I'm working with the affine representations of points of the Secp256k1 elliptic curve (from Bitcoin).
I've read many papers that show that computing some functions, like $f(P)=3P$ can be computed faster than the standard way. Other papers say that with some pre-computation, the field inversion can be amortized if $F^1(P) ... F^k(P)$ must be computed.
I need the fastest function $F(P)$ that, when applied to the last result iteratively, generates a sequences of points whose average period is large (I don't need any proof, it can be just large in practice). To be fast I suppose it should be computed without field inversions. I don't mind to pre-compute some values.
For example, it could be $F(P) = 1.5P+4Q$ for a fixed $Q$. It doesn't matter which function it is, because I need it to generate random points in the curve. The probability distribution doesn't matter either. (notation: $1.5$ is the point halving of $3P$)
Motivation: Solutions to this problem may be helpful for generating vanity addresses.