On the computation of rational points of a hypersurface over a finite field

Abstract

We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average, less than two searches suffice to obtain a rational point. We also analyze the probability distribution of outputs, using the notion of Shannon entropy, and prove that the algorithm is somewhat close to any "ideal" equidistributed algorithm.