Counting points on genus-3 hyperelliptic curves with explicit real multiplication

@article{Abelard2018CountingPO,
  title={Counting points on genus-3 hyperelliptic curves with explicit real multiplication},
  author={Simon Abelard and Pierrick Gaudry and Pierre-Jean Spaenlehauer},
  journal={ArXiv},
  year={2018},
  volume={abs/1806.05834}
}
We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $Fq$, with explicit real multiplication by an order $Z[η]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $O((log q) 6)$ bit-operations, where the constant in the $O()$ depends on the ring $Z[η]$ and on the degrees of polynomials representing the endomorphism $η$. As a proof-of-concept, we compute the… Expand
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