/// <summary>
/// Return the greatest common divisor of X and Y
/// </summary>
///
/// <param name="X">Operand 1, must be greater than zero</param>
/// <param name="Y">Operand 2, must be greater than zero</param>
///
/// <returns>Returns <c>GCD(X, Y)</c></returns>
internal static BigInteger GcdBinary(BigInteger X, BigInteger Y)
{
// Divide both number the maximal possible times by 2 without rounding * gcd(2*a, 2*b) = 2 * gcd(a,b)
int lsb1 = X.LowestSetBit;
int lsb2 = Y.LowestSetBit;
int pow2Count = System.Math.Min(lsb1, lsb2);
BitLevel.InplaceShiftRight(X, lsb1);
BitLevel.InplaceShiftRight(Y, lsb2);
BigInteger swap;
// I want op2 > op1
if (X.CompareTo(Y) == BigInteger.GREATER)
{
swap = X;
X = Y;
Y = swap;
}
do
{ // INV: op2 >= op1 && both are odd unless op1 = 0
// Optimization for small operands (op2.bitLength() < 64) implies by INV (op1.bitLength() < 64)
if ((Y.m_numberLength == 1) || ((Y.m_numberLength == 2) && (Y.m_digits[1] > 0)))
{
Y = BigInteger.ValueOf(Division.GcdBinary(X.ToInt64(), Y.ToInt64()));
break;
}
// Implements one step of the Euclidean algorithm
// To reduce one operand if it's much smaller than the other one
if (Y.m_numberLength > X.m_numberLength * 1.2)
{
Y = Y.Remainder(X);
if (Y.Signum() != 0)
{
BitLevel.InplaceShiftRight(Y, Y.LowestSetBit);
}
}
else
{
// Use Knuth's algorithm of successive subtract and shifting
do
{
Elementary.InplaceSubtract(Y, X); // both are odd
BitLevel.InplaceShiftRight(Y, Y.LowestSetBit); // op2 is even
} while (Y.CompareTo(X) >= BigInteger.EQUALS);
}
// now op1 >= op2
swap = Y;
Y = X;
X = swap;
} while (X.m_sign != 0);
return(Y.ShiftLeft(pow2Count));
}
private static BigInteger ModInverseLorencz(BigInteger X, BigInteger Modulo)
{
// Based on "New Algorithm for Classical Modular Inverse" Róbert Lórencz. LNCS 2523 (2002)
// PRE: a is coprime with modulo, a < modulo
int max = System.Math.Max(X.m_numberLength, Modulo.m_numberLength);
int[] uDigits = new int[max + 1]; // enough place to make all the inplace operation
int[] vDigits = new int[max + 1];
Array.Copy(Modulo.m_digits, 0, uDigits, 0, Modulo.m_numberLength);
Array.Copy(X.m_digits, 0, vDigits, 0, X.m_numberLength);
BigInteger u = new BigInteger(Modulo.m_sign, Modulo.m_numberLength, uDigits);
BigInteger v = new BigInteger(X.m_sign, X.m_numberLength, vDigits);
BigInteger r = new BigInteger(0, 1, new int[max + 1]); // BigInteger.ZERO;
BigInteger s = new BigInteger(1, 1, new int[max + 1]);
s.m_digits[0] = 1;
// r == 0 && s == 1, but with enough place
int coefU = 0, coefV = 0;
int n = Modulo.BitLength;
int k;
while (!IsPowerOfTwo(u, coefU) && !IsPowerOfTwo(v, coefV))
{
// modification of original algorithm: I calculate how many times the algorithm will enter in the same branch of if
k = HowManyIterations(u, n);
if (k != 0)
{
BitLevel.InplaceShiftLeft(u, k);
if (coefU >= coefV)
{
BitLevel.InplaceShiftLeft(r, k);
}
else
{
BitLevel.InplaceShiftRight(s, System.Math.Min(coefV - coefU, k));
if (k - (coefV - coefU) > 0)
{
BitLevel.InplaceShiftLeft(r, k - coefV + coefU);
}
}
coefU += k;
}
k = HowManyIterations(v, n);
if (k != 0)
{
BitLevel.InplaceShiftLeft(v, k);
if (coefV >= coefU)
{
BitLevel.InplaceShiftLeft(s, k);
}
else
{
