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The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb{Z}^2$, we define a family of homomorphisms $\Phi_{u,v}: E(\mathbb{Q}) \rightarrow \mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $\Psi_E$ be the set of suitable fundamental discriminants $-D 0$, we show that as $X \to \infty$, we have $$\#\, \left\{-X < -D < 0: -D \in \Psi_E \right \} \, \gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$$ In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm{tor}}(\mathbb{Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ is $\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups., Comment: 26 pages, 3 tables (final version)

Here we study the twists of the genus 2 curve given by the hyperelliptic equation Y 2 = X 6 + 1 over any field of characteristic different from 2, 3 or 5. Since any curve of genus 2 with group of automorphisms of order 24 is isomorphic (over an algebraically closed field) to the given one, the study of this set of twists is equivalent to the classification, up to isomorphisms defined over the base field, of curves of genus 2 with that number of automorphisms. This contribution closes the series of articles on the classification of twists of curves of genus 2. The knowledge of these twists can be of interest in a wide range of arithmetical questions, such as the Sato-Tate or the Strong Lang conjectures among others.

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum, and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace t becomes close to uniform even when t is averaged over very short intervals inside the Hasse–Weil interval. This result is based on a new form of the large sieve inequality for sparse sequences.

This paper explores a new way of instantiating isogeny-based cryptography in which parties can work in both the \((p+1)\)-torsion of a set of supersingular curves and in the \((p-1)\)-torsion corresponding to the set of their quadratic twists. Although the isomorphism between a given supersingular curve and its quadratic twist is not defined over \(\mathbb {F}_{p^2}\) in general, restricting operations to the x-lines of both sets of twists allows all arithmetic to be carried out over \(\mathbb {F}_{p^2}\) as usual. Furthermore, since supersingular twists always have the same \(\mathbb {F}_{p^2}\)-rational j-invariant, the SIDH protocol remains unchanged when Alice and Bob are free to work in both sets of twists.

The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist., Comment: Corrected formulation of Theorem 7.4; to appear in Transactions of the AMS

For an abelian variety $A$ over a number field $F$ , we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by- $3$ -isogeny on $A$ factors as a composition of $3$ -isogenies over $F$ . This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than $1$ . In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension $1$ , we deduce that if $E/F$ is an elliptic curve admitting a $3$ -isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank $0$ and a positive proportion have $3$ -Selmer rank $1$ . These results on bounded average ranks in families of quadratic twists represent new progress toward Goldfeld’s conjecture—which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$ —and the first progress toward the analogous conjecture over number fields other than $\mathbb{Q}$ . Our results follow from a computation of the average size of the $\phi $ -Selmer group in the family of quadratic twists of an abelian variety admitting a $3$ -isogeny $\phi $ .

Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K, then there is a natural bijection between the set of equivalence classes of elliptic subcovers of C/K and the set of elliptic subgroups of its Jacobian JC/K.

Let K = Q ( − 3 ) or Q ( − 1 ) and let C n denote the cyclic group of order n. We study how the torsion part of an elliptic curve over K grows in a quadratic extension of K. In the case E ( K ) [ 2 ] ≈ C 2 ⊕ C 2 we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.

We consider a collection of L-functions of elliptic curves twisted by a Dirichlet character modulo q (q is a prime number), and prove for this collection a joint limit theorem for weakly convergent probability measures in the space of analytic functions as q → ∞. The limit measure is given explicitly.

International audience; We prove lower bounds on the degree of polynomials interpolating the Naor–Reingold pseudo-random function over a finite field and over the group of points on an elliptic curve over a finite field.

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modular curve $X_0(3p)$ has infinitely many $F$-rational points the method applies to ``most'' elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0(3p)$ has only finitely many points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate-Shafarevich groups. The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p \equiv 1 \pmod 9$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate-Shafarevich groups.

In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas-Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case., Minor revision. Accepted version

Let $E$ be the elliptic curve $y^2=x^3+(i-2)x^2+x$ over the imaginary quadratic field $\mathbb{Q}(i)$. In this paper, we investigate the supersingular primes of $E$. We introduce the curve $C$ of genus two over $\mathbb{Q}$ covering a quotient of $E$ and for any prime number $p$, we state a condition (over $\mathbb{F}_p$) about the reduction of the jacobian variety of $C$ modulo $p$ which is equivalent to the existence of a supersingular prime of $E$ lying over $p$ (Theorem 5.10).

On the quadratic twist of a GLV curve, we explore faster scalar multiplication on its x-coordinate system utilizing three-dimensional GLV method. We construct and implement two kinds of three-dimensional differential addition chains, one of which is uniform and the other is non-uniform but runs faster. Implementations show that at about 254-bit security level, the triple scalar multiplication using our second differential addition chains runs about \(26\%\) faster than the straightforward computing using Montgomery ladder, and about \(6\%\) faster that the double scalar multiplication using DJB chains.

Estimates of the complexity of the point division operation into two for twisted Edwards curve are obtained in comparison with the doubling of the point. One of the applications of the divisibility properties of a point into two is considered to determine the order of a point in a cryptosystem. The cryptological security of the pseudo-random sequence generator proposed by the author is shown on the basis of a curve in the form of Edwards. A new generation scheme and a new one-sided function of a pseudo-random cryptological security sequence based on these curves are proposed. The degree of embedding of these curves into a finite field for pairing on friendly elliptic curves of prime order or almost prime order is investigated. Pairingfriendly curves of prime or near-prime order are absolutely essential in certain pairing-based schemes like short signatures with longer useful life. For this goal we construct friendly curves on base of family of twisted Edwards curves. The possibility of constructing a twisted Edwards order curve, that is, one that has a minimal cofactor 4, has been found. A solution for the inverse doubling problem is obtained for quasi-elliptic curves that represented in the twisted Edwards form. Also its application to the proving of cryptographic pseudo-random sequence generator. It makes it possible to prove the cryptological security of the pseudo-random sequence we developed.

