/**
* Computes the norm of an element <code>λ</code> of
* <code><b>Z</b>[τ]</code>.
* @param mu The parameter <code>μ</code> of the elliptic curve.
* @param lambda The element <code>λ</code> of
* <code><b>Z</b>[τ]</code>.
* @return The norm of <code>λ</code>.
*/
public static BigInteger Norm(sbyte mu, ZTauElement lambda)
{
BigInteger norm;
// s1 = u^2
BigInteger s1 = lambda.u.Multiply(lambda.u);
// s2 = u * v
BigInteger s2 = lambda.u.Multiply(lambda.v);
// s3 = 2 * v^2
BigInteger s3 = lambda.v.Multiply(lambda.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);
}
/**
* Partial modular reduction modulo
* <code>(τ<sup>m</sup> - 1)/(τ - 1)</code>.
* @param k The integer to be reduced.
* @param m The bitlength of the underlying finite field.
* @param a The parameter <code>a</code> of the elliptic curve.
* @param s The auxiliary values <code>s<sub>0</sub></code> and
* <code>s<sub>1</sub></code>.
* @param mu The parameter μ of the elliptic curve.
* @param c The precision (number of bits of accuracy) of the partial
* modular reduction.
* @return <code>ρ := k partmod (τ<sup>m</sup> - 1)/(τ - 1)</code>
*/
public static ZTauElement PartModReduction(BigInteger k, int m, sbyte a,
BigInteger[] s, sbyte mu, sbyte c)
{
// d0 = s[0] + mu*s[1]; mu is either 1 or -1
BigInteger d0;
if (mu == 1)
{
d0 = s[0].Add(s[1]);
}
else
{
d0 = s[0].Subtract(s[1]);
}
BigInteger[] v = GetLucas(mu, m, true);
BigInteger vm = v[1];
SimpleBigDecimal lambda0 = ApproximateDivisionByN(
k, s[0], vm, a, m, c);
SimpleBigDecimal lambda1 = ApproximateDivisionByN(
k, s[1], vm, a, m, c);
ZTauElement q = Round(lambda0, lambda1, mu);
// r0 = n - d0*q0 - 2*s1*q1
BigInteger r0 = k.Subtract(d0.Multiply(q.u)).Subtract(
BigInteger.ValueOf(2).Multiply(s[1]).Multiply(q.v));
// r1 = s1*q0 - s0*q1
BigInteger r1 = s[1].Multiply(q.u).Subtract(s[0].Multiply(q.v));
return(new ZTauElement(r0, r1));
}
/**
* Multiplies a {@link Essensoft.AspNetCore.Security.math.ec.AbstractF2mPoint AbstractF2mPoint}
* by an element <code>λ</code> of <code><b>Z</b>[τ]</code>
* using the <code>τ</code>-adic NAF (TNAF) method.
* @param p The AbstractF2mPoint to Multiply.
* @param lambda The element <code>λ</code> of
* <code><b>Z</b>[τ]</code>.
* @return <code>λ * p</code>
*/
public static AbstractF2mPoint MultiplyTnaf(AbstractF2mPoint p, ZTauElement lambda)
{
AbstractF2mCurve curve = (AbstractF2mCurve)p.Curve;
sbyte mu = GetMu(curve.A);
sbyte[] u = TauAdicNaf(mu, lambda);
AbstractF2mPoint q = MultiplyFromTnaf(p, u);
return(q);
}
/**
* Multiplies a {@link Essensoft.AspNetCore.Security.math.ec.AbstractF2mPoint AbstractF2mPoint}
* by a <code>BigInteger</code> using the reduced <code>τ</code>-adic
* NAF (RTNAF) method.
* @param p The AbstractF2mPoint to Multiply.
* @param k The <code>BigInteger</code> by which to Multiply <code>p</code>.
* @return <code>k * p</code>
*/
public static AbstractF2mPoint MultiplyRTnaf(AbstractF2mPoint p, BigInteger k)
{
AbstractF2mCurve curve = (AbstractF2mCurve)p.Curve;
int m = curve.FieldSize;
int a = curve.A.ToBigInteger().IntValue;
sbyte mu = GetMu(a);
BigInteger[] s = curve.GetSi();
ZTauElement rho = PartModReduction(k, m, (sbyte)a, s, mu, (sbyte)10);
return(MultiplyTnaf(p, rho));
}
/**
* Computes the <code>[τ]</code>-adic window NAF of an element
* <code>λ</code> of <code><b>Z</b>[τ]</code>.
* @param mu The parameter μ of the elliptic curve.
* @param lambda The element <code>λ</code> of
* <code><b>Z</b>[τ]</code> of which to compute the
* <code>[τ]</code>-adic NAF.
* @param width The window width of the resulting WNAF.
* @param pow2w 2<sup>width</sup>.
* @param tw The auxiliary value <code>t<sub>w</sub></code>.
