 1 ? s : s*Jacobi(b%a1, a1);
}
//***********************************************************************
// Overloading of subtraction operator
//***********************************************************************
public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
{
var result = new BigInteger
{DataLength = (bi1.DataLength > bi2.DataLength) ? bi1.DataLength : bi2.DataLength};
long carryIn = 0;
for (int i = 0; i < result.DataLength; i++)
{
long diff = bi1._data[i] - (long) bi2._data[i] - carryIn;
result._data[i] = (uint) (diff & 0xFFFFFFFF);
carryIn = diff < 0 ? 1 : 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
const 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;
}
//***********************************************************************
// 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;
}
//***********************************************************************
// Overloading of multiplication operator
//***********************************************************************
public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
{
const 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
{
}
var 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 = (bi1._data[i]*(ulong) bi2._data[j]) +
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;
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;
}
//***********************************************************************
// 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;
}
//***********************************************************************
// Overloading of unary >> operators
//***********************************************************************
public static BigInteger operator >>(BigInteger bi1, int shiftVal)
{
var 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;
}
//***********************************************************************
// Returns the modulo inverse of this. Throws ArithmeticException if
// the inverse does not exist. (i.e. gcd(this, modulus) != 1)
//***********************************************************************
public BigInteger ModInverse(BigInteger modulus)
{
BigInteger[] p = {0, 1};
var 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))
{
var quotient = new BigInteger();
var 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;
}
//***********************************************************************
// Overloading of the NEGATE operator (2's complement)
//***********************************************************************
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());
var 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 carry = 1;
int index = 0;
while (carry != 0 && index < MaxLength)
{
long 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;
}
//***********************************************************************
// Overloading of modulus operator
//***********************************************************************
public static BigInteger operator %(BigInteger bi1, BigInteger bi2)
{
var quotient = new BigInteger();
var remainder = new BigInteger(bi1);
const int lastPos = MaxLength - 1;
bool dividendNeg = false;
if ((bi1._data[lastPos] & 0x80000000) != 0) // bi1 negative
{
bi1 = -bi1;
dividendNeg = true;
}
if ((bi2._data[lastPos] & 0x80000000) != 0) // bi2 negative
bi2 = -bi2;
if (bi1 < bi2)
{
return remainder;
}
if (bi2.DataLength == 1)
SingleByteDivide(bi1, bi2, quotient, remainder);
else
MultiByteDivide(bi1, bi2, quotient, remainder);
if (dividendNeg)
return -remainder;
return remainder;
}