Trisection for non-supersingular genus 2 curves in characteristic 2

Abstract

We study division by 3 in Jacobians of genus 2 curves over binary fields with a 2-torsion subgroup of rank 1 or 2. We characterize the 3-torsion divisors and provide, for every $D\in \mathrm{Jac}\left(\mathrm{C}\right)\left({\mathbb{F}}_{2^m}\right)$, a formula for the coordinates of the divisors in the set $\{D^{\prime }\in \mathrm{Jac}(\mathrm{C}\left)\right({\mathbb{F}}_{2^m})\mid 3D^{\prime }=D\}$.

Keywords:

• genus 2 curves
• binary fields
• Jacobian
• trisection
• cryptography

2010 AMS Subject Classifications:

• 11G20
• 14H40
• 14H45

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jordi Pujolàs http://orcid.org/0000-0002-1192-8192

Additional information

Funding

Research of the authors was supported in part by grants [MTM2013-46949-P] (Spanish Ministerio de Ciencia e Innovación), [2014SGR-1666] (Generalitat de Catalunya), FONDECYT [1110578] and Anillo [ACT56] (Chile), CONICYT/Chile doctoral scholarship.