internal static BigInteger Pow(BigInteger value, uint p)
{
var b = value;
for (var i = 0; i < p; i++)
value = value*b;
return value;
}
internal BigInteger Xor(BigInteger other)
{
var len = Math.Min(data.Length, other.data.Length);
var result = new uint[len];
for (var i = 0; i < len; i++)
result[i] = data[i] ^ other.data[i];
return new BigInteger(result);
}
/// <summary>
/// Performs n / d and n % d in one operation.
/// </summary>
/// <param name="n">A BigInteger, upon exit this will hold n / d</param>
/// <param name="d">The divisor</param>
/// <returns>n % d</returns>
public static uint SingleByteDivideInPlace(BigInteger n, uint d)
{
ulong r = 0;
var i = n.length;
while (i-- > 0)
{
r <<= 32;
r |= n.data[i];
n.data[i] = (uint) (r/d);
r %= d;
}
n.Normalize();
return (uint) r;
}
public BigInteger(BigInteger bi, uint len)
{
data = new uint[len];
for (uint i = 0; i < bi.length; i++)
data[i] = bi.data[i];
length = bi.length;
}
public static BigInteger modInverse(BigInteger bi, BigInteger modulus)
{
if (modulus.length == 1) return modInverse(bi, modulus.data[0]);
BigInteger[] p = {0, 1};
var q = new BigInteger[2]; // quotients
BigInteger[] r = {0, 0}; // remainders
var step = 0;
var a = modulus;
var b = bi;
var mr = new ModulusRing(modulus);
while (b != 0)
{
if (step > 1)
{
var pval = mr.Difference(p[0], p[1]*q[0]);
p[0] = p[1];
p[1] = pval;
}
var divret = multiByteDivide(a, b);
q[0] = q[1];
q[1] = divret[0];
r[0] = r[1];
r[1] = divret[1];
a = b;
b = divret[1];
step++;
}
if (r[0] != 1)
throw (new ArithmeticException("No inverse!"));
return mr.Difference(p[0], p[1]*q[0]);
}
public static BigInteger RightShift(BigInteger bi, int n)
{
if (n == 0) return new BigInteger(bi);
var w = n >> 5;
var s = n & ((1 << 5) - 1);
var ret = new BigInteger(Sign.Positive, bi.length - (uint) w + 1);
var l = (uint) ret.data.Length - 1;
if (s != 0)
{
uint x, carry = 0;
while (l-- > 0)
{
x = bi.data[l + w];
ret.data[l] = (x >> n) | carry;
carry = x << (32 - n);
}
}
else
{
while (l-- > 0)
ret.data[l] = bi.data[l + w];
}
ret.Normalize();
return ret;
}
public static BigInteger LeftShift(BigInteger bi, int n)
{
if (n == 0) return new BigInteger(bi, bi.length + 1);
var w = n >> 5;
n &= ((1 << 5) - 1);
var ret = new BigInteger(Sign.Positive, bi.length + 1 + (uint) w);
uint i = 0, l = bi.length;
if (n != 0)
{
uint x, carry = 0;
while (i < l)
{
x = bi.data[i];
ret.data[i + w] = (x << n) | carry;
carry = x >> (32 - n);
i++;
}
ret.data[i + w] = carry;
}
else
{
while (i < l)
{
ret.data[i + w] = bi.data[i];
i++;
}
}
ret.Normalize();
return ret;
}
public static BigInteger[] multiByteDivide(BigInteger bi1, BigInteger bi2)
{
if (Compare(bi1, bi2) == Sign.Negative)
return new BigInteger[2] {0, new BigInteger(bi1)};
bi1.Normalize();
bi2.Normalize();
if (bi2.length == 1)
return DwordDivMod(bi1, bi2.data[0]);
var remainderLen = bi1.length + 1;
var divisorLen = (int) bi2.length + 1;
var mask = 0x80000000;
var val = bi2.data[bi2.length - 1];
var shift = 0;
var resultPos = (int) bi1.length - (int) bi2.length;
while (mask != 0 && (val & mask) == 0)
{
shift++;
mask >>= 1;
}
var quot = new BigInteger(Sign.Positive, bi1.length - bi2.length + 1);
var rem = (bi1 << shift);
var remainder = rem.data;
bi2 = bi2 << shift;
var j = (int) (remainderLen - bi2.length);
var pos = (int) remainderLen - 1;
var firstDivisorByte = bi2.data[bi2.length - 1];
ulong secondDivisorByte = bi2.data[bi2.length - 2];
while (j > 0)
{
var dividend = ((ulong) remainder[pos] << 32) + remainder[pos - 1];
var q_hat = dividend/firstDivisorByte;
var r_hat = dividend%firstDivisorByte;
do
{
if (q_hat == 0x100000000 ||
(q_hat*secondDivisorByte) > ((r_hat << 32) + remainder[pos - 2]))
{
q_hat--;
r_hat += firstDivisorByte;
if (r_hat < 0x100000000)
continue;
}
break;
} while (true);
//
// At this point, q_hat is either exact, or one too large
// (more likely to be exact) so, we attempt to multiply the
// divisor by q_hat, if we get a borrow, we just subtract
// one from q_hat and add the divisor back.
