Paper 2007/174
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 $(3641/2880)q^5+O(q^4)$.
Metadata
- Available format(s)
-
PDF PS
- Category
- Public-key cryptography
- Publication info
- Published elsewhere. Unknown where it was published
- Keywords
- hyperelliptic cryptosystems
- Contact author(s)
- nart @ mat uab cat
- History
- 2007-05-20: received
- Short URL
- https://ia.cr/2007/174
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2007/174, author = {Cevahir Demirkiran and Enric Nart}, title = {Counting hyperelliptic curves that admit a Koblitz model}, howpublished = {Cryptology {ePrint} Archive, Paper 2007/174}, year = {2007}, url = {https://eprint.iacr.org/2007/174} }