//***********************************************************************
// 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,
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;
}
//***********************************************************************
// 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 ((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
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 = 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;
}