/** * 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 + 2^190 + 2^94 uint[] x1 = this.x; if (Nat256.IsZero(x1) || Nat256.IsOne(x1)) { return(this); } uint[] t1 = Nat256.Create(); uint[] t2 = Nat256.Create(); SecP256R1Field.Square(x1, t1); SecP256R1Field.Multiply(t1, x1, t1); SecP256R1Field.SquareN(t1, 2, t2); SecP256R1Field.Multiply(t2, t1, t2); SecP256R1Field.SquareN(t2, 4, t1); SecP256R1Field.Multiply(t1, t2, t1); SecP256R1Field.SquareN(t1, 8, t2); SecP256R1Field.Multiply(t2, t1, t2); SecP256R1Field.SquareN(t2, 16, t1); SecP256R1Field.Multiply(t1, t2, t1); SecP256R1Field.SquareN(t1, 32, t1); SecP256R1Field.Multiply(t1, x1, t1); SecP256R1Field.SquareN(t1, 96, t1); SecP256R1Field.Multiply(t1, x1, t1); SecP256R1Field.SquareN(t1, 94, t1); SecP256R1Field.Multiply(t1, t1, t2); return(Nat256.Eq(x1, t2) ? new SecP256R1FieldElement(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 2^254 - 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); }