/**
* 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);
}
