scholarly journals Hyperelliptic Curves for the Vector Decomposition Problem over Fields of Even Characteristic

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Seungkook Park
Keyword(s):  
Finite Field ◽  
Infinite Family ◽  
Hyperelliptic Curves ◽  
Decomposition Problem ◽  
Vector Decomposition ◽  
Genus Two

We present an infinite family of hyperelliptic curves of genus two over a finite field of even characteristic which are suitable for the vector decomposition problem.

scholarly journals Hyperelliptic classes are rigid and extremal in genus two

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Vance Blankers
Keyword(s):  
Infinite Family ◽  
Hyperelliptic Curves ◽  
Weierstrass Points ◽  
High Codimension ◽  
Genus Two ◽  
Marked Points

We show that the class of the locus of hyperelliptic curves with $\ell$ marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free marked points is rigid and extremal in the cone of effective codimension-($\ell + m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension. Comment: Published version

scholarly journals The point decomposition problem over hyperelliptic curves

2017 ◽  
Vol 86 (10) ◽  
pp. 2279-2314
Author(s):  
Jean-Charles Faugère ◽  
Alexandre Wallet
Keyword(s):  
Hyperelliptic Curves ◽  
Decomposition Problem

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

2013 ◽  
Vol 12 (3) ◽  
pp. 651-676 ◽  
Author(s):  
Bryden Cais ◽  
Jordan S. Ellenberg ◽  
David Zureick-Brown
Keyword(s):  
Probability Distribution ◽  
Finite Field ◽  
Random Matrix ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Plane Curves ◽  
Random Curves ◽  
Isomorphism Classes ◽  
Curves Over Finite Fields ◽  
Divisible Groups

AbstractWe describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

scholarly journals Traces, high powers and one level density for families of curves over finite fields

2017 ◽  
Vol 165 (2) ◽  
pp. 225-248 ◽  
Author(s):  
ALINA BUCUR ◽  
EDGAR COSTA ◽  
CHANTAL DAVID ◽  
JOÃO GUERREIRO ◽  
DAVID LOWRY–DUDA
Keyword(s):  
Finite Field ◽  
Moduli Spaces ◽  
Function Field ◽  
Level Density ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Unitary Matrix ◽  
Moduli Spaces Of Curves ◽  
A New Technique ◽  
The One

AbstractThe zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

scholarly journals Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

2017 ◽  
pp. 63-94 ◽  
Author(s):  
Sean Ballentine ◽  
Aurore Guillevic ◽  
Elisa Lorenzo García ◽  
Chloe Martindale ◽  
Maike Massierer ◽  
...  
Keyword(s):  
Hyperelliptic Curves ◽  
Point Counting ◽  
Real Multiplication ◽  
Genus Two
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