How are the unified addition formulae in extended twisted Edwards coordinates derived from the affine addition formulae?

Question

I am trying to understand point addition in RFC8032 section 5.1.4, which references the paper "Twisted Edwards Curves Revisited" (https://eprint.iacr.org/2008/522.pdf) to describe the quick way they perform twisted Edwards curve addition with fewer inversions.

I am reading through the paper and it makes sense until I get to the statement in section 3.1 under figure (5), which says that the unified addition formulae for extended coordinates are somehow derived from the affine addition formulae in figure (1). I cannot for the life of me understand how the formula $$ X_3\:=\:\left(X_1Y_2\:+\:Y_1X_2\right)\left(Z_1Z_2-dT_1T_2\right)$$

can be derived from the formula $$X_3\:=\:\frac{\left(X_1Y_2\:+\:Y_1X_2\right)}{\left(1+dT_1T_2\right)}$$ even with the extended coordinates, and dividing every X1 or Y1 by Z1. Can anyone explain their derivation to me?

Answer 1

To convert extended twisted Edwards coordinates to extended projective coordinates one needs to divide

$$X_3 = (X_1Y_2 + Y_1X_2)(Z_1Z_2 − d T_1T_2)$$

by

$$Z3 = (Z_1Z_2 − d T_1T_2)(Z_1Z_2 + d T_1T_2)$$

\begin{align}\frac{X_3}{Z_3} & = \frac{(X_1Y_2 + Y_1X_2)\color{red}{(Z_1Z_2 − d T_1T_2)}}{\color{red}{(Z_1Z_2 − d T_1T_2)}(Z_1Z_2 + d T_1T_2)} & \text{//cancels}\\[1.5ex] & = \frac{(X_1Y_2 + Y_1X_2)}{(Z_1Z_2 + d T_1T_2)}\\[1.5ex] & = \frac{\frac{(X_1Y_2 + Y_1X_2)}{\color{blue}{Z_1Z_2}}}{\frac{(Z_1Z_2 + d T_1T_2)}{\color{blue}{Z_1Z_2}}} & \text{//divide by } Z_1Z_2 \tag{1}\label{lab1}\\[1.5ex] & = \frac{\frac{\color{green}{X_1}Y_2}{\color{green}{Z_1}Z_2} + \frac{Y_1X_2}{Z_1Z_2} } {\frac{Z_1Z_2}{Z_1Z_2} + \frac{d T_1T_2}{Z_1Z_2} } & { use } X/Z \to Z, Y/Z \to Z\\[1.5ex] & = \frac{(X_1Y_2 + Y_1X_2)}{(1 + d T_1T_2)} \end{align}

Step $\ref{lab1}$ is the confusing step, the coordinates are still in the extended twisted Edwards coordinates need to be translated.