/// <summary>
/// Computes the state at the specified position in the pseudorandom stream relative to a given initial state.
/// </summary>
/// <param name="state">The state at position zero in the stream.</param>
/// <param name="offset">The number of states to skip.</param>
/// <param name="multiplier">The linear congruential generator's multiplier constant, or its multiplicative inverse modulo <paramref name="divisor"/> to go backwards.</param>
/// <param name="increment">The linear congruential generator's increment constant,
/// or its additive inverse modulo <paramref name="divisor"/> multiplied by the multiplicative inverse to go backwards.</param>
/// <param name="divisor">The linear congruential generator's modulus constant.</param>
/// <returns>The state at position <paramref name="offset"/> in the stream.</returns>
protected static ulong SeekState(ulong state, ulong offset, ulong multiplier, ulong increment, ulong divisor)
{
// NewState = (Multiplier**Offset * OldState) + Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
// caller can substitute Multiplier**-1 and -Increment * Multiplier**-1 to go backwards
if (offset == 0)
{
return(state);
}
if (multiplier == 0)
{
return(increment);
}
else if (multiplier == 1)
{
MyUInt256 product = new MyUInt256(increment);
product.Multiply(offset);
if (divisor != 0)
{
product.Modulo(divisor);
}
return(product.ToUInt64());
}
MyUInt256 compdivisor = new MyUInt256(divisor);
if (divisor == 0)
{
compdivisor.Set(0, 1);
}
compdivisor.Multiply(multiplier - 1);
MyUInt256 aexpn = new MyUInt256(multiplier);
aexpn.ExpMod(offset, compdivisor); // = Multiplier**Offset
MyUInt256 fraction = new MyUInt256(aexpn);
fraction.Subtract(1);
fraction.Divide(multiplier - 1); // should always divide evenly; that's why divisor has to include (Multiplier - 1) factor though
fraction.Multiply(increment); // = Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
MyUInt256 newstate = new MyUInt256(state);
newstate.Multiply(aexpn); // = Multiplier**Offset * OldState
newstate.Add(fraction); // = (Multiplier**Offset * OldState) + Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
if (divisor != 0)
{
newstate.Modulo(divisor); // now that we've divided by (Multiplier - 1), we can go back to using 'divisor' instead of 'compdivisor'
}
return(newstate.ToUInt64());
} //PrngLcgBase.SeekState
} //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 state at the specified position in the pseudorandom stream relative to a given initial state.
/// </summary>
/// <param name="state">The state at position zero in the stream.</param>
/// <param name="offset">The number of states to skip.</param>
/// <param name="multiplier">The linear congruential generator's multiplier constant, or its multiplicative inverse modulo <paramref name="divisor"/> to go backwards.</param>
/// <param name="increment">The linear congruential generator's increment constant,
/// or its additive inverse modulo <paramref name="divisor"/> multiplied by the multiplicative inverse to go backwards.</param>
/// <param name="divisor">The linear congruential generator's modulus constant.</param>
/// <returns>The state at position <paramref name="offset"/> in the stream.</returns>
protected static ulong SeekState(ulong state, ulong offset, ulong multiplier, ulong increment, ulong divisor)
{
// NewState = (Multiplier**Offset * OldState) + Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
// caller can substitute Multiplier**-1 and -Increment * Multiplier**-1 to go backwards
if (offset == 0)
return state;
if (multiplier == 0)
{
return increment;
}
else if (multiplier == 1)
{
MyUInt256 product = new MyUInt256(increment);
product.Multiply(offset);
if (divisor != 0)
product.Modulo(divisor);
return product.ToUInt64();
}
MyUInt256 compdivisor = new MyUInt256(divisor);
if (divisor == 0)
compdivisor.Set(0, 1);
compdivisor.Multiply(multiplier - 1);
MyUInt256 aexpn = new MyUInt256(multiplier);
aexpn.ExpMod(offset, compdivisor); // = Multiplier**Offset
MyUInt256 fraction = new MyUInt256(aexpn);
fraction.Subtract(1);
fraction.Divide(multiplier - 1); // should always divide evenly; that's why divisor has to include (Multiplier - 1) factor though
fraction.Multiply(increment); // = Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
MyUInt256 newstate = new MyUInt256(state);
newstate.Multiply(aexpn); // = Multiplier**Offset * OldState
newstate.Add(fraction); // = (Multiplier**Offset * OldState) + Increment * (Multiplier**Offset - 1) / (Multiplier - 1)
if (divisor != 0)
newstate.Modulo(divisor); // now that we've divided by (Multiplier - 1), we can go back to using 'divisor' instead of 'compdivisor'
return newstate.ToUInt64();
}
/// <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;
}