The number of the isomorphism classes of hyperelliptic curves of genus four over finite fields

Hyperelliptic curves of genus ≤ 3 over finite fields have been researched and recommended for cryptography for about twenty years. Though the hyperelliptic curves over finite fields of genus four may been not secure for general cryptographic applications, such as digital signature systems, but some special hyperelliptic curves of genus four may have some privileges when they are applied in pairing-based cryptosystems. Hyperelliptic curve classification based on isomorphism is helpful for the selection of secure hyperelliptic curves over finite fields for practical cryptosystems. The isomorphism classes of hypereilliptic curves of genus 2 or 3 over finite fields have been studied in previous works. Here, the number of isomorphism classes of hyperelliptic curves of genus 4 over finite fields with the characteristics larger than three is given.