/**
* 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 2^252 - 2^1 (i.e. m + 1)
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 251 1s } { 1 0s }
*
* Therefore we need an addition chain containing 251 (the lengths of the repunits)
* We use: 1, 2, 3, 4, 7, 11, 15, 30, 60, 120, 131, [251]
*/
uint[] x1 = this.x;
if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat256.Create();
Curve25519Field.Square(x1, x2);
Curve25519Field.Multiply(x2, x1, x2);
uint[] x3 = x2;
Curve25519Field.Square(x2, x3);
Curve25519Field.Multiply(x3, x1, x3);
uint[] x4 = Nat256.Create();
Curve25519Field.Square(x3, x4);
Curve25519Field.Multiply(x4, x1, x4);
uint[] x7 = Nat256.Create();
Curve25519Field.SquareN(x4, 3, x7);
Curve25519Field.Multiply(x7, x3, x7);
uint[] x11 = x3;
Curve25519Field.SquareN(x7, 4, x11);
Curve25519Field.Multiply(x11, x4, x11);
uint[] x15 = x7;
Curve25519Field.SquareN(x11, 4, x15);
Curve25519Field.Multiply(x15, x4, x15);
uint[] x30 = x4;
Curve25519Field.SquareN(x15, 15, x30);
Curve25519Field.Multiply(x30, x15, x30);
uint[] x60 = x15;
Curve25519Field.SquareN(x30, 30, x60);
Curve25519Field.Multiply(x60, x30, x60);
uint[] x120 = x30;
Curve25519Field.SquareN(x60, 60, x120);
Curve25519Field.Multiply(x120, x60, x120);
uint[] x131 = x60;
Curve25519Field.SquareN(x120, 11, x131);
Curve25519Field.Multiply(x131, x11, x131);
uint[] x251 = x11;
Curve25519Field.SquareN(x131, 120, x251);
Curve25519Field.Multiply(x251, x120, x251);
uint[] t1 = x251;
Curve25519Field.Square(t1, t1);
uint[] t2 = x120;
Curve25519Field.Square(t1, t2);
if (Nat256.Eq(x1, t2))
{
return(new Curve25519FieldElement(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
*/
Curve25519Field.Multiply(t1, PRECOMP_POW2, t1);
Curve25519Field.Square(t1, t2);
if (Nat256.Eq(x1, t2))
{
return(new Curve25519FieldElement(t1));
}
return(null);
}
/**
* return a sqrt root - the routine verifies that the calculation returns the right value - if
* none exists it returns null.
*/
public override ECFieldElement Sqrt()
{
/*
* Raise this element to the exponent 2^254 - 2^30 - 2^7 - 2^6 - 2^5 - 2^4 - 2^2
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 223 1s } { 1 0s } { 22 1s } { 4 0s } { 2 1s } { 2 0s}
*
* Therefore we need an addition chain containing 2, 22, 223 (the lengths of the repunits)
* We use: 1, [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
*/
uint[] x1 = this.x;
if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat256.Create();
SecP256K1Field.Square(x1, x2);
SecP256K1Field.Multiply(x2, x1, x2);
uint[] x3 = Nat256.Create();
SecP256K1Field.Square(x2, x3);
SecP256K1Field.Multiply(x3, x1, x3);
uint[] x6 = Nat256.Create();
SecP256K1Field.SquareN(x3, 3, x6);
SecP256K1Field.Multiply(x6, x3, x6);
uint[] x9 = x6;
SecP256K1Field.SquareN(x6, 3, x9);
SecP256K1Field.Multiply(x9, x3, x9);
uint[] x11 = x9;
SecP256K1Field.SquareN(x9, 2, x11);
SecP256K1Field.Multiply(x11, x2, x11);
uint[] x22 = Nat256.Create();
SecP256K1Field.SquareN(x11, 11, x22);
SecP256K1Field.Multiply(x22, x11, x22);
uint[] x44 = x11;
SecP256K1Field.SquareN(x22, 22, x44);
SecP256K1Field.Multiply(x44, x22, x44);
uint[] x88 = Nat256.Create();
SecP256K1Field.SquareN(x44, 44, x88);
SecP256K1Field.Multiply(x88, x44, x88);
uint[] x176 = Nat256.Create();
SecP256K1Field.SquareN(x88, 88, x176);
SecP256K1Field.Multiply(x176, x88, x176);
uint[] x220 = x88;
SecP256K1Field.SquareN(x176, 44, x220);
SecP256K1Field.Multiply(x220, x44, x220);
uint[] x223 = x44;
SecP256K1Field.SquareN(x220, 3, x223);
SecP256K1Field.Multiply(x223, x3, x223);
uint[] t1 = x223;
SecP256K1Field.SquareN(t1, 23, t1);
SecP256K1Field.Multiply(t1, x22, t1);
SecP256K1Field.SquareN(t1, 6, t1);
SecP256K1Field.Multiply(t1, x2, t1);
SecP256K1Field.SquareN(t1, 2, t1);
uint[] t2 = x2;
SecP256K1Field.Square(t1, t2);
return(Nat256.Eq(x1, t2) ? new SecP256K1FieldElement(t1) : null);
}
