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

TL;DR

This paper presents an algorithm for efficiently finding rational points on hypersurfaces over finite fields, demonstrating that typically fewer than two searches are needed, and analyzing its probabilistic properties.

Contribution

The paper introduces a novel search-based algorithm for rational points on hypersurfaces over finite fields, with average search counts and entropy analysis.

Findings

Less than two searches on average to find a rational point

Algorithm's output distribution is close to ideal entropy-based distribution

Provides probabilistic analysis of the algorithm's efficiency

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.