Counting hyperelliptic curves that admit a Koblitz model

Cevahir Demirkiran; Enric Nart

Paper 2007/174

Counting hyperelliptic curves that admit a Koblitz model

Cevahir Demirkiran and Enric Nart

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) of cryptographic applications is asymptotically $(1 - e^{- 1}) 2 q^{2 g - 1}$ , and not $2 q^{2 g - 1}$ 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}
}