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

  title={Counting points on genus-3 hyperelliptic curves with explicit real multiplication},
  author={Simon Abelard and Pierrick Gaudry and Pierre-Jean Spaenlehauer},
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
Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus
  • S. Abelard
  • Computer Science, Mathematics
  • J. Complex.
  • 2020
We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an orderExpand
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  • Jonathan Lee
  • Computer Science, Mathematics
  • IACR Cryptol. ePrint Arch.
  • 2020
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