We propose a Las Vegas probabilistic algorithm to compute the zeta function of a
genus-$3$
hyperelliptic curve defined over a finite field
${\mathbb{F}}_{q}$, 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
$\tilde{O}\left({\left(logq\right)}^{6}\right)$ bit-operations, where
the constant in the
$\tilde{O}\left(\phantom{\rule{2.77626pt}{0ex}}\right)$
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 compute the zeta function of a curve
defined over a 64-bit prime field, with explicit real multiplication by
$\mathbb{Z}\left[2cos\left(2\pi \u22157\right)\right]$.

Keywords

point-counting, hyperelliptic curves, Schoof's algorithm,
real multiplication