private static GroupElementP1 Madd(ref GroupElementP3 p, ref GroupElementP4 q)
{
var t0 = FieldElementOperations.Add(ref p.Z, ref p.Z); /* D = 2*Z1 */
var r = new GroupElementP1();
/* YpX1 = Y1+X1 */
r.X = FieldElementOperations.Add(ref p.Y, ref p.X);
/* YmX1 = Y1-X1 */
r.Y = FieldElementOperations.Sub(ref p.Y, ref p.X);
/* A = YpX1*ypx2 */
r.Z = FieldElementOperations.Multiplication(ref r.X, ref q.YplusX);
/* B = YmX1*ymx2 */
r.Y = FieldElementOperations.Multiplication(ref r.Y, ref q.YminusX);
/* C = xy2d2*T1 */
r.T = FieldElementOperations.Multiplication(ref q.XY2D, ref p.T);
/* X3 = A-B */
r.X = FieldElementOperations.Sub(ref r.Z, ref r.Y);
/* Y3 = A+B */
r.Y = FieldElementOperations.Add(ref r.Z, ref r.Y);
/* Z3 = D+C */
r.Z = FieldElementOperations.Add(ref t0, ref r.T);
/* T3 = D-C */
r.T = FieldElementOperations.Sub(ref t0, ref r.T);
return(r);
}
private static GroupElementP1 P2ToP1(ref GroupElementP2 p)
{
var r = new GroupElementP1();
/* XX=X1^2 */
r.X = FieldElementOperations.Squared(ref p.X);
/* YY=Y1^2 */
r.Z = FieldElementOperations.Squared(ref p.Y);
/* B=2*Z1^2 */
r.T = FieldElementOperations.DoubleSquare(ref p.Z);
/* A=X1+Y1 */
r.Y = FieldElementOperations.Add(ref p.X, ref p.Y);
/* AA=A^2 */
var t0 = FieldElementOperations.Squared(ref r.Y);
/* Y3=YY+XX */
r.Y = FieldElementOperations.Add(ref r.Z, ref r.X);
/* Z3=YY-XX */
r.Z = FieldElementOperations.Sub(ref r.Z, ref r.X);
/* X3=AA-Y3 */
r.X = FieldElementOperations.Sub(ref t0, ref r.Y);
/* T3=B-Z3 */
r.T = FieldElementOperations.Sub(ref r.T, ref r.Z);
return(r);
}
private static GroupElementP3 P1ToP3(ref GroupElementP1 p) => new GroupElementP3
{
X = FieldElementOperations.Multiplication(ref p.X, ref p.T),
Y = FieldElementOperations.Multiplication(ref p.Y, ref p.Z),
Z = FieldElementOperations.Multiplication(ref p.Z, ref p.T),
T = FieldElementOperations.Multiplication(ref p.X, ref p.Y)
};
private static GroupElementP4 Select(int pos, sbyte b)
{
GroupElementP4 t;
GroupElementP4 minust;
var bnegative = Negative(b);
var babs = (byte)(b - ((-bnegative & b) << 1));
t = new GroupElementP4
{
YplusX = FieldElementOperations.Set1(),
YminusX = FieldElementOperations.Set1(),
XY2D = FieldElementOperations.Set0()
};
var table = LookupTables.Base[pos];
Cmov(ref t, ref table[0], Equal(babs, 1));
Cmov(ref t, ref table[1], Equal(babs, 2));
Cmov(ref t, ref table[2], Equal(babs, 3));
Cmov(ref t, ref table[3], Equal(babs, 4));
Cmov(ref t, ref table[4], Equal(babs, 5));
Cmov(ref t, ref table[5], Equal(babs, 6));
Cmov(ref t, ref table[6], Equal(babs, 7));
Cmov(ref t, ref table[7], Equal(babs, 8));
minust.YplusX = t.YminusX;
minust.YminusX = t.YplusX;
minust.XY2D = FieldElementOperations.Negate(ref t.XY2D);
Cmov(ref t, ref minust, bnegative);
return(t);
}
/// <summary>
/// Curve25519 uses a so-called differential-addition chain proposed by Montgomery to multiply a point,
/// identified only by its x-coordinate, by a scalar
/// </summary>
/// <param name="n"></param>
/// <param name="p"></param>
/// <param name="qSize"></param>
/// <returns></returns>
internal static byte[] ScalarMultiplication(byte[] n, byte[] p, int qSize)
{
var q = new byte[qSize];
var p0 = FieldElementOperations.FromBytes(p);
var q0 = CalculateLadderStep(n, ref p0);
FieldElementOperations.ToBytes(q, ref q0);
return(q);
}
/// <summary>
/// h = a * B
/// where a = a[0]+256*a[1]+...+256^31 a[31]
/// B is the Ed25519 base point (x,4/5) with x positive.