* @param alpha The <code>α<sub>u</sub></code>'s for the window width.
* @return The <code>[τ]</code>-adic window NAF of
* <code>λ</code>.
*/
public static sbyte[] TauAdicWNaf(sbyte mu, ZTauElement lambda,
sbyte width, BigInteger pow2w, BigInteger tw, ZTauElement[] alpha)
{
if (!((mu == 1) || (mu == -1)))
{
throw new ArgumentException("mu must be 1 or -1");
}
BigInteger norm = Norm(mu, lambda);
// Ceiling of log2 of the norm
int log2Norm = norm.BitLength;
// If length(TNAF) > 30, then length(TNAF) < log2Norm + 3.52
int maxLength = log2Norm > 30 ? log2Norm + 4 + width : 34 + width;
// The array holding the TNAF
sbyte[] u = new sbyte[maxLength];
// 2^(width - 1)
BigInteger pow2wMin1 = pow2w.ShiftRight(1);
// Split lambda into two BigIntegers to simplify calculations
BigInteger r0 = lambda.u;
BigInteger r1 = lambda.v;
int i = 0;
// while lambda <> (0, 0)
while (!((r0.Equals(BigInteger.Zero)) && (r1.Equals(BigInteger.Zero))))
{
// if r0 is odd
if (r0.TestBit(0))
{
// uUnMod = r0 + r1*tw Mod 2^width
BigInteger uUnMod
= r0.Add(r1.Multiply(tw)).Mod(pow2w);
sbyte uLocal;
// if uUnMod >= 2^(width - 1)
if (uUnMod.CompareTo(pow2wMin1) >= 0)
{
uLocal = (sbyte)uUnMod.Subtract(pow2w).IntValue;
}
else
{
uLocal = (sbyte)uUnMod.IntValue;
}
// uLocal is now in [-2^(width-1), 2^(width-1)-1]
u[i] = uLocal;
bool s = true;
if (uLocal < 0)
{
s = false;
uLocal = (sbyte)-uLocal;
}
// uLocal is now >= 0
if (s)
{
r0 = r0.Subtract(alpha[uLocal].u);
r1 = r1.Subtract(alpha[uLocal].v);
}
else
{
r0 = r0.Add(alpha[uLocal].u);
r1 = r1.Add(alpha[uLocal].v);
}
}
else
{
u[i] = 0;
}
BigInteger t = r0;
if (mu == 1)
{
r0 = r1.Add(r0.ShiftRight(1));
}
else
{
// mu == -1
r0 = r1.Subtract(r0.ShiftRight(1));
}
r1 = t.ShiftRight(1).Negate();
i++;
}
return(u);
}
/**
* Computes the <code>τ</code>-adic NAF (non-adjacent form) of an
* element <code>λ</code> of <code><b>Z</b>[τ]</code>.
* @param mu The parameter <code>μ</code> of the elliptic curve.
* @param lambda The element <code>λ</code> of
* <code><b>Z</b>[τ]</code>.
* @return The <code>τ</code>-adic NAF of <code>λ</code>.
*/
public static sbyte[] TauAdicNaf(sbyte mu, ZTauElement lambda)
{
if (!((mu == 1) || (mu == -1)))
{
throw new ArgumentException("mu must be 1 or -1");
}
BigInteger norm = Norm(mu, lambda);
// Ceiling of log2 of the norm
int log2Norm = norm.BitLength;
// If length(TNAF) > 30, then length(TNAF) < log2Norm + 3.52
int maxLength = log2Norm > 30 ? log2Norm + 4 : 34;
// The array holding the TNAF
sbyte[] u = new sbyte[maxLength];
int i = 0;
// The actual length of the TNAF
int length = 0;
BigInteger r0 = lambda.u;
BigInteger r1 = lambda.v;
while (!((r0.Equals(BigInteger.Zero)) && (r1.Equals(BigInteger.Zero))))
{
// If r0 is odd
if (r0.TestBit(0))
{
u[i] = (sbyte)BigInteger.Two.Subtract((r0.Subtract(r1.ShiftLeft(1))).Mod(Four)).IntValue;
// r0 = r0 - u[i]
if (u[i] == 1)
{
r0 = r0.ClearBit(0);
}
else
{
// u[i] == -1
r0 = r0.Add(BigInteger.One);
}
length = i;
}
else
{
u[i] = 0;
}
BigInteger t = r0;
BigInteger s = r0.ShiftRight(1);
if (mu == 1)
{
r0 = r1.Add(s);
}
else
{
// mu == -1
r0 = r1.Subtract(s);
}
r1 = t.ShiftRight(1).Negate();
i++;
}
length++;
// Reduce the TNAF array to its actual length
sbyte[] tnaf = new sbyte[length];
Array.Copy(u, 0, tnaf, 0, length);
return(tnaf);
}