Cryptology ePrint Archive: Report 2016/328

Constructing genus 3 hyperelliptic Jacobians with CM

Jennifer Balakrishnan and Sorina Ionica and 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.

Category / Keywords: hyperelliptic curve, CM method

Date: received 24 Mar 2016, last revised 27 May 2016

Contact author: sorina ionica at m4x org

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Version: 20160527:093134 (All versions of this report)

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