### Abstract

Let k = F q be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in q with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q - 1 and q + 1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) for cryptographic applications is asymptotically (1 - e -1)2q 2g-1, and not 2q 2g-1 as it was believed. The curves of genus g = 2 and g = 3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q 3 + O(q 2) and (3641/2880)q 5 + O(q 4). © 2008 de Gruyter.

Original language | English |
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Pages (from-to) | 163-179 |

Journal | Journal of Mathematical Cryptology |

Volume | 2 |

DOIs | |

Publication status | Published - 1 Jul 2008 |

### Keywords

- Finite field
- Hyperelliptic cryptosystem
- Hyperelliptic curve
- Koblitz model
- Rational n-set
- Weierstrass point

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## Cite this

Demirkiran, C., & Nart, E. (2008). Counting hyperelliptic curves that admit a Koblitz model.

*Journal of Mathematical Cryptology*,*2*, 163-179. https://doi.org/10.1515/JMC.2008.008