Spiritual

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

I played around in Sage with generating some curves and then looking at the point count for for , relating it back to the base curve . I found that there was definitely a pattern, but I couldn’t quite pinpoint it1. At this point I turned to Google and found some paper about point counting that mentioned Lucas sequences and slapped myself. It was something like2:

  1. 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.

  2. modulo any mistakes, I’m on the train without internet, so writing from memory currently.