///
/// Preconditions:
/// a[31] <= 127
/// </summary>
/// <param name="a"></param>
/// <param name="offset"></param>
/// <returns></returns>
internal static GroupElementP3 ScalarMultiplicationBase(byte[] a, int offset = 0)
{
GroupElementP3 h;
var e = new sbyte[64];
sbyte carry;
GroupElementP1 r;
GroupElementP2 s;
GroupElementP4 t;
int i;
for (i = 0; i < 32; ++i)
{
e[2 * i + 0] = (sbyte)((a[offset + i] >> 0) & 15);
e[2 * i + 1] = (sbyte)((a[offset + i] >> 4) & 15);
}
/* each e[i] is between 0 and 15 */
/* e[63] is between 0 and 7 */
carry = 0;
for (i = 0; i < 63; ++i)
{
e[i] += carry;
carry = (sbyte)(e[i] + 8);
carry >>= 4;
e[i] -= (sbyte)(carry << 4);
}
e[63] += carry;
/* each e[i] is between -8 and 8 */
h = new GroupElementP3
{
X = FieldElementOperations.Set0(),
Y = FieldElementOperations.Set1(),
Z = FieldElementOperations.Set1(),
T = FieldElementOperations.Set0()
};
for (i = 1; i < 64; i += 2)
{
t = Select(i / 2, e[i]);
r = Madd(ref h, ref t);
h = P1ToP3(ref r);
}
r = P3ToP1(ref h);
s = P1ToP2(ref r);
r = P2ToP1(ref s);
s = P1ToP2(ref r);
r = P2ToP1(ref s);
s = P1ToP2(ref r);
r = P2ToP1(ref s);
h = P1ToP3(ref r);
for (i = 0; i < 64; i += 2)
{
t = Select(i / 2, e[i]);
r = Madd(ref h, ref t);
h = P1ToP3(ref r);
}
return(h);
}
private static void Cmov(ref GroupElementP4 t, ref GroupElementP4 u, byte b)
{
FieldElementOperations.Mov(ref t.YplusX, ref u.YplusX, b);
FieldElementOperations.Mov(ref t.YminusX, ref u.YminusX, b);
FieldElementOperations.Mov(ref t.XY2D, ref u.XY2D, b);
}
private static FieldElement CalculateLadderStep(byte[] n, ref FieldElement p, int noffset = 0)
{
var e = new byte[32];
uint i;
FieldElement x1;
FieldElement x2;
FieldElement z2;
FieldElement x3;
FieldElement z3;
int pos;
uint swap;
for (i = 0; i < 32; ++i)
{
e[i] = n[noffset + i];
}
ClampOperation.Clamp(e, 0);
x1 = p;
x2 = FieldElementOperations.Set1();
z2 = FieldElementOperations.Set0();
x3 = x1;
z3 = FieldElementOperations.Set1();
swap = 0;
for (pos = 254; pos >= 0; --pos)
{
var b = (uint)(e[pos / 8] >> (pos & 7));
b &= 1;
swap ^= b;
FieldElementOperations.Swap(ref x2, ref x3, swap);
FieldElementOperations.Swap(ref z2, ref z3, swap);
swap = b;
/* D = X3-Z3 */
var tmp0 = FieldElementOperations.Sub(ref x3, ref z3);
/* B = X2-Z2 */
var tmp1 = FieldElementOperations.Sub(ref x2, ref z2);
/* A = X2+Z2 */
x2 = FieldElementOperations.Add(ref x2, ref z2);
/* C = X3+Z3 */
z2 = FieldElementOperations.Add(ref x3, ref z3);
/* DA = D*A */
z3 = FieldElementOperations.Multiplication(ref tmp0, ref x2);
/* CB = C*B */
z2 = FieldElementOperations.Multiplication(ref z2, ref tmp1);
/* BB = B^2 */
tmp0 = FieldElementOperations.Squared(ref tmp1);
/* AA = A^2 */
tmp1 = FieldElementOperations.Squared(ref x2);
/* t0 = DA+CB */
x3 = FieldElementOperations.Add(ref z3, ref z2);
/* t1 = DA-CB */
z2 = FieldElementOperations.Sub(ref z3, ref z2);
/* X4 = AA*BB */
x2 = FieldElementOperations.Multiplication(ref tmp1, ref tmp0);
/* E = AA-BB */
tmp1 = FieldElementOperations.Sub(ref tmp1, ref tmp0);
/* t2 = t1^2 */
z2 = FieldElementOperations.Squared(ref z2);
/* t3 = a24*E */
z3 = FieldElementOperations.Multiply121666(ref tmp1);
/* X5 = t0^2 */
x3 = FieldElementOperations.Squared(ref x3);
/* t4 = BB+t3 */
tmp0 = FieldElementOperations.Add(ref tmp0, ref z3);
/* Z5 = X1*t2 */
z3 = FieldElementOperations.Multiplication(ref x1, ref z2);
/* Z4 = E*t4 */
z2 = FieldElementOperations.Multiplication(ref tmp1, ref tmp0);
}
FieldElementOperations.Swap(ref x2, ref x3, swap);
FieldElementOperations.Swap(ref z2, ref z3, swap);
z2 = FieldElementOperations.Invert(ref z2);
x2 = FieldElementOperations.Multiplication(ref x2, ref z2);
var q = x2;
Array.Clear(e, 0, e.Length);
return(q);
}