A CRT algorithm for constructing genus 2 curves over finite fields

TL;DR

This paper introduces a new CRT-based algorithm for constructing genus 2 curves over finite fields with specified Jacobian point counts, offering an alternative to the traditional CM method with cryptographic applications.

Contribution

It presents a novel CRT algorithm for genus 2 curve construction and an endomorphism ring determination method, expanding tools for cryptographic curve generation.

Findings

Successfully computes Igusa class polynomials modulo small primes

Constructs genus 2 curves with desired Jacobian properties

Generalizes endomorphism ring determination to genus 2 Jacobians

Abstract

We present a new method for constructing genus 2 curves over a finite field with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discrete-log based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves.