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In this paper, we are going to study the twisted Hessian curves on the local ring $$\mathbb {F}_{q}[\epsilon ]$$ , $$\epsilon ^{4}=0$$ , with $$\mathbb {F}_{q}$$ is a finite field of order $$q=p^{b}$$ , where p is a prime number $$ \ge 5$$ and $$b\in \mathbb {N}^{*}$$ . In a first time, we study the arithmetic of the ring $$\mathbb {F}_{q}[\epsilon ]$$ , $$\epsilon ^{4}=0$$ , which will be used in the remainder of this work. After, we define the twisted Hessian curves $$H^{4}_{a,d}$$ over this ring and we give essential properties and the classification of these elements. In addition, we define the group extension $$H^{4}_{a,d}$$ of $$H_{a_{0},d_{0}}$$ by $$Ker \ \tilde{\pi }$$ . We finish this work by introducing a new public key cryptosystem which is a variant of Cramer-Shoup public key cryptosystem on a twisted Hessian curves and study its security and efficiency. Our future work will focus on the generalist these studies for all integers $$n>4$$ , $$\epsilon ^{n}=0$$ , which is beneficial and interesting in cryptography.

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.

In this paper we study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained, in general. We focus especially on such cases in which one can identify a rational mapping which is a double cover of a rational curve. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable. We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves. Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. Our goal is not to give a comprehensive list of all such operations but to present at least selected interesting cases originated in geometric modelling and to show a unifying feature of the formulated method. We demonstrate our algorithm on a number of examples.

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.

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.

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.

Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the $j$-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence., Comment: 13 pages and 11 figures. Compared with the first version, the important remark 5.2 was added. To appear in "Noncommutativity and Singularity", Proceedings of French-Japanese Symposia, IHES, 2006. J. P. Bourguignon, M.Kotani, Y.Maeda, N.Tose eds, Advanced Studies in Pure Maths 55, 2009

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.

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.

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.

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.

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.

We present some numerical results on the distribution of nonzero central values of elliptic curve L -functions. We compare the family of elliptic curves E a , b : y 2 = x 3 + a x + b with ( a , b ) ranging over a large box, and Cremona's list of all elliptic curves with conductor 340 000 ≤ N 350 000 . We also present a non-rigorous method for calculating central values that is faster than the approximate functional equation by a logarithmic factor.

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.

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.

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.

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.

In this paper, we de ne the directional associated curve andthe self-associated curve of a null curve in Minkowski 3-space. We studythe properties and relations between the null curve, its directional asso-ciated curve and its self-associated curve. At the same time, by solvingcertain di erential equations, we get the explicit representations of somenull curves. 1. IntroductionIn the history of di erential geometry, the theory of the associated curveof a given curve has been one of interesting topics. Many geometers have in-vestigated this problem from various viewpoints: For instance, in Euclidean3-space, the Bertrand partner and the Mannheim partner curves are two im-portant kinds of associated curves. They are characterized by the curvatureand the torsion, namely, a Bertrand curve satis es + ˝= 1 and a Mannheimcurve has = ( 2 + ˝ 2 ), where is the curvature and ˝the torsion.In Minkowski 3-space, there are three kinds of typical curves, that is, space-like, time-like and null (light-like) curves. Among them, a null curve is quitedi erent from other types of curves. Motivated by the idea about partnercurves, we naturally consider the associated curves of a null curve in Minkowski3-space.In [4], the authors de ned the W-direction curve ~rof a null curve rin sucha way that the tangent vector eld of ~ris W, where W is a vector eld alongthe given null curve. In this case, the curve ris called the W-direction donorcurve of ~r.

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.

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] ).

In this paper, we aim at sustaining the claim that curve-based cryptographic schemes over finite fields of characteristic 2 do not provide enough security. We present algorithms to find all the possible supersingular elliptic curves which can be embedded into a predefined finite field. We also consider the case of hyperelliptic curves with genus 2, including both supersingular and ordinary cases. As computational examples, we show even the DLP on a 3060-bit elliptic curve and the DLP on Jacobians of a 255-bit hyperelliptic curve can be solved by embedding to a 6120-bit extension field.

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 $K$.

In this application, example methods for performing quantum Montgomery arithmetic are disclosed. Additionally, circuit implementations are disclosed for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. This application also shows that elliptic curve discrete logarithms on an elliptic curve defined over an n-bit prime field can be computed on a quantum computer with at most 9n+2┌log 2 (n)┐+10 qubits using a quantum circuit of at most 512n 3 log 2 (n)+3572n 3 Toffoli gates.

International audience; We provide a new family of elliptic curves that results in a one to two percent performance improvement of the elliptic curve integer factoriza-tion method. The speedup is confirmed by extensive tests for factors ranging from 15 to 63 bits.

