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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.

We prove an identity between Mahler measures of polynomials that was originally conjectured by Boyd. The combination of this identity with a result of Zudilin leads to a formula involving a Mahler measure of a Weierstrass form of conductor 17 given in terms of [Formula: see text]. Our proof involves a non-trivial identity between regulators which leads to the elliptic curve [Formula: see text]-function being expressed in terms of the regulator evaluated in a non-rational non-torsion point.

Currently, a lot of standardized elliptic curve cryptographic algorithms are widely used. Due to the constantly increasing execution speed demands, implementation optimization problem become actual. Using elliptic curve arithmetic on a different information security devices require to form an mathematical programming problem of elliptic curves calculation. Solution of this problem allows to find optimal way of calculation in the current condition. This approach increases the speed of cryptographic algorithms on information security devices. The paper present one of the possible ways to optimize computations in elliptic curve arithmetic by using mathematical programming. An optimization problem is formed that describes the calculation of scalar multiplication. Operations composition and multiplication methods composition are introduced to form the corresponding optimization problem. Adduction of elliptic curve calculations to the mathematical programming problem allow to research applicability of known optimization methods to computations in elliptic curves.

How can the projective invariant of the cubic curve approximating the river bed near its meander be calculated? A well-known approach uses the Weierstrass normal form. However, it is important to find this form by means of calculations tolerant to curve representation errors and, in particular, using calculations that do not require computation of tangent lines or inflection points. A new algorithm is proposed for calculation of the projective invariant of the cubic curve. This algorithm can be used to describe river meanders.

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.

Elliptic Curve Scalar multiplication is the method of adding a point on the curve to itself every time[1]. In recent years, research in Scalar Multiplication over Elliptic curves (EC) over a finite fields attracted many researchers, working in field of cryptography, to find out how elliptic curves cryptography (ECC) can be implemented and how to reduce its complexity [4]. The efficient techniques used in Elliptic curve cryptography are Elliptic curve scalar multiplication using pointhalving algorithm [2], then double-base (DB) chain algorithm, and after that step multi-base representation (SMBR), but these techniques have their drawbacks. So, it become imperative to find out a new approach which can be efficiently used for implementation of ECC and further reducing its complexity. The paper proposes a new algorithm Treble algorithm for affine coordinates. We continued doing work using the binary concept or double and add operation with the help of treble approach to make it more efficient which relates to the use of all input values in producing any type of output, including how much time and energy are required The results show that our contribution can significantly enhance EC scalar multiplication. Index Term—Elliptic curve cryptography, double and add operation, affine coordinates.

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.

Improvement in the implementation of elliptic curves cryptography and reducing its complexity are still being actively researched. The representation of integers in non adjacent form has been the subject of various investigations in slightly different contexts. The n -digit Non Adjacent Form (NAF) representation of an integer has no two consecutive non zero digits. Fewer non zero elements mean fewer point additions and therefore more efficient when we need to compute the most popular operation: elliptic curve scalar multiplication. In the present paper we reduce the number of required operations using this representation.

High-performance and fast implementation of point multiplication is crucial for elliptic curve cryptographic systems. Recently, considerable research has investigated the implementation of point multiplication on different curves over binary extension fields. In this paper, we propose efficient and high speed architectures to implement point multiplication on binary Edwards and generalized Hessian curves. We perform a data-flow analysis and investigate maximum number of parallel multipliers to be employed to reduce the latency of point multiplication on these curves. Then, we modify the addition and doubling formulations and employ a newly proposed digit-level hybrid-double Gaussian normal basis multiplier to remove the data dependencies and hence reduce the latency of point multiplication. To the best of our knowledge, this is the first time that one employs hybrid-double multiplication technique to reduce the computation time of point multiplication. Moreover, we have implemented our proposed architectures for point multiplication on FPGA and obtained the results of timing and area. Our results indicate that the proposed scheme is one step forward to improve the performance of point multiplication on binary Edward and generalized Hessian curves.

Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself k times. It is used in elliptic curve cryptography (ECC) as a means of producing a trapdoor function. In this paper, algorithms to compute the elliptic curve scalar multiplication using a special form for integers will introduce, and then two types of signed digit representation will use. The signed digit form of the scalar is calculated by many types of algorithms such as binary , non adjacent form and direct recoding. The results indicate that the proposed methods perform better to compute the scalar multiplication on elliptic curves and it is more efficient than the existing methods.

We compute the F-pure threshold of a degree three homogeneous polynomial in three variables with an isolated singularity. The computation uses elementary methods to prove a known result of Bhatt and Singh (from [2] ).

