Counting points on genus3 hyperelliptic curves with explicit real multiplication
Abstract
We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bitoperations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[\eta]$ and on the degrees of polynomials representing the endomorphism $\eta$. As a proofofconcept, we compute the zeta function of a curve defined over a 64bit prime field, with explicit real multiplication by $\mathbb Z[2\cos(2\pi/7)]$.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.05834
 Bibcode:
 2018arXiv180605834A
 Keywords:

 Mathematics  Number Theory;
 Computer Science  Symbolic Computation;
 Mathematics  Algebraic Geometry
 EPrint:
 Proceedings of the ANTSXIII conference (Thirteenth Algorithmic Number Theory Symposium)