/*
* Mixed comparison: P1 is in Jacobian coordinates, P2 is in
* affine coordinates.
*/
uint EqCTMixed(MutableECPointPrime P1, MutableECPointPrime P2)
{
/*
* If either P1 or P2 is infinity, then they are equal
* if and only if they both are infinity.
*
* If neither is infinity, then we must check the following:
* X1 = X2*(Z1^2)
* Y1 = Y2*(Z1^3)
* Beware that X1, Y1 and Z1 are in Montgomery representation,
* while X2 and Y2 are not.
*/
mt1.Set(P1.mz);
mt1.MontySquare();
mt2.Set(P2.mx);
mt2.MontyMul(mt1);
mt3.Set(P1.mx);
mt3.FromMonty();
uint r = mt2.EqCT(mt3);
mt1.MontyMul(P1.mz);
mt1.MontyMul(P2.my);
mt2.Set(P1.my);
mt2.FromMonty();
r &= mt1.EqCT(mt2);
uint z1z = P1.mz.IsZeroCT;
uint z2z = P2.mz.IsZeroCT;
return((r & ~(z1z | z2z)) ^ (z1z & z2z));
}
internal override MutableECPoint Dup()
{
MutableECPointPrime Q = new MutableECPointPrime(curve);
Q.Set(this);
return(Q);
}
internal override MutableECPoint Decode(byte[] enc)
{
MutableECPointPrime P = new MutableECPointPrime(this);
P.Decode(enc);
return(P);
}
void SetMuxInner(uint ctl,
MutableECPointPrime P1, MutableECPointPrime P2)
{
mx.CopyMux(ctl, P1.mx, P2.mx);
my.CopyMux(ctl, P1.my, P2.my);
mz.CopyMux(ctl, P1.mz, P2.mz);
affine = P2.affine ^ (ctl & (P1.affine ^ P2.affine));
}
internal override void Set(MutableECPoint Q, uint ctl)
{
MutableECPointPrime R = SameCurve(Q);
mx.CondCopy(R.mx, ctl);
my.CondCopy(R.my, ctl);
mz.CondCopy(R.mz, ctl);
affine ^= ctl & (affine ^ R.affine);
}
internal override void Set(MutableECPoint Q)
{
MutableECPointPrime R = SameCurve(Q);
mx.Set(R.mx);
my.Set(R.my);
mz.Set(R.mz);
affine = R.affine;
}
MutableECPointPrime SameCurve(MutableECPoint Q)
{
MutableECPointPrime R = Q as MutableECPointPrime;
if (R == null || !curve.Equals(R.curve))
{
throw new CryptoException("Mixed curves");
}
return(R);
}
internal override uint EqCT(MutableECPoint Q)
{
MutableECPointPrime R = SameCurve(Q);
if (affine != 0)
{
if (R.affine != 0)
{
return(mx.EqCT(R.mx)
& my.EqCT(R.my)
& mz.EqCT(R.mz));
}
else
{
return(EqCTMixed(R, this));
}
}
else if (R.affine != 0)
{
return(EqCTMixed(this, R));
}
/*
* Both points are in Jacobian coordinates.
* If Z1 and Z2 are non-zero, then equality is
* achieved if and only if both following equations
* are true:
* X1*(Z2^2) = X2*(Z1^2)
* Y1*(Z2^3) = Y2*(Z1^3)
* If Z1 or Z2 is zero, then equality is achieved
* if and only if both are zero.
*/
mt1.Set(mz);
mt1.MontySquare();
mt2.Set(R.mz);
mt2.MontySquare();
mt3.Set(mx);
mt3.MontyMul(mt2);
mt4.Set(R.mx);
mt4.MontyMul(mt1);
uint r = mt3.EqCT(mt4);
mt1.MontyMul(mz);
mt2.MontyMul(R.mz);
mt3.Set(my);
mt3.MontyMul(mt2);
mt4.Set(R.my);
mt4.MontyMul(mt1);
r &= mt3.EqCT(mt4);
uint z1z = mz.IsZeroCT;
uint z2z = R.mz.IsZeroCT;
return((r & ~(z1z | z2z)) ^ (z1z & z2z));
}
internal override MutableECPoint MakeGenerator()
{
/*
* We do not have to check the generator, since
* it was already done in the constructor.
