Pairs Of Elliptic Curves Research Articles

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

Let E and E′ be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if E(Fp)≃E′(Fp) but E(Fp2)≄E′(Fp2). Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

On 12-congruences of elliptic curves

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over [Formula: see text] with [Formula: see text]-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of [Formula: see text]-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the [Formula: see text], [Formula: see text] or [Formula: see text]-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the [Formula: see text]-invariants of the pair of elliptic curves.

Elliptic curves with common torsion 𝑥-coordinates and hyperelliptic torsion packets

We establish a connection between torsion packets on curves of genus 2 2 and pairs of elliptic curves realized as double covers of the projective line P x 1 \mathbb {P}^1_x that have many common torsion x x -coordinates. This can be used to show that the set of common torsion x x -coordinates has size at least 22 22 infinitely often and has 34 34 elements in some cases. We also explain how we obtained the current record example of a hyperelliptic torsion packet on a genus 2 2 curve.

Elliptic Curve Pairings

Elliptic curve pairings are a powerful tool and a popular way to construct zero-knowledge proofs, which are beginning to be used in blockchains as a way to provide privacy in the transaction ledger.

On correctness of conditions for the CSIDH algorithm implementation on Edwards curves

Incorrect formulation and incorrect solution of the problem of the CSIDH algorithm implementation on Edwards curves was revealed in one of the famous works. The purpose of this paper is to present a detailed critique of such concept with a proof of its inconsistency. Specific properties of three non-isomorphic classes of super singular curves in the generalized Edwards form are considered: full, quadratic, and twisted Edwards curves. Conditions for existence of curves of all 3 classes with order of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees is based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be built both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems that state that there is a solution within one class of curves with a parameter which is a square. The critical analysis of theorems, lemmas, erroneous statements in this work is carried out. Theorem 2 on quadratic twist in classes of Edwards curves is proved. The CSIDH algorithm modification based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered for.

HOW TO CONSTRUCT CSIDH ON QUADRATIC AND TWISTED EDWARDS CURVES

In one of the famous works, an incorrect formulation and an incorrect solution of the implementation problem of the CSIDH algorithm on Edwards curves is discovered. A detailed critique of this work with a proof of the fallacy of its concept is given. Specific properties of three non-isomorphic classes of supersingular curves in the generalized Edwards form is considered: complete, quadratic, and twisted Edwards curves. Conditions for the existence of curves of all classes with the order p+1 of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be construct both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems with statement about existing a solution within one class of curves with a parameter that is a square. The critical analysis of theorems, lemmas, and erroneous statements in this work is given. Theorem 2 on quadratic twist in classes of Edwards curves is proved. A modification of the CSIDH algorithm based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered.

Rational Equivalences on Products of Elliptic Curves in a Family

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1\times E_2$.

A Low-Power BLS12-381 Pairing Cryptoprocessor for Internet-of-Things Security Applications

We present the first BLS12-381 elliptic curve pairing crypto-processor for Internet-of-Things (IoT) security applications. Efficient finite field arithmetic and algorithm-architecture co-optimizations together enable two orders of magnitude energy savings. We implement several countermeasures against timing and power side-channel attacks. Our crypto-processor is programmable to provide the flexibility to accelerate various elliptic curve and pairing-based protocols such as signature aggregation and functional encryption.

Exceptional isogenies between reductions of pairs of elliptic curves

Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether-Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

Elliptic curves with large intersection of projective torsion points

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

High rank quadratic twists of pairs of elliptic curves

Given a pair of elliptic curves E1 and E2 over the rational field Q whose j-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals d such that the d-quadratic twists of E1 and E2 are both of positive rank. We construct infinite families of pairs of elliptic curves E1 and E2 over Q such that for each pair there exist infinitely many square-free rationals d for which the d-quadratic twists of E1 and E2 are both of rank at least 2.

ELLIPTIC K3 SURFACES ASSOCIATED WITH THE PRODUCT OF TWO ELLIPTIC CURVES: MORDELL–WEIL LATTICES AND THEIR FIELDS OF DEFINITION

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Masato Kuwata. We give an explicit description of the geometric Mordell–Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the nongeneric case, we describe a method to calculate explicitly a finite index subgroup of the Mordell–Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.

Algebraic Description of Jacobians Isogeneous to Certain Prym Varieties with Polarization (1,2)

ABSTRACTFor a class of non-hyperelliptic genus 3 curves C which are twofold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated with such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C → E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of three different elliptic curves. Our description is accompanied with explicit algorithms of calculation of periods of the Prym varieties and of absolute invariants of genus 2 curves. They are followed by numerical examples, which experimentally confirm main results of the article.

Secure and Efficient Secret Sharing Scheme with General Access Structures Based on Elliptic Curve and Pairing

Secret sharing techniques allow a secret to be shared among n participants in such a way that a specified subset of participant can reconstruct the secret by combining their shares. Secret sharing techniques are now the building blocks of several security protocols. Shamir's (t, n) threshold secret sharing scheme is one in which t or more participant can join together to retrieve the secret from the set of n participants. Traditional single secret sharing schemes are modified and generalized to share multiple secrets. Use of Elliptic Curve and Pairing in secret sharing is gaining more importance. The use of Elliptic Curve helps to improve the security and also the computational complexity is reduced. In this paper we propose a secret sharing scheme with a monotone generalized access structure. The scheme makes use of Shamir's scheme and Elliptic Curve pairing. The shares are chosen by the participant itself. So the consistency of the shares are ensured in this scheme. The participant shares remain secret during the reconstruction phase and this provides multi use facility where the same share can be used for the reconstruction of multiple secret. The shared secret, access structure or the participant set can be modified without updating the secret shadow of the participant. This provides dynamism and adds more flexibility to the scheme. The combiner can also verify the shares submitted by the participants during the reconstruction phase in order to identify the cheaters. The cheating detection and cheater identification is done by using bilinear pairing. This proposed scheme is simple and easy to implement compared with other generalized multi secret sharing scheme with extended capabilities using pairing.

Elliptic curves with maximally disjoint division fields

One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, E/Q, is its full-torsion representation ρE : Gal(Q/Q)→ GL2(Ẑ). Generalizing this idea, one can create another full-torsion Galois representation, ρ(E1,E2) : Gal(Q/Q)→ ( GL2(Ẑ) )2 associated to a pair (E1, E2) of elliptic curves defined over Q. The goal of this paper is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation ρ(E1,E2) has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by E1 and E2. The representation ρ(E1,E2) has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.

FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves

FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves

Pairs of elliptic curves with maximal Galois representations

Using a multi-dimensional large sieve inequality, we prove that, for almost all pairs (or indeed almost all k -tuples) of elliptic curves, the associated Galois representation on their torsion has maximal image. This generalizes the authorʼs previous work and provides evidence for an affirmative answer to a higher-dimensional analogue of Serreʼs uniformity question for single elliptic curves. Furthermore, as a consequence of our main theorem, one deduces the triviality of the Brauer group of the Kummer surface attached to almost all pairs of elliptic curves.

Explicit isogeny descent on elliptic curves

In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.

VARIATIONS ON HIGHER TWISTS OF PAIRS OF ELLIPTIC CURVES

In this note we show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v, w, t] such that the sextic twists of the curves E1, E2 by D(u, v, w, t) have rank ≥ 2 over the field ℚ(u, v, w, t). A similar result is proved for simultaneous quartic twists of pairs of elliptic curves with j-invariant 1728.