Constructing genus 3 hyperelliptic Jacobians with CM

Jennifer Balakrishnan; Sorina Ionica; Kristin Lauter; Christelle Vincent

Paper 2016/328

Constructing genus 3 hyperelliptic Jacobians with CM

Jennifer Balakrishnan, Sorina Ionica, Kristin Lauter, and Christelle Vincent

Abstract

Given a sextic CM field K, we give an explicit method for finding all genus 3 hyperelliptic curves defined over the complex whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field K, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field F_p with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo p.

Metadata

Available format(s): PDF
Publication info: Preprint. MINOR revision.
Keywords: hyperelliptic curve CM method
Contact author(s): sorina ionica @ m4x org
History: 2016-05-27: revised; 2016-03-25: received; See all versions
Short URL: https://ia.cr/2016/328
License: CC BY

BibTeX

@misc{cryptoeprint:2016/328,
      author = {Jennifer Balakrishnan and Sorina Ionica and Kristin Lauter and Christelle Vincent},
      title = {Constructing genus 3 hyperelliptic Jacobians with CM},
      howpublished = {Cryptology ePrint Archive, Paper 2016/328},
      year = {2016},
      note = {\url{https://eprint.iacr.org/2016/328}},
      url = {https://eprint.iacr.org/2016/328}
}