*/
MutableECPointPrime G = new MutableECPointPrime(this);
G.Set(gx, gy, false);
return(G);
}
/*
* Extra checks:
* -- modulus is prime
* -- subgroup order is prime
* -- generator indeed generates subgroup
*/
public override void CheckValid()
{
/*
* Check that the modulus is prime.
*/
if (!BigInt.IsPrime(mod))
{
throw new CryptoException(
"Invalid curve: modulus is not prime");
}
/*
* Check that the subgroup order is prime.
*/
if (!BigInt.IsPrime(SubgroupOrder))
{
throw new CryptoException(
"Invalid curve: subgroup order is not prime");
}
/*
* Check that the G point is indeed a generator of the
* subgroup. Note that since it has explicit coordinates,
* it cannot be the point at infinity; it suffices to
* verify that, when multiplied by the subgroup order,
* it yields infinity.
*/
MutableECPointPrime G = new MutableECPointPrime(this);
G.Set(gx, gy, false);
if (G.MulSpecCT(SubgroupOrder) == 0 || !G.IsInfinity)
{
throw new CryptoException(
"Invalid curve: generator does not match"
+ " subgroup order");
}
/*
* TODO: check cofactor.
*
* If the cofactor is small, then we can simply compute
* the complete curve order by multiplying the cofactor
* with the subgroup order, and see whether it is in the
* proper range with regards to the field cardinal (by
* using Hasse's theorem). However, if the cofactor is
* larger than the subgroup order, then detecting a
* wrong cofactor value is a bit more complex. We could
* generate a few random points and multiply them by
* the computed order, but this may be expensive.
*/
}
internal override uint AddCT(MutableECPoint Q)
{
MutableECPointPrime P2 = SameCurve(Q);
if (P2.affine != 0)
{
ms4.Set(P2.mx);
ms5.Set(P2.my);
ms6.Set(P2.mz);
ms4.ToMonty();
ms5.ToMonty();
ms6.SetMonty(~ms6.IsZeroCT);
return(AddCTInner(ms4, ms5, ms6, true));
}
else
{
return(AddCTInner(P2.mx, P2.my, P2.mz, false));
}
}
internal override uint MulSpecCT(byte[] n)
{
uint good = 0xFFFFFFFF;
/*
* Create and populate window.
*
* If this instance is 0, then we only add 0 to 0 and
* double 0, for which DoubleCT() and AddCT() work
* properly.
*
* If this instance (P) is non-zero, then x*P for all
* x in the 1..16 range shall be non-zero and distinct,
* since the subgroup order is prime and at least 17.
* Thus, we never add two equal points together in the
* window construction.
*
* We MUST ensure that all points are in the same
* coordinate convention (affine or Jacobian) to ensure
* constant-time execution. TODO: measure to see which
* is best: all affine or all Jacobian. All affine implies
* 14 or 15 extra divisions, but saves a few hundreds of
* multiplications.
*/
MutableECPointPrime[] w = new MutableECPointPrime[16];
w[0] = new MutableECPointPrime(curve);
w[0].ToJacobian();
w[1] = new MutableECPointPrime(curve);
w[1].Set(this);
w[1].ToJacobian();
for (int i = 2; (i + 1) < w.Length; i += 2)
{
w[i] = new MutableECPointPrime(curve);
w[i].Set(w[i >> 1]);
w[i].DoubleCT();
w[i + 1] = new MutableECPointPrime(curve);
w[i + 1].Set(w[i]);
good &= w[i + 1].AddCT(this);
}
/* obsolete
* for (int i = 0; i < w.Length; i ++) {
* w[i].ToAffine();
* w[i].Print("Win " + i);
* w[i].ToJacobian();
* }
* Console.WriteLine("good = {0}", (int)good);
*/
/*
* Set this value to 0. We also set it already to
* Jacobian coordinates, since it will be done that
* way anyway. This instance will serve as accumulator.
