// D.1.4 91
/**
* 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 2^158 - 2^29
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 129 1s } { 29 0s }
*
* Therefore we need an addition chain containing 129 (the length of the repunit) We use:
* 1, 2, 4, 8, 16, 32, 64, 128, [129]
*/
uint[] x1 = this.x;
if (Nat160.IsZero(x1) || Nat160.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat160.Create();
SecP160R1Field.Square(x1, x2);
SecP160R1Field.Multiply(x2, x1, x2);
uint[] x4 = Nat160.Create();
SecP160R1Field.SquareN(x2, 2, x4);
SecP160R1Field.Multiply(x4, x2, x4);
uint[] x8 = x2;
SecP160R1Field.SquareN(x4, 4, x8);
SecP160R1Field.Multiply(x8, x4, x8);
uint[] x16 = x4;
SecP160R1Field.SquareN(x8, 8, x16);
SecP160R1Field.Multiply(x16, x8, x16);
uint[] x32 = x8;
SecP160R1Field.SquareN(x16, 16, x32);
SecP160R1Field.Multiply(x32, x16, x32);
uint[] x64 = x16;
SecP160R1Field.SquareN(x32, 32, x64);
SecP160R1Field.Multiply(x64, x32, x64);
uint[] x128 = x32;
SecP160R1Field.SquareN(x64, 64, x128);
SecP160R1Field.Multiply(x128, x64, x128);
uint[] x129 = x64;
SecP160R1Field.Square(x128, x129);
SecP160R1Field.Multiply(x129, x1, x129);
uint[] t1 = x129;
SecP160R1Field.SquareN(t1, 29, t1);
uint[] t2 = x128;
SecP160R1Field.Square(t1, t2);
return(Nat160.Eq(x1, t2) ? new SecP160R1FieldElement(t1) : null);
}
