} //ModularMath.MultiplicativeInverse(long,long)
/// <summary>
/// Computes the multiplicative inverse of <paramref name="a"/> modulo <paramref name="modulus"/>.
/// </summary>
/// <param name="a">A non-negative integer less than <paramref name="modulus"/>.</param>
/// <param name="modulus">A non-negative integer defining the integer ring in which <paramref name="a"/> resides.</param>
/// <returns>The multiplicative inverse of <paramref name="a"/> modulo <paramref name="modulus"/>,
/// or null if no such multiplicative inverse exists.</returns>
public static MyUInt256 MultiplicativeInverse(MyUInt256 a, MyUInt256 modulus)
{
if (a.IsMsbSet() || a.Compare(modulus) >= 0)
{
throw new ArgumentOutOfRangeException("a must be a non-negative integer less than modulus");
}
if (a.IsZero())
{
return(null);
}
MyUInt256 mi;
ExtendedEuclidean(a, modulus, out mi);
return(mi);
} //ModularMath.MultiplicativeInverse(MyUInt256,MyUInt256)
} //ModularMath.ExtendedEuclidean(long,long,out ulong)
/// <summary>
/// Computes the greatest common divisor (GCD) of <paramref name="a"/> and <paramref name="b"/>,
/// as well as the multiplicative inverse of <paramref name="a"/> modulo <paramref name="b"/>.
/// </summary>
/// <param name="a">A positive integer less than <paramref name="b"/>.</param>
/// <param name="b">A positive integer greater than <paramref name="a"/>.</param>
/// <param name="mulInverse">Receives the multiplicative inverse of <paramref name="a"/> modulo <paramref name="b"/>
/// if <paramref name="a"/> and <paramref name="b"/> are coprime, or null otherwise.</param>
/// <returns>The greatest common divisor of <paramref name="a"/> and <paramref name="b"/>.</returns>
public static MyUInt256 ExtendedEuclidean(MyUInt256 a, MyUInt256 b, out MyUInt256 mulInverse)
{
if (a.IsZero() || a.Compare(b) >= 0)
{
throw new ArgumentOutOfRangeException("a must be a positive integer less than b");
}
MyUInt256 r0 = new MyUInt256(b), r1 = new MyUInt256(a); // ordered this way because b > a
MyUInt256 q = new MyUInt256(), r = new MyUInt256();
MyUInt256 x0 = new MyUInt256(0), x1 = new MyUInt256(1); // x is a's coefficient; we don't maintain b's coefficient
MyUInt256 x = new MyUInt256(1); // return multiplicative inverse of 1 if loop breaks before x is assigned and if GCD (q) is 1
MyUInt256 tmp = new MyUInt256();
for (; ;)
{
r.Set(r0); // r = r0 % r1
r.Modulo(r1);
if (r.IsZero())
{
break; // break before overwriting previous value of q
}
q.Set(r0); // q = r0 / r1
q.Divide(r1);
x.Set(x0); // x = x0 - q * x1
tmp.Set(q);
tmp.Multiply(x1);
x.Subtract(tmp);
r0.Set(r1); r1.Set(r);
x0.Set(x1); x1.Set(x);
}
if (x.IsMsbSet())
{
x.Add(b);
}
mulInverse = (r1.Compare(1) == 0 ? x : null);
return(r1);
} //ModularMath.ExtendedEuclidean(MyUInt256,MyUInt256,out MyUInt256)
/// <summary>
/// Computes the greatest common divisor (GCD) of <paramref name="a"/> and <paramref name="b"/>,
/// as well as the multiplicative inverse of <paramref name="a"/> modulo <paramref name="b"/>.
/// </summary>
/// <param name="a">A positive integer less than <paramref name="b"/>.</param>
/// <param name="b">A positive integer greater than <paramref name="a"/>.</param>
/// <param name="mulInverse">Receives the multiplicative inverse of <paramref name="a"/> modulo <paramref name="b"/>
/// if <paramref name="a"/> and <paramref name="b"/> are coprime, or null otherwise.</param>
/// <returns>The greatest common divisor of <paramref name="a"/> and <paramref name="b"/>.</returns>
public static MyUInt256 ExtendedEuclidean(MyUInt256 a, MyUInt256 b, out MyUInt256 mulInverse)
{
if (a.IsZero() || a.Compare(b) >= 0)
throw new ArgumentOutOfRangeException("a must be a positive integer less than b");
MyUInt256 r0 = new MyUInt256(b), r1 = new MyUInt256(a); // ordered this way because b > a
MyUInt256 q = new MyUInt256(), r = new MyUInt256();
MyUInt256 x0 = new MyUInt256(0), x1 = new MyUInt256(1); // x is a's coefficient; we don't maintain b's coefficient
MyUInt256 x = new MyUInt256(1); // return multiplicative inverse of 1 if loop breaks before x is assigned and if GCD (q) is 1
MyUInt256 tmp = new MyUInt256();
for (; ; )
{
r.Set(r0); // r = r0 % r1
r.Modulo(r1);
if (r.IsZero()) break; // break before overwriting previous value of q
q.Set(r0); // q = r0 / r1
q.Divide(r1);
x.Set(x0); // x = x0 - q * x1
tmp.Set(q);
tmp.Multiply(x1);
x.Subtract(tmp);
r0.Set(r1); r1.Set(r);
x0.Set(x1); x1.Set(x);
}
if (x.IsMsbSet()) x.Add(b);
mulInverse = (r1.Compare(1) == 0 ? x : null);
return r1;
}
/// <summary>
/// Computes the multiplicative inverse of <paramref name="a"/> modulo <paramref name="modulus"/>.
/// </summary>
/// <param name="a">A non-negative integer less than <paramref name="modulus"/>.</param>
/// <param name="modulus">A non-negative integer defining the integer ring in which <paramref name="a"/> resides.</param>
/// <returns>The multiplicative inverse of <paramref name="a"/> modulo <paramref name="modulus"/>,
/// or null if no such multiplicative inverse exists.</returns>
public static MyUInt256 MultiplicativeInverse(MyUInt256 a, MyUInt256 modulus)
{
if (a.IsMsbSet() || a.Compare(modulus) >= 0)
throw new ArgumentOutOfRangeException("a must be a non-negative integer less than modulus");
if (a.IsZero())
return null;
MyUInt256 mi;
ExtendedEuclidean(a, modulus, out mi);
return mi;
}