International audience; Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery's x-only arithmetic and Ladder algorithm, x-only Diffie–Hellman, y-coordinate recovery, and 2-dimensional and Euclidean differential addition chains such as Montgomery's PRAC algorithm.

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 \(p \ne 2,3\).

Twisted Edwards curves over finite fields have attracted great interest for their efficient and unified addition formula. In this paper, we consider twisted Edwards curves over local fields and introduce a cryptosystem based on quotient groups of twisted Edwards curves over local fields. From the study of formal groups of twisted Edwards curves and twisted Edwards curves over local fields, we give the choice of cryptographic groups. An element in these groups can be uniformly represented by two n digit p-adic numbers, whereas an element in the elliptic curves in Weierstrass form over local fields is represented by a 3 n -2 digit p-adic number and a 4 n -3 digit p-adic number. In the cryptography on elliptic curves in Weierstrass form over local fields, five cases for different input point pairs in computing points addition have to be considered and sometimes points have to be lifted. In the cryptography on twisted Edwards curves over local fields, the addition formula is simple, unified, and complete, which is efficient, does not need lifting points, and is against the side channel analysis. Finally, a speedy point multiplication algorithm and some concrete instances are given.

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.

We prove the -invariant of an irreducible elliptic curve associated with a 3-by-3 or 4-by-4 matrix is real and greater than or equal to 1. Additionally, we correct an erroneous statement appearing in Kippenhahn’s argument about the boundary generating curves of the numerical ranges of matrices.

There is considerable interest in the properties of PH curves in geometric modeling and CAGD because PH curves can be computed at speeds comparable to polynomial curves and used to calculate curve arc lengths and isometric lines. The purpose of this paper is to develop a general approximation of the H-Bezier curve based on PH curves. We call the resulting approximations PHH-Bezier curves for convenience. First, a necessary and sufficient condition for a cubic plane H-Bezier curve to be a PH curve is obtained. Second, based on the H-Bezier curve, the control polygon of the cubic PHH-Bezier curve is constructed from an orderly triangle. The cubic PHH-Bezier curve is then introduced and a new algorithm for their construction is proposed. According to geometric modeling, the error analysis between H-Bezier curve and PHH-Bezier curve is estimated. Finally, the proposed algorithm is verified experimentally. It is demonstrated that cubic PHH-Bezier curves can accurately approximate H-Bezier curves but that the selection of the middle two control points of the PHH-Bezier curve has a profound impact on the quality of the approximation. Error analyses demonstrate that when the middle two control points of the PHH-Bezier curve are mixed with the corresponding original control points, a good approximation is achieved. The new algorithm may thus have considerable potential for use in geometry modeling.

PH curve has the properties that its arc-length is a polynomial of its parameter, and its offset is a rational expression, PH curve is used extensively for CAD and CAGD. In this paper, the planar curves are analytical curves. The arc-length of a curve is geometric and intuitive parameter, for G 1 Hermite data taken from the endpoints of a given planar curve and the arc-length of the curve, we construct a PH quintic curve which preserves the arc-length of the curve. The PH curve is the approximation of the planar curve and keeps the geometric property of the original curve. The scheme is further used to create a planar G 1 spline by joining segments of PH quintic curves after the planar curve is subdivided into segments. With the help of Frenet-Serret formulas and Taylor expansion of planar curves, we deduce the power series of approximation error of segment with arc-length. The approximation order with respect to the given curve by this solution is equal to five and approximation order with respect to its offsets is four. Some related coefficients of terms are listed.

Recently elliptic curve cryptosystems are widely accepted for security applications key generation, signature and verification. Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. it is possible to use smaller primes, or smaller finite fields, with elliptic curves and achieve a level of security comparable to that for much larger integers. Koblitz curves, also known as anomalous binary curves, are elliptic curves defined over F2. The primary advantage of these curves is that point multiplication algorithms can be devised that do not use any point doublings. The ElGamal cryptosystem, which is based on the Discrete Logarithm problem can be implemented in any group. In this paper, we propose the ElGamal over Koblitz Curve Scheme by applying the arithmetic on Koblitz curve to the ElGamal cryptosystem. The advantage of this scheme is that point multiplication algorithms can be speeded up the scalar multiplication in the affine coodinate of the curves using Frobenius map. It has characteristic two, therefore it’s arithmetic can be designed in any computer hardware. Moreover, it has more efficient to employ the TNAF method for scalar multiplication on Koblitz curves to decrease the number of nonzero digits. It’s security relies on the inability of a forger, who does not know a private key, to compute elliptic curve discrete logarithm.

In this paper we study some problems in Hessian topology. We prove that certain real plane curves satisfy the requirements of the Hessian curve of a differential function. The real plane curves we consider are those with k outer ovals and also those which only have one nest of depth k, with \({k \in \mathbb{N}}\).