An Improved Addition Formula on Elliptic Curves Given by Weierstrass Normal Form
Masaaki Shirase · 2013
An improved addition formula for an elliptic curve given by Weierstrass form is proposed. First, the coordinate is converted so that P = (0, y1) and Q = (x2, y2), and then the equation of the elliptic curve becomes y2= x3+ ax2+ bx + c. The proposed formula is thus “x-coordinate of P + Q= (b - 2λy1)/x2”, where λ is the slope of the line through P and Q. The proposed formula can be derived by the geometric definition of point addition. Applying the proposed formula reduces the cost of adding point by about 20% on a system using the mixed coordinate of affine + projective = projective. However, it increases the cost of doubling point, and so we require a further improvement in the future.