/// <summary>
/// Extended GCD algorithm. Finds U and V, that A*U + B*V = GCD(A, B)
/// </summary>
/// <param name="inputA">Number A</param>
/// <param name="inputB">Number B</param>
/// <param name="U">U</param>
/// <param name="V">V</param>
/// <returns>GCD of A and B</returns>
public static SLongIntB eXtendedGCD(SLongIntB inputA, SLongIntB inputB, out SLongIntB U, out SLongIntB V)
{
// GCD == A*U + B*V
SLongIntB A = inputA;
SLongIntB B = inputB;
U = (SLongIntB)1;
SLongIntB D = new SLongIntB(A);
if (B.IsZero())
{
V = (SLongIntB)0;
return D;
}
SLongIntB v1 = (SLongIntB)0;
SLongIntB v3 = new SLongIntB(B);
SLongIntB Q = null;
SLongIntB t3 = null;
SLongIntB t1 = null;
while (!v3.IsZero())
{
LongMath.Divide(D, v3, out Q, out t3);
t1 = U - (Q*v1);
U.Assign(v1);
D.Assign(v3);
v1.Assign(t1);
v3.Assign(t3);
}
V = D - (U*A);
V /= B;
return D;
}
/// <summary>
/// Calculates Jacobi symbol for long numbers A and B - (A/B)
/// </summary>
/// <param name="A"></param>
/// <param name="B"></param>
/// <returns></returns>
public static int JacobiSymbol(SLongIntB a, SLongIntB b)
{
if (b.IsZero())
{
if (a == 1 || a == -1)
return 1;
else
return 0;
}
if (LongMath.IsEven(a) && LongMath.IsEven(b))
return 0;
SLongIntB A = new SLongIntB(a);
SLongIntB B = new SLongIntB(b);
sbyte[] mods = new sbyte[] { 0, 1, 0, -1, 0, -1, 0, 1 };
int v = 0;
SelfTwoFact(B, out v);
int k = 1;
if (v % 2 == 1)
k = mods[A[0] & 7];
LongMath.SelfAbs(B);
if (A < 0)
k = -k;
while (A > 0)
{
SelfTwoFact(A, out v);
if (v % 2 == 1)
k = mods[B[0] & 7] * k;
if ((A[0] & B[0] & 2) != 0)
k = -k;
SLongIntB r = LongMath.Abs(A);
A.Assign(B % r);
B.Assign(r);
}
if (B > 1)
return 0;
else
return k;
}
/// <summary>
/// Uses extended GCD Algorithm to find inverted by modulo element
/// </summary>
/// <param name="A">Number</param>
/// <param name="N">Modulo</param>
/// <param name="U">Inverted element. Can be less, than zero.
/// To get always positive number, use eXGCDInvernedSafe instead.</param>
/// <returns>GCD of A and N</returns>
public static SLongIntB eXGCDInverted(SLongIntB A, SLongIntB N, out SLongIntB U)
{
// 1 == A*U + N*V
if (A.IsZero())
{
U = new SLongIntB();
if (A.IsZero())
return new SLongIntB(N);
else
return new SLongIntB(A);
}
U = (SLongIntB)1;
SLongIntB G = new SLongIntB(A);
SLongIntB v1 = (SLongIntB)0;
SLongIntB v3 = new SLongIntB(N);
SLongIntB Q = null;
SLongIntB t3 = null;
SLongIntB t1 = null;
while (!v3.IsZero())
{
LongMath.Divide(G, v3, out Q, out t3);
t1 = U - (Q*v1);
U.Assign(v1);
G.Assign(v3);
v1.Assign(t1);
v3.Assign(t3);
}
return G;
}