*/
mx.Set(0);
my.Set(0);
mz.Set(0);
affine = 0x00000000;
/*
* We process the multiplier by 4-bit nibbles, starting
* with the most-significant one (the high nibble of the
* first byte, since we use big-endian notation).
*
* For each nibble, we perform a constant-time lookup
* in the window, to obtain the point to add to the
* current value of the accumulator. Thanks to the
* conditions on the operands (prime subgroup order and
* so on), all the additions below must work.
*/
MutableECPointPrime t = new MutableECPointPrime(curve);
for (int i = (n.Length << 1) - 1; i >= 0; i--)
{
int b = n[n.Length - 1 - (i >> 1)];
int j = (b >> ((i & 1) << 2)) & 0x0F;
for (int k = 0; k < 16; k++)
{
t.Set(w[k], ~(uint)(((j - k) | (k - j)) >> 31));
}
good &= AddCT(t);
if (i > 0)
{
DoubleCT();
DoubleCT();
DoubleCT();
DoubleCT();
}
}
return(good);
}
/*
* Checks enforced by the constructor:
* -- modulus is odd and at least 80-bit long
* -- subgroup order is odd and at least 30-bit long
* -- parameters a[] and b[] are lower than modulus
* -- coordinates gx and gy are lower than modulus
* -- coordinates gx and gy match curve equation
*/
internal ECCurvePrime(string name, byte[] mod, byte[] a, byte[] b,
byte[] gx, byte[] gy, byte[] subgroupOrder, byte[] cofactor)
: base(subgroupOrder, cofactor)
{
this.mod = mod = BigInt.NormalizeBE(mod);
int modLen = BigInt.BitLength(mod);
if (modLen < 80)
{
throw new CryptoException(
"Invalid curve: modulus is too small");
}
if ((mod[mod.Length - 1] & 0x01) == 0)
{
throw new CryptoException(
"Invalid curve: modulus is even");
}
int sgLen = BigInt.BitLength(subgroupOrder);
if (sgLen < 30)
{
throw new CryptoException(
"Invalid curve: subgroup is too small");
}
if ((subgroupOrder[subgroupOrder.Length - 1] & 0x01) == 0)
{
throw new CryptoException(
"Invalid curve: subgroup order is even");
}
mp = new ModInt(mod);
flen = (modLen + 7) >> 3;
pMod4 = mod[mod.Length - 1] & 3;
this.a = a = BigInt.NormalizeBE(a);
this.b = b = BigInt.NormalizeBE(b);
if (BigInt.CompareCT(a, mod) >= 0 ||
BigInt.CompareCT(b, mod) >= 0)
{
throw new CryptoException(
"Invalid curve: out-of-range parameter");
}
ma = mp.Dup();
ma.Decode(a);
ma.Add(3);
aIsM3 = ma.IsZero;
ma.Sub(3);
mb = mp.Dup();
mb.Decode(b);
this.gx = gx = BigInt.NormalizeBE(gx);
this.gy = gy = BigInt.NormalizeBE(gy);
if (BigInt.CompareCT(gx, mod) >= 0 ||
BigInt.CompareCT(gy, mod) >= 0)
{
throw new CryptoException(
"Invalid curve: out-of-range coordinates");
}
MutableECPointPrime G = new MutableECPointPrime(this);
G.Set(gx, gy, true);
hashCode = (int)(BigInt.HashInt(mod)
^ BigInt.HashInt(a) ^ BigInt.HashInt(b)
^ BigInt.HashInt(gx) ^ BigInt.HashInt(gy));
if (name == null)
{
name = string.Format("generic prime {0}/{1}",
modLen, sgLen);
}
this.name = name;
}
