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Security and efficiency for resource constrained devices using lightweight cryptography techniques is still a problem. It is important to make these devices secure from the intruders and communicate with them securely. Elliptic Curve Cryptography (ECC) comes out to be a major contender to solve this problem as it provides the security using smaller key length. Since the conventional ECC works on predefined and fixed curves, the elliptic curves are analysed thoroughly since the emergence of ECC. This paper, proposes an implementation of dynamic elliptic curves which are secure and feasible for Internet of Things (IoT) devices and other devices which require lightweight cryptography for security. The experimental results show the feasibility and efficiency of the algorithm used to generate dynamic curve for cryptography. Also, a new curve will be generated at each runtime. It would be nearly impossible for intruders to exploit and retract information from the devices. Hence, the proposed technique is more secure and practically efficient for resource constrained devices.

In this paper, we generalize the direct method of lines for elliptic problems in star-shaped domains. We assume that the boundary of the star-shaped domain is a closed Lipschitz curve that can be parameterized by the angular variable, so that an appropriate transformation of coordinates can be introduced. Then the elliptic problem is reduced to a variational–differential problem on a semi-infinite strip in the new coordinates. We discretize the reduced problem with respect to the angular variable and obtain a semi-discrete approximation. Then a direct method is adopted to solve the semi-discrete problem analytically. Finally, the optimal error estimate of the semi-discrete approximation is given and several numerical examples are presented to show that our method is feasible and effective for a wide range of elliptic problems.

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 E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G, In this new version we fix some errors in Table 5

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.

Scalar multiplication is the most important and expensive operation in elliptic curve cryptosystems. In this study, the authors improve the efficiency of the elliptic net algorithm to compute scalar multiplication by using the equivalence of elliptic nets. The proposed method saves four multiplications by a constant in each iteration loop. Experimental results also indicate that the proposed algorithm will be more efficient than the previously known results on this line while it is still slower than the state-of-the-art algorithm to compute scalar multiplication.

The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step giant-step algorithm (BSGS) or Pollard rho. Montgomery's simultaneous modular inversion can also be used to speed up Pollard rho when running many walks in parallel. We generalize these ideas and exploit the fact that for any two elliptic curve points X and Y, we can efficiently get X-Y when we compute X+Y. We apply these ideas to speed up the baby-step giant-step algorithm. Compared to the previous methods, the new methods can achieve a significant speedup for computing elliptic curve discrete logarithms in small groups or small intervals. Another contribution of our paper is to give an analysis of the average-case running time of Bernstein and Lange's "grumpy giants and a baby" algorithm, and also to consider this algorithm in the case of groups with efficient inversion. Our conclusion is that, in the fully-optimised context, both the interleaved BSGS and grumpy-giants algorithms have superior average-case running time compared with Pollard rho. Furthermore, for the discrete logarithm problem in an interval, the interleaved BSGS algorithm is considerably faster than the Pollard kangaroo or Gaudry-Schost methods.

In this paper, using the technique of the divided difference operators, we introduce a $4n$-point elliptic interpolation formula. Some applications of our interpolation formula to elliptic and basic hypergeometric series are given.

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.

We study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

In this paper, the randomness of binary sequences generated from elliptic curves over a finite field of characteristic 2 is studied. A scheme of construction based on the Chaos-Driven Elliptic Curve Pseudo-random Number Generator (C-D ECPRNG) is proposed. The generators based of this scheme are verified by using tests from the NIST Statistical Test Suite to analyze their statistical properties. An elliptic curve used in the numerical example is defined over F 2 8. The investigations which made for the generated series of two output sequences of the lengths of 210 and 220 bits shown that 14 generators working according to our general scheme exhibit good randomness properties. Next, the binary sequences generated by these 14 schemes were used for encrypting a 256×256 grayscale Lena image as an application example and the security analysis of the ciphered images was carried out.

In this paper we propose a modified Elliptic Net algorithm to compute pairings. By reducing the number of the intermediate variables which should be updated in the iteration loop of the Elliptic Net algorithm, we speed up the computation of pairings. Experimental results show that the proposed method is more efficient than the original Elliptic Net algorithm on Barreto-Naehrig (BN) curves which are the best choice for implementing pairings at 128-bit security level.

We use Masser’s counting theorem to prove a lower bound for the canonical height in powers of elliptic curves. We also prove the Galois case of the elliptic Lehmer problem, combining Kummer theory and Masser’s result with bounds on the rank and torsion of some groups of rational points on an elliptic curve.

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.

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.

In 2006, Mazur, Stein, and Tate [5] gave an algorithm for computing p -adic heights on elliptic curves over Q for good, ordinary primes p ≥ 5 . In this paper, we extend their algorithm to the case of p = 3 . We also discuss the 3-adic precision that must be maintained throughout the computation, following the work of Harvey [2] . We conclude by giving examples of 3-adic regulators and their compatibility with the 3-adic Birch and Swinnerton-Dyer conjecture, computed using Sage [9] .

Given two elliptic curves E 1 E_1 and E 2 E_2 defined over the field of rational numbers Q \mathbb {Q} that have good and ordinary reduction at an odd prime p p , and have equivalent, irreducible mod p p Galois representations, we study the variation of the parity of Selmer ranks and analytic ranks of E 1 E_1 and E 2 E_2 over certain number fields.

