Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Fq

We present a simple and direct method of counting the number of the isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic different from 5. In this case it turns out that the number of isomorphism classes of genus-2 hyperelliptic curve over a given field \( \mathbb{F}_q \) is on the order of q 3. These results have applications to hyperelliptic curve cryptography.

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