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.

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 On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

Linearized systems and a generalized Cayley–Hamilton theorem

2017 ◽  
Vol 16 (06) ◽  
pp. 1750120
Author(s):  
Jeffrey Lang ◽  
Daniel Newland
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Minimum Distance ◽  
Solution Space ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Systems Of Equations ◽  
Square Matrix

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

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 The Characteristic Polynomials of Symmetric Graphs

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 582
Author(s):  
Nafaa Chbili ◽  
Shamma Al Dhaheri ◽  
Mei Tahnon ◽  
Amna Abunamous
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Finite Graph ◽  
Characteristic Polynomials ◽  
Induced Subgraph ◽  
P Elements ◽  
Fixed Set ◽  
Algebraic Invariants

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ .

scholarly journals Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

2009 ◽  
Vol 146 (1) ◽  
pp. 81-101 ◽  
Author(s):  
Dmitry Faifman ◽  
Zeév Rudnick
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Finite Field ◽  
Central Limit ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Distribution Of Zeros ◽  
Expected Number ◽  
Large Genus

AbstractWe study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

scholarly journals Moments of zeta functions associated to hyperelliptic curves over finite fields

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140307 ◽  
Author(s):  
Michael O. Rubinstein ◽  
Kaiyu Wu
Keyword(s):  
Asymptotic Formula ◽  
Finite Fields ◽  
Remainder Term ◽  
Prime Power ◽  
Zeta Functions ◽  
Number Fields ◽  
Hyperelliptic Curves ◽  
Central Point ◽  
Characteristic Polynomials ◽  
Curves Over Finite Fields

Let q be an odd prime power, and denote the set of square-free monic polynomials D ( x )∈ F q [ x ] of degree d . Katz and Sarnak showed that the moments, over , of the zeta functions associated to the curves y 2 = D ( x ), evaluated at the central point, tend, as , to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp (2⌊( d −1)/2⌋). Using techniques that were originally developed for studying moments of L -functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and . We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

scholarly journals Cryptographic aspects of real hyperelliptic curves

2010 ◽  
Vol 47 (1) ◽  
pp. 31-65 ◽  
Author(s):  
Michael J. Jacobson ◽  
Renate Scheidler ◽  
Andreas Stein
Keyword(s):  
Hyperelliptic Curve ◽  
Discrete Logarithm ◽  
Mathematical Structure ◽  
Hyperelliptic Curves ◽  
Cryptographic Protocols ◽  
Discrete Logarithm Problem ◽  
Explicit Formulas ◽  
Genus 2

Abstract In this paper, we give an overview of cryptographic applications using real hyperelliptic curves. We review previously proposed cryptographic protocols and discuss the infrastructure of a real hyperelliptic curve, the mathematical structure underlying all these protocols. We then describe recent improvements to infrastructure arithmetic, including explicit formulas for divisor arithmetic in genus 2, and advances in solving the infrastructure discrete logarithm problem, whose presumed intractability is the basis of security for the related cryptographic protocols.

scholarly journals Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

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