private static void PowCore(uint power, ref FastReducer reducer,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for the last or
// the only power limb using square-and-multiply.
// NOTE: we're using a special reducer here,
// since it's additional overhead does pay off.
while (power != 0)
{
if ((power & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(ref reducer);
}
if (power != 1)
{
value.SquareSelf(ref temp);
value.Reduce(ref reducer);
}
power = power >> 1;
}
}
private static void PowCore(uint[] power, ref FastReducer reducer,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for all but
// the last power limb using square-and-multiply.
// NOTE: we're using a special reducer here,
// since it's additional overhead does pay off.
for (int i = 0; i < power.Length - 1; i++)
{
uint p = power[i];
for (int j = 0; j < 32; j++)
{
if ((p & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(ref reducer);
}
value.SquareSelf(ref temp);
value.Reduce(ref reducer);
p = p >> 1;
}
}
PowCore(power[power.Length - 1], ref reducer, ref value, ref result,
ref temp);
}
private static void PowCore(uint power, uint[] modulus,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for the last or
// the only power limb using square-and-multiply.
// NOTE: we're using an ordinary remainder here,
// since the reducer overhead doesn't pay off.
while (power != 0)
{
if ((power & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(modulus);
}
if (power != 1)
{
value.SquareSelf(ref temp);
value.Reduce(modulus);
}
power = power >> 1;
}
}
private static void Gcd(ref BitsBuffer left, ref BitsBuffer right)
{
Debug.Assert(left.GetLength() >= 2);
Debug.Assert(right.GetLength() >= 2);
Debug.Assert(left.GetLength() >= right.GetLength());
// Executes Lehmer's gcd algorithm, but uses the most
// significant bits to work with 64-bit (not 32-bit) values.
// Furthermore we're using an optimized version due to Jebelean.
// http://cacr.uwaterloo.ca/hac/about/chap14.pdf (see 14.4.2)
// ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz
while (right.GetLength() > 2)
{
ulong x, y;
ExtractDigits(ref left, ref right, out x, out y);
uint a = 1U, b = 0U;
uint c = 0U, d = 1U;
int iteration = 0;
// Lehmer's guessing
while (y != 0)
{
ulong q, r, s, t;
// odd iteration
q = x / y;
if (q > 0xFFFFFFFF)
break;
r = a + q * c;
s = b + q * d;
t = x - q * y;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
break;
if (t < s || t + r > y - c)
break;
a = (uint)r;
b = (uint)s;
x = t;
++iteration;
if (x == b)
break;
// even iteration
q = y / x;
if (q > 0xFFFFFFFF)
break;
r = d + q * b;
s = c + q * a;
t = y - q * x;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
break;
if (t < s || t + r > x - b)
break;
d = (uint)r;
c = (uint)s;
y = t;
++iteration;
if (y == c)
break;
}
if (b == 0)
{
// Euclid's step
left.Reduce(ref right);
BitsBuffer temp = left;
left = right;
right = temp;
}
else
{
// Lehmer's step
LehmerCore(ref left, ref right, a, b, c, d);
if (iteration % 2 == 1)
{
// ensure left is larger than right
BitsBuffer temp = left;
left = right;
right = temp;
}
}
}
if (right.GetLength() > 0)
{
// Euclid's step
left.Reduce(ref right);
uint[] xBits = right.GetBits();
uint[] yBits = left.GetBits();
ulong x = ((ulong)xBits[1] << 32) | xBits[0];
ulong y = ((ulong)yBits[1] << 32) | yBits[0];
left.Overwrite(Gcd(x, y));
right.Overwrite(0);
}
}
private static void Gcd(ref BitsBuffer left, ref BitsBuffer right)
{
Debug.Assert(left.GetLength() >= 2);
Debug.Assert(right.GetLength() >= 2);
Debug.Assert(left.GetLength() >= right.GetLength());
// Executes Lehmer's gcd algorithm, but uses the most
// significant bits to work with 64-bit (not 32-bit) values.
// Furthermore we're using an optimized version due to Jebelean.
// http://cacr.uwaterloo.ca/hac/about/chap14.pdf (see 14.4.2)
// ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz
while (right.GetLength() > 2)
{
ulong x, y;
ExtractDigits(ref left, ref right, out x, out y);
uint a = 1U, b = 0U;
uint c = 0U, d = 1U;
int iteration = 0;
// Lehmer's guessing
while (y != 0)
{
ulong q, r, s, t;
// Odd iteration
q = x / y;
if (q > 0xFFFFFFFF)
{
break;
}
r = a + q * c;
s = b + q * d;
t = x - q * y;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
{
break;
}
if (t < s || t + r > y - c)
{
break;
}
a = (uint)r;
b = (uint)s;
x = t;
++iteration;
if (x == b)
{
break;
}
// Even iteration
q = y / x;
if (q > 0xFFFFFFFF)
{
break;
}
r = d + q * b;
s = c + q * a;
t = y - q * x;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
{
break;
}
if (t < s || t + r > x - b)
{
break;
}
d = (uint)r;
c = (uint)s;
y = t;
++iteration;
if (y == c)
{
break;
}
}
if (b == 0)
{
// Euclid's step
left.Reduce(ref right);
BitsBuffer temp = left;
left = right;
right = temp;
}
else
{
// Lehmer's step
LehmerCore(ref left, ref right, a, b, c, d);
if (iteration % 2 == 1)
{
// Ensure left is larger than right
BitsBuffer temp = left;
left = right;
right = temp;
}
}
}
if (right.GetLength() > 0)
{
// Euclid's step
left.Reduce(ref right);
uint[] xBits = right.GetBits();
uint[] yBits = left.GetBits();
ulong x = ((ulong)xBits[1] << 32) | xBits[0];
ulong y = ((ulong)yBits[1] << 32) | yBits[0];
left.Overwrite(Gcd(x, y));
right.Overwrite(0);
}
}
private static void PowCore(uint power, ref FastReducer reducer,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for the last or
// the only power limb using square-and-multiply.
// NOTE: we're using a special reducer here,
// since it's additional overhead does pay off.
while (power != 0)
{
if ((power & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(ref reducer);
}
if (power != 1)
{
value.SquareSelf(ref temp);
value.Reduce(ref reducer);
}
power = power >> 1;
}
}
private static void PowCore(uint[] power, ref FastReducer reducer,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for all but
// the last power limb using square-and-multiply.
// NOTE: we're using a special reducer here,
// since it's additional overhead does pay off.
for (int i = 0; i < power.Length - 1; i++)
{
uint p = power[i];
for (int j = 0; j < 32; j++)
{
if ((p & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(ref reducer);
}
value.SquareSelf(ref temp);
value.Reduce(ref reducer);
p = p >> 1;
}
}
PowCore(power[power.Length - 1], ref reducer, ref value, ref result,
ref temp);
}
private static void PowCore(uint power, uint[] modulus,
ref BitsBuffer value, ref BitsBuffer result,
ref BitsBuffer temp)
{
// The big modulus pow algorithm for the last or
// the only power limb using square-and-multiply.
// NOTE: we're using an ordinary remainder here,
// since the reducer overhead doesn't pay off.
while (power != 0)
{
if ((power & 1) == 1)
{
result.MultiplySelf(ref value, ref temp);
result.Reduce(modulus);
}
if (power != 1)
{
value.SquareSelf(ref temp);
value.Reduce(modulus);
}
power = power >> 1;
}
}