Let $E: y^2 = x^3 + bx$ be an elliptic curve for some nonzero integer $b$. Also consider $K$ be a number field with $4 \nmid [K : \mathbb{Q}]$. Then in this paper, we obtain a necessary and sufficient condition for $E$ having rank $0$ over $K$.

The field of elliptic curve cryptography has recently experienced a deployment of new models of elliptic curves, such as Montgomery or twisted Edwards. Computations on these curves have been proven to be exception-free and easy to make constant-time. Unfortunately many standards define elliptic curves in the short Weierstrass model, where the above properties are harder to achieve. This is especially true when scalar blinding, a simple but widely deployed side-channel attacks countermeasure, is adopted. In this paper we analyze previously undisclosed exceptional cases of popular scalar multiplication algorithms, highlighting the need for proofs of correctness. Then, with the final goal of providing a compact ECC hardware accelerator for embedded platforms, suitable to offload computations on all elliptic curve models, we present a constant-time adaptation of the Montgomery ladder, leveraging addition formulas by Izu and Takagi, that we prove return the correct result for any input point, any scalar value, on all elliptic curves in Weierstrass form defined over \(\mathbb {F}_p\) with \(p \ne 2,3\).

It is proved in this paper that, for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times that of its y-coordinate.

Fix a smooth curve C over a field K and n distinct points P 1 , … , P n ∈ C ( K ) . Here we prove several results on the gap sequence of the n -ple ( P 1 , … , P n ) in two very different set-ups (either as a subset of N n or, as introduced by Beelen and Tutas, as a subset of Z n ). Most results are when K is algebraically closed and large.

We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves., Comment: v2: 10 pages, major revision due to errors in the main result. v1: 14 pages, submitted to Mathematics of Computation

In the study of Diophantine equations elliptic curves play a vital role due to its application in proof of famous Fermat's Las t Theorem (FLT) and were further developed with its applications in factoring and primality. In this paper we define elliptic curve E(K) over the field K and describe the arithmetic on elliptic curve and give the group law with respect to the characteristic of K.. Due to its group structure and its analogue nature to multiplicative group of a finite field , elliptic curves find their way in enormous applications in cryptography. In this paper we also describe the group law for elliptic curve over a finite ring and propose a public key encryption with elliptic curve over the ring Z pq for p, q are primes.

This paper presents an approach for point addition and doubling on elliptic curves over finite fields F p which is based on one variable polynomial division. This is achieved by identifying the plane F p × F p with the extension field F p2 and transforming the elliptic curve equation as well as line equations arising in point addition or doubling into polynomials in one variable. Hence the intersection of the line with the curve is analogous to roots of the gcd between these polynomials. We show that this approach to arithmetic involves considerable scope for decomposition and parallel computation.

Elliptic curve cryptosystems are built on an underlying additive group, with an addition operation defined as the group operation. The aim of the elliptic curve addition operation is to render an elliptic curve point on the underlying elliptic curve when two ECC points are taken as inputs. However ECC addition formula may not be complete in nature, and may contain exceptional points, for which the addition formula may fail to produce a valid third point. The addition formula for prime order NIST curves were in fact not complete, till Renes et. al. proposed a complete addition formula for the class of prime order NIST curves in their Eurocrypt 2016 paper. The property of completeness ensures a valid third ECC point for any two chosen input points, and thus provides the advantage of using the same formula for both addition and doubling operations. Consequently it is assumed to be inherently side-channel secure, however any practical validation against side-channel protection is not yet present in the literature. In this work we analyse the side-channel protection for this newly constructed unified formula against two horizontal attacks. We show although this new construction is resistant against HCCA, it may be vulnerable to the ROSETTA attack, which exploits inner collisions within field multiplication operations.

Scalar point multiplication operation on elliptic curve is the most expensive part of the elliptic curve cryptosystem, also has always been the hot spot of the research. Recoding the positive integer and reducing the amount of inversion in the operation are the two main ideas. In this article, we use the balanced ternary form to recode the positive integer, at the same time, improve the part of calculation way of, reducing the amount of inversion, decreasing operation cost, and improving operation efficiency

Based on previous work, which presented theorems and algorithms for generating a new type of generalized Ball curve (Partial Said-Ball curve), the present study supplements the transformation formula from a Bézier curve to a Partial Said-Ball curve and prove the corresponding theorem. All the results are useful for the application of generalized Ball curves and their popularization in Computer Aided Geometric Design.

Based on Elliptic Curve cryptosystem, a threshold signature scheme characterized by (k,l) joint verification for (t,n) signature is put forward. After being signed by a signer company employing (t, n) threshold signature scheme, the informationmis transmitted to a particular verifier company, and then the signature is verified through the cooperation ofkones from the verifier company withlmembers, so as to realize a directional transmission between different companies. Finally, the application examples of the company encryption communication system, the generating polynomial of company private key and public key were given. The security of this scheme is based on Shamir threshold scheme and Elliptic Curve system, and due to the advantages of Elliptic Curve, the scheme enjoys wider application in practice.

