Abstract

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 $\tilde{O}\left(\right)$ 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 $\mathbb{Z}\left[2cos\left(2\pi ∕7\right)\right]$.

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Mathematical Subject Classification 2010

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