|
|||||
 |
|
|
|
|
|
|
|
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 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 |
|
