Are there any clever (fast) methods for adding the basepoint (generator) to an arbitrary point on elliptic curve, finally ending in affine coordinates? I.e. if G is the generator for a group on the curve (e.g. 25519) and P is an arbitrary point, is there a faster-than-naive way to find P + G in affine coordinates?

I'm interested in quickly enumerating the points on a curve, starting with G, then 2G, 3G, etc, with final values in affine coordinates.

My current solution starts with G and P in Edwards coordinates and then converts G + P to affine coordinates (which is slow). But I figure there might be some magic that 1) takes advantage of G being known or 2) uses the work previously done when computing the sequence G, 2G, 3G, etc.

Thank you!

EDIT: To be clear, I'm not interested in scalar multiplication of an elliptic curve point by a number (I don't want to compute nG for arbitrary n). Instead I want to "increment" a point P by adding it with the generator G.



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