private static bool IsSquare(uint[] x)
{
uint[] t1 = Nat224.Create();
uint[] t2 = Nat224.Create();
Nat224.Copy(x, t1);
for (int i = 0; i < 7; ++i)
{
Nat224.Copy(t1, t2);
SecP224R1Field.SquareN(t1, 1 << i, t1);
SecP224R1Field.Multiply(t1, t2, t1);
}
SecP224R1Field.SquareN(t1, 95, t1);
return(Nat224.IsOne(t1));
}
/**
* return a sqrt root - the routine verifies that the calculation returns the right value - if
* none exists it returns null.
*/
public override ECFieldElement Sqrt()
{
/*
* Q == 8m + 5, so we use Pocklington's method for this case.
*
* First, raise this element to the exponent 2^221 - 2^29 - 2^9 - 2^8 - 2^6 - 2^4 - 2^1 (i.e. m + 1)
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 191 1s } { 1 0s } { 19 1s } { 2 0s } { 1 1s } { 1 0s} { 1 1s } { 1 0s} { 3 1s } { 1 0s}
*
* Therefore we need an addition chain containing 1, 3, 19, 191 (the lengths of the repunits)
* We use: [1], 2, [3], 4, 8, 11, [19], 23, 42, 84, 107, [191]
*/
uint[] x1 = this.x;
if (Nat224.IsZero(x1) || Nat224.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat224.Create();
SecP224K1Field.Square(x1, x2);
SecP224K1Field.Multiply(x2, x1, x2);
uint[] x3 = x2;
SecP224K1Field.Square(x2, x3);
SecP224K1Field.Multiply(x3, x1, x3);
uint[] x4 = Nat224.Create();
SecP224K1Field.Square(x3, x4);
SecP224K1Field.Multiply(x4, x1, x4);
uint[] x8 = Nat224.Create();
SecP224K1Field.SquareN(x4, 4, x8);
SecP224K1Field.Multiply(x8, x4, x8);
uint[] x11 = Nat224.Create();
SecP224K1Field.SquareN(x8, 3, x11);
SecP224K1Field.Multiply(x11, x3, x11);
uint[] x19 = x11;
SecP224K1Field.SquareN(x11, 8, x19);
SecP224K1Field.Multiply(x19, x8, x19);
uint[] x23 = x8;
SecP224K1Field.SquareN(x19, 4, x23);
SecP224K1Field.Multiply(x23, x4, x23);
uint[] x42 = x4;
SecP224K1Field.SquareN(x23, 19, x42);
SecP224K1Field.Multiply(x42, x19, x42);
uint[] x84 = Nat224.Create();
SecP224K1Field.SquareN(x42, 42, x84);
SecP224K1Field.Multiply(x84, x42, x84);
uint[] x107 = x42;
SecP224K1Field.SquareN(x84, 23, x107);
SecP224K1Field.Multiply(x107, x23, x107);
uint[] x191 = x23;
SecP224K1Field.SquareN(x107, 84, x191);
SecP224K1Field.Multiply(x191, x84, x191);
uint[] t1 = x191;
SecP224K1Field.SquareN(t1, 20, t1);
SecP224K1Field.Multiply(t1, x19, t1);
SecP224K1Field.SquareN(t1, 3, t1);
SecP224K1Field.Multiply(t1, x1, t1);
SecP224K1Field.SquareN(t1, 2, t1);
SecP224K1Field.Multiply(t1, x1, t1);
SecP224K1Field.SquareN(t1, 4, t1);
SecP224K1Field.Multiply(t1, x3, t1);
SecP224K1Field.Square(t1, t1);
uint[] t2 = x84;
SecP224K1Field.Square(t1, t2);
if (Nat224.Eq(x1, t2))
{
return(new SecP224K1FieldElement(t1));
}
/*
* If the first guess is incorrect, we multiply by a precomputed power of 2 to get the second guess,
* which is ((4x)^(m + 1))/2 mod Q
*/
SecP224K1Field.Multiply(t1, PRECOMP_POW2, t1);
SecP224K1Field.Square(t1, t2);
if (Nat224.Eq(x1, t2))
{
return(new SecP224K1FieldElement(t1));
}
return(null);
}
public override ECFieldElement Sqrt()
{
uint[] y = x;
if (Nat224.IsZero(y) || Nat224.IsOne(y))
{
return(this);
}
uint[] array = Nat224.Create();
SecP224K1Field.Square(y, array);
SecP224K1Field.Multiply(array, y, array);
uint[] array2 = array;
SecP224K1Field.Square(array, array2);
SecP224K1Field.Multiply(array2, y, array2);
uint[] array3 = Nat224.Create();
SecP224K1Field.Square(array2, array3);
SecP224K1Field.Multiply(array3, y, array3);
uint[] array4 = Nat224.Create();
SecP224K1Field.SquareN(array3, 4, array4);
SecP224K1Field.Multiply(array4, array3, array4);
uint[] array5 = Nat224.Create();
SecP224K1Field.SquareN(array4, 3, array5);
SecP224K1Field.Multiply(array5, array2, array5);
uint[] array6 = array5;
SecP224K1Field.SquareN(array5, 8, array6);
SecP224K1Field.Multiply(array6, array4, array6);
uint[] array7 = array4;
SecP224K1Field.SquareN(array6, 4, array7);
SecP224K1Field.Multiply(array7, array3, array7);
uint[] array8 = array3;
SecP224K1Field.SquareN(array7, 19, array8);
SecP224K1Field.Multiply(array8, array6, array8);
uint[] array9 = Nat224.Create();
SecP224K1Field.SquareN(array8, 42, array9);
SecP224K1Field.Multiply(array9, array8, array9);
uint[] z = array8;
SecP224K1Field.SquareN(array9, 23, z);
SecP224K1Field.Multiply(z, array7, z);
uint[] array10 = array7;
SecP224K1Field.SquareN(z, 84, array10);
SecP224K1Field.Multiply(array10, array9, array10);
uint[] z2 = array10;
SecP224K1Field.SquareN(z2, 20, z2);
SecP224K1Field.Multiply(z2, array6, z2);
SecP224K1Field.SquareN(z2, 3, z2);
SecP224K1Field.Multiply(z2, y, z2);
SecP224K1Field.SquareN(z2, 2, z2);
SecP224K1Field.Multiply(z2, y, z2);
SecP224K1Field.SquareN(z2, 4, z2);
SecP224K1Field.Multiply(z2, array2, z2);
SecP224K1Field.Square(z2, z2);
uint[] array11 = array9;
SecP224K1Field.Square(z2, array11);
if (Nat224.Eq(y, array11))
{
return(new SecP224K1FieldElement(z2));
}
SecP224K1Field.Multiply(z2, PRECOMP_POW2, z2);
SecP224K1Field.Square(z2, array11);
if (Nat224.Eq(y, array11))
{
return(new SecP224K1FieldElement(z2));
}
return(null);
}
