scholarly journals Spanning the isogeny class of a power of an elliptic curve

2021 ◽  
Vol 91 (333) ◽  
pp. 401-449
Author(s):  
Markus Kirschmer ◽  
Fabien Narbonne ◽  
Christophe Ritzenthaler ◽  
Damien Robert
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Finite Fields ◽  
Modular Forms ◽  
Null Point ◽  
Siegel Modular Forms ◽  
Content Type ◽  
Careful Choice ◽  
Isomorphism Classes

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

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.

scholarly journals Finding roots in with the successive resultants algorithm

2014 ◽  
Vol 17 (A) ◽  
pp. 203-217 ◽  
Author(s):  
Christophe Petit
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Finite Fields ◽  
Coding Theory ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Polynomial Equations ◽  
Preliminary Results ◽  
Characteristic Case ◽  
Practical Security

AbstractThe problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.Supplementary materials are available with this article.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

THE ITERATION DIGRAPHS OF GROUP RINGS OVER FINITE FIELDS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350162 ◽  
Author(s):  
YANGJIANG WEI ◽  
GAOHUA TANG ◽  
JIZHU NAN
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Directed Edge ◽  
Cycle Structure ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cn over a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCn and k such that G(𝔽prCn, k) is semiregular.

scholarly journals Elliptic curves with a given number of points over finite fields

2012 ◽  
Vol 149 (2) ◽  
pp. 175-203 ◽  
Author(s):  
Chantal David ◽  
Ethan Smith
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Distribution Of Primes ◽  
Primes In Arithmetic Progressions ◽  
Interval Distribution ◽  
Better Than

AbstractGiven an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

scholarly journals Counting Quiver Representations over Finite Fields Via Graph Enumeration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Geir Helleloid ◽  
Fernando Rodriguez-Villegas
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Compte Rendu ◽  
Isomorphism Classes ◽  
International Audience ◽  
Derivatives Of ◽  
Nous Utilisons ◽  
Complete Account

International audience Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication. Soit $\Gamma$ un carquois sur $n$ sommets $ v_1, v_2, \ldots , v_n$ avec $g_{ij}$ arêtes entre $v_i$ et $v_j$, et soit $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua a donné une formule pour $A_{\Gamma}(\boldsymbol{\alpha}, q)$, le nombre de classes d'isomorphisme absolument indécomposables de représentations de $\Gamma$ sur le corps fini $\mathbb{F}_q$ avec vecteur de dimension $\boldsymbol{\alpha}$. Nous utilisons la formule de Hua pour montrer que les dérivées de $A_{\Gamma}(\boldsymbol{\alpha}, q)$ par rapport à $q$, alors évaluée à $q=1$, sont des polynômes dans les variables $g_{ij}$, et on peut calculer les termes de plus haut degré de ces polynômes. Les formules pour ces coefficients dépendent de l'énumération de certaines familles de graphes connectés. Cette note donne simplement un aperçu de ces résultats, un compte rendu complet de cette recherche est disponible sur arXiv et a été soumis pour publication.

scholarly journals The existence of primitive normal elements of quadratic forms over finite fields

2020 ◽  
pp. 2250068
Author(s):  
Himangshu Hazarika ◽  
Dhiren Kumar Basnet ◽  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Primitive Element ◽  
Sufficient Condition ◽  
Extension Field ◽  
Normal Element

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

Linear Recurring Sequences and Explicit Factors of x2nd−1 in 𝔽q[x]

2020 ◽  
Vol 27 (03) ◽  
pp. 563-574
Author(s):  
Manjit Singh
Keyword(s):  
Field Theory ◽  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Second Order ◽  
Order Linear ◽  
Linear Recurring Sequences

Let 𝔽q be a finite field of odd characteristic containing q elements, and n be a positive integer. An important problem in finite field theory is to factorize xn − 1 into the product of irreducible factors over a finite field. Beyond the realm of theoretical needs, the availability of coefficients of irreducible factors over finite fields is also very important for applications. In this paper, we introduce second order linear recurring sequences in 𝔽q and reformulate the explicit factorization of [Formula: see text] over 𝔽q in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q + 1.

scholarly journals A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

2015 ◽  
Vol 11 (08) ◽  
pp. 2431-2450 ◽  
Author(s):  
Dermot McCarthy ◽  
Matthew A. Papanikolas
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Modular Forms ◽  
Hypergeometric Functions ◽  
Siegel Modular Forms ◽  
Hecke Operator ◽  
Special Value ◽  
Forms Of Higher Degree

Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function.

scholarly journals Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

2021 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
A. Boulbot ◽  
Abdelhakim Chillali ◽  
A Mouhib
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Finite Ring ◽  
Idempotent Element ◽  
Characteristic 3 ◽  
Specific Equation ◽  
Weierstrass Equation

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}).  In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

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