//
uint t;
uint dPos = 0;
var nPos = pos - divisorLen + 1;
ulong mc = 0;
var uint_q_hat = (uint) q_hat;
do
{
mc += bi2.data[dPos]*(ulong) uint_q_hat;
t = remainder[nPos];
remainder[nPos] -= (uint) mc;
mc >>= 32;
if (remainder[nPos] > t) mc++;
dPos++;
nPos++;
} while (dPos < divisorLen);
nPos = pos - divisorLen + 1;
dPos = 0;
// Overestimate
if (mc != 0)
{
uint_q_hat--;
ulong sum = 0;
do
{
sum = remainder[nPos] + ((ulong) bi2.data[dPos]) + sum;
remainder[nPos] = (uint) sum;
sum >>= 32;
dPos++;
nPos++;
} while (dPos < divisorLen);
}
quot.data[resultPos--] = uint_q_hat;
pos--;
j--;
}
quot.Normalize();
rem.Normalize();
var ret = new BigInteger[2] {quot, rem};
if (shift != 0)
ret[1] >>= shift;
return ret;
}
public BigInteger Pow(BigInteger a, BigInteger k)
{
var b = new BigInteger(1);
if (k == 0)
return b;
var A = a;
if (k.TestBit(0))
b = a;
var bitCount = k.BitCount();
for (var i = 1; i < bitCount; i++)
{
A = Multiply(A, A);
if (k.TestBit(i))
b = Multiply(A, b);
}
return b;
}
public BigInteger Difference(BigInteger a, BigInteger b)
{
var cmp = Kernel.Compare(a, b);
BigInteger diff;
switch (cmp)
{
case Sign.Zero:
return 0;
case Sign.Positive:
diff = a - b;
break;
case Sign.Negative:
diff = b - a;
break;
default:
throw new Exception();
}
if (diff >= mod)
{
if (diff.length >= mod.length << 1)
diff %= mod;
else
BarrettReduction(diff);
}
if (cmp == Sign.Negative)
diff = mod - diff;
return diff;
}
public BigInteger Multiply(BigInteger a, BigInteger b)
{
if (a == 0 || b == 0) return 0;
if (a > mod)
a %= mod;
if (b > mod)
b %= mod;
var ret = a*b;
BarrettReduction(ret);
return ret;
}
public string ToString(uint radix, string characterSet)
{
if (characterSet.Length < radix)
throw new ArgumentException("charSet length less than radix", "characterSet");
if (radix == 1)
throw new ArgumentException("There is no such thing as radix one notation", "radix");
if (this == 0) return "0";
if (this == 1) return "1";
var result = "";
var a = new BigInteger(this);
while (a != 0)
{
var rem = Kernel.SingleByteDivideInPlace(a, radix);
result = characterSet[(int) rem] + result;
}
return result;
}
public Sign Compare(BigInteger bi)
{
return Kernel.Compare(this, bi);
}
public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
{
if (bi1 == 0 || bi2 == 0) return 0;
//
// Validate pointers
//
if (bi1.data.Length < bi1.length) throw new IndexOutOfRangeException("bi1 out of range");
if (bi2.data.Length < bi2.length) throw new IndexOutOfRangeException("bi2 out of range");
var ret = new BigInteger(Sign.Positive, bi1.length + bi2.length);
Kernel.Multiply(bi1.data, 0, bi1.length, bi2.data, 0, bi2.length, ret.data, 0);
ret.Normalize();
return ret;
}
public void BarrettReduction(BigInteger x)
{
var n = mod;
uint k = n.length,
kPlusOne = k + 1,
kMinusOne = k - 1;
// x < mod, so nothing to do.
