/**
* 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 - 2^222 - 2^94 + 2^62
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 31 1s } { 1 0s } { 128 1s } { 31 0s } { 1 1s } { 62 0s}
*
* We use an addition chain for the beginning: [1], 2, 3, 6, 12, [24], 30, [31]
*/
uint[] x1 = this.x;
if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat256.Create();
SM2P256V1Field.Square(x1, x2);
SM2P256V1Field.Multiply(x2, x1, x2);
uint[] x4 = Nat256.Create();
SM2P256V1Field.SquareN(x2, 2, x4);
SM2P256V1Field.Multiply(x4, x2, x4);
uint[] x6 = Nat256.Create();
SM2P256V1Field.SquareN(x4, 2, x6);
SM2P256V1Field.Multiply(x6, x2, x6);
uint[] x12 = x2;
SM2P256V1Field.SquareN(x6, 6, x12);
SM2P256V1Field.Multiply(x12, x6, x12);
uint[] x24 = Nat256.Create();
SM2P256V1Field.SquareN(x12, 12, x24);
SM2P256V1Field.Multiply(x24, x12, x24);
uint[] x30 = x12;
SM2P256V1Field.SquareN(x24, 6, x30);
SM2P256V1Field.Multiply(x30, x6, x30);
uint[] x31 = x6;
SM2P256V1Field.Square(x30, x31);
SM2P256V1Field.Multiply(x31, x1, x31);
uint[] t1 = x24;
SM2P256V1Field.SquareN(x31, 31, t1);
uint[] x62 = x30;
SM2P256V1Field.Multiply(t1, x31, x62);
SM2P256V1Field.SquareN(t1, 32, t1);
SM2P256V1Field.Multiply(t1, x62, t1);
SM2P256V1Field.SquareN(t1, 62, t1);
SM2P256V1Field.Multiply(t1, x62, t1);
SM2P256V1Field.SquareN(t1, 4, t1);
SM2P256V1Field.Multiply(t1, x4, t1);
SM2P256V1Field.SquareN(t1, 32, t1);
SM2P256V1Field.Multiply(t1, x1, t1);
SM2P256V1Field.SquareN(t1, 62, t1);
uint[] t2 = x4;
SM2P256V1Field.Square(t1, t2);
return(Nat256.Eq(x1, t2) ? new SM2P256V1FieldElement(t1) : null);
}