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.

Point compression is an essential technique to save bandwidth and memory when deploying elliptic curve based security solutions in wireless communication systems. In this contribution, we provide new linear algebra (LA) based compression algorithms for multiple points on elliptic curves, that are compression algorithms which only make use of LA (with a constant number of field multiplications and at most one inversion, with no quadratic or higher degree polynomial root finding). In particular, we extend the results of Khabbazian et al. (IEEE Trans Comput 56(3):305---313, 2007) to four (resp. five) points on elliptic curves by generically storing five (resp. six) field elements and provide an asymptotic generalization to any number n of points on a curve $$y^2=f(x)$$y2=f(x) by generically storing $$n+1$$n+1 values.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$., To appear in the Journal of Number Theory

A counterexample is given to a conjecture from the comments to Arnold’s problem 1989–11 about the existence of a tangent foliation in a zero type neighborhood of an elliptic curve.

The paper is devoted to the algebraic and arithmetic structures related to the two-body problem and discuss the possible generalizations. The role of the points of finite order on the elliptic curves is emphasized.

We fi nd the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X_0(n). We show that for n=14, 15, 20, 21, 49 there exist infinitely many quartic fields K such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞ . In the case n=27 we prove that there are infinitely many sextic fields such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞.

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

We propose a modification of the addition law on the Edwards curve over a prime field. We prove three theorems on properties of coordinates of high-order points and on a degenerate pair of twisted curves. We propose an algorithm for reconstructing all unknown points kP of the Edwards curve when only 1/8 of the points are known.

Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X can be obtained by intersecting the dual of the corresponding Cayley cubic with the dual of the quadric containing C. We take this construction to its limit, studying all smooth degenerations and proving that the construction, with appropriate modifications, extends to the complement of a specific divisor in moduli. We work over an arbitrary field of characteristic different from two in order to facilitate arithmetic applications., Comment: 30 pages; Some expository changes; removed erroneous (old) Thm 4.11 and changed (old) Thm 4.23 into (new) Thm 4.17

In this paper, we compute explicit equations for the twists of all the smooth plane quartic curves defined over a number field [Formula: see text]. Since the plane quartic curves are non-hyperelliptic curves of genus [Formula: see text] we can apply the method developed by the author in a previous paper. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to [Formula: see text]-isomorphism.

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set $H^1\big(\operatorname{G}_{\overline{K}/K}, \operatorname{Aut}_{\overline{K}}(E)\big)$. The results are illustrated by examples.

New wireless technologies and approaches enable to connect even the simplest sensors with limited computational power to the global network. The need for efficient and secure solutions is growing with the wider use of these devices. This paper provides a new method for speed optimization of Elliptic Curve Cryptography operations which are frequently used in the light-weight secure communication algorithms. This method is based on the anomalous behaviour of specific elliptic curves. We analyse more than 60 curves of various international standards. Further, our method is less complex, easy to deploy and comparable effective as ordinary, more complex methods. Last but not least, we show the importance of future research in the area of elliptic curve parameterization.DOI: http://dx.doi.org/10.5755/j01.eie.23.5.19248

Elliptic curves over a finite field Fq with j-invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with fixed embedding degree with respect to a subgroup of prime order . For k = 1, 2, these conditions give parameterizations of q in terms of and two integers m, n. We show several examples of families with infinitely many curves. Similar parameterizations for k ? 3 need a fixed kth root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.

Isogenies of elliptic curves have been well-studied, in part because there are several cryptographic applications. Using Velu’s formula, isogenies can be evaluated explicitly given their kernel. However, Velu’s formula applies to elliptic curves given by a Weierstrass equation. In this paper we show how to similarly evaluate isogenies on Edwards curves and Huff curves. Edwards and Huff curves are new normal forms for elliptic curves, different than the traditional Weierstrass form.

Let F be the cubic field of discriminant −23 and its ring of integers. Let Γ be the arithmetic group , and for every ideal , let be the congruence subgroup of level . In [Gunnells and Yasaki 13], the cohomology of various was computed, along with the action of the Hecke operators. The goal of that work was to test the modularity of elliptic curves over F. In the present article, we complement and extend those results in two ways. First, we tabulate more elliptic curves than were found earlier using various heuristics (“old and new” cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona–Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, ...

In this work, we are going to study the elliptic curves over the ring A 3 = ( F 2 d [ X ] ) / ( X 3 ) , precisely we will establish the short exact sequence which is split, and we will define the group extension E a , b ( A 3 ) of E a 0 , b 0 ( F 2 d ) by Ker ( π 3 ) which seems to be beneficial and interesting in ECC.

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3, \mathbb{A}_\mathbb{Q})$ by certain $L$-values of $p$-power twists.

Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of density properties of rational points on the Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence of our results is the following. Fix an abelian variety A over k, and suppose that for some r > 0 the Brauer-Manin obstruction to weak approximation on the Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at least r. Hence if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded. This relates two significant open questions., Comment: 12 pages; final version