The Numbers Of The Isomorphism Classes Of Two Classes Of Hyperelliptic Curves

Since elliptic curve cryptosystems(ECC) were proposed in around 1985, ECC has been studied for about twenty years and has been recently applied in many commercial purposes. In 1989, Koblitz generalized the concept of elliptic curve cryptosystems to hyperelliptic curves of higher genus. Using the Jacobian of a hyperelliptic curve defined over a finite field instead of an elliptic curve over a finite field, we can further reduce the key size while maintaining the same level of security. One can use a hyperelliptic curve of genus 2 over a finite field Fq , where q≈2⁸⁰, and achieve the same level of security as when an elliptic curve group E ( F_q ) is used, where q≈2¹⁶⁰ or a group F_q with q≈2¹⁰²⁴ is used.Hyperelliptic curves over finite fields have been studied for cryptography for twenty years and the hyperelliptic curves of genus larger than 4 have been proved unsafe and unsuitable for practical cryptosystems. For security considerations, hyperelliptic curves used for cryptosystems should be of genus no large than 4. In order to know how many suitable choices of curves for cryptographic purposes there are, it may be useful to classify the isomorphism classes of hyperelliptic curves over finite fields. The number of isomorphism classes of hyperelliptic curves of genus 2 or 3 over F_q had been studied in previous works.In this paper, the number of isomorphism classes of hyperelliptic curves of genus 4 over a finite field with characteristic different from 2, 3 is given, what's more, we also give the number of the isomorphism classes of Picard curves defined over finite fields F_pⁿ(p≠2,3), and these classifications are of interest in designing and implementing hyperelliptic curve cryptosystems.