Montgomery Ladder with affin/projective Coordinates
Maybe one day someone finds this Post again and has the same questions. To you: I hope you're having a great day!


Questions: The projective Arithmetic is faster, because there are only multiplications, squarings, additions to do. The affine Artihmetic is slower, because it takes a lot of time to compute the division. Especially for the big numbers used in modern cryptographie. BUT: That is not the answer to the constant time ladder. Yes affin coordinates with division can't be computed with constant time, but the reason for the projective ladder is a different. You can tranform different types of elliptic curve into the projective form, but for them you can't always use the montgomery ladder. For a long time ( at least when montgomery published his work ) elliptic curve scalar multiplikation on weierstraß was only possible with double and add algorithms. Those algorithms tranformed the scalar into the binary form and had different operations for 1 and 0. Therefor one could approximate how many 1 and 0 there are. The Montgomery ladder has the same operations for both. So there is no difference in time.
Question: With projective coordinates it is only possible to add/double multiples of the same point. The montgomery ladder starts with a given scalar $n$ and Point $P$. In each step two results are calculates $R0$ and $R1$. The important point here is, that those results are either of the form $(n')R0$, $(n'+1)R1$ or $(n'+1)R0$, $(n')R1$. That means the difference between them is always 1. (When you look at the definition for the projective Arithmetic it is clear what that means). For the ladder this means, that $x1$ is always the x coordinate of the starting Point $P$. Therefore it is always the same and must not be computed!


Note: I still can't say, whether my pseudocode is correct or not.