Coordinate Forms

Affine $(x, y)$

Affine form refers to the regular $(x,y)$ form of coordinates on an $xy$-plane.

Problem

In affine form, every addition or doubling requires field inversion when computing fractions (e.g., $\frac{1}{1 + dx_1x_2y_1y_2}$). Like with Weierstrass curves, this becomes the bottleneck for performance.

Extended Affine $(x, y, xy)$

Extended Affine form is a lightweight optimization over the basic affine coordinates. It explicitly stores the product $xy$, which is used repeatedly in twisted Edwards point addition formulas. This avoids redundant multiplications and slightly speeds up scalar multiplication loops.

Point Representation:

A point is stored as:

$$(x, y, xy)$$

This means we store:

• $x$: the affine x-coordinate

• $y$: the affine y-coordinate

• $t = xy$: the product of x and y

Problem

In standard affine form, when computing twisted Edwards additions, the term $xy$ appears frequently. Without optimization, it must be recomputed each time.

Solution

By storing $t = xy$ explicitly as a third coordinate, we reduce the number of field multiplications per operation. Though this doesn't eliminate field inversions like projective coordinates do, it provides a minimal memory overhead with noticeable performance gain in systems where full projective coordinates are unnecessary or costly.

Affine $\Longrightarrow$ Extended Affine

$$(x, y) \Longrightarrow (x, y, xy)$$

Extended Affine $\Longrightarrow$ Affine

Simply discard the third value:

$$(x, y, t) \Longrightarrow (x, y)$$

Extended Projective $(X : Y : Z : T)$

To eliminate the need for inverses during addition and doubling, extended projective coordinates are widely used. This is the most efficient coordinate system for twisted Edwards curves.

Point Representation:

$$x = \frac{X}{Z}, \quad y = \frac{Y}{Z}, \quad T = \frac{XY}{Z}$$

So, the point is stored as:

$$(X, Y, Z, T)$$

The extra coordinate $T$ allows for more efficient addition formulas and better reuse of intermediate terms.

Elliptic Curve Equation:

In this form, the twisted Edwards curve becomes:

$$(aX^2 + Y^2)Z^2 = Z^4 + dX^2Y^2$$

However, in practice, point operations are implemented through optimized formulas that directly compute the result without explicitly evaluating this equation.

Affine $\Longrightarrow$ Extended Projective

$$(x, y) \Longrightarrow (X, Y, Z, T) = (x, y, 1, xy)$$

Extended Projective $\Longrightarrow$ Affine

$$(X, Y, Z, T) \Longrightarrow \left( \frac{X}{Z}, \frac{Y}{Z} \right)$$

Calculating Operations

Refer to Hyperelliptic for Extended Projective points on the best practices on how to calculate addition and double operations on Projective coordinates.

Summary of all forms:

Form	Point Form in Affine Form	Storage of values	Elliptic Curve Equation
Affine	$(x, y)$	$(x, y)$	$ax^2 + y^2 = 1 + dx^2y^2$
Extended Affine	$(x, y)$	$(x, y, t)$	$ax^2 + y^2 = 1 + dx^2y^2$
Extended Projective	$\left( \frac{X}{Z}, \frac{Y}{Z}\right)$	$(X, Y, Z, T)$	$(aX^2 + Y^2)Z^2 = Z^4 + dX^2Y^2$

References

• https://www.hyperelliptic.org/EFD/g1p/auto-twisted.html