// ***********************************************************************
// 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;
}
/// <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;
}