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n2 − 1 and 12n2 + 1 are odd primes, then the general elliptic curve y2 = x3+(36n2−9)x−2(36n2−5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y2 = x3 + 27x − 62 really is an unusual elliptic curve which has large integral points.

In this paper. We introduce some fundamental results of the rational elliptic curve to determine its exact characterization and properties. After we create an rational elliptic curve with a element of infinite order.[5, 6, 7]

Let $\tilde{C}$ be a non-singular plane curve of degree d ≥ 8 with an involution σ over an algebraically closed field of characteristic 0 and $\tilde{P}$ a point of $\tilde{C}$ fixed by σ. Let π : $\tilde{C}$ → C = $\tilde{C}$/$/\langle\sigma\rangle $be the double covering. We set P = π($\tilde{P}$). When the intersection multiplicity at $\tilde{P}$ of the curve $\tilde{C}$ and the tangent line at $\tilde{P}$ is equal to d − 3 or d − 4, we determine the Weierstrass gap sequence at P on C using blowing-ups and blowing-downs of some rational surfaces.

In the present paper, we prove the {\it finiteness} of the set of {\it moderate} rational points of a once-punctured elliptic curve over a number field. This {\it finiteness} may be regarded as an analogue for a once-punctured elliptic curve of the well-known {\it finiteness} of the set of torsion rational points of an abelian variety over a number field. In order to obtain the {\it finiteness}, we discuss the {\it center} of the image of the pro-$l$ outer Galois action associated to a hyperbolic curve. In particular, we give, under the assumption that $l$ is {\it odd}, a {\it necessary and sufficient condition} for a certain hyperbolic curve over a generalized sub-$l$-adic field to have {\it trivial center}.

In elliptic curve cryptography (ECC), a scalar multiplication for rational point is the most time consuming operation. This paper proposes an efficient calculation for a scalar multiplication by applying Frobenious Mapping. Particularly, this paper deals with Barreto-Naehrig curve defined over extension field Fq2, where q = p6 and p is a large prime.

This paper proposes a coordinate system in which a point (x, y) is represented as (X, Y, Z, W), where x = X/Z and y = Y/W are satisfied. In the proposed coordinate system, the cost of scalar multiplication, which is the dominant operation of elliptic curve cryptosystems, on the twisted Edwards curve ±x2 + y2 = 1 + dx2y2 is the same as the existing lowest cost in a mixed coordinate system, and implementation of scalar multiplication becomes easier.

In modern days security issues play a vital and integral part of information technology and its related applications such as ATM and smart cards. Cryptography has evolved as a important element of security process. Elliptic curve cryptography is used to develop a variety of scheme for security purpose. Point addition and point doubling arithmetic used in elliptic curve is applicable for prime field. When same arithmetic performs over extension field it increases the complexity of the application but enhances the security in the field of communication technology. This paper communicates Weierstrass equation as the root of elliptic curve. Also described the group operation on elliptic curve that is point addition and point doubling used for encryption application based on prime field Fp. The proposed work is based on elliptic curve group operations solved in extension field F p 2. This new approach is verified using System for Algebra and Geometry and Experimentation (SAGE) open source software.

This text discusses triangles with the property that a bisector at one vertex, the median at another, and the altitude at the third vertex are concurrent. It turns out that since the 1930s, such triangles appeared in the problem sections of various journals. We recall their well known relation with points on a certain elliptic curve, and we present an elementary proof of the classical result providing the group structure on the real points of such an elliptic curve. A consequence of this answers a question posed by John P. Hoyt in 1991.

Elliptic Curve Cryptography provides similar strength of protection comparing other public key cryptosystems but requires significantly smaller key size. This paper proposes a new faster scalar multiplication algorithm aiming at a more secured Elliptic Curve Cryptography scheme. This paper also proposes a novel Elliptic Curve Cryptography scheme where maximum length random sequence generation method is utilized as data mapping technique on elliptic curve over a finite field. The proposed scheme is tested on various bits length of prime field and key sizes. The numerical experiments demonstrate that the proposed scheme reduces the computation time compared to conventional scheme and shows very high strength against cryptanalytic attack particularly random walk attack.

