Elliptic Scheme Research Articles

Monodromy of double elliptic logarithms

We determine the relative monodromy group of abelian logarithms with respect to periods in the cases of fibered products of elliptic schemes. This gives rise to a result stronger than a theorem due to Y. André and implies in particular the algebraic independence of the logarithm of any non-torsion section and the periods. We then conjecture an analogous result for the general case of an abelian scheme of arbitrary relative dimension. This generalizes a theorem of Corvaja and Zannier which determines the said group in the case of a single elliptic scheme.

A Lightweight Attribute-Based Security Scheme for Fog-Enabled Cyber Physical Systems

In this paper, a lightweight attribute-based security scheme based on elliptic curve cryptography (ECC) is proposed for fog-enabled cyber physical systems (Fog-CPS). A novel aspect of the proposed scheme is that the communication between Fog-CPS entities is secure even when the certification authority (CA) is compromised. This is achieved by dividing the attributes into two sets, namely, secret and shared, and subsequently generating two key pairs, referred to as the partial and final key pairs, for each entity of the Fog-CPS system. Unlike existing attribute-based encryption (ABE) and identity-based encryption schemes, in the proposed scheme, each entity calculates the final public key of the communicating CPS devices without the need of generating and transmitting digital certificates. Moreover, the proposed security scheme considers an efficient and secure key pair update approach in which the calculation overhead is limited to one group element. To show the effectiveness of the proposed scheme, we have calculated and compared the memory and processing complexity with other bilinear and elliptic curve schemes. We have also implemented our scheme in a Raspberry Pi (3B+ model) for CPS simulations. The proposed scheme guarantees the confidentiality, integrity, privacy, and authenticity in Fog-CPS systems.

Unlikely intersections with isogeny orbits in a product of elliptic schemes

Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-R\'emond inequality with the Pila-Zannier strategy.

Cyclotomic torsion points in elliptic schemes

An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin–Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.

Kleptographic Attack on Elliptic Curve Based Cryptographic Protocols

Kleptography is the study of pilfering secure data secretly and subliminally. The concept of inserting backdoors was introduced two decades ago by Young and Yung. However, still it is a serious threat for modern cryptography. Different studies have proved that exploiting implementation errors of cryptographic algorithms needs less effort as compared to attacking its mathematical structure. Inserting the backdoor modifies the standard method of generating public and private key pairs in such away that the public information is meaningful for the attacker. This paper presents the kleptographic attack on cryptographic algorithm based on Elliptic curves. We show the technique of implementing backdoor against Edwards-curve Digital Signature Algorithm, Elliptic curve Diffie-Hellman key exchange scheme, Elliptic curve Digital Signature Algorithm, Elliptic curve Integrated Encryption Scheme, Elliptic curve ElGamal Encryption and Elliptic curve Qu-Vanstone implicit certificate scheme. In practical approach, backdoors are inserted in such a way that their identification is impossible by analyzing the output of an algorithm. Detection of kleptographic implementation is a complex task and very few studies can be found in this direction. We have explored the possibility of detection of malicious code inside the Elliptic curve based algorithms by using the idea of running time analysis. We have shown that by implementing strong Secretly Embedded Trapdoor with Universal Protection (SETUP) attack against Elliptic curve based algorithms, one can detect the presence of backdoor by analyzing the time difference in execution of an honest vs malicious version of code. We also modified the backdoor insertion mechanism in Edwards-curve Digital Signature Algorithm that results in negligible time difference making it impossible for the user to detect backdoor presence.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.

Comparative Analysis of Surface Plants of the Ground Sealing Composition in the Implementation of Construction Works

Processes of layer-by-layer consolidation of soil at construction of the bulk engineering constructions built by method of continuous consolidation are considered. Ways of realization of these processes on the basis of two technological schemes - elliptic and shuttle are shown. The assessment of key parameters is made during the operation of units on the classical elliptic scheme of continuous consolidation of soil and on the elliptic scheme with use of a ring method of a turn of the unit when in the course of construction of a road bed automobile and the railroads the movement of the condensing unit is made from roadsides to an axis. It is shown that working platform* more effective andleast critical to topographical and geometrical parameters, the shuttle scheme of consolidation at which the condensing cars make the back and forth motions accompanied with lateral shift on an adjacent working strip after achievement of the demanded density on the previous strip is. The analsis of the major factors proving application of this or that technological scheme of soil consolidation is made. Operating conditions of units depending on type of the used condensing unit and the accepted production technology of works on elliptic and shuttle schemes which realization allows to reach the greatest production effect are studied

CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity.Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency.The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model.The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary.

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

In recent papers we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples by Bertrand. Furthermore there are applications to the study of Pell equations over polynomial rings; for example we deduce that there are at most finitely many complex t for which there exist A,B \neq 0 in {\mathbb C}[X] with A^2 – DB^2 = 1 for D = X^6 + X + t . We also consider equations A^2 – DB^2 = c'X + c , where the situation is quite different.

Torsion points on families of products of elliptic curves

In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.

Unlikely Intersections in Poincaré Biextensions over Elliptic Schemes

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincare biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.

Numerical methods for a fluid mixture model

SUMMARY In this paper, we firstly apply generalized difference methods to solve a fluid mixture model. The model is usually used to describe the tissue deformations and contains a nonlinear hyperbolic equation and an elliptic equation. Most people have used finite difference methods for solving the elliptic equation and other schemes for solving the hyperbolic equation. It is well known that the accuracy of traditional finite difference method is not high. This may be a serious disadvantage in the fluid mixture model, which describes cell movements and tissue deformations. The numerical methods we propose to improve accuracy are based on generalized Galerkin methods and dual decomposition. By choosing suitable trial function space and test function space, our generalized upwind difference schemes exhibit second-order convergence in space for smooth problems and can eliminate numerical oscillations for discontinuous problems. Some numerical results are presented to demonstrate the advantages of our methods. Copyright © 2012 John Wiley & Sons, Ltd.