if (x.length < k) return;
BigInteger q3;
//
// Validate pointers
//
if (x.data.Length < x.length) throw new IndexOutOfRangeException("x out of range");
// q1 = x / b^ (k-1)
// q2 = q1 * constant
// q3 = q2 / b^ (k+1), Needs to be accessed with an offset of kPlusOne
// TODO: We should the method in HAC p 604 to do this (14.45)
q3 = new BigInteger(Sign.Positive, x.length - kMinusOne + constant.length);
Kernel.Multiply(x.data, kMinusOne, x.length - kMinusOne, constant.data, 0, constant.length, q3.data, 0);
// r1 = x mod b^ (k+1)
// i.e. keep the lowest (k+1) words
var lengthToCopy = (x.length > kPlusOne) ? kPlusOne : x.length;
x.length = lengthToCopy;
x.Normalize();
// r2 = (q3 * n) mod b^ (k+1)
// partial multiplication of q3 and n
var r2 = new BigInteger(Sign.Positive, kPlusOne);
Kernel.MultiplyMod2p32pmod(q3.data, (int) kPlusOne, (int) q3.length - (int) kPlusOne, n.data, 0,
(int) n.length, r2.data, 0, (int) kPlusOne);
r2.Normalize();
if (r2 <= x)
{
Kernel.MinusEq(x, r2);
}
else
{
var val = new BigInteger(Sign.Positive, kPlusOne + 1);
val.data[kPlusOne] = 0x00000001;
Kernel.MinusEq(val, r2);
Kernel.PlusEq(x, val);
}
while (x >= n)
Kernel.MinusEq(x, n);
}
public static uint DwordMod(BigInteger n, uint d)
{
ulong r = 0;
var i = n.length;
while (i-- > 0)
{
r <<= 32;
r |= n.data[i];
r %= d;
}
return (uint) r;
}
public static BigInteger[] DwordDivMod(BigInteger n, uint d)
{
var ret = new BigInteger(Sign.Positive, n.length);
ulong r = 0;
var i = n.length;
while (i-- > 0)
{
r <<= 32;
r |= n.data[i];
ret.data[i] = (uint) (r/d);
r %= d;
}
ret.Normalize();
BigInteger rem = (uint) r;
return new[] {ret, rem};
}
public BigInteger Pow(uint b, BigInteger exp)
{
return Pow(new BigInteger(b), exp);
}
public BigInteger ModPow(BigInteger exp, BigInteger n)
{
var mr = new ModulusRing(n);
return mr.Pow(this, exp);
}
/// <summary>
/// Compares two BigInteger
/// </summary>
/// <param name="bi1">A BigInteger</param>
/// <param name="bi2">A BigInteger</param>
/// <returns>The sign of bi1 - bi2</returns>
public static Sign Compare(BigInteger bi1, BigInteger bi2)
{
//
// Step 1. Compare the lengths
//
uint l1 = bi1.length, l2 = bi2.length;
while (l1 > 0 && bi1.data[l1 - 1] == 0) l1--;
while (l2 > 0 && bi2.data[l2 - 1] == 0) l2--;
if (l1 == 0 && l2 == 0) return Sign.Zero;
// bi1 len < bi2 len
if (l1 < l2) return Sign.Negative;
// bi1 len > bi2 len
if (l1 > l2) return Sign.Positive;
//
// Step 2. Compare the bits
//
var pos = l1 - 1;
while (pos != 0 && bi1.data[pos] == bi2.data[pos]) pos--;
if (bi1.data[pos] < bi2.data[pos])
return Sign.Negative;
if (bi1.data[pos] > bi2.data[pos])
return Sign.Positive;
return Sign.Zero;
}
/// <summary>
/// Adds two numbers with the same sign.