BitLevel.InplaceShiftRight(r, System.Math.Min(coefU - coefV, k));
if (k - (coefU - coefV) > 0)
{
BitLevel.InplaceShiftLeft(s, k - coefU + coefV);
}
}
coefV += k;
}
if (u.Signum() == v.Signum())
{
if (coefU <= coefV)
{
Elementary.CompleteInPlaceSubtract(u, v);
Elementary.CompleteInPlaceSubtract(r, s);
}
else
{
Elementary.CompleteInPlaceSubtract(v, u);
Elementary.CompleteInPlaceSubtract(s, r);
}
}
else
{
if (coefU <= coefV)
{
Elementary.CompleteInPlaceAdd(u, v);
Elementary.CompleteInPlaceAdd(r, s);
}
else
{
Elementary.CompleteInPlaceAdd(v, u);
Elementary.CompleteInPlaceAdd(s, r);
}
}
if (v.Signum() == 0 || u.Signum() == 0)
{
throw new ArithmeticException("BigInteger not invertible");
}
}
if (IsPowerOfTwo(v, coefV))
{
r = s;
if (v.Signum() != u.Signum())
{
u = u.Negate();
}
}
if (u.TestBit(n))
{
if (r.Signum() < 0)
{
r = r.Negate();
}
else
{
r = Modulo.Subtract(r);
}
}
if (r.Signum() < 0)
{
r = r.Add(Modulo);
}
return(r);
}
/// <summary>
/// Calculates x.modInverse(p) Based on: Savas, E; Koc, C "The Montgomery Modular Inverse - Revised"
/// </summary>
///
/// <param name="X">BigInteger X</param>
/// <param name="P">BigInteger P</param>
///
/// <returns>Returns <c>1/X Mod M</c></returns>
internal static BigInteger ModInverseMontgomery(BigInteger X, BigInteger P)
{
// ZERO hasn't inverse
if (X.m_sign == 0)
{
throw new ArithmeticException("BigInteger not invertible!");
}
// montgomery inverse require even modulo
if (!P.TestBit(0))
{
return(ModInverseLorencz(X, P));
}
int m = P.m_numberLength * 32;
// PRE: a \in [1, p - 1]
BigInteger u, v, r, s;
u = P.Copy(); // make copy to use inplace method
v = X.Copy();
int max = System.Math.Max(v.m_numberLength, u.m_numberLength);
r = new BigInteger(1, 1, new int[max + 1]);
s = new BigInteger(1, 1, new int[max + 1]);
s.m_digits[0] = 1;
int k = 0;
int lsbu = u.LowestSetBit;
int lsbv = v.LowestSetBit;
int toShift;
if (lsbu > lsbv)
{
BitLevel.InplaceShiftRight(u, lsbu);
BitLevel.InplaceShiftRight(v, lsbv);
BitLevel.InplaceShiftLeft(r, lsbv);
k += lsbu - lsbv;
}
else
{
BitLevel.InplaceShiftRight(u, lsbu);
BitLevel.InplaceShiftRight(v, lsbv);
BitLevel.InplaceShiftLeft(s, lsbu);
k += lsbv - lsbu;
}
r.m_sign = 1;
while (v.Signum() > 0)
{
// INV v >= 0, u >= 0, v odd, u odd (except last iteration when v is even (0))
while (u.CompareTo(v) > BigInteger.EQUALS)
{
Elementary.InplaceSubtract(u, v);
toShift = u.LowestSetBit;
BitLevel.InplaceShiftRight(u, toShift);
Elementary.InplaceAdd(r, s);
BitLevel.InplaceShiftLeft(s, toShift);
k += toShift;
}
while (u.CompareTo(v) <= BigInteger.EQUALS)
{
Elementary.InplaceSubtract(v, u);
if (v.Signum() == 0)
{
break;
}
toShift = v.LowestSetBit;
BitLevel.InplaceShiftRight(v, toShift);
Elementary.InplaceAdd(s, r);
BitLevel.InplaceShiftLeft(r, toShift);
k += toShift;
}
}
// in u is stored the gcd
if (!u.IsOne())
{
throw new ArithmeticException("BigInteger not invertible.");
}
if (r.CompareTo(P) >= BigInteger.EQUALS)
{
Elementary.InplaceSubtract(r, P);
}
r = P.Subtract(r);
// Have pair: ((BigInteger)r, (Integer)k) where r == a^(-1) * 2^k mod (module)
int n1 = CalcN(P);
if (k > m)
{
r = MonPro(r, BigInteger.One, P, n1);
k = k - m;
}
r = MonPro(r, BigInteger.GetPowerOfTwo(m - k), P, n1);
return(r);
}