Uniform boundedness of p-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves. More precisely, we prove the following. Denote by Γ k the absolute Galois group of k. For an abelian variety A over k and a character $\chi:\Gamma_{k}\rightarrow\mathbb{Z}_{p}^{\ast}$ , define A[p ∞](χ) to be the module of p-primary torsion of $A(\overline{k})$ on which Γ k acts as χ-multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞](χ) is always finite, but the exponent of A[p ∞](χ) may depend on A, a priori. Our main result is about the uniform boundedness of A[p ∞](χ) when A varies in a 1-dimensional family. More precisely, if S is a curve over k and A is an abelian scheme over S, then there exists an integer N:=N(A,S,k,p,χ), such that A s [p ∞](χ)⊂A s [p N ] holds for any s∈S(k). This arithmetic result is obtained as a corollary of the following geometric result on the p-primary torsion of abelian varieties over function fields of curves, combined with Mordell’s conjecture (Faltings’ theorem). Let K be the function field of a curve over an algebraically closed field of characteristic 0 and A an abelian variety over K. Assume for simplicity that A contains no nontrivial isotrivial subvariety. Then, for any c≥0, there exists an integer N:=N(c,A,S,k,p)≥0 such that A[p ∞](K′)⊂A[p N ] for all finite extension K′/K with K′ of genus ≤c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulo p n of p-adic analytic homogeneous spaces. Our uniform boundedness result when χ is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A s , s∈S(k) alluded to above. When χ is the p-adic cyclotomic character, together with certain descent methods, it also yields a proof of the 1-dimensional case of (a generalized variant of) the modular tower conjecture, which was, actually, the original motivation for this work. This is a conjecture arising from the regular inverse Galois problem, whose original form was posed by M. Fried in the early 1990s.

Torsion points on families of squares of elliptic curves

In a recent paper we proved that there are at most finitely many complex numbers λ ≠ 0,1 such that the points $${(2,\sqrt{2(2-\lambda)})}$$ and $${(3, \sqrt{6(3-\lambda)})}$$ are both torsion on the elliptic curve defined by Y 2 = X(X − 1)(X − λ). Here we give a generalization to any two points with coordinates algebraic over the field Q(λ) and even over C(λ). This implies a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes.

Thermal analysis of a diesel engine operating with diesel–biodiesel blends

In this paper we study the transient heat conduction in a piston of a diesel engine, subjected to a periodic boundary condition on the surface in contact with the combustion gases. The heat transfer coefficient at the top surface was modeled taking into account the temperature and pressure inside the combustion chamber. Such instantaneous pressure was measured using a special probe for an engine operating with several blends of diesel and biodiesel, and the temperature was obtained through a First Law analysis. The physical properties, including the cetane number were evaluated experimentally for all diesel/biodiesel blends used in this work. An elliptic scheme of numerical grid generation was used, so that the irregular shaped piston in the physical domain was transformed into a cylinder in a computational domain. The timewise variations of the temperature of several points in the piston were examined for different piston materials and various load conditions.

Log-elastographic and non-marching full inversion schemes for shear modulus recovery from single frequency elastographic data

The goal of elastography is to image the shear stiffness of tissue for cancer diagnosis and the focus of this paper is on single frequency elastographic data. Assuming that the measured displacement of the propagating shear wave satisfies the acoustic wave equation, the shear modulus μ can be recovered by solving a first-order partial differential equation in the inverse problem. To capture possible exponential growth and decay of the targeted parameter μ numerically in a stable manner, we propose a log-elastographic nonlinear scheme and a linear finite difference based elliptic scheme. Both methods are shown to be convergent at first order and their performances are compared with the performances of a semi-implicit upwind scheme and the direct inversion model previously investigated (see [23]). We present shear modulus reconstructions from synthetic data with and without noise.

Structured mesh generation by kriging with local refinement with a new elliptic scheme

Accurate results in finite element analysis are strongly related to mesh quality. In this paper, an automatic quadrilateral mesh generation methodology using kriging interpolation is described, a quality mesh study is conducted, and the development of a new local refinement scheme, called the elliptic scheme, is presented. The new elliptic refinement scheme is evaluated using four standard structural cases, and it is shown that it compares very well with octree-based refinement schemes and other local refinement methods.

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration $${\cal R}$$ (q) on the set $${\cal L}$$ (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $${\cal R}$$ +(q) and $${\cal R}$$ ?(q) of $${\cal R}$$ (q) to the set $${\cal L}$$ +(q) of secant (hyperbolic) lines and to the set $${\cal L}$$ ?(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $${\cal R}$$ ?(q) is pseudocyclic. We further show that the coherent configurations $${\cal R}$$ (q 2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $${\cal R}$$ +(q 2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $${\cal R}$$ +(q 2) and $${\cal R}$$ ?(q 2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.

An algebraic–elliptic algorithm for boundary orthogonal grid generation

To produce grids conforming to the boundary of a physical domain with boundary orthogonality features, algebraic methods like transfinite interpolating schemes can be profitably used. Moreover, the coupling of Hermite-type (also interpolating prescribed boundary direction) schemes with elliptic methods turns out to be effective to overcome the drawback of both algebraic and elliptic strategies. Thus, in this paper, we present an algorithm for the generation of boundary orthogonal grids which couples a mixed Hermite algebraic method with a boundary orthogonal elliptic scheme. Numerical tests on domains with classical geometries show satisfactory performances of the algorithm and coupling effectiveness in achieving grid boundary orthogonality and smoothness.