# Spiritual

I don’t have the exact text of the task, but you connect to a service and are told $∣E(Z_{p})∣$ for some given $p$. The curve itself is unknown (i.e. you don’t get the parameters $a,b$). The goal is to find out how many points is in the corresponding curve over $Z_{p_{k}}$ for a given $k$.

I played around in Sage with generating some curves and then looking at the
point count for $Z_{k}$ for $k=1,2,3,4,⋯$, relating it back
to the base curve $k=1$. I found that there was definitely a pattern, but I
couldn’t quite pinpoint it^{1}. At this point I turned to Google and found some
paper about point counting that mentioned Lucas sequences and slapped myself. It was
something like^{2}:
$∣E(F_{p_{k}})∣=p_{k}+1−V_{k}(p+1−∣E(F_{p})∣,p)$

Yeah, yeah, even though I solved LagLeg prior to this and

*was*able to figure it out there. In one ear, out the other, I don’t have much to say in my defense. Plus I confused myself by looking at direct polynomial relationships and patterns in their factors, without taking away the first and last terms.↩modulo any mistakes, I’m on the train without internet, so writing from memory currently.↩