Isomorphism classes of hyperelliptic curves of genus 2 over F-q

Abstract

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 F-q is on the order of q(3). These results have applications to hyperelliptic curve cryptography.