Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.1423

Counting hyperelliptic curves that admit a Koblitz model

Cevahir Demirkiran·Enric Nart
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let k 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 the cardinality q of k, 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).

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cevahir Demirkiran, Enric Nart. 2007-05-10. Counting hyperelliptic curves that admit a Koblitz model. https://arxiv.org/abs/0705.1423

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Series expansions by generalized Bessel functions for functions related to the lattice point problems for the $p$-circle

The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term using the generalized Bessel functions by E. Krätzel and the results of G. Kuba. On the other hand, for the cases $0<p<2$, the method via this series representation cannot make progress. Therefore, in such cases, it is necessary to consider another method. In this paper, we prove that certain functions closely related to the problems can be displayed as series by newly generalized Bessel functions based on the property $p$-radial, generalization of spherical symmetry, and highlight the possibility that attempts to solve the problems via this display are suitable especially for the cases $0<p\leq1$. This study is based on the harmonic-analytic method by S. Kuratsubo and E. Nakai, using certain functions generalizing the error term of the circle problem by variables and series representation of the functions by the Bessel functions.

math.NT↗
arXiv2026

Density results for $r$-gaps between zeros of the Riemann zeta-function

Let $0<γ_1\leq γ_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $Θ, \vartheta>0$ such that for each $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\geq 1+\fracΘ{r^α} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\leq 1-\frac{\vartheta}{r^α} \] where $α$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh.

math.NT↗
arXiv2026

Stevens' Approach to Kloosterman Sums on Split Groups

We develop a uniform framework for Stevens' approach to Kloosterman sums on split groups over $p$-adic fields. The main result reduces the bounding of Kloosterman sums to that for the so-called Plücker-constrained sets, which are certain subsets of affine spaces constrained by conditions giving by the Plücker coordinates. We then give a nontrivial bound for the Kloosterman sums on the exceptional group $G_2$. As an application, we obtain the regularity of Bessel distributions associated with generic representations on $G_2$.

math.NT↗