/**
* 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 2^190 - 2^62
uint[] x1 = this.x;
if (Nat192.IsZero(x1) || Nat192.IsOne(x1))
{
return(this);
}
uint[] t1 = Nat192.Create();
uint[] t2 = Nat192.Create();
SecP192R1Field.Square(x1, t1);
SecP192R1Field.Multiply(t1, x1, t1);
SecP192R1Field.SquareN(t1, 2, t2);
SecP192R1Field.Multiply(t2, t1, t2);
SecP192R1Field.SquareN(t2, 4, t1);
SecP192R1Field.Multiply(t1, t2, t1);
SecP192R1Field.SquareN(t1, 8, t2);
SecP192R1Field.Multiply(t2, t1, t2);
SecP192R1Field.SquareN(t2, 16, t1);
SecP192R1Field.Multiply(t1, t2, t1);
SecP192R1Field.SquareN(t1, 32, t2);
SecP192R1Field.Multiply(t2, t1, t2);
SecP192R1Field.SquareN(t2, 64, t1);
SecP192R1Field.Multiply(t1, t2, t1);
SecP192R1Field.SquareN(t1, 62, t1);
SecP192R1Field.Square(t1, t2);
return(Nat192.Eq(x1, t2) ? new SecP192R1FieldElement(t1) : 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 2^190 - 2^30 - 2^10 - 2^6 - 2^5 - 2^4 - 2^1
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 159 1s } { 1 0s } { 19 1s } { 1 0s } { 3 1s } { 3 0s} { 3 1s } { 1 0s }
*
* Therefore we need an addition chain containing 3, 19, 159 (the lengths of the repunits)
* We use: 1, 2, [3], 6, 8, 16, [19], 35, 70, 140, [159]
*/
uint[] x1 = this.x;
if (Nat192.IsZero(x1) || Nat192.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat192.Create();
SecP192K1Field.Square(x1, x2);
SecP192K1Field.Multiply(x2, x1, x2);
uint[] x3 = Nat192.Create();
SecP192K1Field.Square(x2, x3);
SecP192K1Field.Multiply(x3, x1, x3);
uint[] x6 = Nat192.Create();
SecP192K1Field.SquareN(x3, 3, x6);
SecP192K1Field.Multiply(x6, x3, x6);
uint[] x8 = x6;
SecP192K1Field.SquareN(x6, 2, x8);
SecP192K1Field.Multiply(x8, x2, x8);
uint[] x16 = x2;
SecP192K1Field.SquareN(x8, 8, x16);
SecP192K1Field.Multiply(x16, x8, x16);
uint[] x19 = x8;
SecP192K1Field.SquareN(x16, 3, x19);
SecP192K1Field.Multiply(x19, x3, x19);
uint[] x35 = Nat192.Create();
SecP192K1Field.SquareN(x19, 16, x35);
SecP192K1Field.Multiply(x35, x16, x35);
uint[] x70 = x16;
SecP192K1Field.SquareN(x35, 35, x70);
SecP192K1Field.Multiply(x70, x35, x70);
uint[] x140 = x35;
SecP192K1Field.SquareN(x70, 70, x140);
SecP192K1Field.Multiply(x140, x70, x140);
uint[] x159 = x70;
SecP192K1Field.SquareN(x140, 19, x159);
SecP192K1Field.Multiply(x159, x19, x159);
uint[] t1 = x159;
SecP192K1Field.SquareN(t1, 20, t1);
SecP192K1Field.Multiply(t1, x19, t1);
SecP192K1Field.SquareN(t1, 4, t1);
SecP192K1Field.Multiply(t1, x3, t1);
SecP192K1Field.SquareN(t1, 6, t1);
SecP192K1Field.Multiply(t1, x3, t1);
SecP192K1Field.Square(t1, t1);
uint[] t2 = x3;
SecP192K1Field.Square(t1, t2);
return(Nat192.Eq(x1, t2) ? new SecP192K1FieldElement(t1) : null);
}