**Counting hyperelliptic curves that admit a Koblitz model**

*Cevahir Demirkiran and Enric Nart*

**Abstract: **Let $k=\mathbb{F}_q$ 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) of 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 $(3641/2880)q^5+O(q^4)$.

**Category / Keywords: **public-key cryptography / hyperelliptic cryptosystems

**Date: **received 10 May 2007

**Contact author: **nart at mat uab cat

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20070520:125411 (All versions of this report)

**Short URL: **ia.cr/2007/174

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