/// </summary>
/// <param name="bi1">A BigInteger</param>
/// <param name="bi2">A BigInteger</param>
/// <returns>bi1 + bi2</returns>
public static BigInteger AddSameSign(BigInteger bi1, BigInteger bi2)
{
uint[] x, y;
uint yMax, xMax, i = 0;
// x should be bigger
if (bi1.length < bi2.length)
{
x = bi2.data;
xMax = bi2.length;
y = bi1.data;
yMax = bi1.length;
}
else
{
x = bi1.data;
xMax = bi1.length;
y = bi2.data;
yMax = bi2.length;
}
var result = new BigInteger(Sign.Positive, xMax + 1);
var r = result.data;
ulong sum = 0;
// Add common parts of both numbers
do
{
sum = x[i] + ((ulong) y[i]) + sum;
r[i] = (uint) sum;
sum >>= 32;
} while (++i < yMax);
// Copy remainder of longer number while carry propagation is required
var carry = (sum != 0);
if (carry)
{
if (i < xMax)
{
do
carry = ((r[i] = x[i] + 1) == 0); while (++i < xMax && carry);
}
if (carry)
{
r[i] = 1;
result.length = ++i;
return result;
}
}
// Copy the rest
if (i < xMax)
{
do
r[i] = x[i]; while (++i < xMax);
}
result.Normalize();
return result;
}
public static BigInteger Subtract(BigInteger big, BigInteger small)
{
var result = new BigInteger(Sign.Positive, big.length);
uint[] r = result.data, b = big.data, s = small.data;
uint i = 0, c = 0;
do
{
var x = s[i];
if (((x += c) < c) | ((r[i] = b[i] - x) > ~x))
c = 1;
else
c = 0;
} while (++i < small.length);
if (i == big.length) goto fixup;
if (c == 1)
{
do
r[i] = b[i] - 1; while (b[i++] == 0 && i < big.length);
if (i == big.length) goto fixup;
}
do
r[i] = b[i]; while (++i < big.length);
fixup:
result.Normalize();
return result;
}
public static BigInteger MultiplyByDword(BigInteger n, uint f)
{
var ret = new BigInteger(Sign.Positive, n.length + 1);
uint i = 0;
ulong c = 0;
do
{
c += n.data[i]*(ulong) f;
ret.data[i] = (uint) c;
c >>= 32;
} while (++i < n.length);
ret.data[i] = (uint) c;
ret.Normalize();
return ret;
}
public static void MinusEq(BigInteger big, BigInteger small)
{
uint[] b = big.data, s = small.data;
uint i = 0, c = 0;
do
{
var x = s[i];
if (((x += c) < c) | ((b[i] -= x) > ~x))
c = 1;
else
c = 0;
} while (++i < small.length);
if (i == big.length) goto fixup;
if (c == 1)
{
do
b[i]--; while (b[i++] == 0 && i < big.length);
}
fixup:
// Normalize length
while (big.length > 0 && big.data[big.length - 1] == 0) big.length--;
// Check for zero
if (big.length == 0)
big.length++;
}
public static BigInteger gcd(BigInteger a, BigInteger b)
{
var x = a;
var y = b;
var g = y;
while (x.length > 1)
{
g = x;
x = y%x;
y = g;
}
if (x == 0) return g;
// TODO: should we have something here if we can convert to long?
//
// Now we can just do it with single precision. I am using the binary gcd method,
// as it should be faster.
//
var yy = x.data[0];
var xx = y%yy;
var t = 0;
while (((xx | yy) & 1) == 0)
{
xx >>= 1;
yy >>= 1;
t++;
}
while (xx != 0)
{
while ((xx & 1) == 0) xx >>= 1;
while ((yy & 1) == 0) yy >>= 1;
if (xx >= yy)
xx = (xx - yy) >> 1;
else
yy = (yy - xx) >> 1;
}
return yy << t;
}
public static void PlusEq(BigInteger bi1, BigInteger bi2)
{
uint[] x, y;
uint yMax, xMax, i = 0;
var flag = false;
// x should be bigger
if (bi1.length < bi2.length)
{
flag = true;
x = bi2.data;
xMax = bi2.length;
y = bi1.data;
yMax = bi1.length;
}
else
{
x = bi1.data;
xMax = bi1.length;
y = bi2.data;
yMax = bi2.length;
}
var r = bi1.data;
ulong sum = 0;
// Add common parts of both numbers
do
{
sum += x[i] + ((ulong) y[i]);
r[i] = (uint) sum;
sum >>= 32;
} while (++i < yMax);
// Copy remainder of longer number while carry propagation is required
var carry = (sum != 0);
if (carry)
{
if (i < xMax)
{
do
carry = ((r[i] = x[i] + 1) == 0); while (++i < xMax && carry);
}
if (carry)
{
r[i] = 1;
bi1.length = ++i;
return;
}
}
// Copy the rest
if (flag && i < xMax - 1)
{
do
r[i] = x[i]; while (++i < xMax);
}
bi1.length = xMax + 1;
bi1.Normalize();
}
public ModulusRing(BigInteger modulus)
{
mod = modulus;
// calculate constant = b^ (2k) / m
var i = mod.length << 1;
constant = new BigInteger(Sign.Positive, i + 1);
constant.data[i] = 0x00000001;
constant = constant/mod;
}
public BigInteger(BigInteger bi)
{
data = (uint[]) bi.data.Clone();
length = bi.length;
}