scholarly journals The 2-adic valuation of the cardinality of Jacobians of genus 2 curves over quadratic towers of finite fields

2019 ◽  
Vol 18 (07) ◽  
pp. 1950135
Author(s):  
Ricard Garra ◽  
Josep M. Miret ◽  
Jordi Pujolàs ◽  
Nicolas Thériault
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sylow Subgroup ◽  
Quadratic Extensions ◽  
Genus 2 ◽  
Adic Valuation ◽  
Genus 2 Curves

Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd characteristic such that [Formula: see text], we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of [Formula: see text]. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of [Formula: see text].

scholarly journals Split abelian surfaces over finite fields and reductions of genus-2 curves

2017 ◽  
Vol 11 (1) ◽  
pp. 39-76 ◽  
Author(s):  
Jeffrey Achter ◽  
Everett Howe
Keyword(s):  
Finite Fields ◽  
Abelian Surfaces ◽  
Genus 2 ◽  
Genus 2 Curves

scholarly journals Genus-2 curves and Jacobians with a given number of points

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Polynomial Time ◽  
Rational Points ◽  
The Other ◽  
Base Field ◽  
Arbitrary Base ◽  
The Family ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

scholarly journals Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

2009 ◽  
Vol 15 (5) ◽  
pp. 569-579 ◽  
Author(s):  
J.M. Miret ◽  
R. Moreno ◽  
J. Pujolàs ◽  
A. Rio
Keyword(s):  
Sylow Subgroup ◽  
Binary Fields ◽  
Genus 2 ◽  
Genus 2 Curves

scholarly journals Explicit 2-power torsion of genus 2 curves over finite fields

2010 ◽  
Vol 4 (2) ◽  
pp. 155-168 ◽  
Author(s):  
Josep Miret ◽  
Jordi Pujolàs ◽  
Anna Rio
Keyword(s):  
Finite Fields ◽  
Genus 2 ◽  
Curves Over Finite Fields ◽  
Genus 2 Curves

scholarly journals EMBEDDING FINITE FIELDS INTO ELLIPTIC CURVES

2019 ◽  
pp. 889
Author(s):  
Amirmehdi Yazdani Kashani ◽  
Hassan Daghigh
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Elliptic Curve Cryptosystems ◽  
General Elliptic ◽  
Genus 2

Many elliptic curve cryptosystems require an encoding function from a finite field Fq into Fq-rational points of an elliptic curve. We propose a uniform encoding to general elliptic curves over Fq. We also discuss about an injective case of SWU encoing for hyperelliptic curves of genus 2. Moreover we discuss about an injective encoding for elliptic curves with a point of order two over a finite field and present a description for these elliptic curves.

scholarly journals Hash functions from superspecial genus-2 curves using Richelot isogenies

2020 ◽  
Vol 14 (1) ◽  
pp. 268-292
Author(s):  
Wouter Castryck ◽  
Thomas Decru ◽  
Benjamin Smith
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Hash Function ◽  
Security Analysis ◽  
Hash Functions ◽  
Square Root ◽  
Square Roots ◽  
Higher Dimensional ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractIn 2018 Takashima proposed a version of Charles, Goren and Lauter’s hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field 𝔽p2. In 2019 Flynn and Ti pointed out that Takashima’s hash function is insecure due to the existence of small isogeny cycles. We revisit the construction and show that it can be repaired by imposing a simple restriction, which moreover clarifies the security analysis. The runtime of the resulting hash function is dominated by the extraction of 3 square roots for every block of 3 bits of the message, as compared to one square root per bit in the elliptic curve case; however in our setting the extractions can be parallelized and are done in a finite field whose bit size is reduced by a factor 3. Along the way we argue that the full supersingular isogeny graph is the wrong context in which to study higher-dimensional analogues of Charles, Goren and Lauter’s hash function, and advocate the use of the superspecial subgraph, which is the natural framework in which to view Takashima’s 𝔽p2-friendly starting curve.

The Fixed Point Locus of the Verschiebung on ℳX(2, 0) for Genus-2 Curves X in Charateristic 2

2014 ◽  
Vol 57 (2) ◽  
pp. 439-448
Author(s):  
YanHong Yang
Keyword(s):  
Fixed Point ◽  
Finite Field ◽  
Fundamental Group ◽  
Genus 2 ◽  
Characteristic 2 ◽  
Genus 2 Curves

Abstract.We prove that for every ordinary genus-2 curve X over a finite field κ of characteristic 2 with Aut(X/κ) = ℤ/2ℤ × S3 there exist SL(2; κ[[s]])-representations of π1(X) such that the image of π1(X̄) is infinite. This result produces a family of examples similar to Y. Laszlo’s counterexample to A. J. de Jong’s question regarding the finiteness of the geometric monodromy of representations of the fundamental group.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

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