# Binary Curve López-Dahab Coordinates

## Introduction

'López-Dahab Coordinates' (short: LD coordinates) are used to
represent elliptic curve points on binary curves y^2 + xy = x^3 + ax^2
+ b. In 'López-Dahab Coordinates' the triple (X, Y, Z) represents the
affine point (X / Z, Y / Z^2).

## Point Doubling (up to 5M)

Let (X, Y, Z) be a point (unequal to the 'point at infinity')
represented in 'López-Dahab Coordinates'. Then its double (X', Y',
Z') can be calculated by

if (X == 0)
return POINT_AT_INFINITY
else
A = X^2
B = Z^2
Z' = A*B
C = A^2
D = b*B^2
X' = C + D
Y' = D*Z' + X'*(a*Z' + Y^2 + D)
return (X', Y', Z')

Note that the total number of field multiplications can be reduced if the curve coefficient a and b are carefully chosen.

## Point Addition (up to 14M)

Let (X1, Y1, Z1) and (X2, Y2, Z2) be two points (both unequal to the
'point at infinity') represented in 'López-Dahab Coordinates'. Then
the sum (X3, Y3, Z3) can be calculated by

A = X1*Z2 + X2*Z1
B = Y1*Z2^2 + Y2*Z1^2
if (A == 0)
if (B != 0)
return POINT_AT_INFINITY
else
return POINT_DOUBLE(X1, Y1, Z1)
C = Z1*A
D = Z2*C
Z3 = D^2
X3 = D*(A^2 + B) + B^2 + a*Z3
E = C*D
F = E^2*Y2
G = X3 + X2*E
Y3 = Z3*X3 + F + B*D*G
return (X3, Y3, Z3)

Note that if curve coefficient a is carefully chosen, the number of
field multiplications can be reduced to 13M.