5.2.4 Point Addition

5.2.4 Point Addition

For point addition, the following method is recommended. A point (x,y) is represented in projective coordinates (X, Y, Z), with x = X/Z, y = Y/Z.

The neutral point is (0,1), or equivalently in projective coordinates (0, Z, Z) for any non-zero Z.

The following formulas for adding two points, (x3,y3) = (x1,y1)+(x2,y2) on untwisted Edwards curve (i.e., a=1) with non-square d, are described in Section 4 of [Faster-ECC] and in [EFD-ADD]. They are complete, i.e., they work for any pair of valid input points.

                A = Z1*Z2
                B = A^2
                C = X1*X2
                D = Y1*Y2
                E = d*C*D
                F = B-E
                G = B+E
                H = (X1+Y1)*(X2+Y2)
                X3 = A*F*(H-C-D)
                Y3 = A*G*(D-C)
                Z3 = F*G

Again, similar to the other curve, doubling formulas can be obtained by substituting equal points, turning four multiplications into squares. However, this is not even nearly optimal; the following formulas described in Section 4 of [Faster-ECC] and in [EFD-DBL] save multiple multiplications.

                B = (X1+Y1)^2
                C = X1^2
                D = Y1^2
                E = C+D
                H = Z1^2
                J = E-2*H
                X3 = (B-E)*J
                Y3 = E*(C-D)
                Z3 = E*J