Isomorphism classes of genus-2 hyperelliptic curves over finite fields

Abstract

We propose a reduced equation for hyperelliptic curves of genus $2$ over finite fields $\fq$ of $q$ elements with characteristic different from $2$ and $5$. We determine the number of isomorphism classes of genus-2 hyperelliptic curves having an $\fq$-rational Weierstrass point. These results have applications to hyperelliptic curve cryptography.