/**
* Computes the norm of an element <code>λ</code> of
* <code><b>R</b>[τ]</code>, where <code>λ = u + vτ</code>
* and <code>u</code> and <code>u</code> are real numbers (elements of
* <code><b>R</b></code>).
* @param mu The parameter <code>μ</code> of the elliptic curve.
* @param u The real part of the element <code>λ</code> of
* <code><b>R</b>[τ]</code>.
* @param v The <code>τ</code>-adic part of the element
* <code>λ</code> of <code><b>R</b>[τ]</code>.
* @return The norm of <code>λ</code>.
*/
public static SimpleBigDecimal Norm(sbyte mu, SimpleBigDecimal u, SimpleBigDecimal v)
{
SimpleBigDecimal norm;
// s1 = u^2
SimpleBigDecimal s1 = u.Multiply(u);
// s2 = u * v
SimpleBigDecimal s2 = u.Multiply(v);
// s3 = 2 * v^2
SimpleBigDecimal s3 = v.Multiply(v).ShiftLeft(1);
if (mu == 1)
{
norm = s1.Add(s2).Add(s3);
}
else if (mu == -1)
{
norm = s1.Subtract(s2).Add(s3);
}
else
{
throw new ArgumentException("mu must be 1 or -1");
}
return(norm);
}
/**
* Rounds an element <code>λ</code> of <code><b>R</b>[τ]</code>
* to an element of <code><b>Z</b>[τ]</code>, such that their difference
* has minimal norm. <code>λ</code> is given as
* <code>λ = λ<sub>0</sub> + λ<sub>1</sub>τ</code>.
* @param lambda0 The component <code>λ<sub>0</sub></code>.
* @param lambda1 The component <code>λ<sub>1</sub></code>.
* @param mu The parameter <code>μ</code> of the elliptic curve. Must
* equal 1 or -1.
* @return The rounded element of <code><b>Z</b>[τ]</code>.
* @throws ArgumentException if <code>lambda0</code> and
* <code>lambda1</code> do not have same scale.
*/
public static ZTauElement Round(SimpleBigDecimal lambda0,
SimpleBigDecimal lambda1, sbyte mu)
{
int scale = lambda0.Scale;
if (lambda1.Scale != scale)
{
throw new ArgumentException("lambda0 and lambda1 do not have same scale");
}
if (!((mu == 1) || (mu == -1)))
{
throw new ArgumentException("mu must be 1 or -1");
}
BigInteger f0 = lambda0.Round();
BigInteger f1 = lambda1.Round();
SimpleBigDecimal eta0 = lambda0.Subtract(f0);
SimpleBigDecimal eta1 = lambda1.Subtract(f1);
// eta = 2*eta0 + mu*eta1
SimpleBigDecimal eta = eta0.Add(eta0);
if (mu == 1)
{
eta = eta.Add(eta1);
}
else
{
// mu == -1
eta = eta.Subtract(eta1);
}
// check1 = eta0 - 3*mu*eta1
// check2 = eta0 + 4*mu*eta1
SimpleBigDecimal threeEta1 = eta1.Add(eta1).Add(eta1);
SimpleBigDecimal fourEta1 = threeEta1.Add(eta1);
SimpleBigDecimal check1;
SimpleBigDecimal check2;
if (mu == 1)
{
check1 = eta0.Subtract(threeEta1);
check2 = eta0.Add(fourEta1);
}
else
{
// mu == -1
check1 = eta0.Add(threeEta1);
check2 = eta0.Subtract(fourEta1);
}
sbyte h0 = 0;
sbyte h1 = 0;
// if eta >= 1
if (eta.CompareTo(BigInteger.One) >= 0)
{
if (check1.CompareTo(MinusOne) < 0)
{
h1 = mu;
}
else
{
h0 = 1;
}
}
else
{
// eta < 1
if (check2.CompareTo(BigInteger.Two) >= 0)
{
h1 = mu;
}
}
// if eta < -1
if (eta.CompareTo(MinusOne) < 0)
{
if (check1.CompareTo(BigInteger.One) >= 0)
{
h1 = (sbyte)-mu;
}
else
{
h0 = -1;
}
}
else
{
// eta >= -1
if (check2.CompareTo(MinusTwo) < 0)
{
h1 = (sbyte)-mu;
}
}
BigInteger q0 = f0.Add(BigInteger.ValueOf(h0));
BigInteger q1 = f1.Add(BigInteger.ValueOf(h1));
return(new ZTauElement(q0, q1));
}
