Vol. 2, 2019

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Constructing Picard curves with complex multiplication using the Chinese remainder theorem

Sonny Arora and Kirsten Eisenträger

Vol. 2 (2019), No. 1, 21–36
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 one to work over smaller fields than the elliptic curve case. 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.

Keywords
Picard curves, Chinese remainder theorem, number theory, arithmetic geometry, constructing curves, genus 3
Mathematical Subject Classification 2010
Primary: 11G10, 11G15
Secondary: 14K22
Milestones
Received: 22 February 2018
Revised: 4 June 2018
Accepted: 10 September 2018
Published: 13 February 2019
Authors
Sonny Arora
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802
United States
Kirsten Eisenträger
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802
United States