/** * 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); }