/**
* Multiplies <code>this</code> by an integer <code>k</code> using the
* Window NAF method.
* @param k The integer by which <code>this</code> is multiplied.
* @return A new <code>ECPoint</code> which equals <code>this</code>
* multiplied by <code>k</code>.
*/
protected override ECPoint MultiplyPositive(ECPoint p, BigInteger k)
{
// Clamp the window width in the range [2, 16]
int width = System.Math.Max(2, System.Math.Min(16, GetWindowSize(k.BitLength)));
WNafPreCompInfo wnafPreCompInfo = WNafUtilities.Precompute(p, width, true);
ECPoint[] preComp = wnafPreCompInfo.PreComp;
ECPoint[] preCompNeg = wnafPreCompInfo.PreCompNeg;
int[] wnaf = WNafUtilities.GenerateCompactWindowNaf(width, k);
ECPoint R = p.Curve.Infinity;
int i = wnaf.Length;
/*
* NOTE: We try to optimize the first window using the precomputed points to substitute an
* addition for 2 or more doublings.
*/
if (i > 1)
{
int wi = wnaf[--i];
int digit = wi >> 16, zeroes = wi & 0xFFFF;
int n = System.Math.Abs(digit);
ECPoint[] table = digit < 0 ? preCompNeg : preComp;
// Optimization can only be used for values in the lower half of the table
if ((n << 2) < (1 << width))
{
int highest = LongArray.BitLengths[n];
// TODO Get addition/doubling cost ratio from curve and compare to 'scale' to see if worth substituting?
int scale = width - highest;
int lowBits = n ^ (1 << (highest - 1));
int i1 = ((1 << (width - 1)) - 1);
int i2 = (lowBits << scale) + 1;
R = table[i1 >> 1].Add(table[i2 >> 1]);
zeroes -= scale;
//Console.WriteLine("Optimized: 2^" + scale + " * " + n + " = " + i1 + " + " + i2);
}
else
{
R = table[n >> 1];
}
R = R.TimesPow2(zeroes);
}
while (i > 0)
{
int wi = wnaf[--i];
int digit = wi >> 16, zeroes = wi & 0xFFFF;
int n = System.Math.Abs(digit);
ECPoint[] table = digit < 0 ? preCompNeg : preComp;
ECPoint r = table[n >> 1];
R = R.TwicePlus(r);
R = R.TimesPow2(zeroes);
}
return(R);
}
/**
* Determine window width to use for a scalar multiplication of the given size.
*
* @param bits the bit-length of the scalar to multiply by
* @return the window size to use
*/
protected virtual int GetWindowSize(int bits)
{
return(WNafUtilities.GetWindowSize(bits));
}
public PreCompInfo Precompute(PreCompInfo existing)
{
WNafPreCompInfo existingWNaf = existing as WNafPreCompInfo;
int reqPreCompLen = 1 << System.Math.Max(0, m_width - 2);
if (CheckExisting(existingWNaf, reqPreCompLen, m_includeNegated))
{
return(existingWNaf);
}
ECCurve c = m_p.Curve;
ECPoint[] preComp = null, preCompNeg = null;
ECPoint twiceP = null;
if (existingWNaf != null)
{
preComp = existingWNaf.PreComp;
preCompNeg = existingWNaf.PreCompNeg;
twiceP = existingWNaf.Twice;
}
int iniPreCompLen = 0;
if (preComp == null)
{
preComp = EMPTY_POINTS;
}
else
{
iniPreCompLen = preComp.Length;
}
if (iniPreCompLen < reqPreCompLen)
{
preComp = WNafUtilities.ResizeTable(preComp, reqPreCompLen);
if (reqPreCompLen == 1)
{
preComp[0] = m_p.Normalize();
}
else
{
int curPreCompLen = iniPreCompLen;
if (curPreCompLen == 0)
{
preComp[0] = m_p;
curPreCompLen = 1;
}
ECFieldElement iso = null;
if (reqPreCompLen == 2)
{
preComp[1] = m_p.ThreeTimes();
}
else
{
ECPoint isoTwiceP = twiceP, last = preComp[curPreCompLen - 1];
if (isoTwiceP == null)
{
isoTwiceP = preComp[0].Twice();
twiceP = isoTwiceP;
/*
* For Fp curves with Jacobian projective coordinates, use a (quasi-)isomorphism
* where 'twiceP' is "affine", so that the subsequent additions are cheaper. This
* also requires scaling the initial point's X, Y coordinates, and reversing the
* isomorphism as part of the subsequent normalization.
*
* NOTE: The correctness of this optimization depends on:
* 1) additions do not use the curve's A, B coefficients.
* 2) no special cases (i.e. Q +/- Q) when calculating 1P, 3P, 5P, ...
*/
if (!twiceP.IsInfinity && ECAlgorithms.IsFpCurve(c) && c.FieldSize >= 64)
{
switch (c.CoordinateSystem)
{
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
iso = twiceP.GetZCoord(0);
isoTwiceP = c.CreatePoint(twiceP.XCoord.ToBigInteger(),
twiceP.YCoord.ToBigInteger());
ECFieldElement iso2 = iso.Square(), iso3 = iso2.Multiply(iso);
last = last.ScaleX(iso2).ScaleY(iso3);
if (iniPreCompLen == 0)
{
preComp[0] = last;
}
break;
}
}
}
}
while (curPreCompLen < reqPreCompLen)
{
/*
* Compute the new ECPoints for the precomputation array. The values 1, 3,
* 5, ..., 2^(width-1)-1 times p are computed
*/
preComp[curPreCompLen++] = last = last.Add(isoTwiceP);
}
}
/*
* Having oft-used operands in affine form makes operations faster.
*/
c.NormalizeAll(preComp, iniPreCompLen, reqPreCompLen - iniPreCompLen, iso);
}
}
if (m_includeNegated)
{
int pos;
if (preCompNeg == null)
{
pos = 0;
preCompNeg = new ECPoint[reqPreCompLen];
}
else
{
pos = preCompNeg.Length;
if (pos < reqPreCompLen)
{
preCompNeg = WNafUtilities.ResizeTable(preCompNeg, reqPreCompLen);
}
}
while (pos < reqPreCompLen)
{
preCompNeg[pos] = preComp[pos].Negate();
++pos;
}
}
WNafPreCompInfo result = new WNafPreCompInfo();
result.PreComp = preComp;
result.PreCompNeg = preCompNeg;
result.Twice = twiceP;
return(result);
}