Geometric primality tests using curves of genus 1 & 2

TL;DR

This paper explores geometric methods for deterministic primality testing using genus 1 and genus 2 curves over finite rings, introducing new tests based on the Jacobian of genus 2 curves.

Contribution

It generalizes existing primality tests with geometric techniques and develops a novel test using genus 2 Jacobians, expanding the toolkit for primality verification.

Findings

Generalized primality tests for integers using genus 1 curves.

Developed a new primality test based on genus 2 Jacobians.

Enhanced understanding of geometric approaches in primality testing.

Abstract

We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.