Section: New Results

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

Participants : Simon Abelard, Pierrick Gaudry [contact] , Pierre-Jean Spaenlehauer [contact] .

In [9], we proposed a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field ${𝔽}_q$, with explicit real multiplication by an order $\mathbb{Z}\left[\eta \right]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $O\left({(logq)}^6\right)$ bit-operations, where the constant in the $O\left(\right)$ depends on the ring $\mathbb{Z}\left[\eta \right]$ and on the degrees of polynomials representing the endomorphism $\eta$. As a proof-of-concept, we computed the zeta function of a curve defined over a 64-bit prime field, with explicit real multiplication by $\mathbb{Z}[2cos(2\pi /7\left)\right]$.