Invalid curve attacks are a well-known class of attacks against implementations of elliptic curve cryptosystems, in which an adversary tricks the cryptographic device into carrying out scalar multiplication not on the expected secure curve, but on some other, weaker elliptic curve of his choosing. In their original form, however, these attacks only affect elliptic curve implementations using addition and doubling formulas that are independent of at least one of the curve parameters. This property is typically satisfied for elliptic curves in Weierstrass form but not for newer models that have gained increasing popularity in recent years, like Edwards and twisted Edwards curves. It has therefore been suggested e.g. in the original paper on invalid curve attacks that such alternate models could protect against those attacks. In this paper, we dispel that belief and present the first attack of this nature against twisted Edwards curves, Jacobi quartics, Jacobi intersections and more. Our attack differs from invalid curve attacks proper in that the cryptographic device is tricked into carrying out a computation not on another elliptic curve, but on a group isomorphic to the multiplicative group of the underlying base field. This often makes it easy to recover the secret scalar with a single invalid computation. We also show how our result can be used constructively, especially on curves over random base fields, as a fault attack countermeasure similar to Shamir's trick.

A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants [12] and to extend our method in [14] for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the 5-Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree., 28 pages

It is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the Pollard rho method. However, the random walks defined on elliptic curve points equivalence class {+/-P} used by Pollard rho will always get trapped in fruitless cycles. We propose an efficient alternative approach to resolve fruitless cycles. Besides the theoretical analysis, we also examine the performance of the new algorithm in experiments with elliptic curve groups. The experiment results show that we can achieve the speedup by a factor extremely close to 2 using the new algorithm with the negation map, which is the best performance reported in the literature.

of products and standards that use public-key cryptography for encryption and digital signature use RSA. The key length for secure RSA has increased over recent years ,and this has put heavier processing load on applications using RSA. Recently, a competing system has begun to challenge RSA: Elliptic curve cryptography (ECC).The principle attraction of ECC, compared to RSA, is that it appears to offer equal security for a far smaller key size, thereby reducing processor overhead. Cryptographers are interested only in elliptic curve that belongs to cyclic abelian group. This paper implements cyclic abelian elliptic curve in MATLAB. The properties of abelian group is proved over the coordinates satisfying the curve. Base points of elliptic curve are generated to prove that the elliptic curve belongs to cyclic abelian group.

The Elliptic Curve Cryptography covers all relevant asymmetric cryptographic primitives like digital signatures and key agreement algorithms. In the present work, we develop a design of elliptic curve operations over binary Fields GF(2 m ). The function used for this purpose is the scalar multiplication kP which is the core operation of ECCs. Wherek is an integer and P is a point on an elliptic curve. The EC Point multiplication processor defined in affine coordinates is achieved by using a dedicated Galois Field arithmetic implemented on FPGA using VHDL language.

Efficient implementation of pairing-based cryptography requires construction of a pairing-friendly curve and its corresponding twisted curve. In this paper, we give a formula which determines the twisted curve. The formula are obtained by using technique in the algebraic number theory such as the complex multiplication theory. Applying the formula, we present an easy method for constructing pairing-friendly curves for BN-family. In fact, our method does not require the process of elliptic curve construction and finite field construction, but only requires searching an integer satisfying some conditions. Using our construction method, we implemented an optimal ate pairing for BN-family which is usable at various security levels, and investigated the time efficiency of the pairing computation for various security levels.

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations., Comment: 17 pages. Submitted

Plane quartic curves given by equations of the form y 2=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form $\int (1/\sqrt{P(x)})\,\mathrm{d}x$ with a quartic polynomial P can be derived directly from this addition law.

Let y2 = x3 + ax + b be an elliptic curve over 𝔽p, p being a prime number greater than 3, and consider a, b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M(p) as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max(a, b) < M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c1p1/2 < M(p) < c2p3/4 with explicit constants c1, c2 > 0.

On the basis of the general Weierstrass model of the cubic curve with parameters µ = (µ1, µ2, µ3, µ4, µ6), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in ℤ[µ] are obtained. The notion of the universal Krichever formal group over the ring A Kr is introduced; its exponential is determined by the Baker-Akhiezer function Φ(t) = Φ(t; τ, g 2, g 3), where τ is a point on the elliptic curve with Weierstrass parameters (g 2, g 3). As a consequence, results on the Krichever genus which takes values in the ring A Kr ⊗ ℚ of polynomials in four variables are obtained. Conditions necessary and sufficient for an elliptic formal group to be a Krichever formal group are found. A quasiperiodic function Ψ(t) = Ψ(t; v,w, µ) is introduced; its logarithmic derivative defines the exponential of the general elliptic formal group law, where v and w are points on the elliptic curve with parameters µ. For w ≠ ±v, this function has the branching points t = v and t = −v, and for w = ±v, it coincides with Φ(t; v, g 2, g 3) and becomes meromorphic. An addition theorem for the function Ψ(t) is obtained. According to this theorem, the function Ψ(t) is the common eigenfunction of differential operators of orders 2 and 3 with doubly periodic coefficients.