Constructing Picard curves with complex multiplication using the Chinese Remainder Theorem

TL;DR

This paper presents a novel algorithm for constructing Picard curves with complex multiplication over finite fields, leveraging class polynomials and the Chinese Remainder Theorem to facilitate cryptographic applications.

Contribution

It introduces a new method for constructing Picard curves with specified endomorphism rings using class polynomials and the Chinese Remainder Theorem, with practical examples.

Findings

Successfully constructed Picard curves with complex multiplication.

Demonstrated the effectiveness of the algorithm for cryptographic curve generation.

Provided explicit examples illustrating the method.

Abstract

We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.