/**
* It requires that all parameters be positive.
*
* @return {@code base<sup>exponent</sup> mod (2<sup>j</sup>)}.
* @see BigInteger#modPow(BigInteger, BigInteger)
*/
private static BigInteger Pow2ModPow(BigInteger b, BigInteger exponent, int j)
{
// PRE: (base > 0), (exponent > 0) and (j > 0)
BigInteger res = BigInteger.One;
BigInteger e = exponent.Copy();
BigInteger baseMod2toN = b.Copy();
BigInteger res2;
/*
* If 'base' is odd then it's coprime with 2^j and phi(2^j) = 2^(j-1);
* so we can reduce reduce the exponent (mod 2^(j-1)).
*/
if (BigInteger.TestBit(b, 0))
{
InplaceModPow2(e, j - 1);
}
InplaceModPow2(baseMod2toN, j);
for (int i = e.BitLength - 1; i >= 0; i--)
{
res2 = res.Copy();
InplaceModPow2(res2, j);
res = res * res2;
if (BitLevel.TestBit(e, i))
{
res = res * baseMod2toN;
InplaceModPow2(res, j);
}
}
InplaceModPow2(res, j);
return(res);
}
/*Implements the Montgomery modular exponentiation based in <i>The sliding windows algorithm and the Mongomery
* Reduction</i>.
*@ar.org.fitc.ref "A. Menezes,P. van Oorschot, S. Vanstone - Handbook of Applied Cryptography";
*@see #oddModPow(BigInteger, BigInteger,
* BigInteger)
*/
private static BigInteger SlidingWindow(BigInteger x2, BigInteger a2, BigInteger exponent, BigInteger modulus, int n2)
{
// fill odd low pows of a2
BigInteger[] pows = new BigInteger[8];
BigInteger res = x2;
int lowexp;
BigInteger x3;
int acc3;
pows[0] = a2;
x3 = MonPro(a2, a2, modulus, n2);
for (int i = 1; i <= 7; i++)
{
pows[i] = MonPro(pows[i - 1], x3, modulus, n2);
}
for (int i = exponent.BitLength - 1; i >= 0; i--)
{
if (BitLevel.TestBit(exponent, i))
{
lowexp = 1;
acc3 = i;
for (int j = System.Math.Max(i - 3, 0); j <= i - 1; j++)
{
if (BitLevel.TestBit(exponent, j))
{
if (j < acc3)
{
acc3 = j;
lowexp = (lowexp << (i - j)) ^ 1;
}
else
{
lowexp = lowexp ^ (1 << (j - acc3));
}
}
}
for (int j = acc3; j <= i; j++)
{
res = MonPro(res, res, modulus, n2);
}
res = MonPro(pows[(lowexp - 1) >> 1], res, modulus, n2);
i = acc3;
}
else
{
res = MonPro(res, res, modulus, n2);
}
}
return(res);
}
private static BigInteger SquareAndMultiply(BigInteger x2, BigInteger a2, BigInteger exponent, BigInteger modulus, int n2)
{
BigInteger res = x2;
for (int i = exponent.BitLength - 1; i >= 0; i--)
{
res = MonPro(res, res, modulus, n2);
if (BitLevel.TestBit(exponent, i))
{
res = MonPro(res, a2, modulus, n2);
}
}
return(res);
}
/**
* @param x an odd positive number.
* @param n the exponent by which 2 is raised.
* @return {@code x<sup>-1</sup> (mod 2<sup>n</sup>)}.
*/
private static BigInteger ModPow2Inverse(BigInteger x, int n)
{
// PRE: (x > 0), (x is odd), and (n > 0)
BigInteger y = new BigInteger(1, new int[1 << n]);
y.numberLength = 1;
y.Digits[0] = 1;
y.Sign = 1;
for (int i = 1; i < n; i++)
{
if (BitLevel.TestBit(x * y, i))
{
// Adding 2^i to y (setting the i-th bit)
y.Digits[i >> 5] |= (1 << (i & 31));
}
}
return(y);
}