An Improved Addition Formula on Elliptic Curves Given by Weierstrass Normal Form

Abstract

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 - 2λ_y1)/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.