In this work, we propose an algorithm to produce the Triple-base chain that optimize the time usage for computing an elliptic curve cryptosystem. Triple-base Chain is a scalar multiplication algorithm, which represents an integer k using three bases {2,3,5}. This paper provides a faster scalar multiplication method of elliptic curve based on {2,3,5} Triple-base Chain. The method proposed by this research speeds up the existing Triple-base Chain algorithm by optimizing the 5P operation of elliptic curve and reordering the operation order of base {2,3,5}. This method can improve the speed of operation from 4 to 6 % compared to the existing {2,3,5} Triple-base Chain.

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.

We establish Liouville-type theorems for the elliptic system of inequalitiesHere satisfy some growth conditions, and is the strongly degenerate operator of the formwhere satisfies some certain conditions. As a direct consequence, we also obtain the Liouville-type theorem for the following elliptic inequality

Elliptic Curve Cryptography recently gained a lot of attention in industry. The principal attraction of ECC compared to RSA is that it offers equal security for a smaller key size. The present paper includes the study of two elliptic curve Ea,b and Ea,-b defined over the ring Zp[i] where i2 = -1. After showing isomorphism between Ea,b and Ea,-b. We define a composition operation (in the form of a mapping) on their union set. Then we have discussed our proposed cryptographic schemes based on the elliptic curve E = Ea,b∪Ea,-b. We also illustrate the coding of points over E, secret key exchange and encryption/decryption methods based on above said elliptic curve. Since our proposed scheme are based on elliptic curve of the particular type therefore the proposed schemes provides a highest strength-per-bit of any cryptosystem known today with smaller key size resulting in faster computations, lower power assumption and memory. Another advantage is that authentication protocols based on ECC are secure enough even if a small key size is used.

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)< ∞.

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.

Since the security of RSA cryptosystem depends on the difficulty of factoring integers, it is the most important problem to factor large integers in RSA cryptosystem. The Lenstra elliptic curve factorization method(ECM) is considered a special purpose factoring algorithm as it is still the best algorithm for divisors not greatly exceeding 20 to 25 digits(64 to 83 bits or so). ECM, however, wastes most time to calculate ∙ mod and so Montgomery and Koyama both give fast methods for implementing ∙ mod . We, in this paper, further analyze Montgomery and Koyama’s methods and propose an efficient algorithm which choose the optimal parameters and reduces the number of multiplications of Montgomery and Koyama’s methods. Consequently, the run time of our algorithm is reduced by 20% or so than that of Montgomery and Koyama’s methods.

In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the group of rational points on an elliptic curve and is thus not a generic algorithm. The algorithm that we describe has some similarities with the most powerful index-calculus algorithm for the discrete logarithm problem over a finite field. The algorithm has no restriction on the finite field over which the elliptic curve is defined.

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.

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.

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.

Pairings on elliptic curves play an important role in cryptography. We provide an explicit formula for vectors of polynomials describing optimal ate pairings over cyclotomic family of elliptic curves. The explicit formula is simple in that it only involves partitioning a certain cyclotomic polynomial. The simplicity of the formula allows us to analyze the sparsity of the vector.

Supersingular elliptic curves have played an important role in the development of elliptic curve crytography. Scott Vanstone introduced the first author to elliptic curve cryptography, a subject that continues to be a rich source of interesting problems, results and applications. One important fact about supersingular elliptic curves is that the number of rational points is tightly constrained to a small number of possible values. In this paper we present some similar results for curves of higher genus. We also present an application to the problem of determining abelian varieties that occur as jacobians.

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.

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

The necessary and sufficient conditions for the parameters of the curve in the canonical form with two points of order 4 are found. Two lemmas about the properties of quadratic residues are proved, using the Gauss scheme for quadratic residues and non-residues. Based on this lemmas, the exact formulas are derived for the number of elliptic curves with non-zero parameters a and b and two points of order 4 that are isomorphic to Edwards curves over the prime field. It is proved that for large fields the share of such curves is close to 1/4.

The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of a certain family of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves. This result can be viewed as a higher dimensional analogue of exotic relations of the elliptic dilogarithm., 25 pages

Elliptic Curve Cryptography (ECC) is a relatively recent branch of cryptography which is based on the arithmetic on elliptic curves and security of the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Elliptic curve cryptographic schemes are public-key mechanisms that provide encryption, digital signature and key exchange capabilities. Elliptic curve a lgorithms are also applied to generation of sequences of pseudo-random numbers. Another recent branch of cryptography is chaotic dynamical systems where security is based on high sensitivity of iterations of maps to initial conditions and parameters. In the present work, we give a short survey describing state-of-the-art of several sugge sted constructions for generating sequences of pseudorandom number generators based on elliptic curves (ECPRNG) over finite fields of prime o rder. In the second part of the paper we propose a method of generating sequences of pseudorandom points on elliptic curves over fin ite fields which is driven by a chaotic map. Such a constructio n improves randomness of the sequence generated since it combines good statistical properties of an ECPRNG and a CPRNG (Chaotic Pseudo- Random Number Generator). The algorithm proposed in this work is of interest for both classical and elliptic curve crypt ography.